Proof. The correction \(c(s,x)\) belongs to \(C\), and \[c(s,tx)c(t,x)=c(st,x).\] Using the cocycle identity for \(\alpha\) and the projective multiplication of \(\gamma\) in Equation (3) gives \[ U^X_s(tx)U^X_t(x)=\alpha(s,t)U^X_{st}(x). \tag{4}\] One can also check all the scalar factors at once in the central extension with multiplication \((z,s)(w,t)=(zw\alpha(s,t),st)\): the representation of its preimage of \(C\) is \((z,c)\mapsto z\gamma(c)\), and the equality \(sr_x=r_{sx}c(s,x)\) gives exactly the ratio in Equation (3). Thus this argument retains, rather than discards, the fixed multiplier.

For explicit independence of representatives, suppose \(r'_x=r_xk_x\), where \(k_x\in C\). The identification from the new coordinate copy of \(V_x\) to the old one is \[F_x=\alpha(r_x,k_x)^{-1}\gamma(k_x).\] Indeed identifying a fibre by \(U_{r'_x}\) instead of \(U_{r_x}\) has precisely this effect. Equation (4) then gives \(F_{sx}U'{}^X_s(x)=U^X_s(x)F_x\).

Now assume that \(C\) is a graph. Both coordinate groups act freely on \(X\). For example, if \((g,e)r_xC=r_xC\), then \(r_x^{-1}(g,e)r_x\in C\) has second coordinate \(e\), so \(g=e\). On an orbit of the first coordinate group choose a fibre \(V_{x_0}\). The unitary identification \[\ell^2(\Gamma)\otimes V_{x_0}\longrightarrow \bigoplus_{g\in\Gamma}V_{(g,e)x_0}, \qquad \delta_g\otimes\xi\longmapsto U^X_{(g,e)}\xi\] intertwines its action with \(\lambda_\mu\otimes1\), since \(U_{(a,e)}U_{(g,e)}=\mu(a,g)U_{(ag,e)}\). Hence it extends normally to \(L_\mu(\Gamma)\). The same argument for the second coordinate gives copies of \(\lambda_{\bar\omega}\), which are the standard normal representations of the generators acting by right adjoints \(\xi\mapsto\xi v_h^*\). The two actions commute, because the product cocycle has trivial mixed-coordinate factors.

The first coordinate orbits are indexed by \(\Lambda/B\), and the second coordinate orbits by \(\Gamma/A\). They are finite in number. Each is a regular module with multiplicity \(\dim_{\mathbb C}V\), proving bifiniteness as well as the dimension formulas below. ◻

Proof. On \(V\otimes L^2(N)\) the subgroup action is \[u_a(\xi\otimes\eta)=\sigma(a)\xi\otimes v_{\delta(a)}\eta.\] Its multiplier is \(\beta(a,b)\omega(\delta(a),\delta(b))=\mu(a,b)\). For normality, choose representatives \(t\) for \(B\backslash\Lambda\). On the summand spanned by \(\xi\otimes v_{\delta(a)}v_t\), the map \[ \xi\otimes v_{\delta(a)}v_t \longmapsto\delta_a\otimes\sigma(a)^*\xi \quad\text{in }\ell^2(A)\otimes V \tag{5}\] is unitary and conjugates the subgroup action to \(\lambda_{\mu|A}\otimes1\). To check the phase, applying \(u_b\) before this map produces \[\omega(\delta(b),\delta(a)) \sigma(ba)^*\sigma(b) =\omega(\delta(b),\delta(a))\beta(b,a)\sigma(a)^* =\mu(b,a)\sigma(a)^*.\] Thus the left subgroup action is normal. Its right \(N\) action is the standard action on the second factor and commutes with it.

Choose representatives \(r\) for \(\Gamma/A\). As a right \(L_{\mu|A}(A)\) module, \(L^2(M)\) has orthonormal free basis \(u_r\); the assertion follows directly from \(E_{L_{\mu|A}(A)}(u_r^*u_{r'})=\delta_{r,r'}1\). The relative product therefore identifies with the orthogonal sum \[ E(A,B,\delta,\sigma) =\bigoplus_{r\in\Gamma/A}u_r\otimes(V\otimes L^2(N)). \tag{6}\] At its identity fibre the graph action is exactly \[U_{(a,\delta(a))}j\xi =u_a\otimes(\xi\otimes v_{\delta(a)}^*) =j\sigma(a)\xi.\] For \(g=ra\), balancing in Equation (6) gives the more explicit formula \[U_{(g,h)}j\xi =\mu(r,a)^{-1}u_r\otimes \bigl(\sigma(a)\xi\otimes v_{\delta(a)}v_h^*\bigr).\] The last Fourier coordinate is \(\delta(a)h^{-1}\), up to its displayed unit scalar. Thus two translated identity fibres are orthogonal unless their labels in \(D/C_\delta\) agree: equality of their first cosets and last Fourier coordinates is exactly that coset equality. Conversely these fibres exhaust every summand in Equation (6). They are an orthogonal, transitive system with graph action \(\gamma\) at the basepoint. Finally \[U_sU_{r_x}j =\frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))} U_{r_{sx}}j\gamma(c(s,x)),\] which proves that the stated unitary has exactly the transport of Equation (3). ◻

Proof. Use Lemma 4 and the orbit decompositions in Lemma 3. Each free coordinate orbit gives \(n\) regular copies, of module dimension \(n\) with the unnormalized matrix trace convention. The respective orbit counts are \([\Lambda:B]\) and \([\Gamma:A]\). ◻

Proof. On \(\ell^2(L)\otimes V\) use the unitary \[T(\delta_g\otimes\xi)=\delta_g\otimes v_g^*\xi.\] Indeed, \[\begin{align*} T(\lambda_\alpha(h)\otimes v_h)T^* (\delta_g\otimes\xi) &=\alpha(h,g)\delta_{hg}\otimes v_{hg}^*v_hv_g\xi\\ &=\alpha(h,g)\nu(h,g)\delta_{hg}\otimes\xi. \end{align*}\] A finite normal module over a finite factor embeds in finitely many copies of the regular module (Jones 2009, Theorems 10.1.1 and 10.2.1). When \(L\) is ICC both twisted group algebras are factors. Applying this unitary to such an embedding proves the second assertion. The formula also proves the last assertion directly. ◻

Proof. Put \(M=L_\mu(\Gamma)\), \(N=L_\omega(\Lambda)\), and \(P=\pi(M)'\). Finite \(M\)-module dimension makes \(P\) a finite factor. Moreover \(N^{\mathrm{op}}\subset P\) has finite index. To see the latter without assuming the endpoint actions are mutual commutants, regard \(K\) as the equivalence module for the mutually commuting factors \(\pi(M)\) and \(P\). It has finite nonzero \(P\)-dimension. Finitely many copies of it therefore contain \(L^2(P)\) as a \(P\)-module. Their restrictions to \(N^{\mathrm{op}}\) are finite, by the original right finiteness of \(K\). Consequently \(L^2(P)\) has finite \(N^{\mathrm{op}}\)-dimension, which is the index assertion. We use the standard finite-module and index facts in this argument in the normal, trace-preserving convention; see (Jones 1983) and (Pimsner and Popa 1986, Proposition 2.1).

Let \(D_c\) be the bounded elements of \(P\) whose \(\mathop{\mathrm{Ad}}\rho\)-orbits are precompact in \(L^2(P)\). The compact vectors of a unitary representation form its closed almost-periodic subspace: they are the closed span of its finite-dimensional subrepresentations. Bounded compact elements are closed under adjoints and products, since \[\|xy-x'y'\|_2 \leq \|x\|_\infty\|y-y'\|_2 +\|x-x'\|_2\|y'\|_\infty\] on uniformly bounded sets. Their bounded balls are strongly closed, because bounded strong convergence in a finite von Neumann algebra implies \(L^2\) convergence. They therefore form a von Neumann subalgebra. They are dense in the compact Hilbert subspace: the nearest-point projection in \(L^2(P)\) onto each closed operator-norm ball is equivariant and nonexpansive, so maps compact orbits to compact orbits, and these projections approximate every \(L^2\) vector.

Twisted conjugation on \(L^2(N)\ominus\mathbb C1\) has no nonzero finite-dimensional subrepresentation. Indeed, the diagonal of a finite-rank invariant orthogonal projection in the Fourier basis is constant on conjugacy classes: scalar conjugation phases disappear from the diagonal. The diagonal is summable. Every nonidentity conjugacy class is infinite, so all its entries vanish. Only the identity vector remains. The trace-preserving expectation \(E_{N^{\mathrm{op}}}:P\to N^{\mathrm{op}}\) is equivariant and \(L^2\)-contractive. It follows that \[E_{N^{\mathrm{op}}}(x)=\operatorname{tr}(x)1 \quad(x\in D_c).\] By the finite-index expectation inequality there is \(\varepsilon>0\) such that \(E_{N^{\mathrm{op}}}(x)\geq\varepsilon x\) for \(x\geq0\). Thus a nonzero projection \(q\in D_c\) has \(\operatorname{tr}(q)\geq\varepsilon\). A finite algebra with such a lower bound has only finitely many orthogonal nonzero projections, and is finite-dimensional.

Restrict \(\Lambda\) to the finite-index subgroup \(\Lambda_0\) fixing all the central blocks of \(D_c\). A central block \(z\) now reduces both endpoint actions. Write \[zK=V_c\otimes K'_c, \qquad zD_c=\mathcal B(V_c)\otimes1.\] The left action commutes with \(D_c\), hence is \(1\otimes\pi'_c\). Choose normalized matrix implementers \(v^c_h\) for \(\mathop{\mathrm{Ad}}\rho(h)|_{zD_c}\). Uniqueness of an implementer up to scalar gives \(v^c_hv^c_k=\nu_c(h,k)v^c_{hk}\), with \(\nu_c\) a normalized cocycle. After removing these implementers, \(\rho(h)\) belongs to the commutant of \(\mathcal B(V_c)\) and therefore has the asserted tensor form. Multiplication gives Equation (8).

Normality and finiteness of \(\pi'_c\) follow by taking a matrix corner of the finite normal left module. For the right action, first embed the original restricted \(\rho\) in a finite multiple of the \(\overline\omega|_{\Lambda_0}\)-regular representation. This is possible because finite-index restriction preserves finite module dimension. Tensor with \(\overline{v^c}\). Lemma  6 gives a finite normal module with multiplier \(\overline{\nu_c}\,\overline\omega\). In \(\overline V_c\otimes V_c\) the vector corresponding to the identity matrix is fixed by \(\overline{v^c_h}\otimes v^c_h\). Tensoring that line with \(K'_c\) exhibits \(\rho'_c\) as a summand of the preceding normal finite representation. This proves its normality for exactly the multiplier in Equation (8). Equivalently, its right factor has cocycle \(\omega|_{\Lambda_0}\nu_c\).

Finally, a compact vector in the reduced commutant embeds as \(1_{V_c}\otimes\xi\) on \(zK\) and as zero on the other blocks. It is compact for \(\Lambda_0\), and hence for \(\Lambda\): the latter orbit is contained in a finite union of translates of the former compact orbit closure. It therefore belongs to \(L^2(D_c)\). Its intersection with the reduced commutant on this block is scalar. This proves Equation (7). ◻

Proof. The \(u_r\) in Equation (9) are an orthonormal right basis for \(L^2(M)\) over the subgroup factor, because \[E_{L_{\mu|\Gamma_0}(\Gamma_0)}(u_r^*u_{r'}) =\delta_{r,r'}1.\] The analogous left-basis identity for the \(v_s\) is \(E_{L_{\omega|\Lambda_0}(\Lambda_0)}(v_sv_{s'}^*) =\delta_{s,s'}1\). The relative tensor-product inner products give the asserted isometries.

For explicit left equivariance, if \(gr=r'a\) with \(a\in\Gamma_0\), then \[u_gu_r=\frac{\mu(g,r)}{\mu(r',a)}u_{r'}u_a.\] Move \(u_a\) across the balanced tensor product. The identity \(u_{r'}^*u_g=(\mu(g,r)/\mu(r',a))u_au_r^*\) shows that the term is exactly the \(r'\)-term of \(J_L(\pi(u_g)\xi)\). The right calculation is the same calculation in the opposite algebra. Explicitly, when \(sh=at\) with \(a\in\Lambda_0\), one has \[v_sv_h=\frac{\omega(s,h)}{\omega(a,t)}v_av_t, \qquad v_hv_t^*=\frac{\omega(s,h)}{\omega(a,t)}v_s^*v_a.\] Balancing \(v_a\) proves right equivariance of Equation (10). The other endpoint action commutes with every operation. Thus the two maps compose to a bimodular isometry into two-sided induction, whose range is a closed summand. The finite basis formulas also show preservation of finite module dimensions.

If an element of \(\Gamma_0\) has finite conjugacy class in \(\Gamma_0\), its conjugacy class in \(\Gamma\) is a finite union of translates of that class. ICC of \(\Gamma\) forces the element to be the identity. This proves the last assertion. ◻

Proof. Embed the omitted slot in a finite regular multiple for its \(\overline\omega\)-projective right representation. At that regular index absorb the diagonal right actions of the remaining slots by Lemma 6. This use of absorption allows an infinite-dimensional auxiliary tensor space. The absorbing unitaries are formed from the other slots’ \(\rho(h)\) and therefore commute with all their \(\pi_j(g)\). After absorption, the \(\Lambda\) action is regular with multiplier \(\overline\omega^{\,m+1}\), and each remaining \(\pi_j\) can independently be embedded in a finite regular multiple for \(L_\mu(\Gamma_j)\). The resulting whole representation is a subrepresentation of a finite regular multiple for exactly Equation (11). The construction is symmetric in the omitted slot.

Write \(P=\pi(M)'\). The final joint commutant is \((P^{\overline\otimes(m+1)})^{\mathop{\mathrm{Ad}}\rho^{\otimes(m+1)}}\). Condition (7) makes this fixed algebra scalar. For completeness, an invariant tensor with a factor in \(L^2(P)\ominus\mathbb C1\) gives a Hilbert–Schmidt intertwiner between that factor’s representation and another unitary representation. The positive compact operator obtained by composing with its adjoint has a nonzero finite-dimensional invariant spectral space, contrary to weak mixing. Decompose each tensor factor as \(\mathbb C1\oplus(L^2(P)\ominus\mathbb C1)\) to apply this argument to every nonconstant summand. ◻

Proof. Translation is continuous in measure on measurable maps on the finite-measure quotient. Applied to the Borel section, this shows that \(c(a,y)\) varies continuously in measure as a discrete-valued map. One can verify the assertion first on compact sets where the section is continuous, by Lusin approximation, and then remove sets of arbitrarily small measure. The corresponding projective unitaries are bounded, so their multiplication operators, together with the base translations, vary strongly on \(\mathcal H\). Conjugation of the corner is inner at each fibre and translation preserves base probability. Thus \(\sigma\) preserves the displayed trace. The first fixed-algebra identity is simply the commutant of all of \(W\) in the decomposable algebra.

For the second identity, undo induction. If \(A(y)\) is a \(\sigma\)-invariant decomposable field, lift it, for \(g=d_0(y)\gamma\), as \[\widetilde A(g)=\pi_0(\gamma)^*A(y)\pi_0(\gamma).\] The cocycle covariance makes this field left \(G\)-invariant. It is therefore almost everywhere constant, with value \(T\), and right \(\Gamma\)-covariance says that \(T\) commutes with \(\pi_0(\Gamma)\). It also commutes with \(W|_{R'_j}\). Conversely every such \(T\) gives a fixed field. Consequently \((\mathcal A_j)^\sigma\) is the commutant of the complementary regular representation obtained by omitting slot \(j\). Lemma 9 makes this commutant finite; its normalized trace is \(\beta_j\)-invariant, since the remaining slot acts by unitaries in that commutant. Including all the slots and using the last assertion of that Lemma gives \(\mathcal N^G=\mathbb C1\). ◻

Proof. Use the Howe–Moore theorem for the noncompact isotropic almost simple root-generated groups and their root-generated lifts from Convention 11 (Howe and Moore 1979); an independent proof with this root-generated group convention is given in (Ciobotaru 2015, Theorem 1.1 and Definition 4.23). We first justify the type-I input for these actual groups. For an isotropic simple factor \(\mathbf H\) over \(k\), choose a nonzero parabolic unipotent radical \(\mathbf U\). The \(k\)-span of the \(\mathbf H(k)\)-conjugates of \(\operatorname{Lie}(\mathbf U)\) is a nonzero invariant Lie ideal, hence all of \(\operatorname{Lie}(\mathbf H)\). Choose a basis \(X_1,\ldots,X_d\) from these conjugate subspaces. The map \[(t_1,\ldots,t_d)\longmapsto\prod_{j=1}^d\exp(t_jX_j)\] has invertible differential at zero and takes values in \(\mathbf H(k)^+\). The local-field inverse function theorem makes this subgroup open; at finite places see (Glöckner 2018, Proposition 1.11). The same proof applies to the root-generated subgroup of an algebraic simply connected cover. At an archimedean place, the generating unipotent groups are connected, so the root-generated group is connected. Being open as well, it is the identity component of the ambient real algebraic group; for a complex group, use restriction of scalars to \(\mathbb R\). The same applies to its algebraic lift. These groups are therefore type I by (Dixmier 1957, Theorem 1). At a finite place, the ambient reductive group of local points is type I (Bernstein 1974), and type I passes to its open root-generated subgroup (Bekka and Harpe 2019, Proposition 6.E.21(1)).

Restrict a separable unitary representation \(\pi\) to one simple factor \(S\). Its type-I decomposition with multiplicity is \[\mathcal H=\int_{\widehat S}^{\oplus} (H_\xi\otimes M_\xi)\,d\eta(\xi),\qquad \pi(s)=\int_{\widehat S}^{\oplus}(\xi(s)\otimes1)\,d\eta(\xi).\] In this central decomposition, every unitary commuting with \(S\) acts fibrewise as \(1\otimes V_\xi\) (Dixmier 1977, Propositions 5.4.7 and 8.6.4, Theorem 8.6.6); see also (Folland 2015, Theorem 7.32). This is the decomposition of the restriction to \(S\), so no invariance of a finer decomposition of the other factors is required. For finite tensor sums in a nontrivial fibre, uniformly over the unitaries \(V_\xi\), \[\left|\left\langle(\xi(s)\otimes V_\xi) \sum_j e_j\otimes u_j,\sum_k f_k\otimes v_k\right\rangle\right| \leq\sum_{j,k}|\langle\xi(s)e_j,f_k\rangle| \|u_j\|\,\|v_k\|.\] Howe–Moore makes the right side tend to zero along escape. Approximation by finite tensor sums is uniform because all operators have norm one. For arbitrary fixed fibre vectors the bound \(\|a_\xi\|\,\|b_\xi\|\) is integrable. Dominated convergence therefore proves the stated uniform decay off the \(S\)-invariants.

Let \(P_i\) project onto the invariants of the \(i\)th simple factor. These projections commute with every group action and with each other. Expand the identity as the finite sum of the mutually orthogonal products obtained from \(\prod_i(P_i+(1-P_i))\). In every sector containing \(1-P_i\), the uniform decay just proved kills its coefficients when that factor escapes, even while all other factor translations vary. Thus for all-factor escape the weak limit is \(\prod_iP_i\), which is the projection onto constants by Proposition 10. For partial escape, expand only over the factors of \(G_0\); the other translations are the identity. The limit is the projection onto \(L^2(\mathcal N)^{G_0}\), namely the trace-preserving conditional expectation. The same argument applies to the simple Levi factors used later. ◻

Proof. At a real place use the Cartan density, a product of the appropriate powers of \(\sinh\alpha(L)\). The upper bound follows on every fixed bounded neighborhood. For the lower bound, even when \(L\) lies on a wall, choose inside the shell one fixed bounded offset box whose simple coordinates are bounded away from zero. On this box every positive-root factor is bounded below by a fixed multiple of its exponential. The box has fixed positive Euclidean volume.

At a finite place the Cartan double coset has measure \([K:K\cap aKa^{-1}]\) times the fixed compact measure. This index is comparable to \(\exp(2\rho(L))\). One can see the uniform comparison in compact root charts: a sufficiently small analytic compact open subgroup has product coordinates in negative-root, centralizer, and positive-root directions. Under conjugation by the dominant translation, its intersection is obtained by shrinking the expanding root lattices. The index is the product of their module indices, with exponent \(\sum_{\alpha>0}m_\alpha\alpha(L)\). Equivariant exponentials on small root lattices and parabolic product uniqueness give this calculation; changing the compact frame changes the small subgroups by a uniformly bounded index. Other Cartan representatives are within bounded distance of the chosen split mesh. This proves both bounds there. Multiplication of Haar measures gives the product assertion. ◻

Proof. Pass to convergent compact Cartan frames. In each highest-weight module, precisely the weights with maximal value on the open face survive projective normalization. They are the submodule generated from the highest line by the complementary Levi. Bounded complementary gaps keep the surviving block and its inverse bounded in those frames. Thus the surviving block acts by an invertible Levi transformation. The same analysis of the inverse matrix identifies the reverse face.

Equivalently, write an ordinary flag in the open Bruhat cell transverse to the reverse face. Conjugating its radical coordinates by the Cartan translation contracts the coordinates belonging to the diverging gaps; the complementary Levi coordinates remain. First project to the reverse refinement space, then apply the surviving Levi transformation. This is the intrinsic residual projectivity. A change of compact frame conjugates these coordinates by the corresponding compact Levi projectivity and leaves the intrinsic map unchanged. Passing to a deeper stratum preserves all preceding image vertices. In a face omitting one type in a higher-rank component, the surviving rank-one split displacement is exactly that omitted coordinate; bounded frames change its length by a uniformly bounded amount. The argument works simultaneously for one omitted type in several components.

In an archimedean module the surviving top block contains the highest lines of all incident flags, and those lines span it by the Levi action. At a finite place this is an algebraic statement about the \(k\)-module only; the later operator-algebra argument instead uses scalar clopen ranges. Exhausting buffered transverse patches gives the assertions on the full stratum. ◻

Proof. Write \(L_i=\alpha_i(y-x)\geq0\) and let \(T_t\) denote the linear map with simple coordinates \(t_iL_i\). For every positive root \(\alpha=\sum_i n_i\alpha_i\), where \(n_i\geq0\), \[0\leq\alpha(T_t(y-x)) =\sum_i n_it_iL_i \leq\sum_i n_iL_i=\alpha(y-x).\] Thus the new point belongs to every affine root half-apartment containing both endpoints: it lies in their root enclosure. Apartment intersections are enclosed. The apartment transition maps agree on this enclosure, and the Weyl stabilizer of a wall displacement also fixes its transformed displacement, since its zero simple coordinates stay zero. This proves independence.

On any apartment and Weyl sector, \(T_t\) has a uniformly bounded Euclidean operator norm, independent of the fractions. Subdivide a segment of varying endpoints into cells and Weyl sectors relative to \(x\), and choose apartments containing \(x\) and those cells. Local finiteness makes this a finite subdivision on a compact segment. Sum the Euclidean bounds and use apartment independence at the interfaces. Reversing the endpoints applies opposition and the complementary fractions, proving the other Lipschitz bound. ◻

Proof. Use the highest-line modules, their normalized matrices and their inverses from Section 3. At archimedean places normalize by the Hilbert–Schmidt norm; at finite places use projective ratio charts with a maximal coordinate. Lemma 15 identifies the surviving Levi blocks when the gap set is \(D\), proves their invertibility on a bounded complementary-gap window, and describes the intrinsic transverse projection followed by residual transport. In each fundamental highest-line coordinate, rank one detects divergence of the corresponding simple gap. These finitely many coordinates therefore record the gap set, its image and reverse flags, and the residual maps. If an additional gap diverges, the previous image vertices persist by that lemma. This proves the stratum and closure assertions in (i), and the residual-map assertion in (ii).

We verify the fixed-multiplication rule that will be used on Fourier indices. For a fixed matrix \(a\), both \(a\) and \(a^{-1}\) have finite norm. Thus projectively normalized multiplication by \(a\) has a denominator bounded away from zero. Applying this to every chosen module and its inverse shows that fixed multiplication changes each simple Cartan gap by a bounded amount. Left multiplication carries the image top space by \(a\), whereas right multiplication leaves it unchanged. Applying the same calculation to inverse matrices gives the reverse-flag rule. The residual map transforms by the corresponding compositions, since its description by transverse projection and surviving Levi transport is intrinsic. This proves (iii).

The last assertion is the spanning and length statement of Lemma 15: the Levi orbit of the highest line spans its surviving top space, and on a panel the only noncompact isotropic factor supplies the omitted simple gap. Compact changes of endpoint frame change that length by a bounded amount. At a finite place the modules remain vector spaces over the local field; no Hilbert-space amplification by them is involved. ◻

Proof. For a fixed shift, normalized matrix multiplication and the inverse formula have denominators bounded away from zero by Lemma 19. Their action on the compact label space is continuous. A diagonal continuous test therefore intertwines the shift in the ultralimit. Equality of the two representations on continuous functions implies equality of their spectral measures, so the assertion extends to Borel projections of the limiting variables. Base multiplication and the reindexing of the base commute with diagonal Fourier tests.

If a Fourier index \(h\) is changed to \(rh\) by a left multiplier, the recorded inverse is changed to \(h^{-1}r^{-1}\). Thus its gap set and image flag are unchanged in the limit. Let \(D\) be a bounded test of those data and \(T\) a bounded left Fourier matrix multiplier. Choose uniformly bounded finite Fourier approximants \(T_l\to T\) in trace \(L^2\). In a form tested on uncut operator-bounded columns, replacing either occurrence of \(T\) by \(T_l\) before moving \(D\) has error bounded by \[C\|D\|\,\|T-T_l\|_2,\] where \(C\) depends only on the fixed column bounds and matrix sizes. For each finite Fourier sum, move \(D\) through its individual shifts by the already proved limiting identity. The remaining product is the uncut Gram factor \(T_l^*T_l\). Replacing this by \(T^*T\) costs another quantity tending to zero, since \[\|T_l^*T_l-T^*T\|_2 \leq (\|T_l\|_\infty+\|T\|_\infty)\|T_l-T\|_2.\] All these errors are estimated before applying the cutoff to the columns. In particular the argument applies to a partial unitary with \(T^*T=e\) and does not assign an operator bound to \(Df\).

For a field reindexed at \(ay\), invariant probability on \(Y\) gives \(\int\|T(ay)-T_l(ay)\|_2^2dy=\|T-T_l\|_2^2\). The same estimates therefore apply after path transport. For a compact family of fixed translations, the cocycle formula and continuity of translations in measure make their matrix fields a compact family in trace \(L^2\). To verify this assertion directly, approximate the Borel section by continuous data on sets of large measure; along a convergent sequence of translations the discrete cocycle labels then agree eventually off sets of small measure. Unitarity bounds the remaining error. A finite net gives uniform finite-Fourier approximation of the compact family. No common finite-Fourier approximation of the escaping paths themselves is used.

Finally, the group-valued section cocycle satisfies its exact composition identity. Applying the projective source representation gives only a scalar function of the base in each composition formula. Pulling a fixed source translation from one end of an array to the other thus multiplies the whole array by one scalar of modulus one. It commutes with diagonal Fourier tests and cancels between the two entries of every sesquilinear form. The same observation applies to a quotient-path calculation: after base reindexing, \(X_tX_a^*\) is the transporter for \(ta^{-1}\) times a common scalar, so its localized Fourier norm is unchanged. Products in these identities are the actual twisted matrix products; their coefficient phases cannot be replaced by absolute values of untwisted convolutions. ◻

Proof. (i) One-ended covariance. Base multipliers and the companion actions commute with the relevant diagonal tests and intertwine the input column map. The compressed image effects therefore lie in \(\mathcal A_j\). A measured target shift changes the inverse Fourier label by the corresponding left geometric action. Lemma 20 consequently gives \(\beta_j\)-equivariance, with the action on functions given by composition with the inverse target element.

Let \(n\) be in the radical contracted by the path. Sliding \(U_n\) through \(U_a\) puts \(U_{ana^{-1}}\) at the far end, up to a common base phase. This translation tends strongly to the identity. In the section model its discrete cocycle is the identity off a set of measure tending to zero, after its pure base shift has been undone. On bounded inputs both descriptions give a vanishing Hilbert error, even with a bounded diagonal test inserted. This proves radical invariance of the full limiting data.

For a fixed centralizing source element \(s\), the corresponding identity slides \(U_s\) to the far end with the same \(s\). Its base reindexing has no effect on an integrated form, and its left corner multiplier can be removed for image-and-stratum tests by Lemma 20. Its uncut Gram factor is the corner identity. Thus the image measurement is centralizer invariant. The radical and this Levi centralizer generate the face parabolic. The same calculation for a conjugated split path removes the end conjugation and leaves precisely source transport of its image marginal.

The mass of a gap stratum is target invariant, hence belongs to \(\mathcal N\). It is invariant under the entire source face parabolic. Every inactive simple factor is in the centralizer. In an active factor the contracted radical contains an escaping sequence. Apply Howe–Moore to the representation of that factor on \(L^2(\mathcal N)\): a radical-invariant vector orthogonal to the factor-invariant subspace would have a matrix coefficient both constant and tending to zero. It must therefore be factor invariant. The mass is invariant under all simple factors and so is scalar, because \(\mathcal N^G=\mathbb C\). This argument applies to a pure vertex path as well as to a mixed path.

(ii) The mixed norm and its polarization. For bounded \(b,f\), tracial cyclicity and the definition of the induced action give \[\|J(\overline b,f)\|^2 =d\lim_a\mathsf T\bigl(ff^*\sigma_{a^{-1}}(bb^*)\bigr).\] On a mixed path, trace mixing makes the last expression \[d^2\tau_{\mathcal N}(ff^*)\tau_{\mathcal N}(bb^*) =\|f\|_2^2\|b\|_2^2.\] Polarization, or the same calculation on finite sums, gives \[\left\|\sum_iJ(\overline{b_i},f_i)\right\|^2 =d^2\sum_{i,l} \tau_{\mathcal N}(f_lf_i^*) \tau_{\mathcal N}(b_ib_l^*).\] This is exactly the norm on \(\overline{\mathcal H}\otimes\mathcal H\). The factor \(\sqrt d\) in (13) is essential for this normalization.

Multiplication of an input by a start base function is represented on arrays by that function at \(y\); an end base function is represented by its value at \(ay\). The diagonal tests commute with both. Their limiting forms are therefore decomposable at the two bases. Moving an independent target shift in either input uses the left or right Fourier transport formula and proves the two covariances. A source radical in either input is slid to its contracted translate in the other input, as above. The fixed translated input converges strongly, its column bound is preserved, and the common phase cancels. Simultaneous centralizing translations use the same exact precomposition and finite-Fourier Gram argument.

For an image-and-stratum event, the ignored end Fourier shifts put each end slice of the compressed effect in \(\overline{\mathcal N}\). Independent reverse-radical invariance makes this slice scalar by the factorwise Howe–Moore argument, since every factor is active. Thus the compressed effect has the form \(1\otimes C\). The frame computation below identifies \(C=\Phi(A)\). Reversing the two inputs proves the reverse statement.

(iii) The nonmixed form. For an escape in \(G_0\), the weak limit of the trace-preserving action on \(L^2(\mathcal N)\) is the projection onto \(L^2(N_0)\). Repeating the norm computation gives (14). For a finite sum the formula is \[ \left\|\sum_iJ(\overline{b_i},f_i)\right\|^2 =d^2\sum_{i,l}\tau_{\mathcal N} \bigl(E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)\bigr). \tag{16}\] It is positive because it is the limit of squared norms. Quotient by its null space and complete when a Hilbert form space is needed. Every covariance proved before the mixed norm calculation remains an equality of these Gram forms. In particular it can be used on the relative-boundedness domains below. The mass calculation in the first part of the proof is separate from this formula and does not replace \(N_0\) by the scalars.

(iv) Frame recovery. For a continuous test of the image and stratum, approximate each left multiplier \(b_r(ay)^*\) by bounded finite Fourier matrices. The inverse-index convention makes the test invariant under their left Fourier shifts. Lemma 20 therefore moves the test past these multipliers. Summing their uncut Gram factors gives \(\sum_r b_r(ay)b_r(ay)^*=e\). The two-column normalization contributes exactly \(d\), proving (15). Equality of the limiting spectral measures extends it to invariant Borel events. The proof used neither factorization of the end variable nor mixed trace mixing.

For a window involving residual or reverse data the left shifts may change the window, so that argument does not apply to the individual window. Suppose instead that all end vectors have been annihilated on each member of an exhaustion of the desired image-and-stratum event. Monotone spectral convergence first gives annihilation on that event, and only then may the finite frame be inserted. This proves the stated order of operations. The same proof covers extra limiting coordinates whenever they too are unchanged by the ignored shifts.

(v) Embedding independence and normal copies. An intertwiner between two regular embeddings is a bounded Fourier matrix on the relevant finite modules. For the image-and-stratum tests, its uncut Gram factor is the appropriate input corner identity. Lemma 20 consequently identifies the two compressed measurements.

For the normal-copy assertion, begin with a regular-multiple embedding for the first two slots and enlarge it by the entire extra slot. At the regular \(\Lambda\)-index \(h\), conjugate the extra slot by the projective inverse of its simultaneous \(\rho(h)\) action. Projective absorption transfers its multiplier to the regular \(\Lambda\) multiplier. This conjugation commutes with the extra \(\Gamma\)-action, which can now be embedded in its own finite regular multiple. Both operations commute with scalar Fourier cutoffs on the original measured \(\Gamma_j\)-coordinate. In the original input coordinates, the one-ended path acts only in the zeroth slot. Its compressed one-ended tests are therefore the original tests tensored with identity on the added slot. The embedding-independence just proved identifies this realization with any subsequent one. This proves the assertion for the measurements, and hence for the normal algebras and uniquely determined operators constructed from them. It makes no assertion that all uncompressed spectral arrays are embedding independent. ◻

Proof. Conditional expectation extends to positive \(L^1\) elements. Put \(A=E_{N_0}(ff^*)\). If \(A\) is bounded, (14) gives \[ \|J(\overline b,f)\|^2 \leq d\|A\|_\infty\|b\|_2^2. \tag{17}\] This proves the first continuous extension from bounded \(b\). The construction always takes the link limit before this extension; it does not define a finite-time product of two arbitrary Hilbert columns.

Let \(c\in N_0\). Along a path in \(G_0\), \(c\) is a fixed source-invariant left multiplier. It can be moved from the input to the end and removed from an image test by the same finite Fourier–Gram argument as a centralizer multiplier. Polarization shows that every such image effect commutes with \(c\). If a bounded operator \(Q\) commutes with \(N_0\), then, for positive \(c\in N_0\), \[\langle Qf,cQf\rangle \leq\|Q\|^2\langle f,cf\rangle.\] The characterization of conditional expectation by these tests implies \[E_{N_0}\bigl((Qf)(Qf)^*\bigr) \leq\|Q\|^2E_{N_0}(ff^*).\] This applies to the generated algebra and its support projections, and also after any fixed finite matrix amplification.

If a closed subspace is invariant under \(N_0\), the spectral projections \(p_n=1_{[0,n]}(E_{N_0}(ff^*))\) preserve it. The columns \(p_nf\) are relatively bounded and converge to \(f\), because \[\|(1-p_n)f\|_2^2 =d\tau_{\mathcal N} \bigl((1-p_n)E_{N_0}(ff^*)\bigr)\longrightarrow0.\] For a relatively bounded \(f\), ordinary left spectral truncation \(f_n=1_{[0,n]}(ff^*)f\) gives operator-bounded columns converging in Hilbert norm, with \(E_{N_0}(f_nf_n^*)\leq E_{N_0}(ff^*)\).

These approximants converge in the link form against a fixed Hilbert \(b\). Indeed first replace \(b\) by a bounded \(b_l\). For that bounded input, (14) bounds the error by \(d\|b_l\|_\infty^2\|f_n-f\|_2^2\). The tails \(b-b_l\) are uniformly controlled by (17), since \[E_{N_0}\bigl((f_n-f)(f_n-f)^*\bigr) \leq 4E_{N_0}(ff^*).\] Take \(n\to\infty\) first and then \(l\to\infty\). This argument, and its version with the two inputs reversed, extends all the bounded-input Gram identities to the stated domains.

Finally, suppose a gap event annihilates every one-column input. For a bounded end column in the two-column link, first approximate its left multiplier by a finite Fourier sum. Each fixed shift preserves the gap event in the limit, so each resulting term vanishes. The approximation error is estimated on the uncut bounded columns by Lemma 20. Taking the approximation limit proves the same inclusion for bounded pairs, with no use of a mixed tensor decomposition. ◻

Proof. For a fixed Fourier shift, the assumed asymptotic identity of bounded tests gives precisely the limiting intertwining identity used in Lemma 20. Its finite-Fourier approximation proof is unchanged. Base decomposability and commutation with companions follow before taking limits. An invariant one-column test therefore gives an element of \(\mathcal N\). The source face-parabolic invariance, followed by the active-factor radical argument and the inactive-factor centralizer argument in Proposition 21, puts this element in \(\mathcal N^G=\mathbb C\).

For mixed links, independent target invariance puts the respective slices in \(\overline{\mathcal N}\) and \(\mathcal N\). Independent contracted-radical invariance makes each slice scalar; polarization and normal slice separation give the scalar joint effect. This reasoning works for a submeasurement with additional spectral slots when its indicated event has these invariances.

For example, fixed left and right multiplications change Cartan coordinates by bounded amounts. If a radial variable is divided by a scale tending to infinity, bounded continuous tests in its chosen compactification are asymptotically unchanged whenever this bounded displacement tends uniformly to zero in that compactified coordinate. For the measurements obtained by adjoining such rescaled radial coordinates to the original face data, the needed source face-parabolic invariance also follows from the construction. A contracted radical preserves the entire enlarged data law: after sliding it to the far end, the near-identity transporter changes the uncut bounded input by a Hilbert-small error, uniformly against all bounded diagonal tests. A fixed centralizer element, including an element of an inactive simple factor, can be slid to that same end. Its base reindexing is removed by invariant probability, and its left multiplier is removed by the finite-Fourier Gram argument. The additional radial test causes no extra term in this argument, because each fixed Fourier shift changes its limiting coordinate by zero. Thus image-and-stratum tests together with these radial coordinates are automatically face-parabolic invariant. In particular their target-invariant masses are scalar even on a pure vertex ray. This conclusion does not require that the nonmixed link norm factor through scalars.

The argument supplies the required invariant limiting tests; no derivative estimate for an arbitrary finite-scale discontinuous mask is asserted. Establish the continuous determining identities first. Their spectral measures then transport identically, giving the assertion for invariant Borel events of the limiting variables. ◻

Proof. Write the projective source multiplier as \(\eta\). The regular representation with multiplier \(\overline\eta\), induced with the same section, has the conjugate of the scalar defect of \(X_a(y)=\pi_0(c(a,y))\). The product of these two fibre transporters consequently satisfies the exact group composition law. The section continuity-in-measure argument gives strong continuity of the resulting induced representation.

The added coordinate is unmeasured. At one base, its two transported vectors have exactly their original inner product, so a base-decomposable compressed form is tensored pointwise with the identity. For a two-ended form the two base algebras commute with its effect, and the same construction is performed independently at the two inputs. The generator identification of minimal spectral dilations gives the corresponding enlarged dilation. In particular a face spectral measure already invariant under the projective transport now commutes with a genuine source representation and can be disintegrated with that representation. All mixing and scalarity statements continue to refer to the original finite algebra \(\mathcal N\); none is transferred to the enlarged commutant. ◻

Proof. We first record the finite-module Bessel estimate used here. For a bounded column \(b\in eL^2(B)^k\) and a Hilbert column \(\xi\), Parseval in the regular coordinates gives \[\sum_{h\in R}|\langle b w_h^*,\xi\rangle|^2 =\|b^*\xi\|_2^2 \leq\|b\|_\infty^2\|\xi\|_2^2.\] The harmless projective phases do not change the squared absolute values. The same bound holds for a subset of translates, in particular the \(R'_j\)-translates. In finite matrix versions one sums this identity over the fixed row and column indices.

For bounded end and start columns, the squared norm of a coefficient at a fixed measured index \(g\in\Gamma_j\) in \(b(ay)^*X_a(y)f(y)\) is the sum over \(R'_j\) of squared frame coefficients of the pair. Equivalently these are inner products of \(X_a(y)f(y)\) with the corresponding partial translates of \(b(ay)\), with \(g\) absorbed into one of the columns. This is just Parseval in the remaining group coordinates, and remains true with their twisted coefficient phases.

Use now the complementary regular realization for \(\Gamma_0\times R'_j\). Project each column to a finite \(\Gamma_0\)-coordinate window. These projections commute with the partial \(R'_j\)-translates, so the Bessel bounds are preserved. Source transport is unitary and commutes with those partial translates, and the fixed measured shift likewise preserves their bounds. Both projected columns converge in integrated Hilbert norm to the original columns as the finite windows increase. The Bessel estimate on the other column bounds the sum of squared coefficient errors by that Hilbert error. Interchanging the two columns gives the estimate for the second error. Thus the finite-window approximation is uniform in the path parameter. Reindexing the end column at \(ay\) does not change its averaged Hilbert error.

For fixed finite windows, the translated supports are disjoint unless the source cocycle \(c(a,y)\) belongs to a fixed finite subset of \(\Gamma_0\). That exceptional event has measure tending to zero. Indeed, for any \(\delta>0\), choose a compact set containing the section values \(d_0(y)\) outside a set of measure \(\delta\). Invariance of quotient probability gives the same assertion for \(d_0(ay)\). On the intersection of the two good sets, if \(c(a,y)\) belongs to a fixed finite set \(L\), then \[a=d_0(ay)c(a,y)d_0(y)^{-1}\] belongs to one fixed compact subset of \(G\). This is impossible eventually on an escaping path. The exceptional measure is at most \(2\delta\), and \(\delta\) is arbitrary.

Off that exceptional set, every partial-translate coefficient of the finite-window pair is zero. On it, their summed squares are uniformly bounded by the Bessel estimate and the column bounds. Integration gives convergence to zero. Remove the finite-window approximation afterward. We have proved vanishing of the norm at every fixed measured \(\Gamma_j\)-index.

A bounded Cartan set is compact in the whole ambient product, so contains only finitely many elements of the discrete lattice. Summing the preceding conclusion gives zero link mass on each bounded Cartan window. Increasing those windows removes the all-bounded stratum. This is a count in the whole lattice and requires no discreteness of a projection to a proper subproduct. Finally apply (15), with its finite corner frame, to obtain the one-column assertion. ◻

Proof. Write \(Z=G/P\) for the full target flag space. Suppose that \(\Phi:C(Z)\to\mathcal A_j\) is such a map. Its mass \(a=\Phi(1)\) belongs to \(\mathcal N^S\): target equivariance makes it commute with the measured target action, and membership in \(\mathcal A_j\) gives the remaining commutations. Choose \(\epsilon>0\) such that \(p=1_{[\epsilon,\infty)}(a)\ne0\). On the reducing space \(p\mathcal H\), the formula \[\Psi(u)=p a^{-1/2}\Phi(u)a^{-1/2}p\] defines a unital completely positive map with the same covariances. A positive map from the commutative algebra \(C(Z)\) is completely positive. The displayed normalization does not require \(a\) to commute with \(\Phi(u)\).

Choose a discrete free subgroup \(F\cong\mathbb F_2\) in \(S\). One obtains it from two sufficiently strong rank-one contractions with disjoint attracting and repelling neighborhoods; at a finite place the same construction uses translations in the rank-one tree. Passing through a finite central cover leaves a free subgroup after choosing the generators in that cover. Let \(\mathcal D\) be the concrete \(C^*\)-algebra on \(p\mathcal H\) generated by the cut base multipliers, the companion actions \(W|_{R'_j}\), and \(U_f\), \(f\in F\). These operators commute with both the measured target action and every value of \(\Psi\).

We first obtain a completely contractive minimum-tensor-norm estimate. We use the minimal Stinespring construction and its commutant lifting (Stinespring 1955) and (Arveson 1969, Theorems 1.1.1 and 1.3.1). In the minimal construction of \(\Psi\), the algebraic vectors are \(u\otimes\xi\), with form \[\left\langle\sum_i u_i\otimes\xi_i, \sum_l v_l\otimes\eta_l\right\rangle =\sum_{i,l}\langle\xi_i,\Psi(\overline{u_i}v_l)\eta_l\rangle.\] The lift of \(D\in\mathcal D\) is \(\widehat D(u\otimes\xi)=u\otimes D\xi\). Commutation with the matrix of values of \(\Psi\) gives \(\|\widehat D\|\leq\|D\|\), with the concrete input norm on the right. Thus the companion representation is controlled before dilation; no larger norm on a newly presented companion algebra is introduced. If \(W_gW_h=\zeta(g,h)W_{gh}\) in the measured copy, the covariant lift \[T_g(u\otimes\xi)=(g\cdot u)\otimes W_g\xi\] has the same multiplier \(\zeta\).

The action of \(\Gamma_j\) on \(Z\) is topologically amenable. By (Anantharaman-Delaroche 2002, Proposition 2.5), choose continuous fields \(\eta_i:Z\to\ell^2(\Gamma_j)\) with finite coordinate support, whose norms tend to one and whose equivariance errors tend to zero uniformly on \(Z\) for each fixed \(g\in\Gamma_j\). Compactness of \(Z\) makes these norms uniformly positive eventually; normalizing pointwise gives continuous unit fields \(\xi_i(z)=\eta_i(z)/\|\eta_i(z)\|\) with the same uniform asymptotic equivariance. If \(\pi_Z\) is the spectral representation of \(C(Z)\) in the dilation, set \[Q_i\eta=\sum_{t\in\Gamma_j}\delta_t\otimes \pi_Z(\xi_i(\cdot)_t)\eta.\] The maps \(Q_i\) are isometries. They intertwine the companion algebra exactly, and \(Q_iT_g-(\lambda_g\otimes T_g)Q_i\) tends to zero in operator norm for each fixed \(g\). Projective absorption is explicit: the unitary \[A(\delta_t\otimes\eta)=\delta_t\otimes T_t^*\eta\] conjugates \(\lambda_g\otimes T_g\) to \(\lambda_{\zeta}(g)\otimes1\). It leaves the companion representation on the second factor unchanged. Taking limits and restricting to the original input proves, at every finite matrix level, \[ \left\|\sum_g W_gD_g\right\| \leq \left\|\sum_g\lambda_{\zeta}(g)\otimes D_g\right\|_{ C^*_{r,\zeta}(\Gamma_j)\otimes_{\min}\mathcal D}. \tag{18}\] This also fixes the multiplier convention: the multiplier in this formula is that of \(W\), and is conjugated if one instead writes the right action with the adjoint generators of \(B\).

Complementary regularity makes \(H\) a subrepresentation of a finite regular multiple for \(\Gamma_0\times R'_j\), with split multiplier. On inducing the \(\Gamma_0\) coordinate, the identification \[(y,h)\longmapsto d_0(y)h\] turns it into the \(G\) coordinate. In these coordinates \(U_f\) acts as left translation on \(G\), times a scalar function of modulus one; the projective regular multiplication only contributes that scalar. The companions \(W|_{R'_j}\) use a separate twisted regular coordinate. Tensoring with the separate measured regular representation in (18), a simple-base finite Fourier matrix field from the full split group algebra of \(R\) acts fibrewise with its original matrix norm.

Consequently, for contraction fields \(C_f\), scalar phases \(\theta_f\), and a finitely supported probability \(\nu\) on \(F\), the operator in this spatial model satisfies \[\left\|\sum_{f\in F}\nu(f)C_f\theta_f L_f\right\| \leq\|\lambda_F(\nu)\|.\] Indeed, pointwise Hilbert norms are bounded by \(\sum_f\nu(f)\|\eta(f^{-1}g)\|\). Disintegrating Haar measure along the left \(F\)-orbits in \(G\) gives the ordinary positive regular convolution bound on each orbit. There is no commutation assumption between \(C_f\) and \(L_f\). For the concrete cut representation this reasoning is applied to the indicated generator polynomials and then restricted to their invariant input subspace; it does not require an extension of every abstract element of the cut companion algebra to the enlarged regular space.

We contradict this estimate on one matrix vector. Regard the matrix \(E(y)=p(y)\) as \(k\) columns, and put, with \(x=f^{-1}y\), \[T_f(y)=p(x)X_f(x)^*p(y).\] This is a contraction. Since \(p\) is \(\sigma_f\)-invariant, \(X_f(x)p(x)X_f(x)^*=p(y)\), and right multiplication after the shift gives \[(R_{T_f}U_f^{(k)}E)(y) =X_f(x)p(x)\,p(x)X_f(x)^*p(y)=p(y).\] Right matrix multiplication has the opposite convention \((R_T)_{\beta\alpha}=R_{T_{\alpha\beta}}\); the transpose of matrix indices implements \(M_k(B)^{\mathrm{op}}\cong M_k(B^{\mathrm{op}})\). In particular it preserves the contraction norm required by the spatial estimate.

For the finitely many \(f\) in the support of \(\nu\), choose contraction fields \(T_{f,n}\), simple in the base and finite in Fourier support, with \(T_{f,n}\to T_f\) in integrated tracial \(L^2\). To see the contractive approximation directly, first approximate in the unit ball by simple-base fields, use Kaplansky density of the reduced group algebra in its group von Neumann algebra, then approximate each reduced-algebra entry in operator norm by a Fourier polynomial and rescale by its at most \(1+o(1)\) norm. On the bounded matrix vector just used, \[\|(R_{T_{f,n}}-R_{T_f})U_f^{(k)}E\|_2 \leq\|T_{f,n}-T_f\|_2.\] Thus \(S_n=\sum_f\nu(f)R_{T_{f,n}}U_f^{(k)}\) satisfies \(S_nE\to E\ne0\). On the other hand, (18) and the spatial estimate give \(\|S_n\|\leq\|\lambda_F(\nu)\|\). For the uniform walk on two free generators and their inverses this norm is at most \(\sqrt3/2<1\), the classical free-group estimate of (Kesten 1959, Theorem 3). For the needed upper bound, the weight \(3^{-|h|/2}\) on the four-regular tree gives this bound by the weighted Schur estimate. The two conclusions are incompatible because \(0<\|E\|_2^2=\mathsf T(p)<\infty\). ◻

Proof. Its scalar mass is some \(m_D>0\), by Proposition 21. Divide the stratum measurement by \(m_D\), and denote the resulting unital full-data measurement by \(\Psi\). If \(D=\Pi\), its image measurement is a nonzero full-flag map fixed by the remaining source component \(S\), contradicting Lemma 26. Thus in the situation under consideration \(D\ne\Pi\).

The marginal inclusion. Let \(\Psi(F)=V_\Omega^*M_FV_\Omega\) be a minimal full-data dilation, and let \(x:\Omega_D\to X_D\) be its image-flag map. Write \(V:\mathcal H\to\mathcal K_X\) for the minimal image dilation. The generator assignment \[M_uVf\longmapsto M_{u\circ x}V_\Omega f\] preserves inner products. It extends to an isometric inclusion \(i:\mathcal K_X\to\mathcal K_\Omega\), with \(iV=V_\Omega\). This inclusion intertwines the image spectral measure, base multiplication, all companions, and the canonical target lifts. Disintegrating over the image variable gives \(\mathcal K_X=\int^\oplus K_x\,d\vartheta(x)\) and isometric fibre inclusions \(i_x\). The base variables and their commuting decompositions are retained in this notation.

Positive density forms. Let \(\mathcal F_x\) be the compact full-refinement space of the image face \(x\), with compact-rotation probability \(\eta_x\). The residual projectivity of a full datum \(\omega\in\Omega_D\) transports the compact-rotation probability on its reverse refinement space to a probability on \(\mathcal F_x\). Write this probability as \[k_\omega(l)\,d\eta_x(l),\qquad k_\omega(l)>0,\qquad \int_{\mathcal F_x}k_\omega(l)\,d\eta_x(l)=1.\] On a compact data window \(C\subset\Omega_D\), the density has bounds \(0<c_C\leq k_\omega(l)\leq C_C<\infty\), uniformly in \(\omega\in C\) and \(l\in\mathcal F_x\). Indeed the residual transformations and their inverses range in a compact set on such a window. Their Jacobians against the compact-rotation measures are positive and uniformly bounded. At a finite place the same conclusion uses the analytic Haar densities in finitely many compact-open charts and bounded absolute Jacobians. No real differential operator is used there.

For fixed \((x,l)\), define bounded increasing operators \[D_{n,x}(l)=i_x^*M_{\min(k_\omega(l),n)}i_x.\] Their limiting quadratic form is \[q_{x,l}(v)=\|M_{\sqrt{k_\omega(l)}}i_xv\|^2.\] This form is closed: multiplication by \(\sqrt{k_\omega(l)}\) is closed and its composition with the isometric inclusion \(i_x\) is closed. Its domain is dense for almost every \((x,l)\). To see this, choose countably many input vectors \(f_r\) such that the sections \(Vf_r(x)\) are total in \(K_x\) almost everywhere. Indeed, the measurable projection onto their fibrewise orthogonal complement kills every angular multiple of \(V\mathcal H\); minimality of the image dilation makes that projection zero. Tonelli and the normalization of \(k\) give \[\int_{X_D}\int_{\mathcal F_x} q_{x,l}(Vf_r(x))\,d\eta_x(l)d\vartheta(x) =\|f_r\|^2.\] Thus their span lies in the form domain almost everywhere, on one common conull set. Let \(D_x(l)\) be the associated positive self-adjoint operator. The bounded measurable approximating forms above, equivalently their resolvents, give its measurable dependence. For every \(v\in K_x\), integration of the nonnegative form yields \[\int_{\mathcal F_x}q_{x,l}(v)\,d\eta_x(l)=\|v\|^2.\] Consequently \[ (Tf)(x,l)=D_x(l)^{1/2}(Vf)(x) \tag{19}\] is a bounded isometry from \(\mathcal H\) to \[\mathcal R= \int_{X_D}^{\oplus}L^2(\mathcal F_x,\eta_x;K_x)\,d\vartheta(x).\]

Equivariance and source lifts. For a target element \(\gamma\), set \[r_{\gamma,x}(l')= \frac{d(\gamma_*\eta_x)}{d\eta_{\gamma x}}(l').\] Transport of the residual probability gives \(k_{\gamma\omega}(\gamma l) =r_{\gamma,x}(\gamma l)k_\omega(l)\). Since \(i\) intertwines the canonical target lifts, the compressed density forms have the same covariance. If \(u_{\gamma,x}:K_x\to K_{\gamma x}\) is the fibre-unitary part of the image lift, their operator form is \[D_{\gamma x}(\gamma l) =r_{\gamma,x}(\gamma l)\, u_{\gamma,x}D_x(l)u_{\gamma,x}^*.\] The target lift on \(\mathcal R\) includes the half-density \(r_{\gamma,x}^{1/2}\), in addition to any factor already present in the image direct integral. Taking square roots proves that \(T\) intertwines the target action. Base and companion operators commute with full-data multiplication and intertwine \(i\), so \(T\) intertwines them as well. Thus \(T\) intertwines base multiplication and all of \(W_R\).

For a source centralizer element \(s\), invariance of the image marginal gives a canonical lift on its minimal dilation, defined on generators by \[U_s^X(M_uVf)=M_uVU_sf.\] It fixes the image coordinate and is strongly continuous, first on these generators and then by density. Extend it to \(\widetilde U_s\) on \(\mathcal R\) by its action on the \(K_x\) fibre, with the induced base reindexing, and the identity on \(l\). It commutes with multiplication by the full refinement flag. These lifts use only the image marginal; no lift preserving the full-data law is being assumed.

The finite-trace intertwiner space. Let \(\mathscr E_0\) be the bounded intertwiners from \(\mathcal H\) to \(\mathcal R\) for base multiplication and all target actions \(W_R\). For \(B,C\in\mathscr E_0\), \(B^*C\in\mathcal N\), so define \[\langle B,C\rangle_{\mathscr E} =\tau_{\mathcal N}(B^*C)\] and take the Hilbert completion \(\mathscr E\). There are two useful exact finite-trace estimates: \[ \|B\|_{\mathscr E}^2 =\frac1d\sum_{r=1}^k\|Bb_r\|^2, \qquad \|Bf\|^2\leq d\|f\|_\infty^2\|B\|_{\mathscr E}^2 \quad(f\text{ operator-bounded}). \tag{20}\] Both follow by testing \(B^*B\) against the corner frame, or against the bounded positive matrix \(ff^*\), respectively.

Source conjugation \(\mathscr U_sB=\widetilde U_sBU_s^*\) is isometric for this norm, because \(\sigma_s\) preserves \(\tau_{\mathcal N}\). Its base-valued projective defects cancel since \(B\) intertwines the base algebra. Hence it is a genuine unitary representation on \(\mathscr E\). Strong continuity follows from the frame equality in (20), strong continuity of the two source lifts, and boundedness of \(B\), then extends by density to the completion.

Here \(S\) is the root-generated Levi group, or its root-generated lift from Convention 11; invariant spaces are taken for that group. This representation has no \(S\)-invariant vector. If it did, choose a bounded intertwiner \(B\) whose orthogonal projection onto the invariant subspace is nonzero. The closed convex hull of its orbit contains that projection. Finite convex combinations \(B_n\) therefore converge to it in \(\mathscr E\), and \(\|B_n\|_{\mathrm{op}}\leq\|B\|_{\mathrm{op}}\). The bounded-column estimate in (20) gives strong convergence on bounded columns. Their density and the common operator bound give a bounded limiting operator \(B_\infty\). It still intertwines the base and target actions, represents the nonzero invariant vector, and is \(S\)-fixed. The map \[u\longmapsto B_\infty^*M_{u(l)}B_\infty\] is then a nonzero positive full-flag map into \(\mathcal A_j\), target-equivariant and \(S\)-invariant. This contradicts Lemma 26.

Howe–Moore now gives weak decay of translates of \(T\) along escape in \(S\), also when multiplied by the additional commuting translations. For precision, disintegrate the representation into type-I irreducibles of \(S\) with multiplicity. There is no trivial summand. Commuting unitaries act on the multiplicity spaces. Approximation by finite tensor sums reduces the assertion to finitely many matrix coefficients of the irreducible \(S\)-representations, which tend to zero uniformly over the multiplicity unitaries. Dominated convergence finishes the argument.

The positive compact-window comparison. For a compact window \(C\subset\Omega_D\), let \(L_C=i^*M_{1_C}i\). This is a decomposable positive contraction on \(\mathcal K_X\), and \(V^*L_CV=\Psi(C)\). Its density lower bound gives \(D_x(l)\geq c_C L_C(x)\) in form order. Operator monotonicity of the square root, followed by \(L_C^{1/2}\geq L_C\), yields \[ D_x(l)^{1/2}\geq\sqrt{c_C}\,L_C(x). \tag{21}\] One may obtain this inequality directly from the bounded \(D_{n,x}(l)\) for \(n\geq c_C\) and then pass to their closed-form limit.

Write the extended path as \(a_t b_s\), with the inner \(t\)-limit first. The exact cocycle formula and the change of base variable \(z=b_sy\) show that the inner full-data law is precomposed by \(U_{b_s}\): \[\Psi_s(F)=U_{b_s}^*\Psi(F)U_{b_s}.\] Its image marginal is unchanged. The appropriate translated intertwiner is therefore \(T^{(s)}=\widetilde U_{b_s}^*TU_{b_s}\). Because the marginal lift fixes \(x,l\), its fibre expression is the positive operator \((U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X\) applied to \(Vf\), with base reindexing understood. Inequality (21) remains valid after this conjugation and bounds the correspondingly translated \(L_C\).

Put \((J_0f)(x,l)=Vf(x)\), the constant-refinement section. For bounded \(f\), the functional \(B\mapsto\langle Bf,J_0f\rangle\) is continuous on \(\mathscr E\), by (20). It is unnecessary that \(J_0\) itself intertwine the target action. Weak decay and the positive inequality give \[0\leq\sqrt{c_C}\,\langle f,\Psi_s(C)f\rangle \leq\langle T^{(s)}f,J_0f\rangle\longrightarrow0.\] The right side is real and nonnegative by its positive fibre expression. Density of bounded columns and \(0\leq\Psi_s(C)\leq1\) extend the vanishing to every input. This proves disappearance of the actual translated window mass, rather than merely weak convergence of an unrelated auxiliary operator.

The entire stratum contribution. Each precomposed law \(\Psi_s\) has mass one and takes values on \(\Omega_D\). Take its outer limit on \(\overline{\Omega_D}\). Every continuous test compactly supported in \(\Omega_D\) has zero limiting effect by the window estimate. A countable compact-window exhaustion therefore shows that the outer law gives no mass to \(\Omega_D\). Lemma 19 places its remaining support in \[\overline{\Omega_D}\setminus\Omega_D \subseteq\bigcup_{D'\supsetneq D}\Omega_{D'}.\] Restoring the scalar factor \(m_D\) proves the assertion for the entire original positive contribution. There are only finitely many gap sets, so the same outer limit tracks all of their contributions. Further subpieces are dominated by these positive total measurements. At the next stage the covariance and scalar-mass statements apply anew to each positive total stratum of the enlarged source face; invariance of the original full-data law was never needed. ◻

Proof. Put \(r=|\Pi|\), and start with a positive target stratum \(D\) on a source face of type \(I\). Append the \(r-|I|\) missing source coordinates at successively slower scales. At each stage the fundamental split vector for a missing root has a nonzero projection to the remaining-Levi component containing that root; its other split parts commute with that component. Thus Lemma 27 applies at each stage. If a full target stratum were reached before the source face were complete, the obstruction in that lemma would already give a contradiction. Otherwise the tracked positive contribution must pass through a strictly increasing chain of gap sets of length \(r-|I|\). Since a target has only \(r\) simple types, \[|D|+r-|I|\leq r,\] which proves \(|D|\leq|I|\). If \(I=\Pi\), this is just the automatic ceiling \(|D|\leq r\), and no remaining-Levi obstruction is invoked.

For a vertex, Lemma 25 excludes zero gaps and the upper bound excludes two or more. The same strict-growth argument, applied successively from that vertex, gives at least \(k\) gaps at stage \(k\), whereas the upper bound gives at most \(k\). This proves the exact lex counts. More generally, an extension from a stage already attaining its upper bound has an exact increment of one. No exact-increment claim from an arbitrary unsaturated simultaneous multi-gap initial path is needed.

Finally choose the common apartment for the two vertex translations and make the faster vertex a dominant fundamental ray \(v\), of type \(j\). If the slower vertex is coaxial, their quotient is again a vertex translation at the faster scale, possibly with the opposed orientation, so the vertex conclusion applies. If its vector is \(w\) and is not coaxial, then it has a nonzero projection to a simple factor of the remaining Levi. Indeed the common kernel of the remaining simple roots \(\Pi\setminus\{j\}\) is precisely \(\mathbb Rv\). Thus a zero projection would force \(w\) to lie on that line, which is the coaxial case. After choosing a dominant chamber inside the remaining Levi, the slower quotient component therefore escapes in a noncompact simple factor; the other split components commute with it. Strict growth from the one-gap faster limit forces at least one additional target gap. All these statements retain the faster-first order of limits. ◻

Proof. We separate the geometric and Fourier parts of the argument.

Common apartments and addition. On compact transverse patches, the two ideal faces admit common apartment or common flat sections whose origins remain in a bounded set. After erasing the bounded thickness, write the two endpoints in such coordinates as \[po=z+A+O(1),\qquad q^{-1}o=z-B+O(1),\] with \(A,B\) in the specified positive faces and \(z\) uniformly bounded. The vector distance between these endpoints is \(A+B+O(1)\). The vector-distance inequality under bounded endpoint perturbations therefore proves the Cartan addition assertion. This argument is made in the whole ambient product; it uses no discreteness of a proper projection of the lattice.

We need the following stability of interpolation. From a fixed endpoint, multiply the simple coordinates \(L_i\) of a dominant vector by arbitrary fractions \(t_i\in[0,1]\). The resulting interpolation is uniformly Lipschitz in both endpoints. In an archimedean polar chart, the angular differential in a positive-root direction is multiplied by \[\frac{\sinh\bigl(\alpha(t\cdot L)\bigr)} {\sinh\bigl(\alpha(L)\bigr)}\leq 1,\] and the radial differential is the uniformly bounded linear map \(L\mapsto t\cdot L\). A wall stabilizer fixes the interpolated vector because zero simple coordinates remain zero. Thus the map extends continuously across the walls; approximation of paths by regular paths gives the same Lipschitz bound there. Reversing the endpoints uses opposition and the complementary fractions.

For a finite-place building, the corresponding apartment-independent interpolation and its uniform Lipschitz bounds in both endpoints are Lemma 17. The product of these real and finite-place bounds gives the assertion for the whole ambient space.

The common shift. Suppose \(qp=q'p'\) and set \(k=p(p')^{-1}\). Then \(k\) sends the primed endpoint pair to the unprimed pair. Write the primed coordinates as \(z'+A',z'-B'\), with the same bounded errors. Cartan addition gives \[A+B=A'+B'+O(1).\] Interpolate from the negative endpoint using the fractions \[t_i=\frac{B_i'}{A_i'+B_i'},\] with any choice in \([0,1]\) when the denominator vanishes. These fractions return \(z'\) in the primed model. The Lipschitz bound and equivariance of interpolation yield \[ ko=z+B'-B+O(1). \tag{24}\] There is no small-denominator loss: the fractions are bounded and are held fixed when applying the Lipschitz estimate.

In case (i), every overlapping coordinate has bounded variation on at least one side. The relation between \(A+B\) and \(A'+B'\) then bounds the corresponding coordinate of \(B'-B\). Outside the overlap, one side is bounded by the thickness, so those coordinates are bounded as well. Thus \(k\) belongs to a fixed compact subset of the whole product \(G\). Its intersection with \(\Gamma\) is finite, uniformly in the locations of the moving width intervals.

In case (ii), an unbounded difference can occur only in the single overlap coordinate \(i\); if there is no overlap we are already in the bounded-shift case. Put \(\delta=B'_i-B_i\). All other coordinates of \(B'-B\) are bounded, and \(A_i-A'_i=\delta+O(1)\). The collision identity gives \[p=kp',\qquad q'=qk,\qquad (q')^{-1}=k^{-1}q^{-1}.\] Apply Equation (24) also to the reversed pair. In the unprimed and primed apartments, respectively, it gives \[ko=z+\delta e_i+O(1),\qquad k^{-1}o=z'-\delta e_i+O(1).\] The origins and both errors are uniformly bounded. Let \(U\) be the patch of type-\(i\) vertices incident to the positive endpoint patch, and \(V\) the patch of type-\(i^*\) vertices incident to the negative endpoint patch. Choose buffered larger patches \(U^+,V^+\) whose every pair is opposite. For sufficiently large positive \(\delta\), the forward vertex of \(k\) lies in \(U^+\) and its reverse vertex, the forward vertex of \(k^{-1}\), lies in \(V^+\). These conclusions follow uniformly from the bounded-distance ray formulas. For large negative \(\delta\) the memberships reverse. The larger patches can be chosen disjoint in the visual boundary, so a fixed sufficiently large \(k\) cannot have both orientations. Bounded \(\delta\) gives bounded \(k\), already treated above. The negative ray has the opposed type \(i^*\); the two types need not coincide.

In the positive orientation the shorter arguments are \(p'\) and \(q^{-1}\), and the length-increasing substitutions are \[p'\longmapsto kp',\qquad q^{-1}\longmapsto k^{-1}q^{-1}.\] The reverse vertex of the first multiplier is in \(V^+\), transverse to the entire buffered positive input patch. The reverse vertex of the second is in \(U^+\), transverse to the entire negative input patch. Thus the witness for \(k\) gives uniform transversality on both whole input patches. The negative orientation interchanges the primed and unprimed indices and replaces \(k\) by \(k^{-1}\).

For clarity, let \(R\) be one of the fixed finite-dimensional representations and put \(S(g)=R(g)/\|R(g)\|\). On either substitution \(r\mapsto ur\), the transverse Cartan-addition estimate applies to every \(r\) in the buffered initial patch. Highest-weight norm comparison gives \[\|S(u)S(r)\|\geq c>0,\qquad \|S(r^{-1})S(u^{-1})\|\geq c>0,\] uniformly in the supported \(u,r\). The second bound applies opposition to Cartan addition and uses \((ur)^{-1}=r^{-1}u^{-1}\). The inverse coordinates are the separately normalized matrices of \(g^{-1}\), so no inverse of a possibly singular limiting matrix is being taken. There are finitely many chosen modules, and the buffer preserves a common lower bound on fixed larger coordinate boxes. Normalized multiplication consequently has uniformly bounded smooth derivatives on those boxes. At a finite place, maximal-coordinate ratio charts have uniformly nonzero denominators; the analytic substitutions pull a fixed clopen test back to a common bounded level.

Gram coefficients and twists. Expand the squared norm in (22) and group the pairs of indices with \(qp=q'p'\) by \(k=p(p')^{-1}\). After trace cyclicity, the summand is the pairing of two Gram coefficients, one made from \(a_{p'}a_p^*\) and the other from \(b_q^*b_{q'}\), with adjoints changed if the opposite orientation is used. Here the lower-case letters are the operator-valued Fourier coefficients. The cocycle factors are exactly the factors in these two actual adjoint products: this follows by associativity of the twisted Fourier monomials, or directly by the scalar cocycle identity. In particular, we do not replace the operator-valued convolution by an absolute-value convolution.

For the finitely many bounded shifts, row and column Cauchy–Schwarz bound each weighted Gram coefficient in operator norm by the appropriate uncut column norm squared. Its trace norm is bounded by the relevant cut Hilbert norm squared. Combining the two bounds symmetrically and then applying tracial Cauchy–Schwarz gives (23), with a factor equal to the uniformly bounded number of shifts. Arbitrary separate bounded cutoffs cause no problem in this part.

For the unbounded shifts, use the orientation just described. When \(p=kp'\) is the length-increased side, the first Gram weight is \[f(kp')\overline{f(p')} =\overline{h_A(p')}\, f(kp')\overline{f_1(p')}.\] Retain \(\overline{h_A(p')}\) and expand the remaining factor in the shorter argument \(p'\). On the other side, retain \(h_B(q)\) in \[g(q)\overline{g(qk)} =h_B(q)\,g_1(q)\overline{g(qk)};\] its shorter geometric argument is \(q^{-1}\). Conjugating a Gram coefficient may conjugate these formulas, with no effect on the cut Hilbert norm. In the opposite orientation interchange primes, retaining \(h_A(p)\) and \(h_B(q')\). The longer-argument occurrence of each retained factor remains inside the expanded expression; its uniform substitution bounds are therefore still required in case (ii).

Extend each remaining expression using an initial-patch cutoff with slack. For the first Gram sequence, choose conjugated basis functions so that its full weight has the form \[ w_k(r)=\sum_n c_n(k)\overline{h(r)u_n(r)},\qquad \|u_n\|_\infty\leq1,\qquad \sum_n\sup_k|c_n(k)|<\infty. \tag{25}\] The last, stronger, order of sum and supremum is essential. It follows from Fourier expansion on a slightly larger real coordinate box and uniform bounds on sufficiently many derivatives. At finite places there is instead a finite basis of fixed-level clopen masks. Products of these constructions cover mixed charts. The other Gram sequence has the same form after the indicated adjunction.

For example, when \(p=kp'\) is the length-increased side, \(h=h_A\) and the first Gram sequence is a sum of coefficients of \(A_0(D_{hu_n}A_0)^*\) with scalar weights \(c_n(k)\). Minkowski, Plancherel, and the Hilbert ideal inequality give \[\left\|\left(\sum_n c_n(k) [A_0(D_{hu_n}A_0)^*]_k\right)_k\right\|_{\ell^2(L^2)} \leq \left(\sum_n\sup_k|c_n(k)|\right) \|A_0\|_\infty\|D_hA_0\|_2.\] On the other side, use the shorter of \(q^{-1},(q')^{-1}\) to obtain the analogous bound for \(B_0^*B_0\). If \(B'-B\) is large and positive, the initial arguments are \(p'\) and \(q^{-1}\); the negative orientation is symmetric. Cauchy–Schwarz in the shift variable proves (23). The estimate also proves the small-factor assertion; a square-root rate for the product norm is sufficient.

Smooth joint symbols are handled by summably separated expansions, retaining the thickness and transverse restrictions with slack in every term. Finally approximate general bounded columns by uniformly bounded Fourier polynomials. Unary products converge in \(L^1\), and lower semicontinuity gives the asserted \(L^2\) bound. No step requires operator-norm control of an angularly cut column. ◻

Proof. First omit \(O\). Source covariance makes the uncut range join a projection over the finite algebra \((\mathcal A_1)^\sigma\). Target covariance transports its range by the finite-dimensional linear target matrices; positive changes in the normalization of \(n(v)\) do not change that range. Induce the target lattice action, with its finite invariant trace, to the ambient group. The induced range field then intertwines a trace-isometric ambient action with the inverse linear transport on \(V\).

We spell out the finite-module fact used here. Let a positive matrix \(A\) have weight spaces \(V_1,\ldots,V_s\) with strictly increasing eigenvalues, and let \(P\) be a projection over a finite algebra. Filter \(\mathop{\mathrm{ran}}P\) by its intersections with \(V_1\oplus\cdots\oplus V_j\). Take the closures of the leading projections of these intersections onto \(V_j\). Polar decomposition shows that their dimensions are the successive differences of the filtration dimensions, so their orthogonal sum \(Q\) has the same trace as \(P\). If a vector belongs to the \(j\)th filtered intersection, its image under \(A^n\), divided by the \(j\)th eigenvalue to the power \(n\), converges to its leading projection. Consequently vectors in \(\mathop{\mathrm{ran}}Q\) are strongly approximated by vectors in \(A^n\mathop{\mathrm{ran}}P\). Since the latter range projections all have the same finite trace, they converge to \(Q\) in trace \(L^2\), equivalently in trace norm for projections.

Apply this pointwise to the induced field and integrate, using the finite trace. A convergent orbit of a trace-isometric action is fixed: the distance between two consecutive orbit points is constant and also tends to zero. The field is therefore invariant under every positive split matrix and its conjugates. These generate the relevant connected simple group. Irreducibility of the complexified real module forces the range to split off the whole matrix factor. The remaining projection descends to the scalar joint fixed algebra, so the uncut join is full. For an amplified real module in a product, use the matrix representation only in that archimedean component and the trivial representation in all others; the same argument applies.

Now restore \(O\). The finite trace dimension of the patch join is constant along target lattice translates, because the transporting linear matrices are invertible. Given any finite scalar test measure on the target flags, choose a regular contracting direction whose attractor is in \(O\) and whose repeller is transverse to almost every point of that measure. Such a repeller exists by the nullity of the proper incidence loci. All-factor mixing on the lattice quotient supplies lattice elements whose two compact frames approximate the chosen frames along increasingly long regular translations: two fixed open frame neighborhoods have positive translated intersection in the quotient, and the neighborhoods may subsequently be shrunk. Contraction away from the repeller implies that suitable translates of \(O\) exhaust the test measure.

Apply this to a faithful scalarization of a countable total family of label-averaged effects. The complements of the translated patch joins then have trace tending to zero. One way to see the last implication is to take a faithful positive element generated by that total family and truncate it spectrally away from zero. The constant trace dimension of the translated joins must therefore be the full trace. This proves patch fullness. The normal-copy assertion follows from the normal ampliation statement in Proposition 21. ◻

Proof. Choose paths \(t,a\) whose inverses tend to the two source vertices in a common apartment, using parabolic invariance to choose the split frames. Take successive limits, with \(t\) faster. Pairing their ordered cut effects between bounded columns and using trace cyclicity gives a pairing of \[(D_AX_t f)(D_BX_a f')^*\] with the quotient-path matrix \(X_tX_a^*\), after the appropriate base reindexing. The latter is the transporter of \(ta^{-1}\) up to a common scalar base phase. Its index product is \(pq^{-1}\), and the intermediate ideal directions are \(p^{-1}o,q^{-1}o\), in accordance with the inverse Fourier convention.

If the source vertices are coaxial, the quotient path has exactly one target gap by Proposition 28. Nondual transverse target types would force two distinct large gaps in the cut product. The bounded-overlap version of Proposition 29 bounds its norm uniformly at fixed angular patches and thickness, independently of the large-gap thresholds. The localized norm of the uncut quotient matrix tends to zero on the forbidden gap strata; this is checked on the finite frame columns. Cauchy–Schwarz therefore makes the pairing zero.

If the source vertices are noncoaxial, the quotient has at least two gaps. Dual transverse target vertex cones put the cut product in a fixed thickened single-vertex cone. The smooth version of Proposition 29, with fixed smooth cuts, again compares it with a forbidden localized quotient norm and gives zero. Fix the smooth thickness and angular cuts first, then exhaust the large-gap and thickness windows. This proves the stated ordered vanishing without a claim about uncut operator norms.

Fullness supplies a nonzero ordered cut product for generic source pairs: successively apply Lemma 31 to the two families. Generic dual source vertices are opposite and hence coaxial, whereas generic nondual source types are noncoaxial. Thus any candidate choices at dual source types have dual target types, and choices at nondual source types have nondual target types. If \(T_i\) is the set of possible target types at source type \(i\), every \(j\in T_i\) and \(k\in T_{i^*}\) satisfy \(k=j^*\). Since the sets are nonempty, each \(T_i\) is a singleton. Preservation of both duality and nonduality makes the resulting map injective, and the finite common type list makes it a permutation.

At equality of higher-rank self-opposite source vertices, the nonopposition exclusion extends by continuity. Indeed such a pair is a limit of distinct nonopposite pairs, obtained by varying a chamber across an adjacent panel in the connected diagram. If the image type were isolated, nonopposition would mean equality. Diagonal support of two unital effects would then make them sharp and equal. The same argument applies along the adjacent chamber changes, and gallery connectivity would make the candidate family constant. It would lie in the finite source-fixed algebra and be target equivariant. Taking its invariant trace would give an invariant target flag probability, contradicting flag contraction as in Lemma 31. Thus a higher-rank self-opposite type cannot map to an isolated type. A non-self-opposite type cannot do so either, by opposition preservation. Since \(p\) is a permutation, it permutes the isolated types. ◻

Proof. Undo source induction. Simultaneous covariance of the base and source label yields measurable unital completely positive maps \[F_x:C(Z)\longrightarrow B(H),\qquad F_{\gamma x}=\pi_0(\gamma)F_x\pi_0(\gamma)^*.\] They are equivariant for \(R\), where the designated measured Gamma factor acts on \(Z\) and the other factors act trivially. The assertion follows either from the lattice cocycle formula or from Haar disintegration on the source homogeneous space. Scalar projective phases disappear in conjugation. The source lattice action on pairs of this boundary is ergodic: the open-pair stabilizer contains a split subgroup escaping in the active factor and all complementary ambient factors, so all-factor mixing on the finite-volume quotient applies.

Let \(\mathfrak C\) be the separable matrix-amplified full twisted crossed product associated with \(C(Z)\) and \(R\). Encode \(F_x\) by its minimal covariant Stinespring dilation (Stinespring 1955) and by the vector obtained from the embedded columns of \(e\), arranged as the matrix \(e\). The resulting positive functional \(\phi_x\) on \(\mathfrak C\) has norm \(d\). The group and matrix operators act on these embedded columns by right multiplication. The regular-normal central summand of the group algebra bidual, mapped normally into the crossed-product bidual, therefore supplies canonical right multipliers by matrices over the regular group von Neumann algebra. The vector functional is supported on the corresponding right-\(e\) corner. This support assertion concerns the embedded matrix frame; angular multiples generating the dilation need not themselves have regular group support.

Source translation acts on \(\phi_x\) by an inner automorphism of this fixed corner. Indeed, in the vector formula for the translated map, \(e\) is replaced by \[\pi_0(\gamma)^*e=e\pi_0(\gamma)^*.\] The right side uses the same canonical right corner unitary for every label and every dilation. Equality on crossed-product polynomials extends normally to the bidual. Extending that corner unitary by the identity on its complementary projection, if desired, gives a unitary of the bidual. In particular every central summand is preserved by these inner automorphisms.

The positive subfunctionals of \(\phi_x\) encode exactly the corresponding equivariant subordinate maps. To verify this, note that the matrix vector is cyclic: its group and matrix translates span the embedded input, and applying angular functions generates the minimal dilation. A dominated positive functional is represented by a positive contraction in this cyclic commutant. It commutes with the matrix index and group representation; compressing weighted angular multiplication gives a subordinate equivariant map on the original input. Conversely, a subordinate map gives the dominated form in the same dilation. This is the covariant form of the CP domination correspondence (Arveson 1969, Theorem 1.4.2). Its coefficients between group and matrix translates of the cyclic vector give uniqueness. This correspondence also intertwines source conjugation by the corner unitaries just described.

Fix the central support \(z\) of one functional representation from the second family. The summand \(z\mathfrak C^{**}\) has separable predual, because that cyclic representation acts on a separable Hilbert space. The components \(\phi_x^z\) form a measurable norm-separable family. Here is the measurable-selection detail. In the Polish norm topology of the positive normal ball of \(z\mathfrak C^{**}\) of radius \(d\), the condition \(0\leq\theta\leq\phi_x\) is Borel: test it on a countable norm-dense family of positive elements of \(\mathfrak C\). Put \(n(x)=\sup\{\|\theta\|:0\leq\theta\leq\phi_x, \ \theta\text{ is }z\text{-normal}\}\). This function is upper semianalytic. For each rational \(q\), the Jankov–von Neumann selection theorem gives, on \(\{n>q\}\), a universally measurable dominated normal functional \(\theta_q(x)\) with norm greater than \(q\) (Kechris 1995). Choose measurable rational \(q_m(x)<n(x)\) with \(n(x)-q_m(x)<1/m\), treating \(n=0\) separately. The maximal normal component \(\phi_x^z\) dominates each such selection, so positivity gives \[\|\phi_x^z-\theta_{q_m(x)}(x)\| =n(x)-\|\theta_{q_m(x)}(x)\|<1/m.\] Thus \(\phi_x^z\) is norm measurable on the completed boundary probability space. No simultaneous measurable choice of the second family’s central supports is required.

The distance between \(\phi_x^z\) and \(\phi_y^z\) is invariant under the diagonal source action, since both are conjugated by the same corner unitary. Double ergodicity makes this distance essentially constant. A measurable map into a separable metric space cannot have a strictly positive constant distance at almost every pair: cover its range by countably many balls of diameter smaller than that constant and take one with positive preimage measure. Thus \(\phi_x^z\) is constant almost everywhere. If it were nonzero, the encoding would give a nonzero source-invariant, target-equivariant positive map into the uninduced finite complementary commutant. Its invariant finite trace would produce an invariant flag probability, which is impossible by target contraction. Hence \(\phi_x^z=0\) almost everywhere.

For each fixed label of the second family, the two functionals are therefore disjoint at almost every first label. Orthogonality is the Borel condition that the distance equals the sum of the norms; the norm is tested on a countable dense set in the separable algebra. Fubini gives the first assertion for almost every independent pair.

Now reinduce and consider the two seminorms in (27). If a nonzero scalar functional were bounded by both, the parallel sum of the two squared forms would be nonzero. Construct it by taking the direct sum of the two form completions and quotienting by the closed span of pairs \((h,-h)\) coming from the common algebraic domain. This subspace is reducing for smooth angular multiplication, base multipliers, and the common target covariances, including their adjoints. Angular multiplication descends, and compression at constant input vectors gives a decomposable equivariant CP map dominated by both original maps. It is nonzero: if every constant input vanished, its angular multiples would vanish, although those vectors generate the quotient. Disintegration gives forbidden subordinate maps on a positive-measure set of base points. This contradiction proves the form assertion. Finally, source covariance transports the generic conclusion over the single transverse orbit. ◻

Proof. Take fast and slow paths towards the same source vertex \(x\). Suppose that two angular effects on disjoint matched target patches have a nonzero ordered product. Choose smooth buffered angular cutoffs with bounded complementary thickness. Write \(A\) for the fast cutoff and \(L\) for its one-ended limiting effect. Let \(B_1\) be the slow cutoff, used on slow-link columns, with its start patch disjoint from that of \(A\). The finite corner-frame identity and exhaustion of fixed windows then give bounded columns \(f,f',b'\) for which the following conjugate-linear functional is nonzero. For \(b(v)=\sum_i h_i(v)b_i\), with smooth \(h_i\) and bounded columns \(b_i\), it is the slow limit of \[ \sum_i\left\langle D_{h_i(q^+)B_1} [b'(ay)^*X_a(y)f'(y)],\, b_i(ay)^*X_a(y)(Lf)(y) \right\rangle. \tag{28}\] Here the inner product is linear in its second argument and \(q^+\) is the image of the ordinary index \(q\), that is, the reverse angular variable in the matched component. Smooth boundary tests on the large-gap windows are extended through the simple top singular line in the normalized proximal matrix chart.

We show that this functional is bounded by the fast uncut form at \(x\). At fixed slow time, replace \(Lf\) by its defining fast weak limit along \(t\). Trace cyclicity pairs \(X_tX_a^*b_i(ay)\) with the transverse product of the cut fast column \(D_AX_tf\) and the adjoint of the first column of (28). Its indices are \(p,q^{-1}\). Transfer \(h_i\) to the pairing column by evaluating it at the image of the inverse product \(qp^{-1}\), which approaches \(q^+\).

The transfer error has a uniform quantitative form. Normalized product formulas are smooth with bounded derivatives on the buffered patches, by Proposition 29. Replace the normalized matrix of \(q\) by its rank-one top truncation. The two angular tests agree exactly on that truncation. Smooth division, or the first-order Taylor formula in a matrix chart, writes their difference as a finite sum of entries of the rank-one defect times symbols with uniformly bounded seminorms, with the original cutoffs retained. The slow cut column multiplied by this defect tends to zero in Hilbert norm, since the slow path has a unique vertex type and the complementary thickness is fixed. The small-factor assertion of Proposition 29 makes the product error tend to zero.

After transfer, the remaining cut product has a uniform Hilbert norm. The squared norm of the summed pairing column has fast limit equal to the fast uncut angular form on \(\sum_i h_i U_a^*b_i\). The split element \(a\) fixes \(x\), so parabolic invariance identifies this with the form on \(b\) at \(x\). Extensions of the transferred symbols away from the tested windows can be chosen to have exactly the prescribed boundary values on every rank-one limit. Cauchy–Schwarz therefore gives the desired form bound.

Independent contracted invariance at the slow reverse end permits simultaneously replacing \(b'\) and every \(b_i\) by \(U_n b'\) and \(U_n b_i\), for a reverse-radical element \(n\). The functional itself is unchanged. The bound just obtained is then the fast form on \(\sum_i h_iU_n b_i\), hence the form on \(b\) at \(n^{-1}x\); the slow translation may be removed because it fixes \(x\). These opposite radical moves cover an open Bruhat cell. All constants may depend on the fixed move, which is harmless. The form bounds extend to arbitrary Hilbert coefficient columns by continuity. If the slow path is nonmixed, \(Lf\) is merely Hilbert, but every end column used before the link limit is operator bounded; the one-bounded-input extension in Lemma 22 is exactly what is used.

Choose two generic moved labels in that cell. The same nonzero functional is bounded by both of their forms, contradicting Lemma 33. Thus candidate products at the same source vertex are supported on equality of target values. For two unital POVMs \(E,F\) with this property, regular approximation gives \(E(S)F(T)=0\) for disjoint Borel sets. Unitality then gives \(E(S)=E(S)F(S)=F(S)\) and \(E(S)^2=E(S)\). Hence the candidates are equal PVMs, as asserted. ◻

Proof. We prove (i) by annihilating the common kernel \[\mathcal C=\bigcap_{z\sim F}\ker\Phi_z(O).\] First choose the domain on which this calculation will be made. If the path is mixed, take any \(f\in\mathcal C\) and operator-bounded columns \(f_l\to f\) in Hilbert norm. If it is a nonmixed vertex path supported in \(G_0\), put \(N_0=\mathcal N^{G_0}\). When \(i\) lies in another component, every type-\(i\) label is incident to \(F\), and ordinary fullness gives \(\mathcal C=0\), proving the inclusion. Otherwise Lemma 22(ii) makes \(\mathcal C\) invariant under \(N_0\). Take \(f\) in its dense relatively bounded part and, by part (iii) of that lemma, choose operator-bounded \(f_l\to f\) satisfying \[E_{N_0}(f_lf_l^*)\leq E_{N_0}(ff^*)\in N_0.\] The approximants need not belong to \(\mathcal C\). In either case, for each \(z\sim F\), \(\langle f_l,\Phi_z(O)f_l\rangle\to0\).

Fix a compact full-data window that sends a buffered transverse end patch \(V_1\) into \(O\). Such windows cover the image event in (i). Fix also \(z\sim F\) and an operator-bounded column \(b_0\). A vertex path \(s^{-1}\) toward \(z\), with a smooth angular cut in \(V_1\) and fixed complementary thickness, realizes its cut end effect by an auxiliary weak limit. Exhausting these cuts and varying \(b_0\) gives a total set in \(\overline{\mathop{\mathrm{ran}}}\Phi_z(V_1)\).

Keep the window, angular cuts, thickness, and bounded testing columns \(b'',f''\) fixed. For each fixed main time \(a\) and approximation index \(l\), insert the auxiliary \(s\)-limit. Trace cyclicity pairs \[ X_s(ay)X_a(y)f_l(y) \tag{30}\] with the product of cut versions of \[X_s(ay)b_0(ay),\qquad b''(ay)^*X_a(y)f''(y).\] If the two product indices are \(q,h\), their observed inverse product is \(h^{-1}q^{-1}\). Transversality and the buffer put its pertinent image in \(O\) once the auxiliary gap is sufficiently large. The effective norm of the pairing column in this inner limit is therefore at most \[\langle f_l,\Phi_z(O)f_l\rangle^{1/2}.\] The main split translation fixes \(z\), because it acts trivially on the Levi chamber refinements incident to \(F\). The bound is thus independent of the main time.

Proposition 29 gives a uniform Hilbert bound for the other, cut product. Its constant depends on the fixed window, thickness, buffered patches, and the bounded columns \(b_0,b'',f''\). It does not depend on \(\|f_l\|_\infty\): \(f_l\) appears only in the pairing column. For each fixed auxiliary cut and \(b_0\), its limiting end vector is fixed. Take the required main-link limits with \(l\) fixed, then let \(l\to\infty\) against that vector. On a mixed path the tensor isometry justifies this passage. On a nonmixed path, the common conditional-expectation bound on \(f_l\) justifies it by Lemma 22. The displayed pairing bound tends to zero.

Now keep \(f\) fixed, exhaust the auxiliary thickness and angular cuts, and take the closed span of the resulting end ranges. The same lemma gives continuity in the entire Hilbert end variable on a nonmixed path; the mixed tensor isometry gives it on a mixed path. Consequently \[E(\mathrm{window})J(\bar b,f)=0 \qquad (b\in\overline{\mathop{\mathrm{ran}}}\Phi_z(V_1)).\] At the fixed-time auxiliary insertion the approximating input \(f_l\) was operator bounded. The subsequent Hilbert-input passages are extensions of the limiting link; no finite-time product of two arbitrary Hilbert columns is used.

For the individual-pair assertion, fix a single \(z\) and repeat the calculation with \(f\in\ker\Phi_z(O)\). In the nonmixed case take \(f\) relatively bounded. Its bounded truncations still have the same expectation bound and need not remain in the kernel; thus this argument applies even when \(i\) is outside the active component. Kernel invariance was needed only for density in the range-inclusion argument. This proves the individual-pair assertion for every Hilbert \(f\) on a mixed path, and every relatively bounded \(f\) on a nonmixed path, with \(b\) any vector in the stated closed end range.

Move the end occurrence also by the independent reverse radical. As \(z\sim F\) and these moves vary, the resulting labels cover a transverse open cell, which is conull. Fullness and strong continuity on fixed input vectors make these end ranges total. The link is therefore annihilated on the window for every end vector.

We make the final frame step explicit, because the nonmixed case cannot use tensor independence. First exhaust the compact full-data windows to the event \(A\) consisting only of the chosen gap stratum and the image condition \(\mathcal I_i(O)\). Monotone spectral continuity gives \(E(A)J(\bar b,f)=0\) for every end vector \(b\). This exhausted event is invariant under the ignored fixed left Fourier shifts; the residual and reverse-window restrictions have been removed. For the finite constant column frame \(b_r\) of \(e\), the frame identity of Proposition 21 is \[c_F\langle f,\Psi_F(\mathcal I_i(O))f\rangle =\frac1d\sum_r\|E(A)J(\overline{b_r},f)\|^2=0.\] Divide by \(c_F>0\), use positivity, and extend from the relatively bounded common-kernel part by density. Orthogonal complements prove (i). An arbitrary Borel output subprojection may be imposed after the buffered ambient tests; no uniform estimate for a discontinuous unexhausted window has been asserted.

For (ii), write \(S(h)\) for the Hilbert-normalized real module matrix at \(h^{-1}\). If \((f_i)\perp L_z\) and \((b_j)\) belongs to the amplified line range on a patch transverse to the output reverse window, the same computation gives \[ E(\mathrm{window})\sum_{ij}E(S_{ij}(h))J(\overline{b_j},f_i)=0. \tag{31}\] Indeed, the auxiliary fast symbol is \(S(q)S(q)^*\) with its cutoffs, and \[S(h)S(q)S(q)^*=\rho(h,q)S(qh)S(q)^*,\] where the normalization ratio \(\rho\) and its needed derivatives are bounded on the transverse supports. Move \(S(qh)^*\) to the full pairing column. Its squared symbol tends in the auxiliary limit to the line-projector effect defining \(L_z\), with the same exact stabilizer translation as above. It therefore vanishes on the approximating kernel input in the limit.

The reverse radical may again move the end occurrence. Amplified fullness now makes the resulting end columns span the whole amplified end space. On a compact stratum window the surviving top block of \(S(h)\) has range containing every incident highest line, by Lemma 19. Bounded local right inverses on its constant-rank image, or finite local spanning columns followed by output matrix multiplication, transfer the annihilation in (31) to the projector \(R_x\). Compressing and using the mixed input independence from Proposition 21 proves (ii). This amplified argument is used only for real Hilbert modules; its finite-place replacement is given below. ◻

Proof. The base failure rule is nonopposition of dual types, supplied by Proposition 32; at equal self-opposite vertices it follows by its continuity argument. Suppose all incident replacements of one or both source vertices satisfy a previously proved failure rule. If an incident pair at the target violates that rule, choose small neighborhoods of the violating vertices. Proposition 36 puts the corresponding output ranges inside sums of the incident source-label ranges. The prior failure rule makes those product ranges orthogonal. Local homogeneous sections ensure that nearby base faces still admit incident values in the chosen neighborhoods. Thus the original ordered product is supported where every such incident target pair obeys the rule. Induction on the word proves failure propagation. This argument uses ordered products and range sums, not a simultaneous Borel calculus for noncommuting measurements.

For the combinatorics, extend a vertex to a chamber in an apartment. Changing that chamber without changing the vertex permits its maximal parabolic Weyl subgroup. An incidence move permits the corresponding successive maximal-parabolic blocks. The largest attainable Bruhat positions are obtained by successive ascent to block maxima, that is, by Demazure multiplication. They are increasing in the initial position by the Coxeter subword rule, and the apartment realizes the maxima. Ending opposite at dual vertex types is equivalent to reaching the longest chamber position after the endpoint extensions. These are the apartment and Bruhat descriptions of parabolic incidence (Borel 1991; Tits 1964), and prove monotonicity of success.

If a word is successful at the least target position, it is successful at every target position. Were its inverse-permuted word to fail at a least source pair, failure propagation would force the unital ordered source product to have empty support. Its total product is the identity, so that is impossible. This gives inclusion of the least-position success languages in one direction. Iterating \(p\), which has finite order on the type list, gives the cyclic chain of inclusions back to the original language. They are therefore all equalities. ◻

Proof. For type \(A\), take an end node. Move the second vertex to an incident vertex of the opposite end type. The word succeeds precisely when a hyperplane containing the second point can avoid the first point, which is equivalent to inequality; the apartment point–hyperplane test gives the same assertion in the building language.

In every other crystallographic diagram choose a self-opposite leaf \(j\). Let \(k\) be its neighbor, write \(W_{\widehat j}\) for the subgroup omitting \(s_j\), and consider round trips from type \(j\) to type \(k\) and back. Omitting \(k\) isolates \(j\), so each round trip permits exactly the Weyl block \[W_{\widehat j}\{1,s_j\}W_{\widehat j}.\] Let \(r\) be the minimum number of occurrences of \(s_j\) in a reduced word for the longest element of this component. Starting at equality, \(r-1\) round trips cannot end opposite: an expression so obtained has at most \(r-1\) occurrences of \(s_j\), and a reduced subword reaching the longest element would contradict the definition of \(r\).

Starting instead at the least nontrivial double coset \(W_{\widehat j}s_jW_{\widehat j}\), the first such block is already available. Choose a reduced longest word with exactly \(r\) occurrences of \(s_j\) and separate it into the intervening \(W_{\widehat j}\) blocks. The remaining \(r-1\) round trips realize it. Every unequal position is above that least nontrivial double coset, so success at all unequal pairs follows from Lemma 37. The self-opposite leaf exists in the non-\(A\) crystallographic list, including the nonreduced cases through their underlying Weyl diagram. ◻

Proof. Apply Lemma 38 in a target higher-rank component, and pull the least-position language back by Lemma 37. The inverse-permuted word fails at equality of source type \(p^{-1}(j)\). Failure propagation therefore puts every ordered product of two candidates at that same source label on the target diagonal. The unital-POVM argument at the end of Proposition 35 makes them equal projections. Together with that proposition, this supplies at least one sharp type in every target component.

Extend any vertex path lexicographically to a full chamber path. Its original marginal is retained. If an already sharp vertex label is incident to that chamber, Proposition 36 puts the corresponding full-path marginal ranges inside that sharp PVM’s ranges. Unitality and the same assertion for complements make the marginal equal to the sharp PVM. A sharp marginal intertwines the minimal dilation: equality in the CP Schwarz inequality on its projections gives the intertwining directly. Thus the tested candidate commutes with the sharp incident measurement, and their ordered product is supported on target incidence.

Let a remaining target type lie in the component of a sharp type. For two candidates at the same source label, suppose their target values \(u,v\) have different incidence families of the sharp type. Choose a sharp-type vertex \(w\) incident to \(u\) but not to \(v\), with small buffered neighborhoods. The first range containment puts the ranges near \(u\) in the sum of the incident source sharp-label ranges near \(w\). Every one of those ranges is orthogonal to the ranges near \(v\), by the full-path incidence rule just proved. This works with two independent full-path choices for the tested candidate. Hence its self-product, and products between choices, are supported where the two sharp-type incidence families agree.

Those families determine the vertex. Their stabilizer contains its maximal parabolic and is proper, because the connected apartment contains nonincident pairs of the relevant types. Bruhat maximality of the maximal parabolic makes that stabilizer precisely the original vertex parabolic. Thus equal families imply \(u=v\). Diagonal support again proves sharpness and uniqueness. Repeating in each component makes all candidates sharp.

We now treat an arbitrary mixed face path of source type \(I\). Fix a positive target stratum with normalized image measurement \(\Psi_F\). For \(i\in I\), the type-\(i\) source vertex \(z_i\) incident to \(F\) is unique. Proposition 36(i) puts the support of \(\Psi_F(\mathcal I_i(O))\) inside \(P_{z_i}(O)\). A target face omitting \(p(i)\) admits two distinct incident vertices of that type. Choose disjoint basic open patches \(O,O'\) around them. The effect \(\Psi_F(\mathcal I_i(O)\cap\mathcal I_i(O'))\) is supported in both orthogonal projections \(P_{z_i}(O),P_{z_i}(O')\), and hence is zero. Countably many such pairs exhaust the omitted-type event. Thus every \(p(i)\), \(i\in I\), occurs. The general upper bound of Proposition 28 forces the stratum type to equal \(p(I)\). On this stratum \(\mathcal I_i(O)\) is precisely the event that its type-\(p(i)\) vertex lies in \(O\). The same support containment and unitality identify its marginal with \(P_{z_i}\). These sharp marginals put the joint face measurement in their common multiplicative domain, giving \(P_F\). This argument uses mixing, but no order among the diverging coordinates.

For lexicographic paths, the full-path marginals are now uniquely identified and commute by the sharp-marginal argument. Their joint laws give \(P_F,P_C\) and the stated incidence support. We also verify that a lexicographic path towards \(F\) has no additional observed vertex type. Extend it to a full path and use persistence and the gap count. An additional sharp marginal would be invariant under the source face parabolic of \(F\) as well as under its own maximal parabolic. These generate the entire relevant simple factor, because the extra source type is not retained by \(F\). For a positive subpiece, its scalar mass times the sharp marginal is its entire angular effect: a POVM dominated by a PVM and having scalar mass \(cI\) equals \(c\) times that PVM, by restriction to a projection and its complement. Normalizing the positive subpiece would therefore produce a source-invariant boundary measurement. Its finite invariant trace is forbidden by flag contraction. Thus precisely \(p(F)\) is observed. In the three-slot system, the two normal copies are marginals of one common full-path dilation. Their sharp incident marginals therefore commute as well. ◻

Proof. Adjacent source chambers share their sharp common panel. Insert its projection between the corresponding consecutive spectral slots. Differences of the two common-panel coordinates therefore annihilate the ordered product, including after multiplication by arbitrary smooth functions expressed by summably separated expansions. The common-panel equations define the gallery space regularly: it is a successive fibre product of the submersions from chambers to panels. They consequently generate its local smooth vanishing ideal, including at stutters. Smooth partitions of unity prove the asserted annihilation. In totally disconnected coordinates, finite clopen partitions insert the common panel projection exactly and give the same conclusion for locally constant tests.

Fullness gives a nonzero opposite-target restriction of the product at some generic, hence opposite, source chamber pair. Source covariance transports this assertion over the opposite orbit. Insert all intermediate PVM identities along any reduced longest source gallery, and use compact partitions of the intermediate target chamber spaces. Nonzero opposite endpoint support requires a gallery of the permuted type connecting opposite target chambers. Both ends use the full spherical Weyl system of the fixed ambient group \(G\). Their longest elements therefore have the same Coxeter length. The permuted word must therefore itself be a reduced longest word; no stutter or nonminimal step is possible.

An alternating rank-two word of Coxeter length \(m_{ij}\) is reduced and extends to a reduced longest word. Its permuted segment is also reduced, so \(m_{ij}\leq m_{p(i)p(j)}\). Cycling the finite permutation \(p\) gives equality. This proves Coxeter compatibility. ◻

Proof. We first work with an archimedean target component. There are two steps, both using ordered spectral slots.

Projection onto the reverse pencil. On the ordinary-end Hilbert space, use the slots in the order: reverse panel, end missing vertex at \(z\), and the observation at the start label, possibly moved by the reverse radical. Denote the value in the last slot by \(v\). A source gallery shortened from the longest by the missing reflection connects the matched chambers. Its target gallery has the same shortened type by Proposition 40. The shortened Weyl position is \(s_iw_0=w_0s_{i^*}\); it has all the other respective left and right descents, so saturation by the parabolics omitting those missing reflections cannot reach \(w_0\). Thus the missing vertices are nonopposite.

A vertex transverse to a panel of the dual missing type has exactly one incident completion of that panel whose missing vertex is nonopposite to it. This is the ordinary rank-one parabolic projection. The ordered gallery equation therefore says that the end missing vertex is exactly \(\operatorname{pr}(v)\) on a transverse window. Integrating the intermediate gallery slots by their identity projections preserves the equation. Other components can be completed arbitrarily.

The line equation. Use (31) at the single start vertex \(z\) and move only its end occurrence by the independent reverse radical. Sharpness permits taking the start column to be the projector \(1-n(P_z)n(P_z)^*\) applied to an arbitrary column, and the end column to be the moved line projector on a transverse patch. Thus, after bounded output localization, the link is annihilated by \[(1-n(u)n(u)^*)S(h)n(v).\] Its two angular inputs commute because they act on the two separate input tensor slots. The identity consequently holds with arbitrary smooth joint angular multipliers, by summable separation. On the stratum, the matrix action is the residual projectivity after transverse projection, so the equations are precisely the line graph \(u=g\operatorname{pr}(v)\). They generate its smooth vanishing ideal on compact transverse windows: the highest-line embedding is immersive, so the derivative in the image-vertex argument has full injective rank. Smooth division is uniform on such windows. Output coordinates may be Borel parameters with locally uniform angular derivative bounds, since separation is performed only in the angular variables.

We now combine the two steps in their necessary order. Preapply at the end a reverse-panel cut followed by a moved-vertex cut, on a buffered transverse rectangle, and leave the start vector arbitrary. Insert the \(Q_z\) spectral slot next to the reverse-panel slot; they commute. The shortened-gallery equation replaces its value by \(\operatorname{pr}(v)\), including after smooth multiplication in the independent start and output variables. Integrate out that slot. Pull the sharp reverse-panel projection through \(J\), using Proposition 21. The line equation then supplies \(u=g\operatorname{pr}(v)=gq\). This proves (32) on those ordered ranges. In particular, we have not applied a discontinuous joint symbol to an arbitrary pair of noncommuting end measurements.

These ordered ranges are total. Their closed join is target invariant. It is fixed by a split source subgroup escaping in every ambient component, fixing the two panels and their respective completions, and normalizing the reverse-radical moves. Its projection is therefore scalar by the invariant-mass and mixing rule of Proposition 21. Some such ranges are nonzero when also varying \(z\): moved start labels then cover generic vertices, and patch fullness applies to a nonzero reverse-panel projection on a uniform transverse target patch. Source Levi transitivity on the completion parameter transports nonzeroness to every \(z\). Hence the join is the whole end space, and the equation holds everywhere.

Finite target places. At a finite target place, replace the Hilbert line argument by the individual-pair assertion of Proposition 36. Choose scalar clopen start and end patches. On a buffered output window carrying the end patch strictly inside the start image patch, a start vector in the kernel of the corresponding \(P_z\) projection and an end vector in the cut range of the same start-label measurement give zero. Since \(P_z\) is sharp, finite clopen partitions imply the exact transverse image substitution. Move the end occurrence by the reverse radical as before. The action on transverse values is the residual projectivity after projection to the reverse pencil.

The shortened-gallery substitution holds in this component by compact clopen partitions and insertion of the common sharp panel projection. Thus the reverse-panel cut followed by the moved-vertex cut again imposes \(u=gq\), including arbitrary locally constant angular multipliers. The same total-range argument applies. This proves the finite-place equation without tensoring the complex Hilbert input with a representation over a nonarchimedean field.

For mixed target coordinates, use finite partitions in the clopen slots and summable smooth separated expansions in the real slots. Individual stratum restrictions are made by Borel output projections after buffered tests in the ambient compactification; no uniform transverse estimate for an arbitrary discontinuous mask in an unbounded-overlap problem is required.

Finally, a source Levi element centralizes the chosen face path. Exact cocycle composition, made before limiting, shows that its simultaneous action at the two inputs changes the array only by base reindexing and a common scalar phase. The tested forms are therefore invariant. If a genuine action is needed, use the conjugate-multiplier regular amplification from Section 4. It tensors the forms by identity and cancels the phase defect at each input, without altering any finite-trace mixing assertion. This proves the stated transport properties. ◻

Proof. The sharp face marginals intertwine the link dilation, so its spectral span disintegrates over the product of the two endpoint spectral measures. The strongly continuous source Levi representations commute with those face PVMs and therefore disintegrate as well. One may obtain continuous fibre representations by disintegrating a countable dense convolution algebra of the group and then using its nondegenerate integrated representations, as in the standard direct-integral construction (Bekka and Harpe 2019, Theorem 1.G.6).

Before removing a null set, choose countably many dense residue labels, compact windows, local graph generators, angular Fourier tests in real charts, fixed-level clopen tests in finite charts, and total input vectors. All the corresponding identities then hold on one conull set. Strong source transport and the uniform angular bounds extend them to every residue label and smooth or locally constant angular test there. The stabilizer invariance at a standard label also disintegrates, first on a countable dense subgroup and then by strong continuity, so the transported family is well defined on the residue quotient. Once a vector identity holds on an individual fibre, any bounded Borel output multiplier is applied directly through that fibre’s spectral calculus. Uniformly summable angular expansions then give tests with Borel output parameters. No countable norm-dense class of all bounded Borel functions is used.

Disintegrate the amplified incidence containment of Proposition 36 on the same determining family. On almost every fibre its closed span contains the Hilbert fibre tensored with the span of all ordinary incident highest lines, and this holds separately for every even highest power. A polynomial with linear Hilbert-vector coefficients that vanishes on all spectral line ranges is orthogonal to their span. It is therefore orthogonal to each ordinary incident highest line, proving polynomial fullness. Norm powers homogenize different even degrees.

For a relation known only on a chart \(\{\Delta\ne0\}\), first clear a common homogeneous denominator and call the resulting numerator \(F_{\mathrm{num}}\). Apply global polynomial fullness to \(\Delta F_{\mathrm{num}}\): it vanishes spectrally on the chart by the relation, and on \(\{\Delta=0\}\) by the extra factor. Divide by \(\Delta\) on the original chart to recover the relation there. Pluecker equations and common local denominators put the stated algebraic membership tests in this form. ◻

Proof. A decomposable operator fixed by left translation by \(U\) is essentially constant on each left coset. Thus there are normal evaluations \[\pi_t^+(d)=d_t^+\in B(H'),\qquad \pi_r^-(d)=d_r^-\in B(I),\qquad Ut,Ur\in U\backslash T, \quad d\in D_*^U.\] They are restrictions of the normal coordinate evaluations on \[\ell^\infty\bigl(U\backslash T; B(H')\oplus B(I)\bigr).\] They are jointly faithful, and \(U\backslash T\) is countable.

Identify \(\overline I\otimes H'\) with \(\operatorname{HS}(I,H')\) by \(\overline\xi\otimes\eta\mapsto |\eta\rangle\langle\xi|\). For fixed \(Ut,Ur\), put \[\Omega_{t,r}=\{g\in T:Utg=Ur\}.\] For every \(\zeta\in\mathcal K\), the Hilbert–Schmidt operator \(K=J^*E(\Omega_{t,r})\zeta\) satisfies \[ d_t^+K=Kd_r^-,\qquad d\in D_*^U. \tag{36}\]

Here is the word-transfer justification of Equation (36). First take a word in generators with real-valued clopen tests and average that word over \(U\). At frame \(u_0t\), the adjoint of the exact graph identity transfers each start test to the corresponding end test at frame \(u_0tg\). On a compact \(g\)-window, take a finite partition into relatively clopen pieces such that each end function \(q\mapsto f(u_0tgq)\) in this finite list is independent of \(g\) within a piece, uniformly for \(u_0\in U\). Indeed the action and the tests are continuous on the compact set of parameters, while a clopen test has finite image. Within a piece every factor therefore intertwines simultaneously. Transferring a word successively gives the reversed order on the conjugate Hilbert space, which under the Hilbert–Schmidt identification is right multiplication by the end word in its ordinary order. All the output cuts remain in place.

On \(\Omega_{t,r}\), write \(tg=u_g r\) with \(u_g\in U\). Haar averaging in \(u_0\) absorbs \(u_g\), so the averaged end word has value \(d_r^-\) and the averaged start word has value \(d_t^+\). This proves Equation (36) for averaged words on compact windows. Linear combinations and adjoints treat complex tests. Kaplansky density approximates a bounded element of \(D_*\) by bounded generator polynomials, with arbitrarily small norm slack. Normal Haar averaging approximates every element of \(D_*^U\) ultraweakly. The evaluations on this fixed algebra are normal, and multiplication by a fixed Hilbert–Schmidt operator is weakly continuous on bounded sets. Thus the identity extends to all \(d\in D_*^U\). Exhausting \(T\) by compact windows removes the temporary localization. The endpoint frames cause no extra evaluation step: their sharp decompositions are retained throughout this calculation.

For fixed \(t\), the sets \(\Omega_{t,r}\) partition the output variable as \(Ur\) varies. Since \(J\) is an isometry, \(J^*\) maps onto \(\operatorname{HS}(I,H')\). The linear span of all the resulting \(K\) is therefore dense in Hilbert–Schmidt norm. The same statement holds with \(r\) fixed and \(t\) varying, because right multiplication by a fixed \(g\) permutes the left cosets. Equation (36) and its adjoint give \[KK^*\in\pi_t^+(D_*^U)',\qquad K^*K\in\pi_r^-(D_*^U)'.\] The nonzero eigenspaces of these compact positive operators are finite-dimensional reducing spaces for the indicated representations. Together they span the ordinary evaluation spaces. For example, a vector orthogonal to all the ranges of the \(K\) would be orthogonal to the ranges of a Hilbert–Schmidt dense family, and hence would vanish.

Restriction to these reducing spaces supplies a jointly faithful family of normal finite-dimensional representations of \(D_*^U\). The central support of each such representation is a finite sum of central atoms, and its algebra on those atoms is a finite sum of finite matrix algebras. Joint faithfulness makes the join of these supports equal to one. Separability of the predual makes the resulting product countable. This proves the first assertion without any restriction on the endpoint Hilbert-space multiplicities.

If \(d\in D_*^T\), its two evaluations are independent of \(t,r\). Equation (36) then holds across every Hilbert–Schmidt operator. If \(AK=KB\) for every rank-one map \(K:I\to H'\) on nonzero Hilbert spaces, testing \(K=|\eta\rangle\langle\xi|\) gives \(A=\lambda1\) and \(B=\lambda1\) with the same scalar. This proves Equation (35).

Finally suppose that \(S\) acted trivially on \(D_*\). Source covariance and transitivity on \(B_S\) would make \(F_{z,f}\) independent of \(z\). For countably many labels and tests, evaluate these explicit decomposable identities at one common generic external frame; the transformed tests still generate \(C(B_T)\). Continuity in the source label then makes the original framed \(P_z\) independent of \(z\) on the whole start space. The actual source Levi cover commutes with the sharp base observation and its frame calculus, so unframing gives the same statement for the original missing vertex.

Choose a chamber completing the source face, fixing completions in the other components. Its full observation is the commuting product of its canonical vertex observations. The retained vertices are fixed by the actual source Levi factor, and the missing vertex has just been shown fixed. Hence the full chamber observation is fixed by this Levi factor as well as by the minimal source parabolic. These groups generate the larger parabolic allowing the omitted simple root. The resulting unital target-equivariant full-boundary observation contradicts Lemma 26. The assertion concerns the original observations and so descends through the identity amplification. In a many-panel link, adjoin the other fixed chamber completions before applying the same argument. Thus the source action is nontrivial in every case used here. ◻

Proof. A continuous action of a connected group fixes every central atom of an atomic von Neumann algebra. To see the relevant discreteness, a normal state supported on one atom takes values zero and one on its distinct translates, so that atom has an open stabilizer. On each finite matrix block the action is a continuous homomorphism into a projective unitary group. A noncompact simple real Lie algebra has no nonzero homomorphism into a compact Lie algebra: simplicity would make it injective, and a positive definite invariant inner product would then make the source Lie algebra compact. A homomorphism with zero differential from a connected Lie group is trivial. Thus the source would fix every \(D_*^U\) pointwise. As compact open \(U\) decreases to the identity, the normal averages \(\int_U\alpha_u(d)\,du\) converge ultraweakly to \(d\). Their ranges therefore generate \(D_*\). This contradicts Proposition 43. The archimedean subset of the finite type list is mapped into itself; finiteness and bijectivity of the type permutation give the assertion for both subsets. ◻

Proof. An automorphism in question fixes \(r_i,r_j\) and their central supports. Work on the common central support of these two projections. The rectangular space \(X=r_iAr_j\) is the full selfdual equivalence bimodule between its endpoint corners: the ultraweak spans of \(XX^*\) and \(X^*X\) are those corners on this support. The restriction of the automorphism to \(X\) is right linear, preserves the right inner product \(x^*y\), and has its inverse as adjoint. It is also left linear.

Adjointable right-module operators on this equivalence bimodule are left multipliers by the left corner. One can verify this directly: finite sums of the rank-one maps \(x\mapsto yz^*x\) approximate its identity strongly; applying an adjointable operator to these sums produces left multipliers, and their bounded strong limit remains in the left corner. Left linearity forces the resulting multiplier to be central. Preservation of the inner product makes it unitary. Thus the automorphisms act on every \(r_iAr_j\) by commuting central unitary multipliers. These multipliers are themselves in corners fixed pointwise by every automorphism in the group, so composition of the actions on each rectangular space is commutative.

It follows that the commutator \(\theta\) of any two such automorphisms fixes every \(r_iAr_j\). For \(a\in A\), \(r_i(\theta(a)-a)r_j=0\) for all \(i,j\). Since the ranges of the \(r_j\) span the representation space, this gives \(\theta(a)=a\). ◻

Proof. This kernel is a closed normal subgroup of \(V\), so its Lie algebra is an ideal in the simple prime-field algebra \(\mathfrak s\). If it were nonzero, it would be all of \(\mathfrak s\), and an open subgroup \(V_0<V\) would fix \(D_0\) pointwise. The source and target actions commute, so \(V_0\) would also fix \(\alpha_t(e)D_*\alpha_t(e)\) pointwise for every \(t\in T\). The join of the projections \(\alpha_t(e)\) is nonzero and \(T\)-fixed; it is one by Proposition 43. A countable subfamily has the same join. Lemma 45 now shows that every commutator from \(V_0\) acts trivially on \(D_*\). There is a nonidentity such commutator because \(\mathop{\mathrm{Lie}}(V_0)=\mathfrak s\) is nonabelian. The kernel of the \(S\)-action is normal in the effective simple group \(S\). It would therefore be all of \(S\), contrary to Proposition 43. ◻

Proof. A union of block orbits would give an invariant central projection, so ergodicity gives transitivity. An element of the chosen block fixed by \(L_n\) extends, by \(U\)-transport, to an invariant element of \(D_n\). Therefore the fixed algebra of \(L_n\) on that block is scalar. In a matrix algebra its automorphism action is implemented projectively, and this scalar-commutant condition is precisely irreducibility. The action on the finite set of blocks is continuous, so \(G_n\) is closed. For each \(v\in V\), transitivity supplies \(u\in U\) carrying the block moved by \(v\) back to the chosen one. Hence \(G_n\to V\) is onto.

An element of \(K_n\) fixes every block label: its permutation commutes with the transitive \(U\)-action and fixes one label. On the chosen block its projective implementer centralizes the projective representation of \(L_n\). Such a projective centralizer is abelian. Indeed, for an implementer \(A\) and projective representation implementers \(W_l\), write \[AW_lA^*=\chi_A(l)W_l.\] The representation multipliers cancel, making \(\chi_A\) a scalar character. The map \([A]\mapsto\chi_A\) is a homomorphism into the abelian character group. Its kernel is trivial by Schur’s lemma. Thus commutators of \(K_n\) act trivially on this block. Commutation with \(U\) and transitivity give the assertion on all blocks. ◻

Proof. If \(p_S\ne p_T\), a closed subgroup of the product of a pro-\(p_S\) group and a pro-\(p_T\) group is the product of its projections. Pass to finite quotients: a subgroup of a finite \(p_S\)-group times a finite \(p_T\)-group has projected groups with no common nontrivial quotient, so is their product. The assertion for closed subgroups follows by inverse limits. Consequently \(V\times1\subset G_n\) for every \(n\). Lemma 47 makes \([V,V]\) act trivially on all \(D_n\) and hence on \(D_0\). The closure of this commutator subgroup has full Lie algebra \([\mathfrak s,\mathfrak s]=\mathfrak s\); for example, take the Lie algebra of the compact abelian quotient by its closure. This contradicts Lemma 46. Applying the assertion at every type proves the last statement. ◻

Proof. In dimension zero, a sufficiently small \(W\) is trivial and \(c=0\) suffices. Assume henceforth that the ambient Lie algebra has positive dimension. Choose a lattice \(\Lambda\) in that Lie algebra \(\mathfrak w\) on which matrix exponential and logarithm are inverse, with \(W=\exp\Lambda\). Shrink it so that \[(\mathop{\mathrm{Ad}}(w)-1)\Lambda\subset p\Lambda\qquad(w\in W).\] Write \(N=\dim_{\mathbb Q_p}\mathfrak w\) and \(M=M_H\). As a submodule of the finite free \(\mathbb Z_p\)-module \(\Lambda\), \(M\) is finitely generated, free, and closed in its rational span. Choose finitely many \(h_1,\ldots,h_a\in H\) whose logarithms generate \(M\). Such a subfamily exists because \(\mathbb Z_p\) is Noetherian.

For \(h\in H\), conjugation by \(h\) and \(h^{-1}\) preserves \(M\). Thus \(A=\mathop{\mathrm{Ad}}(h)-1\) is an endomorphism of \(M\). Its characteristic polynomial on the fixed ambient space has the form \[x^N+a_{N-1}x^{N-1}+\cdots+a_0,\qquad a_j\in p\mathbb Z_p.\] Cayley–Hamilton applied on \(M\), rather than just on the ambient lattice, gives \[ A^NM\subset pM,\qquad A^kM\subset p^{\lfloor k/N\rfloor}M. \tag{40}\] Set \[b=\max\left(0,\sup_{k\geq1} \{v_p(k)-\lfloor k/N\rfloor\}\right)<\infty.\] The logarithm series and Equation (40) give \[\log(1+A)M\subset p^{-b}M.\] In the fixed matrix logarithm chart, \(\log\mathop{\mathrm{Ad}}(h)=\operatorname{ad}(\log h)\). Extending over the chosen generators of \(M\) therefore proves \[[M,M]\subset p^{-b}M.\] Choose \(c\geq b+2\) and put \(L=p^cM\). Then \([L,L]\subset p^2L\).

We check the BCH estimate at every prime. In Dynkin’s formula a homogeneous term of degree \(m\) has a denominator of the form \(km\prod_i r_i!s_i!\), where the \(k\) blocks are nonempty and \(\sum_i(r_i+s_i)=m\). Its \(p\)-valuation is at most \(m-1+v_p(m)\): the factorial terms contribute at most \(m-k\), and \(v_p(k)\leq k-1\). A bracket of length \(m\) in \(L\) lies in \(p^{2(m-1)}L\). For \(m\geq3\) the resulting term therefore lies in \(p^{m-1-v_p(m)}L\subset pL\), with its exponent tending to infinity. The degree-two term is \([X,Y]/2\in pL\), including when \(p=2\). Thus BCH converges, preserves \(L\), and satisfies \[ \operatorname{BCH}(X,Y)\equiv X+Y\pmod{pL}. \tag{41}\] Every nonlinear term contains both variables. The same estimates therefore give, for \(r,s\geq0\), \[\operatorname{BCH}(X,Y)-X-Y\in p^{r+s+1}L \quad(X\in p^rL, Y\in p^sL).\] In particular \(\exp L\) and \(\exp(p^jL)\) are groups.

At level \(j\), the elements \(h_i^{p^{c+j}}\in H\) have logarithms \(p^{c+j}\log h_i\) spanning \(p^jL/p^{j+1}L\). Given \(x\in L\), choose a product of these powers at level zero whose logarithm agrees with \(x\) modulo \(pL\). Its inverse times \(\exp x\) has logarithm in \(pL\). Correct this error with a product at level one, and continue. Equation (41) and its refined estimate improve the error by one level at every step. The correcting factors tend to one, and their successive products in \(H\) converge to \(\exp x\). Closedness of \(H\) proves Equation (39). Every constant was chosen before \(H\), as required. ◻

Proof. Its Lie algebra is invariant under \(\mathop{\mathrm{Ad}}(V)\), because \(G_\infty\to V\) is onto. It is thus an ideal of the simple algebra \(\mathfrak s\). If nonzero it equals \(\mathfrak s\), and \(G_\infty\) contains \(V_0\times1\) for an open subgroup \(V_0<V\). Set \(d=\dim_{\mathbb Q_p}\mathfrak t\). Choose finitely many \(v_1,\ldots,v_a\in V_0\) such that \[ \sum_{i=1}^a \mathop{\mathrm{ran}}\bigl((\mathop{\mathrm{Ad}}(v_i)-1)^d\bigr)=\mathfrak s. \tag{42}\] For this choice, start with a noncentral sufficiently small element of a split torus inside \(V_0\). Its adjoint action is semisimple over a splitting field, so the nonzero image of \(\mathop{\mathrm{Ad}}(v)-1\) is unchanged upon taking a positive power. The span of its conjugates by \(V_0\) is invariant under this open subgroup, hence under \(\mathfrak s\). It is a nonzero ideal and equals \(\mathfrak s\). Finitely many conjugates suffice and give Equation (42).

Pass to the chosen convergent subsequence, denoting its levels again by \(n\). Hausdorff convergence gives \[(v_{i,n},w_{i,n})\in G_n,\qquad (v_{i,n},w_{i,n})\longrightarrow(v_i,1).\] Let \[M_n=\operatorname{span}_{\mathbb Z_p}\log G_n,\qquad M_V=\operatorname{span}_{\mathbb Z_p}\log V.\] Surjectivity of \(G_n\to V\) gives the exact equality \(\operatorname{pr}_{\mathfrak s}M_n=M_V\). All these modules lie in fixed logarithm lattices. Put \[\chi_{i,n}(X)=\det\bigl(X1-(\mathop{\mathrm{Ad}}(w_{i,n})-1)|_{\mathfrak t}\bigr), \qquad B_{i,n}=\chi_{i,n}(\mathop{\mathrm{Ad}}(v_{i,n})-1).\] The polynomials \(\chi_{i,n}\) have integral coefficients and converge coefficientwise to \(X^d\). Conjugation by \((v_{i,n},w_{i,n})\) preserves \(M_n\). Therefore \(\chi_{i,n}(\mathop{\mathrm{Ad}}(v_{i,n},w_{i,n})-1)\) preserves \(M_n\), and Cayley–Hamilton annihilates its target component exactly. For every \(x\in M_V\) choose a lift \((x,y)\in M_n\) and apply this polynomial. It follows that \[ \left(\sum_i B_{i,n}M_V\right)\times\{0\}\subset M_n,\qquad B_{i,n}\longrightarrow(\mathop{\mathrm{Ad}}(v_i)-1)^d. \tag{43}\]

Choose a basis of the fixed lattice \(M_V\) and concatenate the matrices \(B_{i,n}\). Equation (42) gives a nonzero full-rank square minor in the limiting concatenation. The corresponding integral determinants at all sufficiently late levels have one fixed finite valuation, say \(a\). The adjugate identity then yields \[p^aM_V\times\{0\} \subset\left(\sum_iB_{i,n}M_V\right)\times\{0\} \subset M_n\] uniformly at those levels. Apply Lemma 49 in the fixed product logarithm group \(V\times U\). With a constant \(c\) independent of \(n\), it follows that \[\exp(p^{c+a}M_V)\times1\subset G_n.\] After increasing \(a\) if needed, the displayed source exponential is a fixed open subgroup \(V_1<V\). Equation (38) says that \([V_1,V_1]\) fixes \(D_n\) at every sufficiently late level of this subsequence. These levels tend to infinity and their algebras exhaust \(D_0\). Thus \([V_1,V_1]\) fixes \(D_0\). Its closure has full Lie algebra \([\mathfrak s,\mathfrak s]=\mathfrak s\), contradicting Lemma 46. ◻

Proof. Let \(G_\infty\) be any limit. Its Lie projection onto \(\mathfrak s\) is surjective. By Lemma 50, its Lie projection into \(\mathfrak t\) is injective. Consequently \[ \dim_{\mathbb Q_p}\mathfrak s \leq\dim_{\mathbb Q_p}\mathfrak t. \tag{44}\] Use the same missing-root construction at every type. Source and target panel dimensions come from the same finite list, and Corollaries 44 and 48 keep each type cycle over a single prime field. Iterating Equation (44) around that cycle forces equality at every step. Only effective prime-field Lie dimensions enter this argument; no ineffective compact factor or complex Hilbert-space dimension is counted.

Both projections of \(\mathop{\mathrm{Lie}}(G_\infty)\) are now isomorphisms. The target-only subgroup \(G_\infty\cap(1\times U)\) has dimension zero. A compact zero-dimensional analytic subgroup is finite, and \(U\) is torsion-free, so \[ G_\infty\cap(1\times U)=\{1\}. \tag{45}\] Suppose a neighborhood of one failed to contain every sufficiently late target block stabilizer. Choose a smaller clopen neighborhood, levels tending to infinity, offending blocks, and stabilizer elements \(u_n\) outside it. Along a subsequence their \(G_n\) converge to \(G_\infty\) and \(u_n\to u\ne1\). Then \((1,u)\in G_\infty\), contradicting Equation (45). This proves uniform shrinking for arbitrary block choices.

Fix \(m\). For all sufficiently large \(n\), the stabilizer of every block of \(D_n\) lies in \(U_m\). It therefore fixes the image of \(D_m\) in that block pointwise. Its projective representation is irreducible by Lemma 47, so that image is scalar. The block representations are jointly faithful, and therefore \(D_m\) is abelian. Exhaustion makes \(D_0\) abelian.

Write the spectrum of \(D_n\) as the finite transitive \(U\)-space \(X_n\). The inclusions give surjective equivariant maps \(X_{n+1}\to X_n\). Averaging a faithful normal state of \(D_0\) over \(U\) gives a faithful invariant state whose restriction to each \(X_n\) is the uniform probability. Thus \[D_0=L^\infty(X,\nu),\qquad X=\varprojlim X_n,\] with these uniform marginals. The compact group \(U\) acts transitively on \(X\): the sets of elements carrying the first \(n\) coordinates of one point to those of another are nonempty nested closed subsets of \(U\). Their intersection is nonempty. The stabilizer of a point belongs to all its finite-level stabilizers, so uniform shrinking makes it trivial. Hence \(X\) is the translation \(U\)-torsor, and \(\nu\) is Haar probability. ◻

Proof. By Proposition 51, \(e\) is an abelian projection. On its central support \(z_e\) the algebra is type I, and the map \[ Z(D_*)z_e\longrightarrow eD_*e,\qquad a\longmapsto ae \tag{47}\] is a normal isomorphism. In the type I decomposition of this central summand, \(e\) has rank one in every factor, and full central support makes the scalar field faithful, which proves the assertion. The map is \(U\)-equivariant because \(U\) fixes \(e\). Countably many \(T\)-translates of \(e\) have join one, so their central supports cover one and \(D_*\) is type I. Its center has a positive-measure \(U\)-invariant chart isomorphic to the translation space \(U\). Its \(T\)-action is ergodic by Equation (35).

Use the standard Borel point realization for a continuous action of a locally compact second countable group on a separable measure algebra (Mackey 1962). Let \(X\) realize the center, and let \(\varphi:U\to X\) be the measurable isomorphism onto the chart. Equivariance holds almost everywhere for each group element. Fubini in the two Haar variables gives a point \(x_0\in X\) with \[\varphi(u)=ux_0\quad\hbox{for Haar-almost every }u\in U.\] Explicitly, choose a generic \(u_0\) such that \(\varphi(vu_0)=v\varphi(u_0)\) for almost every \(v\), and put \(x_0=u_0^{-1}\varphi(u_0)\). Let \(L=\operatorname{Stab}_T(x_0)\). If \(h\in L\cap U\), compare \(u\) and \(uh\) in a common conull injectivity set of \(\varphi\); their images agree, so \(h=1\). A subgroup meeting a neighborhood of one only at one is discrete and closed. The orbit map \(T/L\to X\) is a Borel injection and has a Borel inverse on its image. Countably many translates of the chart cover the center modulo null sets, so this orbit is conull. The measure class is quotient Haar class, as follows on the chart and its translates. We have obtained \[Z(D_*)\cong L^\infty(T/L).\]

The measurable automorphisms of \(T/L\) commuting with left translation are exactly right translations from \(N_T(L)/L\). For such an automorphism \(F\), the function \(g\mapsto g^{-1}F(gL)\) is essentially left invariant and is therefore essentially constant, say \(nL\). Thus \(F(gL)=gnL\) almost everywhere. Right-\(L\) consistency gives \(n^{-1}Ln\subset L\), and the same argument applied to \(F^{-1}\) gives equality. Conversely a normalizer coset defines such an automorphism. The injection from \(N_T(L)/L\) into the measure-class automorphism group is Borel, with Borel inverse onto its image. Indeed its action is measurable on a countable determining collection of sets, and the Borel inverse follows from injectivity between standard Borel spaces. The commuting source action consequently gives a measurable homomorphism \[\eta:S\longrightarrow N_T(L)/L,\] where the point action uses inverse right translation so that the algebra convention is Equation (46). The normalizer is closed. Measurable homomorphisms between locally compact second countable groups are continuous, so \(\eta\) is continuous and analytic over \(\mathbb Q_p\).

Its differential is injective. Otherwise its kernel has positive Lie dimension; a corresponding small subgroup of \(V\) acts trivially on the center. Since \(V\) fixes \(e\), Equation (47) would make that subgroup act trivially on \(eD_*e\), contradicting Lemma 46. Because \(L\) is discrete and \(\dim\mathfrak s=\dim\mathfrak t\), the closed Lie subgroup \(N_T(L)\) has the full dimension of \(T\) and is open. For each \(l\in L\), conjugation from \(N_T(L)\) into the discrete group \(L\) is continuous. The element \(l\) therefore centralizes an open subgroup. The effective open-centralizer assertion from Section 3 forces \(l=1\). Thus \(L=\{1\}\), the center is the \(T\)-torsor, and \(d\eta\) is an isomorphism.

It remains to remove matrix fibres. The type I decomposition and ergodicity of the center give \[D_*\cong L^\infty(T;B(\mathcal V))\] with one separable Hilbert space \(\mathcal V\) of constant dimension. In a measurable trivialization the action is \[(\alpha_a d)(x)=c(a,x)\bigl(d(a^{-1}x)\bigr),\qquad c(ab,x)=c(a,x)c(b,a^{-1}x),\] where \(c\) takes values in \(\operatorname{Aut}(B(\mathcal V))\). No unitary lift of this automorphism cocycle is needed. Changing variables in its cocycle identity and applying Fubini allows one to freeze a generic starting point \(x_0\) and define \[b(x)=c(xx_0^{-1},x).\] The identity becomes \[c(a,x)=b(x)b(a^{-1}x)^{-1} \quad\hbox{for almost every }(a,x).\] Changing the fibre trivialization by \(b\) removes the cocycle. Continuity of the action extends the equality from almost every \(a\) to every \(a\). Pure translation has fixed algebra the constant fields \(B(\mathcal V)\). Equation (35) therefore forces \(\dim\mathcal V=1\). ◻

Proof. The commutativity of \(D_*\) makes the commutators of its paired generators vanish at almost every external frame. Use a common conull frame set for countably many labels and clopen tests. At one such frame the transformed tests still generate \(C(B_T)\), so the original \(P_z\) commute, as do the \(Q_z\), for the determining labels. Continuity in the source label extends commutation to every fixed pair of labels.

In the intrinsic coordinates \(D_*=L^\infty(T)\) of Proposition 52, the spectral values of the generators give a boundary-valued measurable function \(F(x,z)\). Countable determining observations and continuity in measure give a jointly measurable version. More explicitly, use a faithful probability in the Haar class and approximate the compact label space by finite simple label choices whose errors in measure are summable. The almost-everywhere limit of the resulting jointly measurable approximations represents each fixed label; the countable boundary tests determine the boundary value. Frame covariance yields \[F(ax,z)=aF(x,z)\] almost everywhere, and Fubini gives \(F(x,z)=xF_0(z)\). The commuting source action then gives \[ F_0(sz)=\eta(s)F_0(z) \tag{50}\] in the ordinary measure class on the source boundary.

We explain why \(d\eta\) identifies the original boundaries. Regard each effective adjoint group over its coefficient field as its restriction of scalars to \(\mathbb Q_p\). The resulting Lie algebra is the prime-field Lie algebra used above. In characteristic zero, all derivations of a semisimple Lie algebra are inner, and the identity component of its algebraic automorphism group is the corresponding adjoint algebraic group. Conjugation by \(A=d\eta\) therefore gives an isomorphism of these adjoint algebraic groups over \(\mathbb Q_p\). Differentiating the homomorphism identity gives \[ A\mathop{\mathrm{Ad}}(s)=\mathop{\mathrm{Ad}}(\eta(s))A,\qquad s\in S. \tag{51}\] In faithful adjoint coordinates the algebraic isomorphism on the root-generated point groups is consequently exactly \(\eta\).

An algebraic group isomorphism carries rational parabolics and their unipotent radicals to the corresponding groups. The rational parabolics of a restriction of scalars are precisely the restrictions of the original rational parabolics. For completeness, after a finite separable splitting extension the restriction of scalars is a product indexed by the embeddings of the coefficient field. A parabolic of a product is a product of parabolics; descent permutes the conjugate factors. Choosing one factor identifies the descent condition with a parabolic over the original field. The same argument applies if the effective almost-simple group itself has several absolutely simple factors in one Galois orbit. Minimal rational parabolics thus recover the original rank-one boundary, with no splitness assumption and no prior equality of coefficient fields. This produces a homogeneous analytic homeomorphism \(\tau:B_S\to B_T\) intertwining \(S\) with \(\eta(S)\).

Write \(B_S=S/P\). By Equation (50), the measurable lift \[s\longmapsto\eta(s)^{-1}F_0(sP)\] is essentially left invariant, hence essentially constant. Right-\(P\) consistency makes this constant fixed by \(\eta(P)\). A minimal parabolic fixes exactly its own point on a rank-one boundary: its unipotent radical fixes that point and acts transitively on the complementary open Bruhat cell, which has more than one point. Therefore this constant is the parabolic point defining \(\tau\), and \(F_0=\tau\) almost everywhere. Both the original generators and the functions \(x\mapsto f(x\tau(z))\) are continuous in measure in \(z\). The generator identity consequently holds in \(D_*\) for every fixed label, not only for almost every label.

We now remove the external frame without using point evaluation on arbitrary elements of a diffuse \(L^\infty\) algebra. Let \(\Omega^+\) be a standard spectrum of the commutative algebra of the original start observations, with a faithful normal probability class. The normal representation of \(D_*=L^\infty(T)\) by its start decomposable fields takes values in \(L^\infty(T_{\mathrm{ext}}\times\Omega^+)\). It is pullback by a measurable map \(h(t,\omega)\) into the intrinsic \(T\) variable. The explicit generator identities read \[t\,p_z(\omega)=h(t,\omega)\tau(z)\] for almost every \((t,\omega)\), simultaneously for countably many labels and tests. Choose one generic external frame \(t_0\) and put \(a(\omega)=t_0^{-1}h(t_0,\omega)\). Then \(p_z(\omega)=a(\omega)\tau(z)\) for the determining labels. Each map \(a(\omega)\tau\) is continuous on the compact source boundary. Bounded convergence of its spectral evaluations and the original label continuity extend the equality to every fixed label. The end representation gives \(b\) in the same way. Only these countably many measurable generator identities were evaluated at \(t_0\); no normal evaluation on all of \(L^\infty(T)\) is asserted.

On the independent input tensor product, \(a\) and \(b\) commute. Substitution of Equation (48) in the exact graph link gives \[a\tau(z)=g\,b\tau(z)\] for countably densely many labels simultaneously. Products of the input spectral restrictions for these labels are now legitimate, since all these input observations commute. The map \(\tau\) is onto and the target action is faithful, so the common zero set of these equations is exactly \(g=ab^{-1}\). On compact sets in \(T^3\), finite clopen partitions and successively more determining labels give the corresponding graph substitution. Exhausting the compact sets and then using normal spectral calculus gives Equation (49). Equivalently, for a clopen output cut its range annihilates the complementary input graph cut and conversely; the two identities give the displayed intertwining, and monotone approximation extends it to Borel cuts.

The map from target projectivities to continuous homogeneous boundary maps is continuous and injective, and its inverse onto its image is Borel. Thus the recovered homeomorphism observations and their inverses are measurable in the usual map spaces. The argument uses just the selected coordinate of the link. Other measured coordinates remain in the coefficient spaces, so the same conclusion holds separately on each coordinate of a many-panel link. Commutation between different coordinates or different measured copies will be established in Section 12; it has not been used here. ◻

Proof. Use the convention in which the translated start observation is \(P_z^{(t)}=P_{s_tz}\) on its original input space and the end observation is \(Q_z\). Thus \(s_t\) is the effective start-label transport; it may be the inverse of the actual completing displacement, and \(\ell_S(s_t)=t\) in either orientation. The permissible mesh ensures that \(s_t\) belongs to the root-generated group \(S\), so \(\eta(s_t)\) is defined. The exact translated-input identity of Proposition 41 and Proposition 53 now give \[a\tau(s_tz)=g\,b\tau(z).\] Since \(\tau(s_tz)=\eta(s_t)\tau(z)\), this identifies \[ g=a\eta(s_t)b^{-1}, \tag{54}\] where \(s_t\) is deterministic with total missing gap \(t\). The Hilbert-space transport uses the actual source element or its actual cover; only its effective label action occurs in Equation (54).

Crucially, \(a\) and \(b\) in this equation are the observations on the original independent input spaces. Source covariance moves the entire translation dependence into the middle factor. All slower translations with nonzero missing coordinate have already been combined in \(s_t\); replacing a completing time \(t_0\) by \(t_0+u\) replaces this factor by \(\eta(s_{t_0+u})\). Other split translations have zero coordinate in this residue and act trivially on its observations. Any remaining bounded effective actions range in a fixed compact set and can be included in the compact multipliers. The sharp endpoint bases and their compact frame functions retain their fixed input laws throughout these translations. These statements hold after the inner limit for each fixed choice of the slower parameters.

With \(A=d\eta\), Equation (51) gives \[\mathop{\mathrm{Ad}}(\eta(s_t))=A\mathop{\mathrm{Ad}}(s_t)A^{-1},\qquad \log\|\mathop{\mathrm{Ad}}(\eta(s_t))\|\leq d_St+C.\] The same estimate holds for \(s_t^{-1}\). Fix an input vector \(\xi\in\overline I\otimes H'\) and \(\varepsilon>0\). The commuting map observations \(a,b\) have a joint scalar spectral measure on this vector. Choose compact sets containing both observations except for squared input norm at most \(\varepsilon\|\xi\|^2\). On that joint cut, \(\|\mathop{\mathrm{Ad}}(a)\|\) and \(\|\mathop{\mathrm{Ad}}(b^{-1})\|\) have fixed bounds independent of \(t\) and all slower parameters. Submultiplicativity, Equation (54), and Equation (52) yield \[d_T\ell_T(g)\leq d_St+C_{\xi,\varepsilon}\] on this cut, with a constant independent of those parameters. The exact graph intertwining and the isometry of the link show that the excluded output squared norm is at most the excluded input squared norm. This proves Equation (53), including for input vectors that are not simple tensors.

For finitely many panel coordinates, apply the same argument to each coordinate’s ordinary input laws and sum their exceptional output masses; no commutation of the different input map systems is needed for this upper-tail union bound. All constants are independent of the slower parameters. Consequently the estimate persists through further prescribed iterated limits with nonnegative total coordinate centers. ◻

Proof. For completeness, the distance formula can be checked directly from the metric. Translate and dilate the first point to \((0,1)\), put \(A=\lambda^2+1+|v|^2/4\), \(C=A^2+|z|^2\), and \(c=\sqrt C/(2\lambda)\). In the orthonormal frame of (55), the three components of the gradient of \(c\) are \[\frac{\lambda A}{\sqrt C}-\frac{\sqrt C}{2\lambda},\qquad \frac{Av+J_zv}{2\sqrt C},\qquad \frac{\lambda z}{\sqrt C}.\] Since \(J_zv\perp v\) and \(|J_zv|^2=|z|^2|v|^2\), the sum of their squared norms is \(c^2-1\). Thus \(\operatorname{arcosh}c\) has gradient norm one away from its unique zero \((0,1)\). It is proper. Its negative gradient curves reach that zero in time \(\operatorname{arcosh}c\); conversely its gradient bound bounds its variation by the length of any curve. It is therefore the distance from \((0,1)\), proving (i).

One convenient Weyl inversion, in these coordinates (Pohl 2010, sec. 2.4), is \[ j(v,z,\lambda)= \left(\frac{(-A_0+J_z)v}{B_0},-\frac z{B_0}, \frac\lambda{\sqrt{B_0}}\right),\quad A_0=\lambda^2+|v|^2/4,\quad B_0=A_0^2+|z|^2. \tag{58}\] The identity \(j\delta_R=\delta_{R^{-1}}j\) follows by substitution. On a compact annulus defined by \((|v|^4/16+|z|^2)^{1/4}+\lambda\asymp1\), the denominator is bounded away from zero. All derivatives of the displayed formula are therefore bounded, including at height zero, and its height ratio is positive and bounded. These statements apply to the Iwasawa groups of the symmetric spaces under consideration; the inversion is their Weyl action. No assertion about arbitrary nonsymmetric two-step groups is needed.

We spell out the coordinate issue in (ii). In a finite Bruhat cell the Gauss factors of a compact rotation times a horospherical translation are smooth, with bounded derivatives on buffered compact sets. Write them as \(n'ma_b\bar n\). Passing \(\bar n\) across \(a_{\log\lambda}\) replaces its negative-root coordinates by \(\lambda Y_{-1}\) and \(\lambda^2Y_{-2}\). At \(o\) the first of these tangent directions has no component in the central horospherical tangent space. Taylor’s formula consequently gives horizontal coordinate \(O(\lambda)\) and central coordinate \(O(\lambda^2)\), with smooth quotients, before the outer dilation by \(b\lambda\). After that dilation the two errors are \(\lambda^2U\) and \(\lambda^4W\). This proves (57), including bounded derivatives and a positive height ratio.

Boundary transitions are contact maps: a nonzero central image of a horizontal tangent would, by (55), turn growth of order \(\lambda^{-2}\) for the squared metric into growth of order \(\lambda^{-4}\), contradicting invariance of the metric. After the contact conclusion, comparison of the leading \(\lambda^{-2}\) terms shows that the horizontal derivative is a positive dilation followed by an orthogonal map. Thus the boundary action is horizontal conformal. After recentring both charts, their central component therefore has no linear horizontal term. Substitute \((v,z,\lambda)=(RV,R^2Z,Rh)\) in Taylor’s formula. The horizontal component is divisible by \(R\), the central component by \(R^2\), and the height by \(R\), with bounded derivatives of every prescribed order. The same argument applied to the inverse gives (ii). The stronger errors in (57) also remain bounded after this substitution.

In compactly rotated pole and image charts, \(g\) is \(\delta_{e^{-L}}j\), up to compact isotropy. Rescale the input annulus by \(q^{-1}\) and the output by \((q')^{-1}\). The resulting map is exactly the restriction of \(j\) to a fixed buffered annulus. The derivative and height-ratio conclusions follow from (58). Applying (ii) to a box of relative radius \(\delta/q\) gives the last assertion of (iii). A finite cover away from the pole proves the exterior assertion. ◻

Proof. Both radii are at least \(b/2\), by the triangle inequality. Choose a compactly rotated finite chart containing both endpoints in its buffered interior. Finitely many such choices cover the compact space of endpoint pairs, so their constants are uniform. Let \(\delta=d_{\partial X}(x^+,y^+)\). The heights of \(x,y\) are comparable to their radial exponentials, and their boundary coordinate errors are at most \(Ce^{-\min(d(o,x),d(o,y))}\), by (57). If \(\delta\) is at most a fixed multiple of this quantity the assertion follows. Otherwise the homogeneous separation of their current-point horizontal coordinates is comparable to \(\delta\), and (56) gives \[d(x,y)\ge d(o,x)+d(o,y)+2\log\delta-C.\] Rearranging proves the assertion. For the last statement apply it to \(x=g^{-1}o\), \(y=ho\). ◻

Proof. Multiply an extension by fixed cutoffs and view it as a smooth function on a fixed torus. Integration by parts with \((1-\Delta)^k\) gives (i) whenever \(2k\ge N\); choose \(N\) larger than the dimension plus two. At a later use requiring derivatives of the unary factors, increase \(N\) by the required derivative order. Thus every assertion here uses a finite, specified number of derivatives, chosen before any scale tends to infinity.

For a fixed Fourier term and a matching of row and column boxes, the kernel is a sum of disjoint rectangular rank-one kernels and has Schur norm bounded by the coefficient bound times the overlap constant. A graph of degree \(b\) is the union of at most \(2b-1\) matchings: greedily color its edges, first for finite subgraphs and then pass to a subsequence of the finite colorings. Split boundedly overlapping supports by assigning each point to one of its boundedly many boxes; the assignment is a row or column multiplier of norm one. Summing the matchings and the absolutely convergent Fourier series proves (ii). Each rank-one estimate is a factorization estimate and is unchanged under matrix amplification. ◻

Proof. Subtract the constant value of \(f\) near infinity. Choose \(0<a<b\) such that \(f\) is constant on \([0,2a]\) and vanishes on \([b,\infty)\). Write \(s=-\log\lambda\), \(s'=-\log\lambda'\). Since \(|s-s'|\le d(x,x')\), only a fixed number of offset bands between height blocks of length \(R\) can contribute. At a fixed offset the block pairs are a matching. Simultaneous dilation permits us to work with \(|s|,|s'|\le C_bR\) in each pair.

Take smooth radial cutoffs \(\chi_j(n^{-1}n')\) at radii \(\epsilon_j=e^{-j}\), equal to one on a smaller ball, and choose \(J=\lceil C'_bR\rceil\), with \(C'_b>C_b+b/2+2\). Formula (56) then implies that \(\chi_{-J}=1\) throughout the support of \(f(d/R)\). The annuli \(\chi_j-\chi_{j+1}\), \(-J\le j<J\), and the innermost ball \(\chi_J\) cover the support of the kernel. On each annulus set \[H_j=\log\frac{\max(\epsilon_j,\lambda,\lambda')^2} {\lambda\lambda'}.\] The following estimates are uniform in \(j,R\): the kernel \(f(d/R)-f(H_j/R)\), restricted to that annulus, has Schur norm \(O(R^{-1})\); the same holds on the innermost ball; and the height kernel \(f(H_j/R)-f(H_{j-1}/R)\) has Schur norm \(O(R^{-1})\). We verify the details, since an uncontrolled height transition would invalidate the assertion.

Put \((v,z)=\delta_{\epsilon_j}(V,Z)\). On an annulus \((V,Z)\) lies in a fixed compact set separated from zero. Each ratio involving only one height and the fixed scale \(\epsilon_j\) is adjoined as an independent compact unary coordinate before Fourier separation; derivatives in \(s/R,s'/R\) keep these auxiliary coordinates fixed. If \(\lambda,\lambda'\le\epsilon_j\), put \(\alpha=\lambda/\epsilon_j\), \(\beta=\lambda'/\epsilon_j\). Equation (56) gives \[2\cosh d=e^{H_j}\sqrt P,\qquad P=(\alpha^2+\beta^2+|V|^2/4)^2+|Z|^2.\] Here \(P\) is bounded above and below by positive constants. If \(\lambda\ge\epsilon_j\ge\lambda'\), instead put \(\alpha=\epsilon_j/\lambda\), \(\beta=\lambda'/\epsilon_j\); the same formula holds with \[H_j=\log(\lambda/\lambda'),\qquad P=(1+(\alpha\beta)^2+\alpha^2|V|^2/4)^2+\alpha^4|Z|^2.\] Again \(P\) and all its derivatives are bounded and \(P\ge1\). The reversed mixed case is identical. The four cuts made so far are rectangular height cuts; their Schur norms are one.

If both heights exceed \(\epsilon_j\), distance differs by a bounded amount from \(|s-s'|\). Where \(|s-s'|\le aR\), both arguments of \(f\) are in its constant interval for sufficiently large \(R\). Outside this interval use smooth sign cutoffs in \((s-s')/R\). On the positive piece, for example, take the larger height as the normalizing scale. The formula for \(P\) is the preceding one with the ratio of the smaller height to the larger height in place of \(\alpha\beta\). In the bounded variables \(s/R,s'/R\) that ratio is \(e^{-R|s/R-s'/R|}\), separated from its nonsmooth transition at zero. Every fixed derivative is uniformly bounded, because \(R^m e^{-aR}\) is bounded. The sign cutoffs themselves have uniformly bounded derivatives and hence bounded Schur norm by smooth separation.

In each of these cases, when \(H_j\ge aR\), write explicitly \[d-H_j=\tfrac12\log P+ \log\left(\frac{1+\sqrt{1-4e^{-2H_j}/P}}2\right).\] The last function is evaluated in a compact interval strictly inside its domain of smoothness. It has bounded derivatives in the compact ratio variables just described and in \(s/R,s'/R\). Where \(H_j\) is smaller the multiplier difference is zero, after a smooth transition still inside the constant interval of \(f\). Thus \[f(d/R)-f(H_j/R) =\frac{d-H_j}{R}\int_0^1 f'\bigl(H_j/R+t(d-H_j)/R\bigr)\,dt\] has normalized \(C^m\) norm at most \(C_m/R\). Use adjacent \(\epsilon_j\)-cells for \(n,n'\). Their relative coordinates \((V,Z)\) have uniform bounds. Lemma 57 gives the required Schur estimate. The innermost ball requires only the large-height cases: \(J\) was chosen larger than all the height exponents. Past the exterior cutoff the distance is at least \(bR\).

For the height increments use \[H_j=s+s'-2\min(j,s,s').\] Split each of \(s,s'\) into \(( -\infty,j-1]\), \([j-1,j]\), and \([j,\infty)\). If either is in the first interval, the increment vanishes. If both are in the last interval, \(H_j\) and \(H_{j-1}\) differ by exactly two; their scaled \(f\)-difference has \(C^m\) norm \(O(R^{-1})\) in \(s/R,s'/R\). If exactly one variable lies in the middle interval, use its unit-scale coordinate and the other variable divided by \(R\), with the same conclusion. If both lie in the middle interval, both arguments of \(f\) are in its constant interval and the difference is zero. These are rectangles, so no triangular projection on height indices occurs.

Finally write \(\phi_j=f(H_j/R)\). The exact telescoping identity is \[\sum_{j=-J}^{J-1}\phi_j(\chi_j-\chi_{j+1})+\phi_J\chi_J =\phi_{-J}\chi_{-J} +\sum_{j=-J+1}^J(\phi_j-\phi_{j-1})\chi_j.\] Our choice of \(C'_b\) also gives \(H_{-J}>bR\), so \(\phi_{-J}=0\). Each ball kernel has uniformly bounded Schur norm: its two horizontal entries belong to adjacent scale cells and its normalized symbol is smooth. There are \(O(R)\) increments and \(O(R)\) annular errors, each costing \(O(R^{-1})\). Summing the fixed number of height-block bands proves the Schur bound.

Restrict the kernel to the orbit points \(h^{-1}o\). The regular matrix of the Fourier multiplier is its entrywise multiplication kernel, up to the scalar phases already present in the twisted regular representation. The factorization proof of the Schur bound is unchanged by these phases or by tensoring with \(B\). Coincident orbit points cause no difficulty. This proves the last assertion, including complete boundedness. ◻

Proof. For one \(\nu\), discard the admissibility restriction after using the multiplicity bound. Plancherel and the operator norm bound for the uncut factor give \[\begin{split} \sum_{g,t}\|[Y(D_{\varphi_{i(g,t),\nu}}Y)^*]_g\|_2^2 &\le b\sum_i\|Y(D_{\varphi_{i,\nu}}Y)^*\|_2^2\\ &\le b\|Y\|_\infty^2 \sum_i\|D_{\varphi_{i,\nu}}Y\|_2^2 \le b^2\|Y\|_\infty^2\|Y\|_2^2. \end{split}\] Minkowski’s inequality sums the absolutely summable frequencies. Taking adjoints proves the other orientation. Notice that no operator norm for a localized column was used. ◻

Proof. We estimate the coefficients of \(A_CA_{D'}^*\) at a fixed shift \(g\). Put \(L=r(g)\). Their weights in the coefficient formula for \(YY^*\) are \[ f_C((gh)^+)\overline{f_{D'}(h^+)}. \tag{61}\] Only pairs with both \(h\) and \(gh\) in the long support matter. Their polar heights are at most \(e^{-l}\). We describe the smooth extensions of these weights; all coordinate boxes below come from fixed covers independent of \(g\).

First suppose that the input box is separated from \(g^-\) by at least a sufficiently large multiple of \(\delta\), and its pole distance is \(q=e^{-u}\). If \(2u\le L+C_1\), Lemma 55(iii) says that the map from this box to the output boundary, including the extra coordinate \(e^{-r(h)}/\delta\), has normalized derivatives bounded by a constant. Its image meets only boundedly many output boxes. Extend on a fixed larger box, multiply by its buffered cutoff, and Fourier-expand there. The weight in (61) is now a unary expansion in \(h\), with coefficients decaying faster than any prescribed fixed power. The basis functions depend on \(D'\), the fixed coordinate chart, and the frequency, and not on \(g\) or \(C\). If \(2u>L+C_1\), inversion gives output pole distance \(e^{-L+u}\); unless the output box is itself a pole box, the inverse map has the same bounded-magnification property. Use the reversed unary estimate in that case. A box intersecting the transition \(2u=L+O(1)\) admits either construction, with a larger fixed buffer. Starting from an output box gives the same alternatives. Bounded \(L\) is included by increasing the constants.

Thus every pair except pairs of boxes lying within \(C_2\delta\) of the two respective poles is handled by Lemma 59. To justify the multiplicity used there, a map with bounded magnification sends one buffered \(\delta\)-box into a ball of radius \(C\delta\), which meets a bounded number of reference boxes. The height variable changes this radius by at most \(Ce^{-l}\). This proves the required multiplicity for each \(g\) and initial box, without a restriction on the number of group elements in it.

There remain only boundedly many exceptional pairs for each \(g\). If such a pair is compatible with the long supports, the two current-point pole sizes have product comparable to \(e^{-L}\), by (58), whereas each is at most \(C\delta\). Consequently \[ L\ge2s-C_3. \tag{62}\] Expand each mask as its value at its pole plus its difference from that value. The constant term costs at most \(C\|YY^*\|_2\). Terms with just one nonconstant mask are covered by Lemma 59; any additional constant term created by this expansion is treated with the first term.

For the product of the two differences, partition the input current-point pole coordinate, including height, into smooth unit-width annuli with size \(e^{-u}\), \(0\le u\le L/2+C_4\). Use the analogous output annuli on the remaining part. A smooth transition at the scale \(e^{-L/2}\) has bounded normalized derivatives on both descriptions, by (58). Exterior annuli mean a fixed finite cover away from the pole.

On an input annulus, the two differences, after substitution, have product seminorm bounded by \[ C_m\min(1,e^{s-u})\min(1,e^{s-L+u}) \le C'_m e^{-|s-u|}. \tag{63}\] Here is the derivative justification for both factors. On scales smaller than a mask box, its difference from its pole value is bounded by the relative scale, by Taylor’s formula; the same bound holds for every normalized derivative on the smaller box. On scales larger than the mask box by a fixed factor, the mask is zero on the long support, because its current-point height is much smaller than \(\delta\). It can therefore be replaced by zero on that part before subtracting its pole value, giving a bounded constant symbol. At comparable scales its given seminorm bound suffices. The output factor has exactly the same three descriptions after the normalized inversion, with scale \(e^{-L+u}\). The collar substitutions have bounded derivatives including the normalized height by Lemma 55(ii). These constructions give smooth extensions on buffered boxes; equality is required only on the two long supports. Finally, if \(u\le s\), use (62) in the second minimum; if \(u\ge s\), use the first minimum. This proves the last inequality in (63).

At a fixed integer depth \(u\), use the reference boundary boxes of radius \(e^{-u}\), and append the height divided by \(e^{-u}\) as an auxiliary unary variable. Only boundedly many such boxes meet the annulus about the given pole. Their basis functions are chosen before \(g\); the coordinates of the pole merely change the expansion coefficients. The number of original exceptional pairs is also bounded. Lemmas 57 and 59 therefore give a bound \(Ce^{-|s-u|}\|Y\|_\infty\|Y\|_2\) in the square sum of all coefficients using this depth. Sum over \(u\) in norm; the geometric series has a uniform sum. The reversed annuli give the same bound. Combining the nonexceptional and exceptional estimates proves the inequality in (60). Its equality is the trace identity \(\tau(A_CA_{D'}^*A_{D'}A_C^*) =\tau(A_C^*A_CA_{D'}^*A_{D'})\), summed over the masks.

For general \(Y\), choose bounded Fourier-polynomial approximations in the strong-star topology and apply a smooth radial cutoff equal to one on \(\{r\ge l\}\) and zero on \(\{r\le l/2\}\). Proposition 58 bounds these approximants uniformly. Increase the constant in the hypothesis on \(l\) to allow this slack. Each masked column converges in \(L^2\), so each product of two masked columns converges in \(L^1\), by tracial Cauchy–Schwarz. For a fixed finite family of pairs, the preceding estimate bounds their products in the Hilbert direct sum of \(L^2(D)\). Weak compactness gives a weak \(L^2\) limit, and the \(L^1\) convergence identifies it with the desired products. Lower semicontinuity proves the bound for this finite family; exhaustion proves it for all pairs. In particular the diagonal terms put every masked column in \(L^4\), so the trace identity in (60) remains valid. This also explains why neither infinite Fourier support nor a nondiscrete projected group changes the proof. ◻

Proof of the right-density assertion. Write \(K_{d_1}(v,w)=\int\theta_{o,d_1}(p,v)\theta_{o,d_1}(p,w)\,dp\). Disintegrate this integral at radial depth \(s=d(o,p)\). Its range is \([{d_1}-C,2{d_1}+C]\). With measure \(ds/{d_1}\), the resulting kernels \(k_{d_1,s}(v,w)\) are supported on adjacent boundary boxes of radius \(e^{-s}\), with normalized \(C^m\) bounds independent of \({d_1},s\), for every fixed \(m\).

Here are the normalization details. The polar volume is \(J(s)\,ds\,d\nu\), where \(J(s)=c(\sinh s)^{\dim V}(\sinh2s)^{\dim Z}\asymp e^{Qs}\) for \(s\ge1\). The shadow of one ray has angular volume \(O(e^{-Qs})\). Formula (64) contributes \({d_1}^{-1}\), removed by the choice of \(ds/{d_1}\). Thus no volume factor remains. To check derivatives, choose smooth compact frames near one endpoint, translate back along its ray by depth \(s\), and rescale neighboring directions by \(e^{-s}\). Their negative-root coordinates have sizes \(e^{-s}\) and \(e^{-2s}\); conjugation by the split translation makes both bounded. The positive-root coordinates contract and the remaining factors stay compact. Lemma 55(ii), followed by (64) on this fixed bounded set, therefore bounds every prescribed normalized derivative. The same argument applies to the polar Jacobian after its factor \(e^{Qs}\) is removed.

Separate \(k_{d_1,s}\) on adjacent boxes. At each frequency the right density is a sum \(\sum_{C,D'}t_{C,D'}A_C^*B_{D'}\), where \((t_{C,D'})\) is a scalar matrix of bounded degree and bounded operator norm, and both mask families satisfy the hypotheses of Lemma 60. The column form of tracial Hölder gives \[\left\|\sum_{C,D'}t_{C,D'}A_C^*B_{D'}\right\|_2 \le \|t\| \left\|\sum_C A_C^*A_C\right\|_2^{1/2} \left\|\sum_{D'}B_{D'}^*B_{D'}\right\|_2^{1/2}.\] Equation (60) bounds this by \(C\|Y\|_\infty\|Y\|_2\). Fourier coefficients are summable; choose their decay order larger than the polynomial derivative growth of the masks. Integration in \(ds/{d_1}\), over an interval of bounded total mass, proves the right-density bound. Finally \[\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_2^2 =\| (\mathbb V_{d_1}Y)^*\mathbb V_{d_1}Y\|_2^2\] by the semifinite trace, first for finite-rank columns and then by monotone approximation. This proves the additional assertion in (i). ◻

Proof. We first prove uniformity in all angular derivatives, including at the two ends of the annular decomposition. By a compact rotation put \(go=(0,\epsilon)\), \(\epsilon=e^{-L}\), and \(g^+=0\). For a finite endpoint \(v=(V,W)\), define \[p(v)=|V|^2/4,\quad D_h(v)=(h^2+p(v))^2+|W|^2,\quad P_h(v)=\left(\frac{(h^2+p(v)+J_W)V}{D_h(v)}, \frac W{D_h(v)}\right).\] Formula (58) shows that \(\delta_{\sqrt{D_1(v)}}L_{P_1(v)^{-1}}jL_{v^{-1}}\) fixes \(o\) and sends \(v\) to infinity. In this chart the displaced point is \((n,e^\beta)\), with \[ n=\delta_{\sqrt{D_1(v)}}(P_1(v)^{-1}P_\epsilon(v)), \qquad e^\beta=\epsilon\sqrt{D_1(v)/D_\epsilon(v)}. \tag{68}\] All products in this formula are products in the real two-step group \(N_b\); no nonassociative scalar multiplication is used.

Put \(v=\delta_Ru\), \(R=e^{-j}\), \(t=\epsilon/R\), on a fixed buffered annulus for \(u=(V,W)\), and write \[\begin{split} \Delta_R(u)&=(1+R^2p(u))^2+R^4|W|^2,\\ Q_R(u)&=\left( \frac{R^2(1+R^2p(u)+R^2J_W)V}{\Delta_R(u)}, \frac{R^4W}{\Delta_R(u)}\right). \end{split}\] Direct substitution in (68) gives \[ \beta=2j-L+B(R,t,u),\quad B=\tfrac12\log(\Delta_R/D_t),\quad \delta_Rn=m(R,t,u) =\delta_{\sqrt{\Delta_R}}(Q_R^{-1}P_t). \tag{69}\] For \(R,t\in[0,1]\), the denominator \(D_t\) is bounded away from zero on the buffered annulus, and \(\Delta_R\ge1\). Thus \(B,m\) have bounded derivatives of every fixed order. As \(R,t\to0\), \(m\to P_0(u)\), uniformly with derivatives; this limit stays away from the identity. Consequently \(m\) is uniformly separated from the identity if both \(j\) and \(L-j\) exceed a fixed constant. For the innermost ball take \(j=L,t=1\): \(D_1\ge1\), so all upper bounds remain valid, including at \(u=0\). The exterior follows from finitely many compact charts away from the pole, with \(\beta=-L+O_{C^m}(1)\), \(n=O_{C^m}(1)\).

Let \(H(m)=\int F(q)F(m^{-1}q)\,dq\). Then \(H\) is smooth and compactly supported, and \(H(e)=1\). Haar substitution in (64) gives the exact formula \[ w_{d_1}(g,v)=\int\psi(t)\psi(t-\beta/{d_1}) H(\delta_{e^{-t d_1}}n)\,dt. \tag{70}\] On a normalized annulus it becomes \[\frac1{d_1}\int\psi(s/{d_1})\psi((s-2j+L-B(u))/{d_1}) H(\delta_{e^{j-s}}m(u))\,ds.\] Replace \(B\) by zero using the mean-value formula. This costs \(O_m({d_1}^{-1})\) after angular differentiation: the extra factor is \({d_1}^{-1}\), and all derivatives of the accompanying \(H\) factor are bounded on the relevant support. Indeed, in the interior annuli, compact support of \(H\) and separation of \(m\) from the identity force \(s-j\ge-C\); near \(j=0\), the first \(\psi\) forces \(s-j\ge {d_1}-C\); near \(j=L\), either of the shifted \(\psi\)’s in the mean-value formula forces the same inequality. The interval of integration has length \(O({d_1})\).

It remains to replace \(H(\delta_{e^{j-s}}m(u))\) by \(\mathbf1_{s>j}\). On the interior annuli, with \(r=s-j\), every angular derivative satisfies the integrable bound \[\left|\partial_u^\gamma \left(H(\delta_{e^{-r}}m(u))-\mathbf1_{r>0}\right)\right| \le C_\gamma\left( \mathbf1_{[-C,0]}(r)+e^{-r}\mathbf1_{[0,\infty)}(r)\right).\] For negative \(r\) this follows from the support of \(H\); for positive \(r\) it follows from Taylor’s formula at the identity, because horizontal and central coordinates and all their \(u\)-derivatives acquire factors \(e^{-r}\) and \(e^{-2r}\). Integration against \(ds/{d_1}\) gives \(O_m({d_1}^{-1})\). In the two end bands the \(\psi\) supports already imply \(r\ge {d_1}-C\), so the same conclusion follows without a lower bound on \(m\). This proves (66) on all pieces.

Finally differentiation under the integral gives \[\partial_ja_{d_1}(j,L)=-\frac1{d_1}\psi(j/{d_1})\psi((L-j)/{d_1}) -\frac2{{d_1}^2}\int_j^\infty\psi(s/{d_1}) \psi'((s-2j+L)/{d_1})\,ds.\] This is \(O({d_1}^{-1})\). At \(j=0\) the integral is \(\int\psi(t)\psi(t+L/{d_1})\,dt\); the fundamental theorem of calculus in \(L^2\) proves the last assertion. ◻

Proof of the left-density assertion in Proposition 61. At a shift \(g\), the left density has the coefficients of \(YY^*\) weighted, term by term, by the overlap of the two rays based at \(o,go\), towards \((gh)^+,gh^+\). There are no such coefficients unless \(L=r(g)\le4{d_1}+C\): both ray functions are supported within distance \(2{d_1}+C\) of their basepoints.

We may replace the second endpoint by \((gh)^+\), with an error whose coefficient \(L^2\) norm is at most \[ C e^{C'{d_1}-l}\|Y\|_\infty\|Y\|_2. \tag{71}\] To verify this uniform bound, regard \(h\) in compact polar coordinates with the additional unary variable \(e^{l-r(h)}\in[0,1]\). Equation (57) gives smooth agreement of its current-point direction and ideal endpoint at height zero. Acting by \(g\), whose length is at most \(4{d_1}+C\), preserves this vanishing and multiplies each fixed derivative bound by at most \(e^{C_m {d_1}}\). One way to see the latter bound directly is to compose at most \(4{d_1}+C+1\) isometries of length at most one and differentiate a fixed finite number of times; compact coordinate transitions have bounded derivatives, and the resulting bounds grow at most exponentially. The half-space charts needed at interior depth at most \(C d_1\) have denominators bounded below by \(e^{-C_m {d_1}}\), as is also explicit in (58). Formula (64) and the volume on its support have the same fixed-order exponential bounds. The overlap difference therefore has a unary smooth extension with seminorm at most \(C_m e^{C_m {d_1}-l}\), on each of a fixed finite set of angular charts and the normalized height interval. Fourier separation and Lemma 59, with the other factor uncut, give (71). There is no summation over the number of shifts in \(\{r\le4{d_1}+C\}\).

For the resulting common-endpoint coefficients, apply Lemma 62. At each annular depth, the error is a unary symbol of seminorm \(O({d_1}^{-1})\) on a bounded number of reference boxes around the pole for that \(g\). Thus its coefficient \(L^2\) norm is at most \(C{d_1}^{-1}\|Y\|_\infty\|Y\|_2\), by Lemma 59. Telescope the constants \(a_{d_1}(j,L)\) against single smooth balls. Their increments have the same \(O({d_1}^{-1})\) bound and are treated by the same unary estimate. The baseline simply multiplies the uncut coefficient by a bounded scalar. There are at most \(C(1+{d_1})\) levels because \(L\le4{d_1}+C\). Summing these norms and (71) proves the asserted uniform bound for the left density.

We record the stronger short-coefficient conclusion needed below. If \(a=o({d_1})\), then on the shifts \(r(g)\le a\) the baseline differs from one by \(O(a/{d_1})\), and only \(C(1+a)\) levels occur. The left-density coefficients restricted to these shifts therefore differ from those of \(YY^*\) by a quantity bounded in \(L^2\) by \[ C\left(\frac{1+a}{d_1}+e^{C'{d_1}-l}\right) \|Y\|_\infty\|Y\|_2. \tag{72}\] This tends to zero when both displayed norms remain bounded, in the scale regime of Proposition 61. For cross densities of two bounded long columns the corresponding statement follows by polarization; the sums and differences of the columns retain the same long support. In particular, \[ \langle\mathbb V_{d_1}Z,\Delta(x)\mathbb V_{d_1}Y\rangle -\langle Z,xY\rangle\longrightarrow0 \tag{73}\] uniformly when the operator and Hilbert norms of the long columns \(Y,Z\), and the Hilbert norm of \(x\), are bounded, with \(\mathop{\mathrm{supp}}\widehat x\subset\{r\le a\}\). Explicitly, polarization and Cauchy–Schwarz bound the absolute value of the difference by \[C\left(\frac{1+a}{d_1}+e^{C'd_1-l}\right)\|x\|_2 (\|Y\|_\infty+\|Z\|_\infty)(\|Y\|_2+\|Z\|_2).\] The Cauchy–Schwarz step pairs the restricted density coefficients with the Fourier coefficients of \(x\). In particular it does not replace \(\|x\|_2\) by an operator bound on an algebra whose trace mass could vary.

For infinite Fourier support, the bounded approximants with slacked radial cutoffs used after (60) apply here as well. The density bounds yield weakly convergent \(L^2\) subsequences, whose trace pairings identify their limits with the vector densities. This proves (i) in full generality. ◻

Proof of the pairing and intertwining assertions in Proposition 61. First prove (iii). The kernel \(K_{d_1}(v,w)\) satisfies \(K_{d_1}(v,v)=1\). For each fixed derivative order \(m\), its derivatives on fixed angular charts are bounded by \(C_m e^{C_m {d_1}}\): use smooth compact frames, the depth restriction \(d(o,p)\le2{d_1}+C\), and the same fixed-order coordinate and volume bounds as in (71). On a pair of adjacent boxes of radius \(e^{-d_2}\), Taylor’s formula on the diagonal and the weighted coordinate scaling imply \[ \|K_{d_1}-1\|_{C^m_{\rm normalized}} \le C_m e^{C_m {d_1}-d_2}. \tag{74}\] For derivatives of positive order, at least one normalized angular derivative supplies a factor \(e^{-d_2}\); the zero-order estimate uses the equality on the diagonal. Central derivatives supply the stronger factor \(e^{-2d_2}\).

Choose \(d_2\) with \({d_1}\ll d_2\ll l'\). The assumed support condition makes every contributing pair adjacent at scale \(e^{-d_2}\). Insert a smooth partition at that scale and separate \(K_{d_1}-1\) on each adjacent pair. At one frequency, Cauchy–Schwarz for the columns of cut vectors gives \[\left|\sum_{C,D'}t_{C,D'}\langle Z_C,Y_{D'}x\rangle\right| \le\|t\|\,\|x\|_\infty \left(\sum_C\|Z_C\|_2^2\right)^{1/2} \left(\sum_{D'}\|Y_{D'}\|_2^2\right)^{1/2}.\] The matrix norm is bounded by its degree times its largest coefficient, which tends to zero by (74). Overlap bounds control the two sums by \(C\|Z\|_2^2,C\|Y\|_2^2\). Choose the finite derivative order in advance sufficiently large for a summable Fourier expansion. Since \(d_2/{d_1}\to\infty\), the sum of the frequency bounds still tends to zero. This proves (65). The proof only needs \(Y,Z\in L^2\); Fourier truncation of those two vectors and continuity extend it with \(x\) remaining bounded.

For right multiplication by a short element we verify the support condition. If \(r(h),r(hk)\ge l-a\) and \(r(k)\le a\), then \(d(ho,hko)\le a\). In a compact finite boundary chart the heights of these two points are comparable to their radial exponentials. Their horizontal separation, measured by a homogeneous gauge, either is of that small height order or, by (56), satisfies \[d(ho,hko)\ge r(h)+r(hk) +2\log d_{\partial X}(h^+,(hk)^+)-C.\] The collar errors are absorbed in the first alternative. A finite chart cover, or a compact rotation placing the pair in one such chart, gives \(d_{\partial X}(h^+,(hk)^+)\le Ce^{-c(l-2a)}\) with fixed \(c>0\). Since \(a/{d_1}\to0\) and \(l/{d_1}\to\infty\), assertion (iii) applies. It also applies with \(xx^*\), supported in \(\{r\le2a\}\), and with the long column \(Yx\). Expanding the square of \(\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\), its three terms tend respectively to \(\|Yx\|_2^2,\|Yx\|_2^2,-2\|Yx\|_2^2\). This proves the right assertion in (ii).

For the left assertion use (73) first with \(x^*x\), then with the cross pair \(Z=xY\) and \(x\). The support of \(x^*x\) lies in \(\{r\le2a\}\), and \(xY\) has support in \(\{r\ge l-a\}\). The same expansion of the squared norm now proves \(\|\mathbb V_{d_1}(xY)-\Delta(x)\mathbb V_{d_1}Y\|_2\to0\).

All proofs used trace identities, Plancherel, and constants uniform in the coefficient algebra. They therefore apply to matrix algebras and, after integration, to measurable fields. To make any support-preserving approximation needed here, first use bounded Fourier-polynomial approximation and then smooth radial cutoffs with fixed slack. Proposition 58 gives uniform operator bounds; doubling a short support and halving a long support preserves all scale separations in the statement. ◻

Proof. Pair the appropriate density from Proposition 61 with \((x-x')^*(x-x')\), or with the reversed product on the right. Tracial Cauchy–Schwarz and \[\|(x-x')^*(x-x')\|_2 \le\|x-x'\|_\infty\|x-x'\|_2\] give, with \(z=x-x'\), the homogeneous bound \[ \max\{\|\Delta(z)\mathbb V_{d_1}Y\|_2^2, \|(\mathbb V_{d_1}Y)z\|_2^2\} \le C\|Y\|_\infty\|Y\|_2\|z\|_\infty\|z\|_2. \tag{75}\] In the amplified case use the density \(\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\) and the same inequality with the semifinite Hilbert norm of \(z\). For measurable fields, integrate this inequality. Uniform operator bounds and Cauchy–Schwarz on the base give the product of the integrated Hilbert norms of \(Y\) and \(z\). Thus integrated Hilbert-small changes suffice; no pointwise error bound is required. ◻

Proof. Choose a compact \(K\subset G\) for which \(|\{y:d_0(y)\notin K\}|<\varepsilon\). Uniformly for \(s\in C\), outside a set of measure at most \(2\varepsilon\), both \(d_0(y)\) and \(d_0(sy)\) belong to \(K\). There \[c(s,y)\in \Gamma\cap K^{-1}CK,\] a finite set. Apply (78) to this finite set. Diagonalization over compact exhaustions of \(G\), followed by Proposition 58, gives (79). More explicitly, use a fixed smooth function equal to one on \([0,1]\) and zero on \([2,\infty)\), at a cutoff \(b_n=o(R_n)\) tending sufficiently slowly to infinity relative to the required finite lists, and put \(a_n=2b_n\). The completely bounded multiplier norm gives the uniform operator bound. The normalized factor traces also give \(\|X_s^{\Theta_n}(y)\|_2=1\); Plancherel bounds the integrated Hilbert norms of the short approximants by the supremum norm of the fixed cutoff. Thus both kinds of bounds required in Proposition 61 are uniform in these applications.

Here is the spectral assertion needed in the last sentence. Multiplication by an element supported in \(\{r_T\le a_n\}\) sends the Fourier support of a radial interval into its \(a_n\)-neighborhood, on either side, because \[ |r_T(gh)-r_T(h)|\le r_T(g),\qquad |r_T(hg)-r_T(h)|\le r_T(g). \tag{80}\] Take two closed rescaled intervals at positive distance. A short multiplier cannot connect their radial cuts for large \(n\). For smooth radial cuts of a uniformly bounded column the errors from replacing an actual transporter by its short approximation are Hilbert-small: one factor is Hilbert-small and the other is uniformly operator-bounded. The radial multiplier bound supplies the latter bound for every cut used here. Finite spectral partitions therefore show that multiplication by the actual unitary transporter intertwines the limiting rescaled length measurement. Approximation of continuous functions on \([0,\infty]\) removes the partitions. Reindexing \(Y\) preserves its probability measure, and the scalar phases in a product of transporters cancel from all these Gram forms. ◻

Proof. Let \((C,\varepsilon)\) be a compact Kazhdan pair for the whole locally compact product \(G\); property \((T)\) provides such a pair (Bekka et al. 2008). Choose a relatively compact neighborhood \(V\) of the identity with positive Haar measure, and a symmetric compactly supported probability density \(\nu\) bounded below on \(V\cup CV\) and their inverses. For a unit vector \(\xi\), writing \(g=(gv)v^{-1}\) and averaging over \(v\in V\) bounds \[\sup_{g\in C}\|\pi(g)\xi-\xi\|^2 \le C_1\int_G\|\pi(h)\xi-\xi\|^2\,d\nu(h).\] Indeed the two terms in the triangle inequality are integrated over \(CV\) and \(V^{-1}\), where the density has a uniform positive lower bound. The Kazhdan inequality thus gives a uniform upper bound \(1-\delta\) for the self-adjoint averaging operator \(\pi(\nu)\) on the complement of the invariant vectors. The lazy average \((1+\pi(\nu))/2\) is positive and has norm at most \(1-\delta/2\) there. Convolve this probability on both sides with a symmetric compactly supported probability density \(\chi\). Its operator is \[\pi(\chi)\frac{1+\pi(\nu)}2\pi(\chi),\] which is positive and has the same uniform upper bound. The resulting probability \(m_0\) is absolutely continuous. ◻

Proof. The inserted path and the order of limits. First pass to a subsequence on which the sum laws converge. Insert a fixed translation \(s\in G\) between the two positive axial segments. Away from the Bruhat wall corresponding to backtracking, Cartan multiplication expresses \(a_{t_n}s a_{u_n}\) as \(a_{t_n+u_n}\) with pre- and post-translations in a compact set depending on \(s\). This includes a segment whose length stays bounded. To verify the assertion when both lengths diverge, write \(s\) in the open opposite-radical–Levi–radical cell for the axis: the two radical factors contract at the respective ends, and the Levi factor has a fixed split component. The omitted wall is a proper lower-dimensional Bruhat cell and has Haar measure zero. The components of \(s\) outside the chosen source factor are fixed compact displacements. The exceptional wall in that factor has Haar measure zero, also after taking its product with the complementary factors. Lemma 65 therefore identifies the uncut law of the inserted product with the limiting sum law, for almost every \(s\).

The Gram computation. Write \(A_n(y)=X^{\Theta_n}_{a_{t_n}}(y)\) and \(B_n(y)=X^{\Theta_n}_{a_{u_n}}(a_{u_n}^{-1}y)\). For smooth radial functions \(f,g\), constant near the two ends of the compactified interval, put \[ A_{n,f}=D_{f(r_T/R_n)}A_n,\qquad B_{n,g}=D_{g(r_T/R_n)}B_n . \tag{82}\] Consider the columns \[ A_{n,f}(sy)X^{\Theta_n}_s(y)B_{n,g}(y). \tag{83}\] They are uniformly operator-bounded. Their mixed Gram forms are pairings between \(A_{n,f'}^*A_{n,f}\) and the translate of \(B_{n,g}B_{n,g'}^*\) under the induced trace action \[F(y)\longmapsto \mathop{\mathrm{Ad}}\Theta_n(c(s,s^{-1}y))F(s^{-1}y).\] Its invariant space consists of constant scalar fields: an invariant equivariant field lifts to a constant field on \(G\), whose value commutes with \(\Theta_n(\Gamma)\). More explicitly, the lifted field is left invariant, hence essentially constant by Haar invariance; its residual right \(\Gamma\)-equivariance imposes precisely that commutation relation. Conversely a scalar constant gives an invariant field. The matrix products appearing in the Gram forms have uniformly bounded integrated Hilbert norms, since each radial piece has both a uniform operator bound and a uniform Hilbert bound. Average \(s\) with \(m_0^{*k}\). Lemma 66 shows that the difference from the product of the scalar Gram forms is \(O(c_0^k)\), uniformly in \(n\).

Take first a Hilbert limit in \(n\), for each fixed \(k\), and then a Hilbert limit in \(k\). Equivalently, form the two successive Hilbert ultraproducts of the spaces with measures \(dy\,dm_0^{*k}(s)\). Polarization of the preceding calculation gives an isometry \[ J:L^2(P_t)\otimes L^2(P_u)\longrightarrow\mathcal K,\qquad f\otimes g\longmapsto \bigl(A_{n,f}(sy)X^{\Theta_n}_s(y)B_{n,g}(y)\bigr)^{\lim}. \tag{84}\] The Fourier length in the output has its own spectral measurement \(E\) on \(\mathcal K\). For each fixed \(k\), the measure \(m_0^{*k}\) has compact support. All short-approximation errors may therefore be sent to zero in \(n\) before increasing \(k\). Dominated convergence identifies the uncut output law for that fixed average with the subsequential sum law. No uniform approximation over the growing supports of all convolution powers is asserted or needed.

Positive finite input windows. We prove that its restriction to the range of \(J\) is addition. An output interval strictly above the sum of two input intervals is excluded by (80) and short approximation of \(X^{\Theta_n}_s\). It remains to exclude a strict deficit. Localize the two input lengths in smooth windows bounded below by \(\varepsilon'R_n\), and localize the output in a window shorter than their sum by \(\varepsilon R_n\), with \(\varepsilon,\varepsilon'>0\). All window boundaries may be enlarged slightly. Write \(A=A_{n,f}\), \(B=B_{n,g}\). In testing the output cut, trace cyclicity gives pairings of \(A(sy)^*\) against \[X^{\Theta_n}_s(y)B(y)C(y,s),\] where \(C\) is uniformly operator-bounded and has the Fourier support of the inverse shortened-output window. In particular one may take \(C\) to be the adjoint of the smooth squared output cut of the product itself.

Choose, by Lemma 65, scales \[a_n\ll d_n\ll R_n .\] On each compact set of \(s\)’s replace \(X^{\Theta_n}_s\) by its short approximation. Lemma 56 implies that every contributing pair of long indices has directions at distance at most \(C e^{-c\varepsilon R_n}\). Proposition 61 therefore permits insertion of the two ray lifts and then intertwines the short left multiplication through the lift. The result, with an error tending to zero, is \[ \tau\bigl(\zeta_n(sy)^* \Delta_T(X^{\Theta_n}_s(y))\xi_n(y)C(y,s)\bigr), \quad \xi_n=\mathbb V_{d_n}B,\quad \zeta_n=\mathbb V_{d_n}(A^*). \tag{85}\] The errors are uniform for \(s\) in a fixed compact set. Replacing the short approximation by the actual transporter on the lifted vector is legitimate because the left density has uniformly bounded tracial \(L^2\)-norm: for a bounded Hilbert-small change \(z\), \[ \|\Delta_T(z)\xi_n\|^2 \le \|z\|_\infty\|z\|_2\|\rho^{\,L}_{\xi_n}\|_2 . \tag{86}\] The integrated version follows from Cauchy–Schwarz on \(Y\).

The exact trace inequality for (85) is \[\begin{align*} &\left|\tau\bigl(\zeta_n(sy)^* \Delta_T(X^{\Theta_n}_s(y))\xi_n(y)C(y,s)\bigr)\right|^2 \\ &\quad\le \|C(y,s)\|_2^2 (\mathop{\mathrm{Tr}}\otimes\tau)\left[ \zeta_n(sy)\zeta_n(sy)^* \mathop{\mathrm{Ad}}\Delta_T(X^{\Theta_n}_s(y)) \bigl(\xi_n(y)\xi_n(y)^*\bigr)\right]. \tag{87}\end{align*}\] Both density fields on the right have uniformly bounded trace-Hilbert norm, by Proposition 61; they are independent of \(s\) apart from the displayed transport. The induced amplified action has no invariant vector, by admissibility and the same equivariant-field lifting argument. Its \(m_0^{*k}\)-average is therefore bounded by \(C c_0^k\). Integrate (87), first let \(n\) tend to infinity, and then let \(k\) tend to infinity. The shortened-output cut vanishes.

Zero and infinite input values. Suppose first that one input value is zero and the other is a finite value \(v\). Restrict the first input to \([0,\delta]\) and the second to \([v-\delta,v+\delta]\cap[0,\infty)\), on the rescaled length axis. The short intermediate transporter changes length by \(o(1)\) on this scale. The forward and reverse triangle inequalities therefore restrict the output to \([v-2\delta-o(1),v+2\delta+o(1)]\cap[0,\infty)\). Every closed output window separated from \(v\) is excluded by choosing \(\delta\) sufficiently small. This includes \(v=0\), where the upper triangle bound suffices. The cut replacements have the same Hilbert-small errors as before.

For an input value \(+\infty\), it suffices to exclude output in a fixed finite interval \([0,M]\). If the other input is finite, choose a bounded neighborhood \([0,B]\) containing it and restrict the infinite input to \([N,\infty]\), with \(N>M+B+1\). The reverse triangle inequality excludes the output window, after the short intermediate replacement. If both input values are infinite, restrict both to \([N,\infty]\), where \(2N>M+1\). The connecting output index then has a deficit at least \((2N-M-o(1))R_n\) from the sum of the two long input lengths. The lifted shortening argument just proved applies: it used a positive lower length bound and a uniform deficit, but no upper bound on either long input. Use smooth tail cutoffs constant near infinity, whose multiplier norms are bounded by Proposition 58. Hold \(N,M\) fixed through the limits in \(n\) and then \(k\); afterward exhaust the finite output intervals and increase the finite lower input cutoffs. This excludes every finite output value at an infinite sum.

Identification of the output measurement. We have proved that a product of supported smooth input functions has no output in a closed window disjoint from the sums of its input supports. Characteristic functions of interval partitions whose boundaries have zero input measure are obtained by Hilbert approximation with such smooth functions. The isometry \(J\) preserves this approximation.

Here is an exact spectral argument that does not assume in advance that \(E\) preserves the range of \(J\). Write \(a(s,t)=s+t\), a continuous map on the compact square, and let \(P_{\rm in}\) be multiplication by Borel subsets of that square on \(L^2(P_t)\otimes L^2(P_u)\). Fix a closed output set \(F\). Product test functions supported on rectangles whose closures lie in the open set \(a^{-1}(F)^c\) have dense span in \(\mathop{\mathrm{ran}}P_{\rm in}(a^{-1}(F)^c)\), by regularity of the product measure and Hilbert approximation with smooth interval cutoffs. For each such rectangle the compact sum support misses \(F\). A finite cover of \(F\) by the excluded output windows and the rectangle support conclusion therefore give \[E(F)J P_{\rm in}(a^{-1}(F)^c)=0, \qquad E(F)J=E(F)J P_{\rm in}(a^{-1}(F)).\] Choose closed sets \(K_j\uparrow F^c\); for example use positive distance at least \(1/j\) from \(F\) in a compatible compact metric. Apply the same conclusion to every \(K_j\). Since \(a^{-1}(K_j)\) is disjoint from \(a^{-1}(F)\), it gives \[E(K_j)J P_{\rm in}(a^{-1}(F))=0.\] Strong convergence of these spectral projections yields \(E(F^c)J P_{\rm in}(a^{-1}(F))=0\). Combining the two identities proves \[E(F)J=J P_{\rm in}(a^{-1}(F)).\] The monotone-class theorem extends this equality to all Borel sets, and spectral integration then gives \[E(\varphi)J=J\,M_{\varphi(s+t)} \qquad(\varphi\in C([0,\infty])).\] On \(J(1\otimes1)\), the output law is the limiting sum law identified in the first paragraph. This proves (81). Compact pre- and post-translations are covered by Lemma 65. Every convergent subsequence of the sum laws has the same limit, so compactness of the space of probability measures on \([0,\infty]\) proves convergence of the original sequence. ◻

Proof. The complementary \(\Lambda\)-factor in the two-slot regular model is diffuse. In a sufficiently large finite matrix amplification choose a projection in that coefficient factor with the same trace as the regular-model projection \(e\). A partial isometry identifying the projections gives the required unital tensor-coordinate representation. Scalar relative commutant follows from Proposition 10.

Choose a full lexicographic chain starting at a vertex and adding one new simple gap at each stage. By Proposition 28, all its target gaps diverge. Apply this to the finite matrix of unit columns, and choose successively finite path times at which the integrated mass in successively larger Cartan windows is small. At each selected time, first make the sum of the errors for the finite matrix and the finitely many prescribed windows smaller than a number tending to zero. Choose a base point where that nonnegative sum is no larger than its integrated bound. Each individual error is then small at the same base point. The resulting lattice cocycle values give \(\gamma_n\).

A fixed matrix corner equivalence preserves this escape. Approximate its entries and their adjoints in trace Hilbert norm by uniformly bounded finite Fourier matrices. Errors are uniformly small on the uncut unitary columns. Each fixed Fourier shift changes every Cartan coordinate by a bounded amount. It therefore preserves escape, and it intertwines rescaled continuous radial tests in the limit. After moving the test through the approximating matrix, its uncut Gram factor converges to the Gram of the partial isometry. The same argument, with slack in an unscaled window, preserves complementary tightness.

For amplified admissibility, finite-rank compressions of \(\lambda_{X_T}(h_T)\) tend to zero when \(r_T(h)\to\infty\). This is also seen by embedding \(L^2(X_T)\) into a multiple of \(L^2(T)\), since the maximal compact stabilizer is compact, and testing compactly supported vectors. Compress \(\Delta_T(\delta(\gamma_n))\) between finite-rank operators in the translation coordinate. Plancherel and the escaping Fourier law make its tracial \(L^2\)-norm tend to zero. Pairings for conjugation between finite-rank, bounded trace-Hilbert tensors consequently tend to zero, by Cauchy–Schwarz; approximation extends this to all trace-Hilbert vectors. An invariant vector would have constant nonzero self-pairing along this sequence. ◻

Proof. Complementary tightness is the pure-ray conclusion of Section 5, transported through the fixed corner equivalence. We prove the lower bound before that equivalence. Complementary regularity supplies a nonzero constant frame column \(f\) for \(\pi_0(\Gamma)\) together with \(W_\Lambda\). If \(e_j\) are the finitely many identity columns in the measured regular model, the frame inequality, applied to \(W_h e_j\), gives \[ \sum_{\gamma\in\Gamma} \|D_{\{h\}}\pi_0(\gamma)f\|_2^2\le C_f \qquad(h\in\Gamma). \tag{89}\] Indeed the left side is the sum, over \(\gamma\), \(\lambda\), and the finite column index \(j\), of the squared coefficients against \(W_hW_\lambda e_j\). Summing first over \((\gamma,\lambda)\) is exactly the complementary frame bound.

Let \(\Omega_t\) be a source Cartan shell of fixed positive thickness about length \(t\) in \(S\), with all compact rotations and with complementary coordinates in a fixed compact neighborhood. Cartan integration gives \(|\Omega_t|\asymp e^{Q_S t}\). Restrict both section values to a compact set \(K\). For fixed \(y\) and \(\gamma\), the times with transporter \(\gamma\) lie in \[K\gamma d_0(y)^{-1},\] so their Haar volume is at most \(|K|\). Integrating (89) therefore bounds the averaged mass at each measured index by a constant independent of that index and of \(t\).

For a fixed complementary compact window, the number of measured lattice indices with \(r_T(h)\le c't\) is at most \(C e^{Q_Tc't}\). To see this without assuming a discrete projection to \(T\), thicken the indices on the right by a small lattice-disjoint neighborhood in the full ambient group. Cartan length changes by a bounded amount and the complementary window remains compact; Haar integration then gives the stated bound. It follows that the restricted low-length mass averaged over \(\Omega_t\), divided by \(|\Omega_t|\), is at most \[ C e^{(Q_Tc'-Q_S)t}. \tag{90}\]

Choose \(c<c'<Q_S/Q_T\). The lost section sets have arbitrarily small measure, uniformly in the translating time, and the norm of the uncut translated constant column is unchanged. Complementary tightness similarly removes the restriction to the fixed measured compact window. Finally Lemma 23 makes every limiting low-length effect scalar, and the compact-change argument identifies its value uniformly throughout the thick shell, with slack from \(c\) to \(c'\). A subsequence violating (88) would thus have positive averaged mass in (90), a contradiction. The corner-change argument of Lemma 68 finishes the proof. ◻

Proof. Smooth radial truncations have a uniform operator bound by Proposition 58. Thus tight elements admit Hilbert approximation by bounded sequences supported in \(r_T\le CL_n\). The triangle inequalities (80) prove closure under multiplication; inversion preserves length and gives closure under adjoint. The tightness condition is closed in Hilbert norm on every bounded operator ball. These facts prove von Neumann closure.

All coefficient-unitary sequences belong to \(Q_0\). If \(z\) is central in \(Q_0\), its representing sequence commutes asymptotically, uniformly, with those unitaries: otherwise choose a violating unitary at each index. The Hilbert closed convex hull of the coefficient conjugation orbit gives conditional expectation onto its commutant. Consequently \(z\) is represented, up to a Hilbert-null sequence, on the group factor alone.

All fixed group labels also belong to \(Q_0\), since \(L_n\ge1\). Conjugation by those labels acts on the orthogonal complement of the identity Fourier coefficient without invariant vectors. In fact the squared moduli of an invariant vector’s coefficients would be constant on conjugacy classes; ICC forces all nonidentity coefficients to vanish. Scalar cocycle phases do not change this argument. A fixed Kazhdan pair for \(\Gamma\) gives a uniform spectral gap for all these conjugation representations, even as the cocycles vary. Approximate commutation therefore forces \(z\) to be scalar.

Apply the same Kazhdan inequality to \(\mathop{\mathrm{Ad}}\Theta_n\) on \(L^2(D_n)\ominus\mathbb C1\). Relative irreducibility says exactly that this representation has no invariant vector. An ultraproduct element commuting with every \(\Theta(\gamma)\) must accordingly be scalar. ◻

Proof. The Kazhdan gap first supplies an invariant finite right module close to the original copy of the tight algebra. Its dimension is close to one, but we need it to be exactly one in order to obtain a conjugating unitary. We will exclude any excess dimension by applying the ray lift to a column orthogonal to the tight algebra; the resulting almost invariant amplified state contradicts admissibility. We then align the unitary with the identity.

A nearby invariant module. Put \(D_\mathcal U=\prod_{\mathcal U}D_n\) and use Lemma 70. In the Jones basic construction \(\langle D_\mathcal U,e_{Q_0}\rangle\), with canonical semifinite trace (Jones 1983), one has \[ \|\Theta(\gamma)e_{Q_0}\Theta(\gamma)^*-e_{Q_0}\|_2^2 =2\|\Theta(\gamma)-E_{Q_0}\Theta(\gamma)\|_2^2 . \tag{93}\] The right side is \(O(\varepsilon_K^2)\): a smooth cutoff equal to one on \([0,L_n]\) produces a tight approximation. Its outer radius may be a fixed larger multiple of \(L_n\), chosen so that the multiplier scale always exceeds the threshold in Proposition 58. This does not increase the error beyond the tail in (92). The Hilbert-space Kazhdan inequality (Bekka et al. 2008) makes the projection of \(e_{Q_0}\) to the invariant Hilbert subspace \(O(\varepsilon_K)\)-close to it. This projection is the Hilbert limit of convex combinations of its conjugates, and is therefore a positive contraction \(z\). Put \(p_0=1_{[1/2,1]}(z)\). Spectral rounding gives \[ \|p_0-e_{Q_0}\|_2\le2\|z-e_{Q_0}\|_2=O(\varepsilon_K). \tag{94}\] For completeness, split the left squared norm into \(\mathop{\mathrm{Tr}}(e_{Q_0}(1-p_0))+\mathop{\mathrm{Tr}}((1-e_{Q_0})p_0)\); the two terms are bounded by four times the corresponding terms with \((1-z)^2\) and \(z^2\). This proves the inequality without a commutation assumption. The projection \(p_0\) commutes with \(\Theta(\Gamma)\), and its trace \(d'\) is close to one.

Take \(\varepsilon_K\) small enough that \(d'<2\). A right \(Q_0\)-module isometry onto \(p_0L^2(D_\mathcal U)\) has initial projection \(q\in M_2(Q_0)\), of trace \(d'\). Let its two basis entries be the row \(V\). Initially these entries lie in \(L^2(D_\mathcal U)\), and \[ Vq=V,\qquad E_{M_2(Q_0)}(V^*V)=q,\qquad \Theta(\gamma)V=Vb_\gamma , \tag{95}\] where \(b_\gamma\) is a projective unitary representation in \(qM_2(Q_0)q\). The positive \(L^1\)-element \(VV^*=\sum_iV_iV_i^*\) is invariant under \(\Theta\). Its spectral projections belong to the scalar relative commutant; hence \[ VV^*=d'1. \tag{96}\] This also proves that every \(V_i\) is bounded. Thus \[ e_1=(d')^{-1}V^*V\le q \tag{97}\] is a projection of trace one and commutes with all \(b_\gamma\).

Excluding excess module dimension. The projection \(q\) has trace \(d'\), whereas \(e_1\) has trace one. Thus \(e_1=q\) will force \(d'=1\) and turn the invariant module into one unitary copy of \(Q_0\). Suppose instead that \(e_1\ne q\), and put \[H_1=e_1-q/d'.\] Then \(H_1\ne0\), \(E_{M_2(Q_0)}H_1=0\), and \(H_1\) commutes with \(b_\gamma\). This is the nonzero column whose long Fourier mass will contradict the amplified gap. Orthogonality to \(Q_0\) implies that representing columns of \(H_1\) have vanishing Fourier mass below every fixed multiple of \(L_n\). Indeed test orthogonality against the smooth positive radial cutoff of the same bounded column; this cutoff belongs to \(Q_0\) and is equal to one on the desired lower window.

Here and below the finite matrix trace only changes constants. Choose bounded representatives \(G_n\) of \(H_1\) with exact support \(r_T\ge l_n\), and short approximations to \(q,b_\gamma\) with support \(r_T\le a_n\), so that \[ L_n\ll a_n\ll d_n\ll l_n,\qquad \|G_n-H_{1,n}\|_2\longrightarrow0. \tag{98}\] All approximations are uniformly operator-bounded. Since these are elements of fixed \(2\times2\) matrix algebras over normalized finite factors, they also satisfy \[\|x_n\|_2\le\sqrt2\,\|x_n\|_\infty.\] Thus both norm bounds in Proposition 61 and Corollary 63 apply to every short or long column used here. One explicit diagonal choice is as follows. For the first \(j\) group labels choose a fixed multiple \(M_jL_n\) that approximates their tight columns within \(1/j\). Require the long column to have mass less than \(1/j\) below \(j^2M_jL_n\), and pass to indices satisfying this and the finitely many algebraic approximations. Then use \(a_n=M_jL_n\), \(d_n=jM_jL_n\), \(l_n=j^2M_jL_n\), after harmless fixed slack in the cutoffs. This construction produces an ordinary subsequence.

Lift the coisometric row \(V/\sqrt{d'}\) to exact coisometric rows \(W_n\). This is obtained by polar decomposition, cutting a small spectral neighborhood of zero, and filling the vanishing trace defects in the finite factors. The corrections tend to zero in Hilbert norm: away from zero polar correction is uniformly continuous, and the discarded support has trace tending to zero. Put \(e_{1,n}=W_n^*W_n\) and \[\Theta'_n(\gamma)=W_n^*\Theta_n(\gamma)W_n .\] These are exact projective representations in the \(e_{1,n}\)-corners. Choose projection lifts \(q_n\) and corner-unitary lifts \(b_{\gamma,n}\). The identities in the ultraproduct give \[ \|\Theta'_n(\gamma)-e_{1,n}b_{\gamma,n}\|_2\longrightarrow0. \tag{99}\] Neither the representatives \(b_{\gamma,n}\) nor their short cuts need satisfy an exact representation law.

Let \(\xi_n=\mathbb V_{d_n}G_n\). Proposition 61, first for short approximations and then using the density error (86), gives \[ \Delta_T(b_{\gamma,n})\xi_n-\xi_nb_{\gamma,n}\to0,\qquad \Delta_T(q_n)\xi_n-\xi_n\to0,\qquad \xi_nq_n-\xi_n\to0 \tag{100}\] in Hilbert norm. These statements use only the ordinary identities \(b_\gamma H_1=H_1b_\gamma\) and \(qH_1=H_1q=H_1\). Put \(\xi'_n=\Delta_T(e_{1,n})\xi_n\). The left density of \(\xi_n\) has Fourier support in \(\{r_T\le4d_n+C\}\), since the two ray tubes can overlap only when their base points are at that distance. Consequently \[\langle\Delta_T(G_n)\xi_n,\xi_n\rangle=0\] for large \(n\). The identity \(e_{1,n}=G_n+q_n/d'+o_{L^2}(1)\) and the uniform density bound now yield \[ \|\xi'_n\|^2=(d')^{-1}\|\xi_n\|^2+o(1) \longrightarrow(d')^{-1}\|H_1\|_2^2>0 . \tag{101}\] Using (99) and (100), we also obtain \[ \Delta_T(\Theta'_n(\gamma))\xi'_n-\xi'_nb_{\gamma,n}\to0. \tag{102}\] Extend \(b_{\gamma,n}\) by the identity off \(q_n\). This does not change the last assertion, since \(\xi'_nq_n-\xi'_n\to0\).

The normalized left vector states defined by \(\xi'_n\) on the semifinite corner with unit \(\Delta_T(e_{1,n})\) are now almost invariant in predual norm. Right multiplication by the extended \(b_{\gamma,n}\) does not change such a state; the error in (102) changes it by at most twice the vector error times the vector norm. Their positive density square roots are almost invariant Hilbert unit vectors by the Powers–Størmer inequality (Powers and Størmer 1970), whose short trace proof we include: \[ \|A^{1/2}-B^{1/2}\|_2^2\le\|A-B\|_1 \quad(A,B\ge0). \tag{103}\] To verify this inequality, put \(S=A^{1/2}\), \(T=B^{1/2}\), and \(D=S-T\). The trace duality inequality and cyclicity give \[\|A-B\|_1\ge \mathop{\mathrm{Tr}}\bigl(\operatorname{sgn}(D)(S^2-T^2)\bigr) =\mathop{\mathrm{Tr}}\bigl(|D|(S+T)\bigr).\] Subtracting \(\mathop{\mathrm{Tr}}(D^2)\) from the last expression leaves \(2\mathop{\mathrm{Tr}}(D_+T)+2\mathop{\mathrm{Tr}}(D_-S)\ge0\). All products are trace-class by Hilbert–Schmidt Cauchy–Schwarz; finite spectral truncation gives the same calculation in the semifinite setting. Conjugation on this corner is unitarily equivalent, via \(\Delta_T(W_n)\), to the original amplified conjugation. Admissibility and the fixed Kazhdan gap contradict these almost invariant unit vectors. This proves \(e_1=q\).

Recovering a near-identity unitary. Taking traces gives \(d'=1\). A matrix equivalence over the factor \(Q_0\) reduces \(q\) to a single unit, and the row \(V\) becomes an actual unitary \(u\) with \(u^*\Theta(\Gamma)u\subset Q_0\) and \(p_0=ue_{Q_0}u^*\). Moreover \[ \|E_{Q_0}u\|_2^2=\mathop{\mathrm{Tr}}(e_{Q_0}p_0). \tag{104}\] Align the polar part of \(E_{Q_0}u\) by a right unitary from \(Q_0\); partial polar isometries extend to unitaries in a finite factor. Since \(E_{Q_0}u\) is a contraction, the aligned unitary satisfies \[ \|u-1\|_2^2 \le2-2\|E_{Q_0}u\|_2^2 =\|p_0-e_{Q_0}\|_2^2. \tag{105}\] Lift \(u\) by unitaries and diagonalize the countably many tightness conditions. Choosing \(\varepsilon_K\) sufficiently small gives the requested bound. Inner conjugacy preserves relative irreducibility; applying \(\Delta_T\) to the conjugating unitary proves preservation of amplified admissibility. ◻

Proof. Write \(\xi\) for the source multiplier of \(\Theta\), and put \[T_h(y)=\Theta(c(h,h^{-1}y)).\] The genuine section-cocycle identity gives \[\begin{align*} T_g(y)T_h(g^{-1}y) &=\nu(g,h)(y)T_{gh}(y),\\ \nu(g,h)(y) &=\xi\bigl(c(g,g^{-1}y), c(h,h^{-1}g^{-1}y)\bigr). \end{align*}\] Thus \(\nu\) is a normalized scalar-function-valued cocycle for the action on \(Y\). In the finite twisted crossed product for this action use shift unitaries \(b_h\) with multiplier \(\beta/\nu\). Tensor this crossed product with the final finite algebra \(D\). Then \[ \Psi(w_h)=T_hb_h,\qquad \Psi(w_g)\Psi(w_h)=\beta(g,h)\Psi(w_{gh}). \tag{106}\] Orthogonality of the shifts makes \(\Psi\) trace-preserving on the twisted group algebra. It extends normally: trace preservation gives equality of every even moment of the absolute value of a group polynomial, hence equality of its operator norm; the resulting trace-preserving isometric map extends by bounded strong approximation. Apply \(\mathop{\mathrm{id}}_{B_0}\otimes\Psi\) to \(\delta\).

Here is the kernel formula. If \(\delta(w_\gamma)=\sum_h a_{\gamma,h}\otimes w_h\), then, for a Borel set \(E\) of final Fourier lengths, its mass after insertion is \[ \sum_{h\in\Gamma}\|a_{\gamma,h}\|_2^2 \int_Y\bigl\|D_{1_{\{r_T\in E\}}} \Theta(c(h,h^{-1}y))\bigr\|_2^2\,dy . \tag{107}\] Distinct \(h\)’s are orthogonal in the middle shift coordinate. Integration over a source spherical transporter gives the same formula for spherical laws. The change \(y\mapsto h^{-1}y\) identifies the inner integral with the spherical kernel at the translation \(h\in G\). In particular all inserted kernels are probability kernels.

It remains to prove admissibility and factoriality of the coefficient tensor, rather than assume them. Use \(\Psi(w_h)\) as a right Fourier basis for the middle crossed product. Its coefficient Hilbert space is \[\mathcal E=L^2(Y;L^2(D)).\] Left multiplication by \(\Psi(w_g)\) acts on this coefficient space by \[ (\mathcal U_gk)(y)=\mathop{\mathrm{Ad}}T_g(y)\,k(g^{-1}y), \tag{108}\] and shifts the basis from \(h\) to \(gh\), with multiplier \(\beta(g,h)\). Right multiplication only shifts the basis from \(h\) to \(hg\), with multiplier \(\beta(h,g)\). The coefficient action is the restriction of the \(G\)-induced trace action of \(\mathop{\mathrm{Ad}}\Theta\). Its invariant space is \(\mathbb C1\). On \(\mathcal E\ominus\mathbb C1\), coefficients tend to zero along sequences escaping in every simple factor, by Howe–Moore and the product invariant-space decomposition of Section 3.

To apply this decay to Fourier-distributed unitaries, take bounded finite basis tests \(b\,k\,\Psi(w_h)\) and \(b'\,k'\,\Psi(w_{h'})\), and write \(A=\sum_i a_iw_i\) for one of the escaping unitaries of \(\delta\). The conjugation pairing is bounded, up to the fixed norms of \(b,b'\), by \[ \sum_{i,j:\,ih=h'j} \|a_i\|_2\|a_j\|_2 |\langle k',\mathcal U_i k\rangle|. \tag{109}\] The relation determines \(j\) uniquely from \(i\). Its sum without the last factor is at most \(\sum_i\|a_i\|_2^2=1\), by Cauchy–Schwarz. Outside fixed large compact windows in every factor, the last coefficient is uniformly small. The remaining terms tend to zero by Fourier escape and Cauchy–Schwarz; fixed shifts in the relation do not change escape. Thus the conjugation pairing tends to zero on the nonscalar coefficient summand. On the scalar coefficient summand it is precisely the conjugation representation of \(\delta\), whose invariant space is scalar. Density proves scalar relative commutant for the inserted representation. The center of its ambient finite tensor lies in that commutant, so the ambient tensor is a factor.

For the amplified assertion replace \(\mathcal E\) by \[\mathcal E_{\rm amp}= L^2\bigl(Y; L^2(B(L^2(X_T))\bar\otimes D,\mathop{\mathrm{Tr}}\otimes\tau)\bigr)\] and replace \(T_g\) by \(\Delta_T(T_g)\). This induced coefficient representation has no invariant vector, by admissibility of \(\Theta\). Finite-rank trace-Hilbert coefficient tests and the calculation (109) now apply on the entire amplified space. They prove absence of invariant vectors there. The coefficient algebra of the resulting tensor consists of \(B_0\), the middle finite crossed product, and the coefficient factor of \(D\); its final group factor is still the original measured factor of \(\Theta\). This proves all assertions. ◻

Proof. Fix an integer \(m\). For large \(n\), \(u_n=mt_n+(u_n-mt_n)\) with nonnegative remainder. Pass to a subsequence on which the laws of these segments converge. Repeated use of Proposition 67 makes the limit at \(u_n\) equal to \(P^{*m}*Q\), where \(P([a,\infty])\ge\alpha\) and \(Q\) is supported in \([0,\infty]\). For a finite \(M\), \[ (P^{*m}*Q)([0,M]) \le \mathbb P\{\operatorname{Bin}(m,\alpha)\le M/a\}. \tag{111}\] The right side tends to zero as \(m\to\infty\). Diagonal extraction over the countably many \(m\)’s justifies taking them after the subsequential limit. This proves (110). Compact changes are covered by Lemma 65; a failure of uniformity would give a violating deterministic sequence. ◻

Proof. Use the representation \(\delta\) of Lemma 68. Suppose first that, on a pure isolated source type \(i\), its matched length is totally superlinear: \[ P_{\delta,t}(t)([0,C])\longrightarrow0 \qquad(C<\infty). \tag{113}\] The contradiction will use the entire permutation cycle through \(i\). Composing the original representation around \[i\longmapsto p(i)\longmapsto\cdots\longmapsto i\] returns to the same source and measured type. Under the assumption (113), its first stage increases length faster than linearly; the entropy bound prevents the later stages from losing that increase. We will insert this cycle before an admissible representation that reaches a large Fourier threshold in nearly the least possible source time. The composition would reach the same relative threshold in a strictly shorter time.

Choosing a nearly earliest witness. Fix numbers \(D'_n\to\infty\). For each \(n\), minimize over all admissible representations with both source and measured type equal to \(i\), all cutoffs \(L\ge1\), and times \(t\ge1\), subject to \[\begin{align*} \|D_{1_{\{r_i>L\}}}\Theta(\gamma)\|_2 &\le\varepsilon_K &&(\gamma\in\mathcal F),\tag{114}\\ P_{\Theta,D'_nL}(t)([1,\infty]) &\ge\tfrac12 .&& \tag{115}\end{align*}\] Here \(L\) controls the Fourier tails of the fixed Kazhdan set, and \(D'_nL\) is the output threshold to be reached. Choose a witness \(t_n\) at most twice the infimum, and write its cutoff as \(L_n\) and its output scale as \(R_n=D'_nL_n\).

The feasible set is nonempty. Use the identity group embedding into \(L_\eta(\Gamma)\), with any fixed cocycle, and take \(L\) containing the finite Kazhdan set in the measured length. Relative irreducibility is ICC; amplified admissibility follows from all-factor escape of lattice elements and translation coefficient decay. For this embedding the transporter length on a pure axis differs from the source length by a bounded amount outside arbitrarily small section sets. Thus sufficiently large \(t\) meets (115).

Tightening the witness and repeating increments. Apply Lemma 71, choosing \(\varepsilon_K\) so small that the resulting conjugacies preserve at least \(1/4\) mass in (115). This preservation is uniform on moving transporters: for every unitary \(x\), \(\|u_n^*xu_n-x\|_2\le2\|u_n-1\|_2\), and the same bound holds after integration over the section cocycle. Rename the tightened representations \(\Theta_n\). Each fixed element is tight at scale \(L_n\), hence short on scale \(R_n\).

After extraction \(t_n\to\infty\). Otherwise Lemma 65, with its proof applied to tightness on scale \(L_n\), would make all bounded-time transporters short on scale \(R_n=D'_nL_n\), contradicting the retained \(1/4\) mass. Lemma 73 therefore gives \[ u_n/t_n\to\infty \quad\Longrightarrow\quad P_{\Theta_n,R_n}(u_n)\longrightarrow\delta_\infty . \tag{116}\]

Shortening the witness by a cycle of insertions. Now insert copies of \(\delta\) before \(\Theta_n\), once for each step around the finite permutation cycle through \(i\), using Lemma 72. The first insertion, viewed from the source, is the totally superlinear one in (113). Choose \(m_n\to\infty\) sufficiently slowly that \[ t'_n=t_n/m_n\to\infty,\qquad P_{\delta,t_n}(t'_n)([m_n,\infty])\longrightarrow1 . \tag{117}\] Such a choice follows directly from total superlinearity: for each integer \(m\), choose a threshold after which the ratio of measured length to source length exceeds \(m^2\) with probability at least \(1-1/m\), and require \(t_n/m\) to exceed that threshold.

Every subsequent insertion retains length much larger than \(t_n\), by Lemma 69. Its complementary coordinates remain tight. To justify this assertion for the random middle indices in (107), first restrict the complementary coordinates to a fixed compact window with arbitrarily large probability. Cartan decomposition then writes those indices as pure translations with compact pre- and post-factors. The entropy estimate is uniform on these windows. After the finitely many insertions, the terminal kernel \(\Theta_n\) is tested on its source type \(i\) at lengths divided by \(t_n\) tending to infinity in probability. Equation (116) and its compact uniformity show that the final measured length divided by \(R_n\) tends to infinity in probability.

For a fixed source group element the preceding finite chain of kernels instead gives a fixed probability measure on the countable set of terminal input labels. Its mass can be restricted to a finite set with arbitrarily small loss. Tightness of the terminal fixed-label laws at scale \(L_n\), and the uniform section approximation for each such label, prove tightness at scale \(L_n\) for every fixed group element of the composite representation. Hence, for a fixed sufficiently large \(C_1\), and after extraction, the composite satisfies (114) with cutoff \(C_1L_n\). Its final escape also gives (115) with threshold \(C_1R_n\), at time \(t'_n\). The ratio of these new scales is still \(D'_n\). Admissibility is preserved by every insertion. Thus this is a feasible witness in the very same minimization, and \[t_n\le2\inf t\le2t'_n,\] contrary to \(t'_n/t_n\to0\).

From a scalar subsequence to the fixed-vector conclusion. Total superlinearity is therefore impossible. There are \(C_0,\alpha_0>0\) and \(T_n\to\infty\) for which the scalar law has mass at least \(\alpha_0\) below \(C_0T_n\). This is uniform for \(t\in[T_n/2,T_n]\), after enlarging \(C_0\) slightly. Indeed, a violating sequence \(t_n\) would, by Proposition 67 applied to the fixed representation \(\delta\) on scale \(T_n\), give a convolution limit for \(T_n=t_n+(T_n-t_n)\). The mass of a nonnegative sum below \(C_0\) cannot exceed the mass of either summand below \(C_0\). Weak-limit boundary issues are removed by the stated slack in the cutoff.

Lemma 23 identifies all these limiting effects as scalar. The corner transfer in Lemma 68 identifies their scalar values in the original regular model. It follows that the same lower bound holds on every fixed bounded column, uniformly over the indicated times. Effect boundedness and Hilbert density extend it to every fixed \(\xi\), proving (112). Tensor density gives normal Hilbert amplifications. The compact perturbation statement follows from the same short approximation and rescaled-effect argument. ◻

Proof. Use the regular corner from the complementary regular realization in Section 3 and amplify its left Fourier action by \(\widehat\Delta(w_{(h,\lambda)})=L_{\hat h}\otimes w_{(h,\lambda)}\). In the rectangular trace Hilbert space with corners \(\widehat\Delta(e),e\), source transport acts by \[Z(y)\longmapsto \widehat\Delta(X_s(s^{-1}y))Z(s^{-1}y)X_s(s^{-1}y)^*.\] The scalar projective phases cancel. We show that every nonzero projection \(d_2\in\mathcal N^S\) contains the right support of a nonzero invariant column.

Choose a unit vector \(\psi\in L^2(\widehat G_T)\) with strictly positive translation coefficient \(q\mapsto\langle L_q\psi,\psi\rangle\), bounded below on each compact set. For example, choose an everywhere positive square-integrable function; its coefficient is positive and continuous. Test the preceding action on \(\widehat\Delta(d_2)(\psi\otimes d_2)\). Reindexing the base and using \(X_s(y)d_2(y)X_s(y)^*=d_2(sy)\) expresses its overlap with its \(s\)-translate as the sum of squared Fourier coefficients of \(X_s(y)d_2(y)\) weighted by \(\langle L_{\hat h}\psi,\psi\rangle\). The total unweighted mass is \(\|d_2\|_2^2\). Complementary tightness for a pure isolated type makes these overlaps bounded below by a positive constant, uniformly in \(s\in S\). The uniformity follows first on a Cartan ray and then on all of \(S\) by Cartan decomposition and the compact-transport approximation used above; bounded Cartan parameters form a strongly continuous compact family.

The element of least norm in the closed convex hull of this orbit is therefore nonzero and invariant. Its right support is contained in \(d_2\). Polar decomposition in the rectangular affiliated-operator space gives an invariant partial-isometry column; its initial projection belongs to \(\mathcal N^S\). Repeat in the orthogonal complement of the initial projection. Separability gives a countable exhaustion. Place the resulting columns in separate multiplicity coordinates. Their direct sum \(V_\Delta\) satisfies \(V_\Delta^*V_\Delta=e\) and the source intertwining relation. In particular it is bounded, and its integrated rectangular trace Hilbert norm squared is \((\mathop{\mathrm{Tr}}\otimes\tau)(e)=d\).

We now remove the left amplification. For a Fourier output index \(k\) and parameter \(P\in\widehat G_T\), use the unitary change of variables \[F_k(P)\longmapsto \widetilde F_k(p)=F_k(\hat k p).\] Left multiplication by \(\widehat\Delta(w_h)\) becomes ordinary left multiplication, while right multiplication by \(w_h^*\) acquires \(L_{\hat h}\) in the new variable. This proves the source and right action relations for the resulting isometry \(V\).

The same change of variables leaves diagonal angular Fourier tests unchanged. To pass such a test through \(V_\Delta\) on a defining pure path, first trim the extra Hilbert coordinate to finite rank and approximate the remaining column by bounded finite Fourier columns in integrated trace Hilbert norm. A fixed left Fourier shift does not change the limiting image direction. Moving each such shift through the angular test leaves its uncut Gram factor. The approximation error tends to zero on bounded input columns because \(V_\Delta^*V_\Delta=e\); Hilbert density then handles every input. Thus compression of the constant-amplified limiting angular measure by \(V\) is \(P_x\). Both measures are sharp. For a spectral projection \(E\) with \(V^*EV=P\), the equality \(\|(EV-VP)\xi\|^2=0\) proves exact intertwining. This proves the angular relation. The assertions about \(W_C\) now follow directly from the displayed intertwinings and monotone exhaustion of \(L^2(\widehat G_T)\). ◻

Proof. We prove a bound \(C_C\delta^{-Q_T}\) in each unit length shell. Choose a sufficiently small relatively compact identity neighborhood \(U\) such that \(hU\), \(h\in\Gamma\), are disjoint. Right thickening changes \(r_T(h)\) by a bounded amount and changes the forward direction \(h^+\) by \(O(e^{-r})\) in a shell \(r\le r_T(h)<r+1\). It enlarges \(PC^{-1}\) to a translate of a fixed compact set. Cartan integration therefore gives, for a ball of radius \(R\ge e^{-r}\), \[ \#\{h:r\le r_T(h)<r+1,\ \hat h\in PC^{-1},\ h^+\in B(z,R)\} \le C_C(1+Re^r)^{Q_T}. \tag{119}\] Indeed the angular ball has measure at most \(C(R+e^{-r})^{Q_T}\), the radial shell has Jacobian at most \(Ce^{Q_Tr}\), and the complementary factor has bounded Haar volume independent of its left translate \(P\). This argument counts full lattice points, not points in a possibly nondiscrete projection.

We record the two boundary estimates used below. Put \(R_0=C'e^{-r}/\delta\). A set at distance at least \(c\delta/2\) from \(h^-\) has image under \(h\) contained in \(B(h^+,R_0)\). Moreover, if \(\mathop{\mathrm{supp}}f_A\) is within \(c\delta/2\) of \(h^-\), then on an annulus at distance \(\rho\ge R_0\) from \(h^+\), \[ \|f_A\circ h^{-1}-f_A(h^-)\|_{C^m(\rho\text{-normalized charts})} \le C_m\frac{R_0}{\rho}. \tag{120}\] Here one may enlarge \(C'\) by a fixed factor. Use the normalized Carnot charts of Lemma 55. To see these estimates, put the two polar endpoints at zero and infinity. The Cartan element is a graded dilation of ratio \(e^{-r}\) in the chart at the attracting endpoint. Chart inversion sends an annulus of radius \(\rho\) to an annulus of radius comparable to \(\rho^{-1}\), with uniformly bounded derivatives after graded normalization. Thus the inverse image of the annulus is within \(O(e^{-r}/\rho)\) of the repelling endpoint. Taylor’s formula for a mask whose derivatives are normalized at scale \(\delta\) proves (120). Every nonconstant normalized derivative has the same bound: its chain-rule factors contain at least one positive-degree dilation factor. Compact polar rotations preserve the chosen horizontal metric, so all constants are uniform in \(h\).

If \(R_0\) is bounded below by a fixed positive number, total shell counting already gives \(C_C\delta^{-Q_T}\). Assume otherwise. From (119), for either one of the individual PVMs \(P_x,P_y\), we obtain \[ \sum_h P_z(1_{B(h^+,bR_0)})\le C_{C,b}\delta^{-Q_T}1 \quad (z=x\text{ or }y), \tag{121}\] where the sum is over the shell and complementary window. This is just the pointwise counting inequality integrated against a single PVM.

Conjugation by \(W_h\) composes the masks with \(h^{-1}\). For those \(A\) whose supports are at distance at least \(c\delta/2\) from \(h^-\), the rightmost masks are supported in \(B(h^+,R_0)\). Bounded overlap gives \[\begin{align*} &\sum_{A,D} \|P_y(g_D\circ h^{-1})P_x(f_A\circ h^{-1})\xi\|^2\\ &\hspace{20mm}\le C\sum_A\|P_x(f_A\circ h^{-1})\xi\|^2 \le C\|P_x(1_{B(h^+,R_0)})\xi\|^2. \end{align*}\] Summing in \(h\) uses (121) for \(P_x\).

Only boundedly many remaining \(A\) occur for each \(h\). Separation implies that all their paired \(g_D\) are far from \(h^-\). Summing the \(g_D\) squares reduces their contribution to \[C\sum_{h,A\text{ remaining}} \|P_y(1_{B(h^+,R_0)})P_x(f_A\circ h^{-1})\xi\|^2.\] Subtract the scalar \(f_A(h^-)\). The scalar term is controlled by (121) for \(P_y\). The remainder in an enlarged pole ball is controlled by that equation for \(P_x\), using its uniformly bounded amplitude. Decompose the rest into annuli of radii \(\rho_j=2^jR_0\). On each annulus, (120) gives normalized size \(O(2^{-j})\).

Here is the summation detail that preserves the order of the two PVMs. Fix a bounded-overlap global cover by cells of scale \(\rho_j\). Only boundedly many cells meet any one of these annuli. Smooth expansion on each normalized cell gives \[F_{h,A,j}= \sum_{E,\nu}c_{h,A,E,\nu}\varphi_{E,\nu},\qquad |c_{h,A,E,\nu}| \le C_m2^{-j}(1+|\nu|)^{-m},\] where \(F_{h,A,j}\) is the annular remainder, \(\varphi_{E,\nu}\) is supported in an enlargement of \(E\), and the functions \(\varphi_{E,\nu}\) do not depend on \(h\) or \(A\). For fixed \(\nu\) they have a uniformly bounded square sum over \(E\). The expansion follows by a smooth cutoff to a unit cube and its Fourier series; taking \(m\) above the chart dimension makes the weights summable. Cauchy–Schwarz over the bounded number of cells for a fixed \(h,A\) and then (121) for \(P_y\) give \[\begin{align*} &\sum_{h,A,E}|c_{h,A,E,\nu}|^2 \|P_y(1_{B(h^+,R_0)})P_x(\varphi_{E,\nu})\xi\|^2\\ &\qquad\le C_m2^{-2j}(1+|\nu|)^{-2m}\delta^{-Q_T} \sum_E\|P_x(\varphi_{E,\nu})\xi\|^2\\ &\qquad\le C_m2^{-2j}(1+|\nu|)^{-2m}\delta^{-Q_T}\|\xi\|^2. \end{align*}\] In the first inequality the vector \(P_x(\varphi_{E,\nu})\xi\) is independent of \(h\). Sum first over frequencies and then over annuli in the Hilbert norm of the \((h,A)\)-indexed family. The sums \(\sum_\nu(1+|\nu|)^{-m}\) and \(\sum_j2^{-j}\) converge. This proves the required shell bound. Summing the \(O(1+L)\) shells proves (118). ◻

Proof. Fix an opposite pair \((x_0,y_0)\) and let \(J_0=MA<S\) be its compact-by-split stabilizer. Temporarily write \(p_\delta=p_\delta(x_0,y_0)\). We first place a Parseval frame below this positive effect, then apply Fourier packing to its complementary localization. The stabilizer average will cancel the length factor, after which we can unfold the estimate over all opposite pairs.

Choose complementary Parseval frame columns \(b_j\in H\) for the joint action \(\pi_0,W_\Lambda\); thus \[\sum_{\gamma,\lambda,j} |W_\lambda\pi_0(\gamma)b_j\rangle \langle W_\lambda\pi_0(\gamma)b_j|=1\] in the strong sense. Such a finite frame of orbit generators is part of the complementary regular realization. Let \(f\ge0\) be bounded and compactly supported on \(G\), with \[ \int_{J_0}f(ag)\,da\le1\qquad(g\in G). \tag{124}\] Using the induction section \(d_0\), define the fixed Hilbert vectors \[v_{\gamma j}(o')= f(d_0(o')\gamma)^{1/2} p_\delta(o')^{1/2}\pi_0(\gamma)b_j.\] The vectors depend on the masks and on \(f\), which remain fixed throughout the later limit in \(n\). Since \(J_0\) fixes the reference pair, \(U_a\) commutes with \(p_\delta\). At each base point, reindexing \(\gamma\) by the section cocycle and using (124) gives the following inequality of positive operators on the fibre \(H\): \[\int_{J_0}\sum_{\lambda,\gamma,j} |W_\lambda(U_av_{\gamma j})(o')\rangle \langle W_\lambda(U_av_{\gamma j})(o')|\,da \le p_\delta(o') \quad\text{for almost every }o'.\] Its global rank-one consequence on \(\mathcal H=L^2(Y;H)\) is \[ \int_{J_0}\sum_{\lambda,\gamma,j} |W_\lambda U_av_{\gamma j}\rangle \langle W_\lambda U_av_{\gamma j}|\,da \le p_\delta. \tag{125}\] Indeed the reindexed scalar is \(f(a^{-1}d_0(o')\gamma')\), where \(\gamma'=c(a,a^{-1}o')\gamma\); its integral is at most one. The remaining sum is the Parseval identity compressed by \(p_\delta^{1/2}\). We retain this pointwise positive inequality for the Fourier-row tests below. It also gives the displayed inequality on the global Hilbert space by first applying \[\left|\int_Y\langle\xi(y),v(y)\rangle\,dy\right|^2 \le \int_Y|\langle\xi(y),v(y)\rangle|^2\,dy\] and then summing and integrating. The projective phases disappear in these pointwise squared coefficients.

We next identify the Fourier norm to which packing applies. Let \(V_\Delta\) be the column in the proof of Lemma 75, before the change of variables that removes its left amplification. At complement parameter \(P\) and base point \(o'\), an adjoint identity row is \(V_\Delta(P,o')^*\mathbf1_{\rm row}\). For a fibre input \(z\in H\), its pairing with \(W_hW_\lambda z\) reads the corresponding output row of the \((h,\lambda)\) Fourier coefficient of \(V_\Delta z\): \[\left\langle V_\Delta(P,o')^*\mathbf1_{\rm row}, W_hW_\lambda z\right\rangle.\] This uses the right-adjoint convention for \(W\). Summing the squared coefficients over rows and \(\lambda\) is therefore the Fourier Hilbert norm in the measured \(h\) coordinate. After the change of variables in the localization lemma, \(P=\hat h p\). Thus \[p\in C\quad\Longleftrightarrow\quad \hat h\in PC^{-1}.\] This is precisely the complementary window in the packing inequality.

Test (118) on these adjoint rows, insert the pointwise frame inequality underlying (125), and integrate in \(P,o'\). The integrated sum of their squared trace norms is \((\operatorname{Tr}\otimes\tau)(e)=d\). The preceding coefficient identity and Parseval therefore give, with \(D_{\le L}\) the measured Fourier projection onto \(r_T(h)\le L\), \[ C_C(1+L)\delta^{-Q_T} \ge\int_{J_0}\sum_{\gamma,j} \|D_{\le L}1_CVU_av_{\gamma j}\|^2\,da. \tag{126}\] The coefficient calculation is first made for bounded columns of finite Fourier support. Approximation and the finite integrated squared norm of the rows extend it to the fixed Hilbert vectors above. Positivity permits all sums and integrals by Tonelli’s theorem.

We now choose the stabilizer interval so that its length cancels the factor \(1+L\). Proposition 74 provides \(T_n\to\infty\), \(A_0<\infty\), and \(\eta_0>0\) such that, with fixed slack in \(A_0\), the cutoff \(r_T\le A_0T_n\) has lower limiting mass at least \(\eta_0\) times the squared norm of each fixed input, uniformly for stabilizer lengths in \([T_n/2,T_n]\). The assertion passes to the constant amplification by tensor density. Since \(1_C\) commutes with the constant-amplified \(U_a\), the inputs here are the fixed vectors \(1_CVv_{\gamma j}\). Restrict (126) to that stabilizer segment, set \(L=A_0T_n\), and divide by \(T_n\). Its Haar measure is comparable to \(T_n\), with the \(M\) factor compact. Fatou’s lemma, first for finite sums of \((\gamma,j)\) and then by monotone exhaustion, yields \[ \sum_{\gamma,j}\langle v_{\gamma j},W_Cv_{\gamma j}\rangle \le C'_C\delta^{-Q_T}. \tag{127}\] The constant is independent of the density \(f\) satisfying (124).

Undo induction by lifting at \(g=d_0(o')\gamma\) with operator conjugation by \(\pi_0(\gamma)^*\). Since \(W_C\) commutes with \(U_S\), its uninduced field \(W_C^0\) depends only on the complementary coordinate \(g'\in\widehat G_S\). Since the pure-vertex family is fixed by complementary source factors, its uninduced effect depends only on the inverse-translated pair. These two operators commute. Unfolding the sum over \(\gamma\) in (127) therefore integrates \[\sum_j\langle b_j, W_C^0(g')p_\delta^0(x,y)b_j\rangle\] against the density induced by \(f\) on \(J_0\backslash G\cong B_S^{(2)}\times\widehat G_S\). Every compactly supported quotient density between zero and one can be obtained by choosing, on finitely many local sections, a nonnegative compactly supported fibre density of integral one and using a partition of unity. Exhausting the quotient and applying Tonelli’s theorem gives \[ \int_{\widehat G_S}\int_{B_S^{(2)}} \sum_j\langle b_j,W_C^0(g')p_\delta^0(x,y)b_j\rangle \,dm(x,y)\,dg' \le C'_C\delta^{-Q_T}. \tag{128}\] The same argument applies to any fixed frame translate \(\pi_0(\gamma_0)W_{\lambda_0}b_j\): its full orbit is again a Parseval family, up to irrelevant scalar phases.

We spell out why this gives a dense supply of constant inputs for (123). Restrict \(o'\) to a measurable set on which \(d_0(o')\) lies in a compact subset \(K\) of \(G\). Haar measure on the section image is restricted Haar, up to the fixed covolume normalization. Projection of its restriction to \(K\) onto \(\widehat G_S\) is dominated by a finite constant times Haar measure: the fibre in \(S\) lies in a fixed compact set of finite Haar volume. For a bounded scalar function \(a\) supported on this set, put \[\xi(o')=a(o')W_C(o')^{1/2} \pi_0(\gamma_0)W_{\lambda_0}b_j.\] Commutation with the angular effects moves \(W_C^{1/2}\) into the positive pairing. Changing the pair variable by the \(S\) coordinate of \(d_0(o')\) preserves \(dm\). The preceding Haar domination and (128) now prove (123) for this fixed \(\xi\). Finite sums are handled by Cauchy–Schwarz. Such vectors span a dense space: compact section restrictions exhaust \(Y\), the frame translates span \(H\), bounded simple functions are dense in the base variable, and \(W_C^{1/2}\to1\) strongly along a compact exhaustion.

Finally, let \(\mathscr E\) be the maximal linear domain of vectors for which (123) holds for every fixed finite set of mask controls of sufficiently high order. The dense supply just obtained is contained in this domain. A bounded operator commuting with all \(P_x(f)\) commutes with \(p_\delta\) and preserves the bound, with constant multiplied by its squared norm. A fixed source transformation preserves \(dm\) and reindexes the pair. A fixed target transformation changes scale, support, separation, and normalized derivatives by fixed constants because it is a smooth conformal map of the compact horizontal boundary. Thus it maps a controlled family into another permitted controlled family. These observations prove all the asserted domain invariances. ◻

Proof. Under the change from \(w\) to \(y\), the density of \(\nu_\epsilon\) with respect to \(dx\,dy\) is at most \(C\epsilon^{-Q_S}\) and is supported where \(d_S(x,y)\le C\epsilon\). Since the invariant pair measure has density comparable to \(d_S(x,y)^{-2Q_S}\), the pullback of any separated target mask effect satisfies \[ \int\langle\xi,p_\delta(x,y)\xi\rangle\,d\nu_\epsilon \le C_\xi\epsilon^{Q_S}\delta^{-Q_T}. \tag{130}\] Its bound by a constant times \(\|\xi\|^2\) also follows from the uniform operator bound for \(p_\delta\) in Definition 76.

Here are the details needed to apply this estimate to a kernel. Choose a smooth dyadic decomposition of the complement of the target diagonal. The piece at scale \(\delta\) is supported where \(c\delta\le d_T(w,v)\le C\delta\). Cover each boundary by cells of radius \(c'\delta\) with \(c'\) small relative to \(c\). The pairs of cells meeting this support have bounded degree: each cell is paired with at most a fixed number of other cells. Their enlarged supports are separated, and their centers have uniformly bounded density at scale \(\delta\).

Suppose \(F\) has horizontal order at least \(m\), where \(m=1\) or \(2\). On each such product of cells, every fixed finite collection of normalized derivatives of the kernel piece is bounded by \(C\delta^m\). For \(m=1\), this is Taylor expansion at the diagonal. For \(m=2\), the constant and horizontal linear terms vanish; the remaining horizontal quadratic terms and the nonhorizontal linear terms both have Carnot degree at least two. Taylor expansion in the rescaled coordinates also gives the same estimate after each of the specified normalized derivatives. Compactness makes the bounds uniform in the cell centers.

Expand the rescaled kernel in a Fourier series on slightly larger coordinate boxes. Integration by parts in both variables, to an order larger than the box dimensions and the required mask orders, gives a summable family of separated terms \[\delta^m c_{A,D;k,l}\, g_{D,k}(w)f_{A,l}(v).\] The coefficients decrease faster than the polynomial growth of the normalized mask seminorms in \(k,l\). For a fixed pair of frequencies, the leftmost supports have bounded overlap. The spectral calculus for \(P_y\), followed by Cauchy–Schwarz for the bounded number of terms at a cell, therefore gives \[\left\|\sum_{A,D}c_{A,D;k,l} P_y(g_{D,k})P_x(f_{A,l})\xi\right\|^2 \le C_{k,l}\sum_{A,D} \|P_y(g_{D,k})P_x(f_{A,l})\xi\|^2.\] Coefficients can be absorbed into the masks; duplicating an index when necessary changes density and overlap by only the bounded degree. Thus (130), followed by the summation over frequencies, bounds this annular piece in root mean square by \[ C_{F,\xi}\delta^m \min\{1,\epsilon^{Q_S/2}\delta^{-Q_T/2}\}. \tag{131}\] This computation uses only one PVM for the leftmost overlap and the ordered positive effect for the remaining factors.

There is also an estimate \(C_F\delta_0^m\|\xi\|\) for the entire kernel piece supported where \(d_T(w,v)\le C\delta_0\). Indeed repeat the cell expansion at scale \(\delta_0\); only boundedly many cells meet a given cell, and the constant mask bound applies without separation.

Put \(a=Q_S/Q_T\) and choose a dyadic \(\delta_0\) comparable to \(\epsilon^a\). For \(m=1\), the triangle inequality and (131) give \[C_{F,\xi}\left(\delta_0+ \epsilon^{Q_S/2}\sum_{\delta\ge\delta_0} \delta^{1-Q_T/2}\right) \le C'_{F,\xi}\epsilon^a,\] because \(Q_T>2\). When \(Q_S=Q_T=Q\) and \(m=2\), take \(\delta_0\asymp\epsilon\) instead. Summing \(\epsilon^{Q/2}\delta^{2-Q/2}\) gives exactly the three rates in the statement, including the \(C\epsilon^2\) diagonal remainder. Each rate is \(o(\epsilon)\). ◻

Proof. For \(F(w,v)=f(w)-f(v)\), ordered evaluation is \(P_y(f)-P_x(f)\). To extract a horizontal derivative, use a smooth compactly supported weight \(\psi_i(w)\) whose horizontal first moments are the \(i\)th coordinate vector. Its absolute value is dominated by a fixed smooth nonnegative weight of the kind used in Lemma 80. The expressions \[\frac1\epsilon\int\psi_i(w) (P_{x\delta_\epsilon w}(f)-P_x(f))\xi\,dw\] are consequently bounded in local \(L^2\) by \(C\epsilon^{Q_S/Q_T-1}\). For a smooth compactly supported test field, change variables from \(x\) to \(x\delta_\epsilon w\) in its adjoint pairing. First-order Taylor expansion of the test field and the smooth measure density shows that the limit is the distributional horizontal derivative: the adjoint test is \(-X\varphi-(\operatorname{div}_{dx}X)\varphi\).

If \(Q_S/Q_T>1\), all these derivatives vanish. Brackets of horizontal fields then annihilate the same distribution, because \([X,Y]u=X(Yu)-Y(Xu)\). Bracket generation implies that every coordinate derivative vanishes, so \(P_x(f)\xi\) is constant on the connected boundary. Density of \(\mathscr E\) and strong continuity make \(P_x\) constant, a contradiction. Hence \(Q_S\le Q_T\). The matching is a permutation of the finite collection of isolated types; going around each cycle forces equality throughout that cycle.

With equality, the displayed difference quotients are bounded in \(L^2\). Weak compactness and their identified distributional limit give \(D_X(f)\xi\in L^2\) in each chart. A finite cover and a smooth partition of unity give the global statement. The linearity assertions follow directly from distributional differentiation. ◻

Proof. Write \(\Delta_{x,y}f=P_y(f)-P_x(f)\). Sharpness gives the exact identity \[(\Delta_{x,y}f)P_x(g)-P_y(g)\Delta_{x,y}f =\mathcal K_{x,y}\bigl((f(w)-f(v))(g(v)-g(w))\bigr).\] The right side has horizontal order two. Its norm on any \(\xi\in\mathscr E\), divided by \(\epsilon\), tends to zero in the averages used to recover a horizontal derivative.

Fix \(\xi,\eta\in\mathscr E\) and real \(g\). In the pairing of the left term with \(\eta\), move \(\Delta_{x,y}f\) to \(\eta\) by adjunction. For the other term, keep the bounded quotient applied to \(\xi\) and move \(P_y(g)\) to the test vector. Strong continuity permits replacing \(P_y(g)\) there by \(P_x(g)\): its difference on a fixed test vector tends to zero in \(L^2\) by dominated convergence, whereas the averaged quotients stay bounded in \(L^2\). The weak derivative limit therefore gives \[ \langle P_x(g)D_X(f)(x)\xi,\eta\rangle =\langle\xi,P_x(g)D_X(f)(x)\eta\rangle \quad\hbox{for almost every }x. \tag{134}\] One obtains the pointwise assertion by first inserting arbitrary bounded scalar parameter tests. Complex weights follow by linearity and adjunction.

Apply (134) with products of angular tests. If \(u=\sum_j a_jP_{\bullet}(g_j)\xi_j\) and \(v=\sum_k b_kP_{\bullet}(h_k)\eta_k\), the formal right side of (132) satisfies \[\langle T_X(f)u,v\rangle_{\mathcal K} =\langle u,T_X(f)v\rangle_{\mathcal K}.\] In particular, if an expression for \(u\) represents zero, its formal image is orthogonal to every \(v\in\mathcal C\). This space is dense, since it includes bounded simple parameter fields with values in \(\mathscr E\). The image is therefore zero. This proves both well-definedness and symmetry, and hence closability. The identical calculation with one more real angular weight proves the final symmetry assertion.

For cancellation, consider \[F(w,v)=\sum_j g_j(w)(f_j(w)-f_j(v)).\] It vanishes on the diagonal, and its second-variable horizontal derivative there is \(-\sum_jg_jd_{\rm hor}f_j=0\). Lemma 80 gives an \(o(\epsilon)\) bound. Taking the same weak derivative limit and replacing left multipliers at \(y\) by those at \(x\) gives \(\sum_jP_x(g_j)D_X(f_j)(x)\xi=0\) for every \(\xi\in\mathscr E\). Commuting the additional multipliers in (132) proves (133) on \(\mathcal C\). The asserted commutation on this domain also follows from that rule. ◻

Proof. Choose finite smooth Parseval horizontal source fields \((X_\alpha)_{\alpha=1}^{n_S}\) and finite smooth Parseval horizontal target covectors \((\theta_r)_{r=1}^{n_T}\). For completeness, start with finitely many smooth spanning sections. Their positive Gram operator is invertible on the horizontal bundle; multiplication by its inverse square root gives the required Parseval system. Smooth coordinate differentials provide the target spanning sections, so each \(\theta_r\) has an expression of the form used above. Define the finite column \[D:\mathcal C\longrightarrow\mathcal K^{n_Sn_T},\qquad Du=(T_{X_\alpha}(\theta_r)u)_{\alpha,r}.\] Every component is symmetric on \(\mathcal C\) by Lemma 82. If \(u_n\to0\) and \(Du_n\to v\), testing each component against \(\mathcal C\) gives \(v=0\). Thus \(D\) is closable. Let \(\overline D\) be its closure and put \(l=|\overline D|=(\overline D^*\overline D)^{1/2}\) on \(\mathcal K\).

Multiplication by a scalar parameter unitary preserves \(\mathcal C\) and commutes with \(D\), acting diagonally on the output. The same is true for \(P_{\bullet}(e^{it g})\), with \(g\) real smooth. Their inverses have the same property. These identities extend to the closed graph, and consequently to the closed quadratic form \(u\mapsto\|\overline D u\|^2\). Therefore \(l\) strongly commutes with the parameter multiplication algebra and the angular spectral algebra. In particular \(l=\int^{\oplus}l_x\,dx\) is decomposable and \(l_x\) strongly commutes with \(P_x\) almost everywhere.

If \(V\) is a unitary commuting with every \(P_x\), invariance of \(\mathscr E\) and distributional differentiation give \(D_X(f)V\xi=VD_X(f)\xi\). This identity extends to \(\mathcal C\) and then to the closed graph. Applying it to the base unitaries and the unmeasured group actions proves affiliation to \(\mathcal A_1\); applying it to the complementary source factors proves their invariance. Separability permits a common determining set for these almost-everywhere commutation assertions.

We give the norm argument for covariance to keep track of closures. The Parseval systems realize isometric embeddings of the horizontal bundles into trivial finite-dimensional bundles. Their Gram matrices are the corresponding orthogonal projections. Linearity in \(X\) and (133) say exactly that \(Du\) belongs to the range of these source and spectral target projections. Thus an isometry between horizontal bundles preserves the sum of squares of the column entries, even if the Parseval systems are redundant. In particular, any two finite Parseval systems give the same squared form on the same intrinsic module core. Their closed squared forms, and hence their operators \(l\), agree. The maximal estimate domain \(\mathscr E\) is also unchanged by replacing the controlled Carnot atlas: a finite refinement and the smooth changes of rescaled coordinates transfer every mask control with a fixed change of its constants, in both directions. Thus the construction depends only on the PVM family and the chosen compact horizontal metrics. The cell expansions used to prove the estimates, and the localizing isometry used to prove density of \(\mathscr E\), do not enter its definition.

Differentiate the identity \(\sigma_aP_x=P_{ax}\) on constant-domain vectors. The chain rule sends a source vector \(X\) at \(x\) to \(da_xX\) at \(ax\). This map is the product of \(\operatorname{dil}(a,x)\) and a horizontal isometry. Similarly, differentiating \(\beta_hP_x(f)=P_x(f\circ h^{-1})\) replaces a target covector by its pullback under \(h^{-1}\). Horizontal cancellation justifies writing this pullback in the chosen target coframe. Its norm factor, evaluated spectrally, is \(\operatorname{dil}(h^{-1},P_x)\); its other factor is a horizontal isometry. The preceding projection identities justify dropping those isometries when taking the sum of squared column norms.

These identities hold on the full core \(\mathcal C\). Indeed source transport reindexes its scalar coefficients, transforms \(\mathscr E\) into itself, and transports \(P_x(g)\) by covariance; measured transport replaces \(g\) by \(g\circ h^{-1}\) and also preserves \(\mathscr E\). On \(\mathcal K\), source change of variables includes the square root of the smooth Radon–Nikodym derivative of \(dx\). This is an allowed scalar multiplier and commutes with \(D\), so it contributes no additional derivative. For each fixed group element, all dilation factors and inverses are bounded on the compact boundaries. Source and target transforms therefore give invertible maps of the graph norms. The identities extend to the closed quadratic forms. The uniqueness of the positive operator associated to such a form proves (135); spectral target dilations commute strongly with \(l\), as already proved.

Finally, if \(l=0\), every column component vanishes on \(\mathcal C\). Spanning by the Parseval fields and covectors, followed by horizontal cancellation, gives \(D_X(f)\xi=0\) for every smooth \(f\), horizontal \(X\), and \(\xi\in\mathscr E\). The bracket argument in Lemma 81 would make \(P_x\) constant. Hence \(l\ne0\). ◻

Proof. We first replace the field in Lemma 83 by an everywhere covariant version. Fix \(x_0\in B_S\) and let \(J\) be its parabolic stabilizer. Pull the field back along \(g\mapsto gx_0\) and set \[F(g)=\operatorname{dil}(g,x_0)\sigma_g^{-1}(l_{gx_0}).\] This is a measurable field of positive self-adjoint operators. One can test measurability through their bounded resolvents, or their closed graph projections. Haar disintegration over \(S/J\) and (135) give \(F(ag)=F(g)\) for almost every \(g\), for each fixed \(a\in S\). Fubini, followed by the nonsingular change of variables \((a,g)\mapsto(ag,g)\), shows that \(F(g_1)=F(g_2)\) for almost every pair. Thus \(F\) has a single essential value \(L_0\). A countable determining family of bounded matrix coefficients of graph projections suffices for this argument.

For each \(k\in J\) the definition and (129) give \[F(gk)=\operatorname{dil}(k,x_0)\sigma_k^{-1}F(g).\] Consequently \(\sigma_kL_0=\operatorname{dil}(k,x_0)L_0\). We may therefore define, independently of the representative \(g\), \[l_{gx_0}=\operatorname{dil}(g,x_0)^{-1}\sigma_gL_0.\] This agrees with the old field almost everywhere and satisfies its source covariance for every \(g,x\). Its other properties hold as well. To see the target law explicitly, commute \(\beta_h\) past \(\sigma_g\) in the pullback. Since \(\sigma_gP_{x_0}=P_{gx_0}\), its spectral dilation becomes the fixed operator \(\operatorname{dil}(h^{-1},P_{x_0})\) on \(F(g)\). The essential value \(L_0\) therefore satisfies the target law at \(x_0\), and transport proves it everywhere. Angular commutation, affiliation, and complementary source invariance pass to \(L_0\) and transport in the same way.

Let \(e_0=\mathop{\mathrm{supp}}L_0\). The target dilation is bounded, positive, invertible, and commutes with \(L_0\). Taking supports in its covariance law therefore gives \(\beta_h(e_0)=e_0\) for all \(h\), so \(e_0\in\mathcal N\). Source stabilizer covariance gives \(\sigma_k(e_0)=e_0\) for \(k\in J\). In addition, \(e_0\) is fixed by every complementary source factor. Choose a split sequence in \(J\) escaping in \(S\), and multiply it by sequences escaping in each complementary factor. Every resulting element fixes \(e_0\), while the all-factor mixing established in Section 3 makes its translates converge weakly to \(\tau(e_0)1\). Hence \(e_0=\tau(e_0)1\). It is nonzero because the derivative column was nonzero; as a projection it must be \(1\). This use of complementary source invariance is essential to the application of mixing.

All transported \(l_x\) thus have zero kernel. Define \(b_x=l_x^{-Q}\) by the spectral calculus. It is densely defined, positive, self-adjoint, injective, and affiliated to \(\mathcal A_1\). Taking inverse powers in the two covariance laws, using strong commutation with the spectral dilation, proves (136). The compact horizontal Jacobian is the \(Q\)th power of the dilation, so these are precisely the asserted inverse-Jacobian laws.

We verify the normal-copy assertion at the level of domains. For an ampliation \(\mathcal H\otimes\mathcal L\), fix an orthonormal basis \((e_n)\) of the separable multiplicity space. If \(\Xi=\sum_n\xi_n\otimes e_n\) belongs to its maximal estimate domain, positivity of every pair effect implies \(\xi_n\in\mathscr E\) for each \(n\): each coordinate energy is bounded by the total energy. Conversely, every finite sum with \(\xi_n\in\mathscr E\) belongs to the amplified estimate domain. The finite-coordinate projections commute with the PVMs, preserve that domain, and commute with its weak derivatives. For a generator \(aP_{\bullet}(g)\Xi\) of the amplified module core, those projections therefore converge in both the input norm and the derivative-column norm. Its projected vectors lie in \(\mathcal C\odot\operatorname{span}\{e_n\}\), and the same holds for finite sums of generators. Thus this algebraic tensor core is graph-dense in the amplified core. On it the derivative column is exactly \(D\otimes1\). Their closures agree, and the spectral calculus gives \(l^{\rm amp}=l\otimes1\) and \(b^{\rm amp}=b\otimes1\). Unitary identification with a normal copy preserves these identities. In particular no mixing assertion on an enlarged commutant is required. ◻

Proof. Opposite nonzero weights pair perfectly for the preserved form, and its zero-weight space is positive. Consequently the positive and nonnegative weight spaces are mutual annihilators and their middle quotient is positive. The assertion passes to spectral stages and Hilbert multiplicities. Right multiplication by \(Q\) preserves a column stage, so inserting that additional symmetry gives the same annihilator relation for the total form.

Here is the algebraic content of the required polynomial tests. In an individual compact face frame, restrict the surviving highest-line embedding of the ambient vertex to the rank-one Levi orbit. The complex stabilizer of this line is the parabolic consisting of the split centralizer and positive root groups. It embeds the corresponding complex parabolic variety projectively. The positive and nonnegative planes of an algebraic covering module are stabilized by that parabolic. So is its top split-weight plane: the split centralizer preserves it, and positive-root operators cannot raise the split weight beyond its maximum. The Plücker coordinates of all these planes are therefore rational, homogeneous coordinates on patches of this embedding. An orthogonal projector is rational in a spanning matrix and its conjugate, by Gram inversion. Thus each projector relation is a rational expression in the real coordinates of the highest line. Clear its denominators on a patch and multiply once more by a denominator that vanishes off that patch. The resulting polynomial vanishes there as well, so it is a global polynomial relation. Multiplication by norm powers makes it homogeneous of an even degree when necessary. This additional clearing factor is essential: no assertion about a rational expression on its zero-denominator set is being made.

Apply the fibrewise line-span statement of Section 5 to these homogeneous polynomial relations with linear Hilbert-vector coefficients. In particular, a vector orthogonal to every measured positive stage would be orthogonal to the positive plane at every ordinary flag. The latter planes span \(V\): their span is a nonzero invariant subspace on each chosen irreducible summand. Tensoring with the coefficient Hilbert space and allowing every matrix column proves dense spanning. Countably many determining labels and patch equations suffice first; strong continuity of the label action then gives every label. ◻

Proof. Set \(C(g)=\mathop{\mathrm{Ad}}\pi_T(g)\). On a relatively compact projectivity window \(U\), take \(v\in E(U)\mathcal K\) and define, with \(a,b\) indexing matrix entries rather than rows of \(V\), \[D_{ab}=J^*E(C_{ab})v\in\operatorname{HS}(I,H'), \qquad A=J^*v.\] The entry convention is explicitly \[\langle\xi',D_{ab}\eta\rangle =\langle\overline\eta\otimes\xi',J^*E(C_{ab})v\rangle.\] For end and start spectral stage projectors \(F(u_-)\) and \(L(u_+)\), pairing a coefficient of \((1-L)DF\) tests the link after multiplication by conjugated entries of \[(1-L(u_+))C(g)F(u_-).\] These entries vanish by (32). Determine the equations first on countable labels and compact windows and then apply bounded output calculus. This proves the two stage inclusions; total adjunction and mutual annihilation prove the reverse inclusions. In particular the argument uses complex matrix entries and does not require a real structure on \(V\).

Conjugation fixes scalar matrices and preserves traceless matrices. These two spaces are also orthogonal for the total form, since \(\mathop{\mathrm{Tr}}(QXQ)=\mathop{\mathrm{Tr}}X\). Thus the splitting reduces \(D\) and \(D^\sharp\), and its scalar block is \(A\).

To obtain the approximation, first fix one compact window of nonzero compression. On it the norms of \(C(g)\) and \(C(g)^{-1}\) have a common finite bound. Subdivide this fixed window into finitely many Borel windows of arbitrarily small coefficient oscillation. At least one subwindow \(U\) still has nonzero compression. Choose a value \(C_0\) on it; it is a total-form isometry with the already fixed norm bound. Put \(b_U=\|J^*E(U)\|>0\) and choose a unit \(v\in E(U)\mathcal K\) nearly attaining this norm, with \(\sqrt a=\|J^*v\|\geq b_U/2\). For a coefficient error \(f\) supported on \(U\), \[\|J^*E(f)v\|_2\leq b_U\|f\|_\infty \leq2\sqrt a\,\|f\|_\infty.\] There are a fixed finite number of matrix coefficients. Therefore, if the window oscillation is \(\delta\), \(D=C_0\otimes A+F\) with \(\|F\|_2\leq C\delta\sqrt a\). The form symmetries have norm one, and \(C_0C_0^\sharp=1\), so the ideal inequality \(\|XY\|_1\leq\|X\|_2\|Y\|_2\) bounds the difference in (140) by \(C'(\delta+\delta^2)a\), where \(C'\) uses only the original compact window and the matrix size. Choose \(\delta\) accordingly before the subdivision to obtain the asserted \(\varepsilon a\) bound. ◻

Proof. Every resolvent away from the nonzero spectrum of a compact operator preserving a closed stage also preserves that stage. In fact the restriction of \(\lambda-K\) to the stage is a scalar multiple of identity minus a compact operator, has Fredholm index zero, and is injective when the ambient operator is invertible. It is therefore onto the stage. Contour integration shows that \(F_0\) preserves both stages. Because \(DD^\sharp\) is total-selfadjoint, grouping conjugate eigenvalues makes \(F_0\) total-selfadjoint. Its range is consequently nondegenerate, with the induced stages still mutual annihilators.

Let \(P_z^{\mathrm{stage}}\) denote the Hilbert orthogonal projector onto the matrix stage \(\mathcal L(z)\). The operator \(F_0P_z^{\mathrm{stage}}\) is an idempotent with image \(W\cap\mathcal L(z)\). It is norm-continuous in \(z\): \(P_z^{\mathrm{stage}}\) and its adjoint are strongly continuous, and \(F_0\) has finite rank. Hence its rank is locally constant and therefore constant on the connected rank-one boundary. The same reasoning will apply to the second cut below.

Annihilation implies that \(F_0^{\sharp_{\mathrm{col}}}\) preserves both stages as well. On the positive Hilbert quotient \(\mathcal D(z)/\mathcal L(z)\), let \(f\) be the induced idempotent of \(F_0\). Its image has dimension \[\dim(W\cap\mathcal D(z))- \dim(W\cap\mathcal L(z))=\dim W-2r.\] Column adjunction descends to Hilbert adjunction on this quotient, so \(F_0F_0^{\sharp_{\mathrm{col}}}\) induces \(ff^*\). Choose \(e_1\) to be the Riesz projection of \(F_0F_0^{\sharp_{\mathrm{col}}}\) onto all of its strictly positive real eigenvalues. Only finite-rank spectra away from zero occur. The projection is column-selfadjoint, preserves the stages, and has range in \(W\). Passage to the quotient retains exactly the positive spectral subspace of \(ff^*\), whose dimension is \(\mathop{\mathrm{rank}}f\). On \(W_1\) the corresponding dimension is \(\dim W_1-2\dim(e_1\mathcal L(z))\). Thus the second cut retains the whole original middle quotient: the nonzero spectral range of \(ff^*\) is \(\mathop{\mathrm{ran}}f\). The dimensions lost in \(W\) are twice those lost from its isotropic stage, proving (141).

Since \(e_1\) is column-selfadjoint, \(W_1\) is column-nondegenerate; \(V_z=W_1\cap\mathcal L(z)\) and its perpendicular in \(W_1\) is \(W_1\cap\mathcal D(z)\). Positivity of the middle quotient is inherited from the ambient one. Applying \(e_1\) to the dense stage span proves that the \(V_z\) span \(W_1\).

It remains to produce actual finite-dimensional transports. Denote the genuine, strongly continuous ambient source transport by \(U_s\); it is a column-form isometry. For \(s\) near the identity, \[A_s=e_1U_s|_{W_1}\] is invertible, by finite rank and strong continuity. It maps \(V_z\) into \(V_{sz}\) for every \(z\), and equal constant dimensions make this equality for every \(z\). Its column adjoint is \(e_1U_{s^{-1}}|_{W_1}\) and carries \(V_{sz}\) back onto \(V_z\). Consequently \(B_s=A_s^{\sharp_{\mathrm{col}}}A_s\) preserves every \(V_z\) and is near identity. Its analytic inverse square root is column-selfadjoint and also preserves every \(V_z\). Thus \[A_s B_s^{-1/2}\] is an exact column-form isometry transporting all these planes simultaneously. A multiplication rule for the compressions is not needed.

The pairs \((s,A)\) such that \(A\) is a column-form isometry of \(W_1\) and \(AV_z=V_{sz}\) for all \(z\) form a closed subgroup of \(S\times U(q|_{W_1})\). Closedness follows from continuity of the plane family. The preceding construction supplies local lifts; hence this Lie group projects onto connected \(S\) and its Lie algebra projects onto \(\mathfrak s\). Its solvable radical maps to zero, since \(\mathfrak s\) is simple. A Levi subalgebra maps onto \(\mathfrak s\), and a complementary simple ideal in that Levi subalgebra maps isomorphically onto \(\mathfrak s\); these are the usual Levi decomposition and complete reducibility facts for finite-dimensional Lie algebras (Knapp 2023, Appendix B, Lemma B.1 and Theorem B.2). Integrating this Lie-algebra section on the simply connected source group yields the asserted representation. ◻

Proof. Use the finite-dimensional source representation from Lemma 89. Fix a split endpoint \(z_0\), write \(h\) for the positive split-weight dimension of \(W_1\), and let \(r_0\) be the negative index of the form on its zero-weight space. Nonzero opposite weight spaces pair perfectly. Since the isotropic \(V_{z_0}\) has positive perpendicular quotient, it exhausts the negative index, and \[\dim V_{z_0}=h+r_0.\] The source parabolic at \(z_0\) preserves this plane, so in particular it has a split-weight decomposition. Its zero-weight part is isotropic in the zero-weight space and has dimension at most \(r_0\).

Let \(u\) be the dimension of the trivial summand of \(W_1\). Projection of \(V_z\) to this summand is independent of \(z\), by transport, and is the whole summand because the planes span \(W_1\). At \(z_0\) it is the image of the zero-weight part, so \(u\leq r_0\). Complete reducibility and (141) give, with nontrivial irreducibles listed with repetitions, \[ \delta_0=(c_S-2)(r-\dim V_{z_0})+(c_Sr_0-u) +\sum_{\sigma\ne1}(c_Sh(V_\sigma)-\dim V_\sigma). \tag{142}\] Every term is nonnegative. Here \(c_S>2\), the first dimension difference is nonnegative, and \(c_Sr_0-u\geq(c_S-1)r_0\).

The uniform ratio gap in Lemma 86 now has a quantitative use. For a non-optimal \(\sigma\), the ratio bound gives explicitly \[c_Sh(V_\sigma)-\dim V_\sigma \geq\frac{\varepsilon_S}{c_S-\varepsilon_S}\dim V_\sigma.\] Thus its contribution to (142) controls its whole dimension. The dimension outside \(W_1\), the entire non-optimal and trivial dimension in \(W_1\), and \(r_0\) are each at most \(C\delta_0\).

Let \(O\) be the optimal isotypic sum and \(N\) its invariant complementary sum. This is an orthogonal decomposition for the preserved form. Indeed each optimal irreducible has a nondegenerate invariant Hermitian form, so its Hermitian contragredient is itself; invariant pairings with inequivalent non-optimal or trivial constituents vanish. Put \(U=\operatorname{pr}_O V_{z_0}\). It is parabolic-invariant and \(\dim U_0\leq r_0\).

We bound its negative-weight part explicitly. Choose a nonzero real simple restricted-root vector and its Cartan reflection. Their bracket lies in the split line: it lies in the centralizer and changes sign under the Cartan involution. After rescaling they form an \(\mathfrak{sl}_2\)-triple with diagonal element \(2H\), where \(H\) is the simple-gap-one generator. Every optimal weight for \(H\) is integral. In a finite-dimensional \(\mathfrak{sl}_2\)-module, raising from weight \(-j\) to zero is injective for each integer \(j>0\). Parabolic invariance therefore gives injections \(U_{-j}\to U_0\). The finite optimal list has at most \(d_S\) distinct negative integral weights, whence \[\dim U_{\leq0}\leq(d_S+1)r_0.\]

Since \(\dim V_{z_0}\geq h(O)\) and \(\dim N\leq C\delta_0\), the positive part of \(V_{z_0}\) has dimension at least \(h(O)-\dim N-(d_S+1)r_0\). Intersecting it with \(O_{>0}\) loses at most another \(\dim N\). Thus \[\operatorname{codim}_{O_{>0}}(O_{>0}\cap V_{z_0}) \leq2\dim N+(d_S+1)r_0\leq C\delta_0.\] Moreover pairing \(O_{\geq0}\) against \(V_{z_0}\) only tests \(U_{\leq0}\), so \[\operatorname{codim}_{O_{\geq0}} (O_{\geq0}\cap V_{z_0}^{\perp})\leq C\delta_0.\]

Write \(O=\bigoplus_\sigma V_\sigma\otimes M_\sigma\) over the finite optimal list. For each basis vector of \((V_\sigma)_{>0}\) impose on \(m\in M_\sigma\) the condition \(v\otimes m\in V_{z_0}\); for each basis vector of \((V_\sigma)_{\geq0}\) impose \(v\otimes m\in V_{z_0}^{\perp}\). Each condition has rank at most the corresponding codimension just bounded. There are a fixed finite number of basis vectors, and their module dimensions are fixed. The resulting multiplicity subspaces therefore remove at most \(C\delta_0\) dimensions from \(O\). Their full-module sum \(G_W\) has the claimed codimension in \(W\). Transport gives both plane conditions at all labels. Finally, \(V_z\subset\mathcal L(z)\), and its perpendicular in \(W_1\) equals \(W_1\cap\mathcal D(z)\) because \(e_1\) is column-selfadjoint. Thus the nonnegative condition also lifts to the ambient measured stage. ◻

Proof. Sum the Riesz ranges \(W\) of \(DD^\sharp\) with weight \(w\) equal to the absolute value of the associated eigenvalue, grouping conjugates with their common absolute value. Lidskii’s theorem (Simon 2005) and (140) give \[\sum_Ww\dim W\geq(n_0^2-\varepsilon)a.\] For the restriction to a fixed invariant positive stage, the sum of absolute eigenvalues, with their Riesz multiplicities \(r\), is bounded above by its trace norm. The compression of \(1\otimes AA^*\) to that stage is positive. If \(P_z^{\mathrm{stage}}\) is the Hilbert stage projector, its partial matrix trace is \[\mathop{\mathrm{Tr}}_{\operatorname{End}(V)}P_z^{\mathrm{stage}} =n_0h(V)\,1_{H'}.\] Indeed each spectral positive-plane projector has rank \(h(V)\), and there are \(n_0\) matrix columns. The compression therefore has trace \(a n_0h(V)=a n_0^2/c_T\). Compressing the error in (140) does not increase its trace norm. Consequently \[ \sum_Ww r\leq(n_0^2/c_T+\varepsilon)a. \tag{143}\] These restriction multiplicities are exactly \(\dim(W\cap\mathcal L(z))\): the resolvent argument in Lemma 89 also identifies the restricted Riesz projections. The inequality \(\dim W\leq c_Sr\) now gives \(c_T\leq c_S\) as \(\varepsilon\downarrow0\). The real non-isolated type set is invariant under the type permutation: Corollary 44 preserves the real types, and Proposition 40 preserves the Coxeter components. Applying this inequality round each of its finite cycles gives equality on every matched pair.

With equality established, (143) and the lower bound above imply \[\sum_W w\delta_0\leq(c_S+1)\varepsilon a.\] The scalar block of \(DD^\sharp\) is exactly \(AA^*\). Put \[S_W=W\cap(\mathbb C1_V\otimes H').\] The nonzero such subspaces are precisely the eigenspaces \(1_V\otimes\ker(AA^*-\mu)\), \(\mu>0\), so \(\sum_Ww\dim S_W=\mathop{\mathrm{Tr}}(AA^*)=a\). If \(G_W\cap S_W=\{0\}\) for every \(W\), the quotient map \(S_W\to W/G_W\) would be injective. Thus \(\dim S_W\leq\operatorname{codim}_W G_W\), and Lemma 90 would give \[a\leq C\sum_Ww\delta_0\leq C(c_S+1)\varepsilon a,\] which is impossible for sufficiently small \(\varepsilon\). Hence one \(G_W\) contains a nonzero scalar matrix \(1_V\otimes\eta\). Projecting its optimal module copies to each matrix column gives the maps \(B\) in the statement. The columns of this scalar matrix are \(v_i\otimes\eta\), proving the assertion for every vector of \(V\). Interchanging start and end proves the other assertion.

If a genuine source amplification was used, apply a coefficient functional in the added Hilbert factor to the maps \(B\) and to the scalar witness. At least one coefficient of a nonzero witness is nonzero, and all plane equations are unchanged because the added factor is a multiplicity for those equations. This yields a witness on an original fibre. ◻

Proof. Fix countably many dense labels and bases of the finitely many optimal modules. Each section condition is a bounded linear equation on \(\mathop{\mathrm{Hom}}(V_\sigma,V\otimes H_0)\): multiply the image of a source-plane projector by the complement of the measured-plane projector. Thus \(\mathscr S_\sigma\) is a closed common kernel. One can realize all conditions as the kernel of a single bounded map to a countable Hilbert sum by inserting summable positive weights. Strong continuity recovers the conditions at every label.

Choose bases \(u_{\sigma,a}\) of \(V_\sigma\) and \(v_i\) of \(V\). Algebraic membership \(v_i\otimes\eta\in\mathcal E\) can be expressed using only one unknown section \(B_{i,\sigma,a}\) per source basis element: \[v_i\otimes\eta=\sum_{\sigma,a}B_{i,\sigma,a}u_{\sigma,a}.\] Indeed finite sums of sections of a fixed type may be combined, including their scalar coefficients. The simultaneous equations for all \(i\) define another closed linear kernel in the product of \(H_0\) and finitely many section spaces. The scalar domain is its coordinate image in \(H_0\). Kernel projections and range-support projections of bounded decomposable maps disintegrate, so the projection onto \(\overline{\mathscr D}\) disintegrates to the fibrewise closure projections. Lemma 91 makes this projection nonzero; no measurable choice of individual witnesses is necessary.

The defining equations commute with base multipliers and all complementary right symmetries. The closure projection consequently belongs to \(\mathcal A_j\). An actual target translation changes compact angular frames by a projectivity depending on the sharp face, but not on the panel label. The corresponding matrix correction \(\pi_T\) is boundedly invertible for each fixed translation: its Levi projection is bounded in compact frames, and the same is true for its inverse. Correct each transported section by this matrix. The scalar columns remain in the span of the corrected sections by combining them with the entries of the inverse matrix. These entries are own-face spectral multipliers, which preserve the section equations. The inverse translation proves equality of the transported scalar domains. No correction is needed for base or complementary right symmetries. It follows that the closure projection is fixed by the measured right action and hence belongs to \(\mathcal N\).

Source panel covariance follows directly as well: if \(B\) is a section, then \(U_sB\sigma(s)^{-1}\) is a section with the labels reindexed. Since \(U_s\) acts only on the Hilbert coefficient, it preserves the scalar domain. Split translations that fix every panel label preserve its defining equations without reindexing. Choose such translations escaping in every factor, as in the original mixed face path. The closure projection \(p\in\mathcal N\) is invariant under this sequence. Mixing in \(\mathcal N\) therefore gives \(\tau(p)=\lim\tau(p\sigma(a)p)=\tau(p)^2\). Since \(p\ne0\), it equals \(1\). This proves density, and also makes \(\mathcal E\) dense in \(V\otimes H_0\). ◻

Proof. Let \(\mathscr A\) be the unital algebra generated by the signs on an irreducible module \(W\). The connected source normalizes \(\mathscr A\). Its Jacobson radical has a nonzero common kernel: a finite-dimensional nilpotent algebra annihilates a nonzero vector. That common kernel is source-invariant because the radical is characteristic. Irreducibility makes it all of \(W\), so the radical is zero. The connected source cannot permute the finitely many central blocks of the semisimple algebra, and their images are invariant subspaces. There is consequently one block, and \[W=E\otimes F,\qquad \mathscr A=\operatorname{End}(E)\otimes1.\] Derivations of a matrix algebra are inner. For each source Lie-algebra element choose the unique traceless implementer of its derivation on \(\operatorname{End}(E)\). These implementers form a Lie representation: the defect in a commutator is both scalar and traceless, hence zero. Subtracting this representation from the action on \(W\) leaves a representation in the commutant \(1\otimes\operatorname{End}(F)\). This gives a tensor decomposition of the Lie representation.

Suppose first that the complexified source algebra is simple. Two nontrivial irreducibles of a simple complex Lie algebra do not have an irreducible tensor product. For completeness, if their highest weights are \(\lambda,\mu\ne0\), the tensor product contains the constituent of highest weight \(\lambda+\mu\). In the Weyl dimension formula each positive coroot contributes \[\frac{(a+r)(b+r)}{r(a+b+r)}\geq1, \quad a=\langle\lambda,\alpha^\vee\rangle,\quad b=\langle\mu,\alpha^\vee\rangle,\quad r=\langle\rho,\alpha^\vee\rangle>0.\] There is a positive coroot pairing positively with both nonzero dominant weights, because the root system is irreducible. Its factor is strictly larger than one. Hence \(\dim V_\lambda\dim V_\mu>\dim V_{\lambda+\mu}\), excluding irreducibility. In the displayed tensor decomposition, irreducibility of \(W\) therefore makes one factor trivial. The sign is nonscalar, so the \(E\) factor is nontrivial; a trivial \(F\) has dimension one by irreducibility. This proves \(\mathscr A=\operatorname{End}(W)\).

The remaining real-simple case is \(\mathfrak{so}(3,1)\), whose complexification has two simple summands. Its only optimal module is the Lorentz vector module. Its split weights are \(1,0,0,-1\), so a split sign is the split generator itself. The linear span of its source conjugates is the image of a nonzero adjoint ideal in the real-simple algebra and is therefore the whole Lie-algebra image. The associative algebra it generates acts irreducibly and equals the full matrix algebra by the same radical and single-block argument, or by Burnside’s theorem.

An intertwiner of all signs has invariant kernel and image for their full matrix algebras. A nonzero such map is therefore an isomorphism, and the space of such maps has dimension one. Conjugating by the source preserves this line, because it permutes the axes. Connected semisimplicity has no nontrivial one-dimensional characters, so the line is fixed pointwise. The map is an ordinary source intertwiner. ◻

Proof. For two sections \(B:V_\sigma\to V\otimes H_0\) and \(C:V_\tau\to V\otimes H_0\), represent their cross pairing by \(K:V_\tau\to V_\sigma\) using the nondegenerate source form: \[q(Bv,Cw)=q_\sigma(v,Kw).\] If \(w\) is positive at an endpoint and \(v\) is nonnegative, their images belong to mutually annihilating measured stages. Hence \(Kw\) is positive at that endpoint. Interchanging positive and nonnegative gives the corresponding nonnegative inclusion. Repeating at the opposite endpoint gives the negative and nonpositive inclusions, and hence exact preservation of the three split signs. Lemma 93 therefore makes \(K\) an ordinary source intertwiner. Schur’s lemma shows that inequivalent types are orthogonal and, within a fixed type, \[q(Bv,Cw)=q_\sigma(v,w)\,b_\sigma(B,C)\] for a Hermitian multiplicity pairing \(b_\sigma\).

This pairing is positive semidefinite. Choose a nonzero source zero-weight vector \(v\); its form is positive by Lemma 86. The vector \(Bv\) lies in the measured nonnegative stage, whose form is nonnegative. Thus \(b_\sigma(B,B)\geq0\), and the same argument applied to arbitrary linear combinations proves matrix positivity for every finite family of sections.

To justify descent rather than assume it, consider the algebraic evaluation map \[T:\bigoplus_\sigma V_\sigma\otimes\mathscr S_\sigma \longrightarrow\mathcal E, \qquad T(v\otimes B)=Bv.\] Its pullback form is the direct sum of \(q_\sigma\otimes b_\sigma\). For a finite collection of sections the radical is precisely the sum of \(V_\sigma\) tensored with the radicals of the positive semidefinite multiplicity forms. A radical vector remains radical after any further finite collection is added: Cauchy–Schwarz for a positive semidefinite form makes its pairing with every newly added vector zero. Its evaluation therefore pairs to zero with all of \(\mathcal E\), and is zero by the density in Lemma 92 and ambient nondegeneracy. Conversely a vector evaluating to zero is in this radical, by pairing with the same finite module span. Thus the kernel of \(T\) is exactly the radical. It is invariant under every source matrix action, proving well-definedness and the adjoint assertion.

Finally \(q_\sigma(v,Q_\sigma w)\) is the compact-form positive Hilbert pairing. The pullback of \(q(x,R(\mathbf Q)y)\) is consequently the tensor product of this positive pairing with \(b_\sigma\), summed over types. Its only null vectors are the radicals just identified with \(\ker T\). This proves the strict inequality (144) for every nonzero evaluated vector. ◻

Proof. Write \(D_t^{lm}\) for the block from input matrix column \(m\) to output matrix column \(l\). The stage calculation in Lemma 88 shows that, for an end section \(B\), \[D_t^{lm}(v)B\sigma(s_t)^{-1}\] is a start section. This gives action exchange on each column block, with \(X_\sigma\) replaced by \(\mathop{\mathrm{Ad}}\sigma(s_t)X_\sigma\). Applying the statement with \(X=1\) to the scalar columns and using scalar/traceless reduction yields \[D_t(v)(1_V\otimes\eta)=1_V\otimes A_t(v)\eta.\] Every output column is a sum of section evaluations, proving that \(A_t(v)\eta\) is in the start scalar domain. The general column-block identity then gives (145) by summation.

To estimate a fixed matrix entry, pair its output against a fixed start scalar-domain test \(\xi\). A Hilbert pairing with a column \(x\) can be written as a column-form pairing with \(Qx\). Move the algebra action on the right side of (145) to this fixed test by Proposition 94. Expand that fixed test in finitely many section expressions and matrix units. On the finite optimal list, \[\|\mathop{\mathrm{Ad}}\sigma(s_t)\|\leq e^{2d_St}\qquad(t\geq0),\] so the norm of the moved action on the fixed test is at most \(C_\xi e^{2d_St}\). If \[f_{t,M}(g)=e^{-2d_T\ell(g)} 1_{\{d_T\ell(g)>d_St+M\}},\] then \[\|A_t(E(f_{t,M})v)\|_2 \leq\|E(f_{t,M})v\| \leq e^{-2d_St-2M}\|v\|,\] because \(J_t\) is an isometry. The remaining fixed input contributes only its Hilbert norm. This proves the bound. In particular the proof estimates the action on a fixed test vector and never the norm of an unbounded action on the moving vector \(A_t(v)\eta\). ◻

Proof of Proposition 85. Take the iterated path limits in their stated order and then the limit \(M\to\infty\). Equation (146) vanishes on the positive infinite-upper-excess summand. Put \[b(g)=e^{-d_T\ell(g)}\pi_T(g).\] Both \(b\) and \(b^*\) are contractions. Right multiplication of the vanishing matrix by \(Q\) uses \[\pi_T(g)^{-1}Q=Q\pi_T(g)^*.\] It changes normalized conjugation into left and right multiplication on \(ZQ\) by \(b,b^*\). Further multiplication on the outside by \(b^*,b\) gives the vanishing matrix with both outside factors \(b^*b\).

Compact Cartan decomposition identifies their limit. In the surviving Levi block, if \(\pi_T(g)=k_1\exp(\ell(g)H_T)k_2\) in compact frames, then \[b^*b=k_2^*\exp\bigl(2\ell(g)(H_T-d_T)\bigr)k_2\] tends, on divergent residual gap, to the projector on the highest input split-weight plane. Denote this projector by \(\Pi\); as a function of the additional reverse angular vertex, it is the \(Q\)-image of the highest plane for the inverse projectivity. Compact factors commute with \(Q\), so these descriptions agree.

All frame passages can be made on continuity sets of arbitrarily large fixed input mass. The sharp endpoint base laws are exactly the same throughout the slower translations, because those translations stabilize the bases. Approximate the Borel compact frames on such sets and then remove their complements. The additional reverse vertex is sharp by the total angular statements recalled at the beginning of the section. Thus \(\Pi\) is available as a sharp input spectral projector in the limit, without a continuity assumption on a moving Borel frame.

Suppose that the infinite-excess summand is nonzero. Fixed target shifts at either input alter inverse-index gaps only by bounded amounts, so they preserve this summand. The independent all-factor contracted symmetries preserve it as well, in the same order of limits as for the mixed link. The scalar-mass argument of Section 4 therefore makes its compression a positive scalar multiple of identity. Write its scalar mass as \(m>0\). The restricted link divided by \(\sqrt m\) is an isometry and retains the sharp angular marginals. Indeed the angular subeffect at a sharp input projection \(P\) is supported on \(P\), while its complementary cut is supported on \(1-P\); their sum is \(mI\), so the first subeffect equals \(mP\). This is the sharp-compression criterion of Proposition 21. We may therefore pull the additional reverse vertex, denoted \(Q_y\), through the restricted conjugate link. Pulling the two outside copies of \(\Pi\) evaluates both at the same end slot, and removal of the scaled isometry and a fixed nonzero start test \(\xi\) gives \[ \left[\Pi(v)Z_\eta(X)Q\Pi(v)\right]_{v=Q_y}=0. \tag{147}\] The spectral multipliers in this formula act on its Hilbert-vector entries. In particular no section action has been moved through either projector.

To apply fibrewise polynomial fullness, we need this identity at every panel label with \(\eta\) and \(X\) fixed. Conjugate the completing axis by a compact source Levi rotation. This conjugates the entire path, moving fixed source transports into its two inputs, and preserves the same scalar excess mass. Its only effect on the section argument is a fixed conjugation of the source reindexing. The scalar domains are preserved by these transports by Lemma 92. To see explicitly that the data can be held fixed, the section transport is \(B\mapsto U_k B\sigma(k)^{-1}\), whence \(U_kR(X)U_k^{-1}=R(\mathop{\mathrm{Ad}}\sigma(k)X)\). For prescribed \(\eta,X\) apply the unrotated argument to \(\eta'=U_k^{-1}\eta\) and \(X'=(\mathop{\mathrm{Ad}}\sigma(k))^{-1}X\). Then \[(1\otimes U_k)Z_{\eta'}(X')=Z_\eta(X).\] Thus Equation (147) holds at every rotated label for the same fixed \(\eta\) and \(X\). Impose countably many dense labels first and extend by strong label continuity. Lemma 87, with its extra denominator clearing, now gives the vanishing at every ordinary residue value \(v\) for fixed \(\eta\) and \(X\).

Fix such an ordinary value and take a constant nonzero \(w\in\mathop{\mathrm{ran}}\Pi(v)\) in the compact frame. Choose any nonzero \(\eta\in\mathscr D\) at the end and set \(X=\mathbf Q\). Pairing (147) with \(w\otimes\eta\) gives \[0=\big\langle w\otimes\eta, R(\mathbf Q)(Qw\otimes\eta)\big\rangle =q\bigl(Qw\otimes\eta, R(\mathbf Q)(Qw\otimes\eta)\bigr).\] The last quantity is strictly positive by (144), since \(Qw\otimes\eta\ne0\). This contradiction proves that the infinite-upper-excess mass is zero and establishes (137).

Every bound used fixed square-integrable scalar-domain data; these data are dense by Lemma 92, so the zero-mass conclusion extends to all fixed inputs. To check the offset assertion, absorb a displacement in this panel’s split direction into the total parameter \(t\) before applying \(\|\mathop{\mathrm{Ad}}\sigma(s_t)\|\leq e^{2d_St}\). For each fixed negative offset, that total parameter is eventually positive; offsets are exhausted only after the ordinary path limits. A split translation \(a\) in another completing direction fixes every label of this panel. It sends a section \(B\) to \(U_aB\), so on \(\mathcal E\) one has \[R(X)U_a=U_aR(X).\] Its unitary action therefore leaves unchanged the norms of the matrix actions moved to fixed tests in Lemma 95. This proves the stated uniformity for other completing translations and the subsequent offset limits. ◻

Proof. The joint raising map \[V[0]\longrightarrow\bigoplus_{j=1}^r V[\alpha_j], \qquad v\longmapsto(E_{\alpha_j}v)_{j=1}^r\] is injective. Indeed, a vector in its kernel is a highest-weight vector of weight zero: the simple raising operators generate the positive nilpotent algebra. The finite-dimensional module generated by such a vector is trivial. To see the last assertion directly, restrict to each simple-root \(\mathfrak{sl}_2\): a highest vector of weight zero is also killed by its lowering operator. Thus all simple raising and lowering operators, and the Cartan algebra, kill the vector. A nonzero such vector would be an invariant vector in the nontrivial irreducible \(V\), which is impossible.

For the second assertion, restrict \(V\) to the root \(\mathfrak{sl}_2\) corresponding to \(\alpha\). In every irreducible \(\mathfrak{sl}_2\) summand, lowering is injective on a positive \(\alpha^\vee\)-eigenspace: only the lowest weight is killed, and that weight is nonpositive. The lowering map is therefore injective on \(V[\mu]\) and takes it into \(V[\mu-\alpha]\). Finally, Weyl-group representatives in the simply connected group give isomorphisms between the weight spaces in a Weyl orbit. ◻

Proof. First let \(n+1=2b+1\) with \(b\geq2\). The absolute root system is \(B_b\), in coordinates \(e_1,\ldots,e_b\), and the split functional extracts the first coordinate. The Weyl group acts by all signed permutations. If a nonzero weight has exactly \(j\) nonzero coordinates, symmetry gives \[\frac{p(\mathcal O)}{|\mathcal O|}=\frac{j}{2b}.\] Consequently every nonzero orbit has positive surplus for \(c=2b+1\). The short-root orbit has size \(2b\), positive count one, and surplus one. The long-root orbit has size \(2b(b-1)\), positive count \(2(b-1)\), and surplus \[(2b+1)2(b-1)-2b(b-1)=2(b^2-1).\] Write \(m_{\mathrm s}\) and \(m_{\mathrm l}\) for their multiplicities. There are \(b-1\) long absolute simple roots and one short one, so Lemma 96 gives \[m_0\leq(b-1)m_{\mathrm l}+m_{\mathrm s}.\] Substituting this bound in Equation (148) leaves the nonnegative expression \[\bigl(2(b^2-1)-(b-1)\bigr)m_{\mathrm l} +\sum_{\substack{\mathcal O\text{ neither root orbit}}} m_{\mathcal O}s_{2b+1}(\mathcal O).\] Every coefficient displayed here is strictly positive. Equality therefore permits only the short-root orbit and full-Cartan zero, with \(m_0=m_{\mathrm s}\). The highest weight must be \(e_1\), which gives the standard module. Its weights are \(0,\pm e_1,\ldots,\pm e_b\), once each, so equality is attained. This reasoning also covers the half-integral weight lattice: those weights have no zero coordinates and hence strictly positive surplus.

Now let \(n+1=2b\) with \(b\geq3\). The absolute root system is \(D_b\). Its even signed permutations give the same fraction \(j/(2b)\). If a zero coordinate is available, its sign can absorb the parity restriction. If every coordinate is nonzero, flipping the first and one other sign exchanges the two signs in the first coordinate, so that case also has positive fraction \(1/2\). Thus \[s_{2b}(\mathcal O)=(j-1)|\mathcal O|\geq0.\] The single root orbit has size \(2b(b-1)\) and surplus \(2b(b-1)>b\). Its multiplicity \(m_{\mathrm r}\) satisfies \(m_0\leq b m_{\mathrm r}\). Equation (148) therefore proves the bound. At equality the root orbit and full-Cartan zero are absent, and every remaining weight has only one nonzero coordinate. The highest weight is consequently \(a e_1\) with an integer \(a\geq1\); half-integral weights cannot have this pattern. If \(a>1\), lowering by \(e_1-e_2\) produces the weight \((a-1)e_1+e_2\) by Lemma 96. This has two nonzero coordinates and positive surplus, a contradiction. Hence \(a=1\), giving the standard module with weights \(\pm e_i\).

For completeness, in \(D_2\) the two root orbits \(\{\pm(e_1-e_2)\}\) and \(\{\pm(e_1+e_2)\}\) each have size two, positive count one, and surplus two for \(c=4\). Each contains one absolute simple root, so each surplus strictly exceeds its contribution to the zero-multiplicity bound. An explicit calculation also handles all modules for the nonsimple complexification. They are \[\operatorname{Sym}^a\mathbb C^2\otimes \operatorname{Sym}^b\mathbb C^2, \qquad a,b\in\mathbb Z_{\geq0},\quad (a,b)\neq(0,0).\] Their split weights are \((a+b-2i-2j)/2\) for \(0\leq i\leq a\), \(0\leq j\leq b\). The dimension is \((a+1)(b+1)\), and the split-zero multiplicity is zero if \(a,b\) have different parity, and \(\min(a,b)+1\) otherwise. In the latter case assume \(a\leq b\). The zero proportion is \(1/(b+1)\leq1/2\), with equality possible only when \(a=b=1\): if \(b=0\) the representation is trivial, and if \(b=1\) equal parity forces \(a=1\). Since the positive and negative weight counts agree, \[\frac{\dim V}{h(V)} =\frac{2}{1-z(V)/\dim V}\leq4,\] with equality exactly for the four-dimensional standard module.

Finally, for \(B_1\), or \(SO(2,1)\), a covering-group irreducible has highest split weight \(a\in\frac12\mathbb Z_{>0}\) and the string \(a,a-1,\ldots,-a\). For integral \(a\) its ratio is \((2a+1)/a=2+1/a\leq3\), with equality only at \(a=1\). For strictly half-integral \(a\) the zero space is absent and the ratio is two. This proves the remaining case, including its spin modules. ◻

Proof. Use the \(A_{k-1}\) coordinates with sum zero and split functional \(\mu\mapsto\mu_1-\mu_2\). Suppose that the distinct coordinate values of a nonzero weight have multiplicities \(r_1,\ldots,r_s\), where \(s\geq2\) and \(\sum r_i=k\). Choosing the ordered first and second coordinates uniformly over a permutation orbit gives \[\frac{p(\mathcal O)}{|\mathcal O|} =\frac{k^2-\sum_i r_i^2}{2k(k-1)}\geq\frac1k.\] Indeed, merging parts increases their sum of squares, and among two positive parts \(r,k-r\) that sum is largest at \(r=1\) or \(k-1\). Thus \(\sum r_i^2\leq(k-1)^2+1\), with equality exactly for the partition \((k-1,1)\). Equality in the orbit bound therefore means that exactly one coordinate is exceptional.

The root orbit consists of the \(k(k-1)\) vectors \(e_i-e_j\). There is one root of split weight two and \(2(k-2)\) roots of split weight one, so its surplus for \(c=k\) is \[k(2k-3)-k(k-1)=k(k-2)>k-1.\] If \(m_{\mathrm r}\) is its multiplicity, then \(m_0\leq(k-1)m_{\mathrm r}\). All other orbit surpluses are nonnegative. Equation (148) proves the bound and shows that equality forces \(m_0=m_{\mathrm r}=0\) and only one-exceptional-coordinate patterns.

A dominant highest weight with that pattern is \(a\omega_1\) or \(a\omega_{k-1}\), where \(a\) is a positive integer. The corresponding modules are \(\operatorname{Sym}^aE\) and its dual. Their weights are obtained from the monomial exponent tuples \((a_1,\ldots,a_k)\), \(a_i\geq0\), \(\sum a_i=a\), by subtracting \(a/k\) from each coordinate. If \(a\geq3\), the tuple \((a-1,1,0,\ldots,0)\) has three distinct coordinate values, and therefore positive surplus. If \(a=2\) and \(k>3\), the tuple \((1,1,0,\ldots,0)\) has two groups of sizes \(2,k-2\), again giving positive surplus. These cases cannot attain equality. For \(a=1\), all weights have the required pattern and \(\dim E=k\), \(h(E)=1\). For \(a=2,k=3\), the only exponent patterns are permutations of \((2,0,0)\) and \((1,1,0)\); both have one exceptional coordinate and neither gives the full-Cartan zero weight. In that case \(\dim\operatorname{Sym}^2E=6\) and its split weights are \(2,1,0,0,-1,-2\), so \(h=2\). Dualizing reverses the split weights and preserves these conclusions. ◻

Proof. The absolute root system is \(C_k\), its full weight lattice is \(\mathbb Z^k\), and the split functional is \(\mu\mapsto\mu_1+\mu_2\). The Weyl group consists of all signed permutations. Suppose that a nonzero weight has \(j\) nonzero coordinates. The probability that just one of its first two coordinates is nonzero is \(2j(k-j)/(k(k-1))\). In that event the split weight is positive with probability \(1/2\). The probability that both are nonzero is \(j(j-1)/(k(k-1))\). Conditional on their absolute values, at least one of the four sign choices has positive sum; exactly one does if and only if those absolute values agree. Consequently \[ \frac{p(\mathcal O)}{|\mathcal O|} \geq\frac{j(k-j)+j(j-1)/4}{k(k-1)}. \tag{149}\] For \(k\geq4\) the numerator is a strictly concave quadratic in \(j\). At the endpoints \(j=1\) and \(j=k\) its values are \(k-1\) and \(k(k-1)/4\), respectively. Thus the right side is at least \(1/k\). Equality can occur only for \(j=1\), or for \(k=j=4\); in the latter case equality in the sign count further requires all absolute values to agree.

Let \(m_{\mathrm s}\) and \(m_{\mathrm l}\) denote the multiplicities of the short-root orbit \(\{\pm e_i\pm e_j:i\neq j\}\) and the long-root orbit \(\{\pm2e_i\}\). Lowering \(2e_1\) by \(e_1-e_2\) gives \(e_1+e_2\), so Lemma 96 gives \(m_{\mathrm l}\leq m_{\mathrm s}\). Since there are \(k-1\) short absolute simple roots and one long one, \[m_0\leq(k-1)m_{\mathrm s}+m_{\mathrm l} \leq k m_{\mathrm s}.\] The short-root orbit has size \(2k(k-1)\). Its positive roots for the split functional are \(e_1+e_2\) and the \(4(k-2)\) roots having one index in \(\{1,2\}\) and the other outside that set. Thus its positive count is \(4k-7\), and its surplus is \[k(4k-7)-2k(k-1)=k(2k-5)>k.\] Together with Equation (149), this proves the bound and shows that equality requires \(m_{\mathrm s}=m_{\mathrm l}=m_0=0\) and only zero-surplus nonzero orbits. A highest weight of the first possible kind is \(a e_1\). If \(a>1\), lowering by \(e_1-e_2\) produces \((a-1)e_1+e_2\), which has strictly positive surplus. The other possible highest weight is \((a,a,a,a)\) when \(k=4\). Lowering by \(e_3+e_4\) produces \((a,a,a-1,a-1)\). For \(a=1\) this has support two, and for \(a>1\) it has unequal nonzero absolute values; in either case its surplus is strictly positive. The only remaining highest weight is \(e_1\). The standard module has dimension \(2k\) and two positive split weights, so it attains the bound.

For \(C_3\), use \(c=7/2\). The following table gives every possible nonzero orbit pattern, its size, its positive count, and its surplus. Letters denote positive integers, and distinct letters denote distinct integers; permutations and all sign choices are included. \[\begin{array}{c|r|r|r} \text{pattern}&|\mathcal O|&p(\mathcal O)& (7/2)p(\mathcal O)-|\mathcal O|\\ \hline (a,0,0)&6&2&1\\ (a,a,0)&12&5&11/2\\ (a,b,0)&24&12&18\\ (a,a,a)&8&2&-1\\ (a,a,b)&24&10&11\\ (a,b,c)&48&24&36 \end{array}\] Here the first row has positive split weight exactly when its nonzero coordinate occupies one of the first two positions and has positive sign. In the second row, placing zero in the third position contributes one positive sign choice, while placing it in either of the first two positions contributes two, for a total of five. Distinct nonzero absolute values in the third and sixth rows prevent a zero sum of the first two coordinates, so exactly half the orbit is positive. In the fourth row the first two signs must both be positive, leaving two choices for the third sign. In the fifth row, the exceptional absolute value occupies the third position in eight orbit points, of which two are positive; in the other sixteen points the first two absolute values differ, so eight are positive. This gives all the entries without a generic-position assumption on the integers.

Write \(t_a\) for the multiplicity of the orbit \((a,a,a)\). Lowering this weight by \(e_1+e_2\) gives \((a-1,a-1,a)\), whose orbit multiplicity is at least \(t_a\). For \(a=1\) this companion orbit is \((1,0,0)\), with surplus one; for \(a>1\) it has pattern \((a,a-1,a-1)\), with surplus eleven. The companion orbits are distinct as \(a\) varies. None is the short-root orbit \((1,1,0)\). Thus each negative contribution \(-t_a\) in Equation (148) is paid for by a distinct companion orbit; the payment leaves at least \(10t_a\) when \(a>1\).

Independently, the earlier lowering comparison of long and short roots gives \(m_0\leq3m_{\mathrm s}\). The short-root surplus is \(11/2>3\), so its contribution pays for full-Cartan zero with strict excess unless \(m_{\mathrm s}=m_0=0\). Every other unassigned surplus is positive. The bound follows. At equality, all \(t_a\) with \(a>1\) vanish, and the only remaining orbits can be \((1,1,1)\) and \((1,0,0)\) with equal multiplicity. The module is nontrivial, so both occur. Since \((1,1,1)-(1,0,0)=e_2+e_3\) is a positive root, its highest weight must be \((1,1,1)\). Its highest-weight multiplicity is one, so both orbit multiplicities are one.

This module is \(\bigwedge^3_0E\). Indeed, the standard weights are \(\pm e_1,\pm e_2,\pm e_3\). In \(\bigwedge^3E\), choosing one signed weight of each index gives all eight \((\pm1,\pm1,\pm1)\) weights once. Choosing an opposite pair and one other signed weight gives each \(\pm e_i\) twice, because either of the other two indices can supply the opposite pair. Hence \(\bigwedge^3E\) has these eight weights and twelve sparse weight vectors. The invariant symplectic bivector gives an injective equivariant map \(E\to\bigwedge^3E\), \(v\mapsto\Omega\wedge v\); on a symplectic basis vector its image is a sum of two distinct nonzero wedge-basis vectors. Removing this standard summand leaves the eight sign weights and the six sparse weights, each once. To verify irreducibility without assuming it, the possible dominant highest weights in this complement are \((1,1,1)\) and \((1,0,0)\). The former occurs once and, by lowering \(e_1+e_2\), its irreducible constituent already contains the sparse orbit. It therefore contains all fourteen available weight vectors. The complement is irreducible, has dimension fourteen and positive count four, as required. ◻

Proof. First fix \(x\in K_c\) such that \(\mathop{\mathrm{Ad}}x\) is not the identity on any simple ideal. Put \[U=(1-\mathop{\mathrm{Ad}}x)\mathfrak k_c.\] The span of \(\mathop{\mathrm{Ad}}(k)U\) over \(k\in K_c\) is an \(\mathop{\mathrm{Ad}}(K_c)\)-invariant ideal. Its projection to every simple ideal is nonzero, and hence it is all of \(\mathfrak k_c\). Choose \(h_1,\ldots,h_N\in K_c\) so that \(\sum_j\mathop{\mathrm{Ad}}(h_j)U=\mathfrak k_c\). Consider the analytic map \[F:K_c^N\longrightarrow K_c, \qquad F(k_1,\ldots,k_N)=\prod_{j=1}^N k_jxk_j^{-1}.\] Write \(p_0=1\) and \(p_j=\prod_{\ell\leq j}k_\ell xk_\ell^{-1}\). For variations \(k_j(t)=k_j\exp(tX_j)\), right trivialization of the differential gives \[dF(X_1,\ldots,X_N)F^{-1} =\sum_{j=1}^N \mathop{\mathrm{Ad}}(p_{j-1}k_j)(1-\mathop{\mathrm{Ad}}x)X_j.\] Choose \(k_j=p_{j-1}^{-1}h_j\) recursively. At that point the displayed differential is onto.

Equip the groups with invariant analytic metrics. In invariant frames the matrix \(B\) of \(dF\) is analytic, and \(\det(BB^{\mathsf T})\) is an analytic function on the connected manifold \(K_c^N\), nonzero at the constructed point. Its zero set has Haar measure zero. On the remaining set, submersion coordinates make \(F\) a coordinate projection. A countable cover by such coordinates and Fubini’s theorem show that the inverse image of any Haar-null subset of \(K_c\) is null there. The critical set is also null. Therefore the pushforward of product Haar measure under \(F\) is absolutely continuous. Equivalently, if \(\nu_x\) is the conjugacy-orbit probability measure of \(x\), then \[\nu_x^{*N}=f_x\,dk,\qquad f_x\in L^1(K_c).\]

For an irreducible unitary \(\pi\), Schur’s lemma gives \[\int\pi(g)\,d\nu_x(g) =a_\pi I,\qquad a_\pi=\frac{\chi_\pi(x)}{d_\pi}.\] Taking normalized traces after \(N\) convolutions yields \[a_\pi^N=\int_{K_c}f_x(g) \frac{\chi_\pi(g)}{d_\pi}\,dk.\] Approximate \(f_x\) in \(L^1\) by a function \(f_{x,M}\in L^2\). Orthogonality of irreducible characters and Bessel’s inequality give \(\int f_{x,M}\chi_\pi\,dk\to0\) as \(\pi\) leaves finite subsets; the conjugate characters form the same orthonormal family. Moreover, \[\left|\int(f_x-f_{x,M})\frac{\chi_\pi}{d_\pi}\,dk\right| \leq\|f_x-f_{x,M}\|_1,\] because \(|\chi_\pi|\leq d_\pi\). First letting \(\pi\) leave finite subsets and then letting \(M\) tend to infinity proves \(a_\pi^N\to0\), and therefore \(a_\pi\to0\).

For each simple ideal choose a nonzero eigenvalue \(i\beta\) of \(\operatorname{ad}A\) on that ideal. Such an eigenvalue exists because the ideal has zero center and the projection of \(A\) is nonzero. If \(\exp(tA)\) acts centrally on that ideal, then \(t\beta\in2\pi\mathbb Z\). The union of these exceptional parameter sets, for \(0\leq t\leq T\), has measure zero. For every other \(t\), the preceding normalized-character decay applies. Averaging a representation over the periodic subgroup projects onto \(\ker d\pi(A)\), so \[\frac{\dim\ker d\pi(A)}{d_\pi} =\frac1T\int_0^T \frac{\chi_\pi(\exp(tA))}{d_\pi}\,dt.\] The integrands are bounded in absolute value by one. Dominated convergence proves the assertion along every sequence leaving finite subsets, hence on the entire irreducible dual. The argument uses all simple ideals at once and therefore also applies when \(K_c\) has several simple factors. ◻

Proof. We use a positive Hilbert norm invariant under a compact real form of the complexified group. In each standard realization, the split generator is Hermitian for this norm. Write an invariant Hermitian form as \(q(v,w)=\langle Qv,w\rangle\), using consistent conventions for the two Hermitian products. Its invariance under \(\exp(tH_R)\) implies \[d\pi(H_R)^*Q+Qd\pi(H_R)=0.\] When the split generator is Hermitian, this says that \(Q\) takes \(V_t\) to \(V_{-t}\). Nondegeneracy therefore pairs opposite weight spaces perfectly and all other pairs trivially. Each pair \(V_t\oplus V_{-t}\) for \(t>0\) has signature \((\dim V_t,\dim V_t)\). It follows that if the negative index of \(q\) is \(h(V)\), its restriction to \(V_0\) is positive definite. We verify that index and the normalization \(Q^2=I\) in all the cases.

For the orthogonal standard module of dimension \(N=n+1\), take the Hermitian extension of the real quadratic form of signature \((N-1,1)\) and the Euclidean compact-form norm. Its symmetry is \(Q=\operatorname{diag}(1,\ldots,1,-1)\). A split generator interchanges one positive basis vector and the negative basis vector, with zero action on their orthogonal complement. Its eigenvalues are \(1,-1\) and \(N-2\) zeros. Thus \(h=1\) is the negative index and the zero space is positive and nonzero. This includes the orthogonal low-rank names used above.

For the unitary standard module of dimension \(k\), use the Hermitian form of signature \((k-1,1)\) and its usual positive compact-form norm. The same two-dimensional boost gives split weights \(1,-1\), and zero with multiplicity \(k-2\). Again the negative index is \(h=1\). On the dual, use the inverse Hermitian form and the dual Hilbert norm; its matrix is the transpose inverse of \(Q\), still a self-adjoint involution, with the same signature and opposite split weights.

For \(SU(2,1)\), the induced form on \(\operatorname{Sym}^2E\) has positive index \(\dim\operatorname{Sym}^2\mathbb C^2+1=4\) and negative index two: the negative vectors are the products of one vector from the positive two-plane with the negative basis vector. The induced symmetry is the restriction of \(Q\otimes Q\) to the symmetric tensors, and hence is a self-adjoint involution for the induced compact-form norm. The induced split generator is Hermitian. Its split spectrum is \(2,1,0,0,-1,-2\), so \(h=2\) equals the negative index and the two-dimensional zero space is positive. The dual argument gives the identical conclusions for \(\operatorname{Sym}^2E^*\).

For the standard \(Sp(n,1)\) module, of complex dimension \(2k=2n+2\), group the orthonormal basis into symplectic pairs \(e_i,f_i\). Give each of the first \(k-1\) pairs sign \(+1\) and the last pair sign \(-1\). The preserved Hermitian form has signature \((2k-2,2)\), its symmetry is diagonal with these signs, and a split generator couples one positive pair to the negative pair. It has weights \(1,-1\), each twice, and zero with multiplicity \(2k-4\). Thus its negative index is \(h=2\) and its zero space is positive. These are integral weights.

It remains to examine the primitive cube for \(Sp(2,1)\). Let \(E=\mathbb C^6\) with compact-orthonormal basis \(e_i,f_i\), \(1\leq i\leq3\), and let \[q(e_i,e_j)=q(f_i,f_j)=s_i\delta_{ij}, \qquad (s_1,s_2,s_3)=(1,1,-1), \qquad \omega(e_i,f_j)=\delta_{ij}.\] All other pairings between distinct displayed basis vectors vanish, with \(\omega\) extended skew-symmetrically. Write \(C e_i=s_i e_i\) and \(C f_i=s_i f_i\). The Hermitian form \(q_3\) on \(\bigwedge^3E\) has signature \[\left(\binom43+4\binom22,\ 2\binom42\right)=(8,12).\] The first number counts wedges with zero or two negative basis vectors, and the second counts wedges with one. Put \(\Omega=\sum_i e_i\wedge f_i\) and \(L(v)=\Omega\wedge v\). For each basis vector, \(L(v)\) contains the two complete pairs of the other indices. Each term has signed squared norm equal to that of \(v\), since the squared sign of a complete pair is one. The wedge basis supports for distinct unpaired basis vectors are disjoint. Consequently \[q_3(Lv,Lw)=2q(v,w).\] In particular \(L\) is injective and its image has signature \((4,2)\).

Let \(c:\bigwedge^3E\to E\) be symplectic contraction, normalized as the adjoint of \(L\) for the positive compact-form norm. Explicitly it contracts each complete \(e_i,f_i\) pair with its wedge sign; equivalently, \[c(u\wedge v\wedge w) =\omega(u,v)w-\omega(u,w)v+\omega(v,w)u.\] This formula shows equivariance under the complex symplectic group, and a basis calculation gives \(cL=2I\). Since \((\bigwedge^2C)\Omega=\Omega\), one has \[(\bigwedge^3C)L=LC.\] It follows both that \(L(E)\) reduces the self-adjoint involution \(\bigwedge^3C\) and that \(q_3(Lv,w)=q(v,cw)\). Therefore \[P=\bigwedge^3_0E=\ker c =L(E)^{\perp_{q_3}}\] is also the compact-form orthogonal complement of \(L(E)\). Its signature for \(q_3\) is \((4,10)\). On \(P\) choose instead \(q_P=-q_3|_P\), whose signature is \((10,4)\) and whose symmetry is \(-(\bigwedge^3C)|_P\), a self-adjoint involution. The compact-form orthogonal decomposition is invariant under the compact real form, hence under its complexified Lie algebra; the split generator remains Hermitian on \(P\).

The weight calculation in Lemma 99 gives the primitive split spectrum \[\begin{array}{c|rrrrr} t&2&1&0&-1&-2\\ \hline \dim P_t&2&2&6&2&2. \end{array}\] Thus \(h(P)=4\) is exactly the negative index of \(q_P\); the general opposite-pair argument already proves positivity on the six-dimensional \(P_0\). One can also see its sign directly. Choose the split generator \[H_Re_2=e_3,\quad H_Re_3=e_2,\qquad H_Rf_2=-f_3,\quad H_Rf_3=-f_2,\qquad H_Re_1=H_Rf_1=0.\] It is Hermitian for the compact norm, preserves \(\omega\) infinitesimally, and anticommutes with \(C\). Define \[a_1=(e_2+e_3)/\sqrt2,\quad a_2=(f_2-f_3)/\sqrt2,\qquad b_1=(e_2-e_3)/\sqrt2,\quad b_2=(f_2+f_3)/\sqrt2.\] The \(a_i\) have split weight one, the \(b_i\) have split weight minus one, and \(q(a_i,b_j)=\delta_{ij}\) while each of their two spans is isotropic. For \(z=e_1,f_1\) put \(w_{ij}(z)=a_i\wedge z\wedge b_j\). These eight vectors span the zero space in the full cube and satisfy \[q_3(w_{ij}(z),w_{kl}(z')) =-\delta_{il}\delta_{jk}\delta_{zz'}.\] For each \(z\), the positive direction is \(w_{12}-w_{21}\); the three negative directions are \(w_{11},w_{22}\) and \(w_{12}+w_{21}\). Moreover, \[\Omega=e_1\wedge f_1+a_1\wedge b_2-a_2\wedge b_1, \qquad L(z)=-(w_{12}(z)-w_{21}(z)).\] Removing \(L(E)_0\) therefore removes exactly the two positive directions, leaving six negative directions for \(q_3\), as asserted. Negating the form makes all of them positive.

In every case all listed weights are integral and the zero spaces just computed are nonempty. The statements about annihilators now follow from the perfect opposite-weight pairings and the positive zero space. To verify the parabolic assertion, use a positive multiple of \(H_R\) that is an integral cocharacter of the simply connected complexified group. A root space whose root is nonnegative on \(H_R\) maps a weight space \(V_t\) into \(V_{t+\alpha(H_R)}\) and therefore preserves both \(V_{>0}\) and \(V_{\geq0}\). The corresponding algebraic parabolic, generated by this Cartan and these root groups, preserves both spaces. This is the complex parabolic attached to the oriented rank-one boundary point. Its translates are algebraic subspaces in the appropriate Grassmannians, hence give the asserted parabolic stages independently of the choice of opposite point. Their middle quotient identifies with \(V_0\) and inherits its positive form. ◻

Proof of Lemma 86. The bounds and their complete equality lists follow from Lemmas 97, 98, and 99; the form and integrality claims follow from Lemma 101. The displayed spectra give \(d_R=2\) in the two stated exceptional cases and \(d_R=1\) in the others.

It remains to obtain one uniform gap for all other irreducibles. Let \(K_c\) be the simply connected compact real form of the complexified Lie algebra and set \(A=iH_R\) in that compact real form. In the coordinate realizations used above, \(A\) has a period after passing to the full covering-group weight lattice. Its projection to every simple ideal is nonzero. This is immediate in the simple cases; for \(SO(3,1)\) the two \(\mathfrak{sl}_2\) components of the displayed split functional are both nonzero. Lemma 100 consequently gives \(z(V)/\dim V\to0\) as the highest weight leaves finite subsets. The split functional and its negative are Weyl conjugate in each of these realizations, so \[h(V)=\frac{\dim V-z(V)}2, \qquad \frac{\dim V}{h(V)} =\frac2{1-z(V)/\dim V}\longrightarrow2.\] Every tabulated \(c_R\) is greater than two. Choose \(\tau\) with \(2<\tau<c_R\). Only finitely many irreducibles have ratio at least \(\tau\). After the maximizing modules are removed, the ratios of this finite set are all strictly smaller than \(c_R\) by the equality classification. The maximum of those remaining ratios and \(\tau\) is therefore a number \(b_R<c_R\). Taking \(\varepsilon_R=c_R-b_R\) proves the uniform gap. Finally, reversing \(H_R\) interchanges positive and negative spaces, preserves all ratios, and preserves the largest absolute weight. Thus \(c_R\) and \(d_R\) are unchanged at the opposite orientation. ◻

Proof. Use the actual dominant translation vector of the centralizer, as fixed in Section 3; it records the exact module of each centralizer character. The remaining centralizer parts and their inverses are uniformly bounded on the weight spaces. Highest-weight strings separate every other block from the highest line by a positive multiple of the pertinent gap. After division by the exact highest-line growth those blocks therefore vanish. The compact factors are isometries for the chosen norms.

In compact frames the limit is the rank-one operator \(\widetilde u\otimes\widetilde w\), with unit lifts of the extremal lines, up to a scalar of absolute value one. Evaluating it on the line at \(v\) gives its normalized pairing with \(w\), proving (150). Multiplying on either side transforms the relevant extremal line; the norm needed to renormalize that line is exactly its \(j\)-factor. At a finite place one can alternatively evaluate on a compact transverse test line and divide by its reverse pairing. Once the gap is large, the ultrametric inequality makes the resulting norm identity exact. All denominators have a uniform lower bound on the buffered patches, so the locally constant tests have uniform levels there. In particular there is no additional bounded but nonvanishing Cartan error in (150).

For the Jacobian assertion, reduce by compact frames to a minimal parabolic element. Its action on the opposite-root chart has determinant given by the product of positive-root modules, including double roots and their multiplicities. This is \(\exp(2\rho)\) and gives the inverse compact-angular Jacobian. At an isolated real vertex the horizontal dilation gives the same Jacobian by its tangent grading. Thus its \(j\) is exactly the cocycle of the height already constructed in Section 8; equality of the homogeneous dimensions in Proposition 84 yields \[\lambda_i^S(e_i)=\lambda_{p(i)}^T(e_{p(i)}) \qquad\text{at isolated positions}.\] ◻

Proof. For a length-increasing substitution \(p\mapsto up\), normalized matrix multiplication gives, componentwise, \[\ell(up)=\ell(u)+\ell(p)+q(S_u,S_p).\] Here \(S\) denotes projectively normalized matrix coordinates, and \(q\) has a smooth extension with uniform derivative bounds on the compact coordinate boxes. Its denominator is bounded below by transversality. For inverse norms use \((up)^{-1}=p^{-1}u^{-1}\), so the normalized product has the reversed order. At finite places \(q\) is bounded and locally constant with a uniform level on the same kind of buffered normalized boxes, by Lemma 102 and the local-field conventions.

Fourier inversion gives \[\begin{align*} \varphi\!\left(\frac{\ell(up)-b}{R}\right) =\int_{\mathbb R^q}\widehat\varphi(\xi) &\exp\!\left(\frac{\mathrm i\xi\cdot(\ell(u)-b)}{R}\right) \exp\!\left(\frac{\mathrm i\xi\cdot\ell(p)}{R}\right)\\ &\hspace{8mm}\cdot \exp\!\left(\frac{\mathrm i\xi\cdot q(S_u,S_p)}{R}\right)\,d\xi, \end{align*}\] with the Fourier normalization absorbed in \(\widehat\varphi\). The first factor is a scalar of modulus one; it contains the unrestricted center \(b\). The second is a bounded unary multiplier on the shorter argument. Its logarithmic argument is not restricted to a compact set and is not differentiated as a rescaled compact coordinate. Only the third factor is expanded in the compact normalized coordinates.

For each fixed derivative order \(r\), that compact factor has seminorm at most \(C_r(1+|\xi|)^r\), uniformly in \(R\geq1\), the center, and the supported shifts. Smooth separated expansions on a fixed larger box therefore have a summable supremum bound of that polynomial size. In a finite-place slot one uses the fixed finite partition into clopen boxes instead. Apply the unary Gram–Plancherel estimate (23) at each frequency, retaining any designated shorter cutoff outside the expansion. Minkowski’s inequality and \[\int_{\mathbb R^q}|\widehat\varphi(\xi)|(1+|\xi|)^r\,d\xi<\infty\] give the required uniform bound. The inequality holds because \(\widehat\varphi\) is Schwartz. A bounded additional character frequency replaces \(\xi/R\) by \(\theta+\xi/R\) in the compact factor, with the same polynomial estimate. Finitely many cutoffs use joint frequency variables and products of Schwartz transforms.

The compact expansion uses a common unary basis with coefficient bound \(\sum_n\sup_u|c_n(u)|<\infty\), as in (25); a bound with the supremum and sum in the opposite order would not suffice. The frequency integral above retains exactly this stronger bound.

The argument uses only scalar unary masks and the coefficient-valued Gram estimate. It thus preserves the normalized-trace, fixed-matrix-size, and homogeneous Hilbert bounds of that estimate. In the bounded-overlap case, separate bounded radial and phase masks are already permitted directly. Finally, ordinary finite-support exhaustion is obtained by unshifted cutoffs in a proper finite list of logarithmic norms, equal to one on increasing Cartan balls. Their supports meet the discrete lattice in finitely many points. Their radii may increase as fast as a given test or approximation requires without changing the bound. ◻

Proof. At fixed path parameters, move the start power of frequency \(\theta_n\) through \(X_a\). Since \(a^{-1}\) has split vector \(L\), its source covariance produces \(\exp(-\mathrm i\theta_n\lambda_i^S(L))\). To read coefficient \((h,r)\), pair against \(\zeta(ay)w_{(h,r)}\), where \(r\) comprises the unmeasured indices. Moving the end action through this right unitary changes \(\zeta\) to \[b^{-\mathrm i\theta_n} j_T(P_{x_i},h^{-1})^{-\mathrm i\theta_n}\zeta.\] Conjugation in the first input gives exactly the signs of (151). The remaining group, base, and coefficient multipliers commute with the power.

On this moved side take a bounded start column before any movement. The end column may be Hilbert: the finite-time bilinear estimate with one bounded input is uniform. Lemma 102 and uniform separated expansions replace the target factor by the pairing phase on transverse patches. Imaginary powers at \(\theta_n\) converge in Hilbert norm on each fixed column to the power at \(\theta\).

We justify the original side without asserting a product estimate for two arbitrary Hilbert columns. Construct the end vector as a cut one-ended effect applied to a bounded column \(\zeta_0\). Its defining auxiliary path is a pure vertex path at \(x_i\) in the isolated case, with smooth angular cut and compact complementary thickness. In the later tuple case it is a mixed path at \(F\) with bounded centered widths. Choose its image patch buffered transverse to the reverse main window. By sharpness the ranges of such vectors exhaust the corresponding angular patches.

First retain finitely many measured indices. Test the main link against bounded-column arrays, with smooth output tests at fixed thickness and sufficiently large prescribed gaps; allow powers of the retained phase as well. At a fixed main time the cut end vector can be replaced by its defining auxiliary weak limit. This remains valid when the other input is the Hilbert error from replacing a height-applied start vector by a bounded approximation: at finite measured support the remaining coefficients are bounded operators, and normality permits bounded tests in the full remaining tracial algebra.

More concretely, let \(f\) be bounded and let \(f_l\) be a bounded-column approximation to \(b^{\mathrm i\theta}f\). After trace rearrangement the start error is paired against a product of the form \[(D_{\mathrm{aux}}X_s(ay)\zeta_0(ay)) (D_{\mathrm{main}}T),\] where \(T\) is the bounded-column testing array. Its intermediate directions are \(q^{-1}o\) and \(ho\). Transversality to the reverse tuple of a full chamber suffices for a tuple patch by opposite-face projection. In the tuple case the overlap widths are bounded; in the isolated case the auxiliary support is a single smooth vertex cone, with compact cuts in the other factors. Hence (22) and Lemma 103 bound this product uniformly. The error is at most that fixed bound times \[\|b^{\mathrm i\theta_n}f-f_l\|_2.\] Common projective transport phases disappear from this tracial estimate. First choose \(l\) and then pass to the parameter limit; the error tends to zero as \(l\to\infty\).

The finite-support tests can use the smooth radius cutoffs of Lemma 103. A phase may be computed from the logarithmic Hilbert norm of the weight matrix: at a divergent pertinent gap the difference from the logarithmic top norm tends to zero by rank one. At a finite place the top growth is computed exactly on a compact transverse input line for all sufficiently large gaps, after division by the bounded locally constant reverse pairing. Thus all phase/radius tests retain the uniform estimate before supports are removed.

At each main time the radii may be made sufficiently large for the moved-side expansions and the bounded start approximant. Their product error bound is independent of those radii. After the bounded start approximation, the end vector may remain Hilbert by the one-bounded-input estimate. Exhaust auxiliary thickness and transverse patches, then the main stratum using ambient continuous tests with slack and a final Borel output projection. Testing in the spectral span generated by these compact variables and the retained phase proves the vector equality. Additional phases not used in this equality need not be included in that span. A second measured leg can remain throughout in the coefficient algebra. This proves (151) with the stated order and domains. ◻

Proof. The closed range sum is target invariant. It is also fixed by an all-factor escaping source subgroup fixing the relevant labels and normalizing the opposite radical moves. Its projection is therefore scalar by the link invariance argument. It is nonzero: simultaneous source transport to generic pairs, followed by patch fullness, supplies a nonzero ordered rectangle. This proves totality. The tuple assertion follows by the same argument on the open opposite-tuple orbit.

For the isolated height put \(\theta_n\to0\) in (151). Pull the sharp reverse angular variable through \(J\). The pairing phase and both imaginary powers tend to one on each buffered rectangle. Hence the retained phase fixes \(J\) on its total range sum and therefore everywhere. If tightness failed, some fixed bounded-column input would retain a positive mass beyond thresholds tending sufficiently slowly to infinity. Averaging \(|e^{\mathrm it r}-1|^2\) over a shrinking interval of \(t\) gives, by the elementary sinc formula, a uniformly positive value on that escaped mass. Choose a frequency in each such interval. These frequencies tend to zero and contradict the just-proved equality. Thresholds and testing vectors can be selected by countable exhaustion along the successive path filters. The same proof permits offsets held after all ordinary times, so the centered isolated widths are tight on those iterated paths too. ◻

Proof. Unfold, for fixed \(y\), the identity \(a d_0(y)=d_0(y')\gamma\). At fixed \(h\) the integrated coefficient sum is, up to the fixed normalization, the sum of \[\big|\langle W_{h,r}^*b_0(y'),\pi_0(\gamma)f_l\rangle\big|^2\] over \(\gamma,r,l\), at the independent base variables \(y,y'\). Move \(W_r\) to the complementary frame index. Complementary Parseval and commutation up to scalar phases make the whole sum equal to one squared norm of \(b_0(y')\). Integration proves the bound. In particular no extra factor comes from summing the unmeasured group indices.

A bounded set of compact rotations and permissible split offsets gives a compact family of translated inputs in Hilbert norm, preserving the operator bound of the bounded column. Approximate the Hilbert column first by bounded ones, and then both translated families by finite compact nets. The one-bounded-input probability estimate controls the errors uniformly in path time. Exact translations therefore permit passage under these bounded parameter averages. Apply the argument successively for the different path times; no exchange with an arbitrary nonuniform ultralimit is needed. ◻

Proof. We first prove the high-coordinate upper bounds. Compare a tuple path \(a\) with a faster opposite many-panel lexicographic face path \(d\). All the parameters of \(d\) are taken to their limits before those of \(a\). The path \(d\) omits exactly the complementary selected high types \(i^*\) and retains full isolated directions. Thus \(da^{-1}\) has split vector \(L+L_{\mathrm{panel}}\), with the panel part faster. Cut the two columns to transverse start angular patches and compact complementary thickness. Also impose bounded centered isolated widths on the tuple column, using Lemma 105, and test a high excess \[\alpha_{p(i)}(\mu(h^{-1}))>s_i t_i+M.\] Trace cyclicity pairs the cut \(X_d\) column times the adjoint of the cut \(X_a\) column against \(X_dX_a^*\), which, after base change and a harmless common phase, is the \(da^{-1}\) column. The product has a uniform Hilbert bound by the bounded-overlap case of (22). Transverse Cartan addition forces excess greater than \(M-C\) in the pertinent residual quotient gap. Apply the local upper-speed estimate, (53) or (137), to the inverse path in the opposite orientation. Its missing-type bound is exactly the needed bound on the ordinary index product. Cauchy–Schwarz therefore makes this pairing tend to zero as \(M\to\infty\).

Exhaust continuity cuts, complementary thickness, and angular restrictions. If positive infinite excess remained, its one-ended angular sublaw would be a positive scalar times \(P_F\). Its ordered products with all transverse opposite many-panel patches could not vanish: simultaneous conjugation of the source tuple and opposite subfaces ranges over their open orbit, and fullness supplies nonzero products on conull such choices. The scalar mass is unchanged by these fixed conjugations, which only cost bounded shifts or tight perturbations. This contradiction proves all high upper bounds. The isolated ones already follow from centered isolated tightness. Both local upper-speed estimates use the total shifted parameters, so this reasoning also applies with the permitted further offsets.

Use the frame and bounded end column of Lemma 106. Around \(L\) integrate over a fixed thick shell, with independent compact rotations and offsets in a fixed box whose complementary coordinates are positive. At a real wall take an interior subbox; the Cartan density then gives the lower bound \[c\exp(2\rho_S(L)).\] At a finite place integrate over the disjoint selected Cartan double cosets in the permissible mesh. Each is bi-compact averaging times a constant comparable to \(\exp(2\rho_S(L+\eta))\), by the root-coordinate index count. The same lower estimate follows. The mixed link isometricity, the just proved upper bounds, and complementary thickness retain a positive mass after sufficiently large fixed cuts. This can be done uniformly over the bounded translation parameters by the compact-net argument of the lemma.

Disjoint fixed neighborhood thickening of lattice points and Cartan integration bound the target count in the corresponding windows by \[C\exp\!\left(\sum_{i\in I}t_i\,2\rho_T(s_i e_{p(i)})\right).\] At finite places include all Cartan entries in the window enlarged by a fixed constant; their multiplicity per unit mesh box is bounded. The coefficient integral bound gives the comparison of these exponentials. Make \(i\) the fastest coordinate to obtain \[2\rho_S(e_i)\leq2\rho_T(s_i e_{p(i)}).\] The lists are two copies of the same type list, and the speed ratios telescope around each permutation cycle. Multiplication around that cycle gives equality at every position.

Suppose next that a selected coordinate has positive mass of infinite deficit from its calibrated center. This sublaw again has scalar mass on the mixed link. Fix complementary and other upper widths large enough to retain a mass \(\varepsilon>0\), and then impose deficit greater than \(M\). The same lower shell estimate and the coefficient integral bound give \[\varepsilon c\exp(2\rho_S(L)) \le C\exp(2\rho_S(L)) \exp\bigl(-M\,2\rho_T(e_{p(i)})\bigr).\] Here the cut widths are fixed before \(M\) is exhausted, so \(c,C>0\) do not depend on \(M\). The final factor is the gain in the target packing bound. In simple coordinates every coefficient \(2\rho_T(e_j)\) is positive, and integration up to each upper bound costs a constant times its endpoint exponential. There is consequently no polynomial length loss. The displayed inequality is impossible for large \(M\), proving tightness of every deficit.

The same argument applies to any purported positive mass in a subsequent offset limit: offsets are fixed during all ordinary path times and are exhausted only afterwards. Finally, \(p\) preserves the Coxeter components by Section 5, and the tuple selects only one type in each component. Evaluating \(\lambda_{p(i)}^T\) in (152), using \(\lambda_{p(i)}^T(e_{p(i)})=2\rho_T(e_{p(i)})\), therefore proves tightness of each \(r_i\). This evaluation requires no cross-weight Cartan identity. ◻

Proof. The compressed characteristic operators of the tight variable \(r_i\) commute with the sharp tuple coordinates. They lie in \(\overline{\mathcal A_j}\bar\otimes\mathcal A_j\), have independent contracted invariances, and transform under independent target shifts by the endpoint line phases of Lemma 102. At a given end and frequency \(t\), a bounded solution \(B_t\), commuting with the pertinent sharp vertex, has the tilted law \[\beta_h B_t=B_tj_T(P_{x_i},h^{-1})^{\mathrm it}.\] For two solutions of the same frequency, both orders of their products with adjoints have untilted target laws and the contracted invariances. They are in \(\mathcal N\) and scalar. A nonzero solution is consequently a scalar multiple of a unitary, and the solution space has dimension at most one.

Normal slices of the characteristic operators give locally nonzero solutions near zero, strongly continuous together with their adjoints. Here tightness supplies strong convergence of the characteristic operators at zero. Normalize the nonzero slices to unitaries. Products of these local solutions give the corresponding solution line at every frequency, and the lines form a continuous projective one-parameter unitary group. They commute with \(P_F\) because the original compressed operators do.

The group consisting of those unitary solution lines has local continuous sections over \(\mathbb R\) and circle fibers, hence is locally a two-dimensional Euclidean group. The locally Euclidean groups theorem makes it a Lie group (Gleason 1952; Montgomery and Zippin 1952). Its Lie-algebra extension over the one-dimensional abelian Lie algebra splits; integrating that section gives a continuous splitting over the simply connected real line. Stone’s theorem (Stone 1932, Theorem B) now writes the resulting group as \(b_{F,i}^{\mathrm it}\) for the exponential of a selfadjoint affiliated operator. Thus \(b_{F,i}\) is positive and injective. One-dimensionality of both end solution lines gives (154). After the genuine unitary groups are removed, its scalar coefficient is continuous and positive definite: in the operator-valued positive-definite matrix tests absorb the inverse unitary group factors into the testing vectors. Its value at zero is one.

We determine the source covariance rather than assuming its character. The source parabolic stabilizer of \(F\) normalizes these intrinsic solution laws and therefore acts on the height by a continuous positive character. Replace \(L\) by \(L+mA_0\) along a selected split-center direction, with the integer \(m\) held after all ordinary path times. Before recentering, exact precomposition translates the characteristic law by \(m\log(\operatorname{char}(A_0))\) in the start slot. Recentering subtracts \(m\lambda_i^S(A_0)\). Proposition 107 gives tightness even when these subsequent multiples are exhausted. A nonzero residual drift would contradict that tightness on each unit input. Hence the two characters agree on every selected split direction. Those directions span the split center of the tuple parabolic. A real logarithmic character vanishes on its radical, compact parts, and derived Levi factors, as explained above; the finite-place mesh determines it there as well. Compact transport now gives exactly (153) at all labels.

At an isolated position retain the previously constructed \(b_{x_i}\). It solves the same one-dimensional laws, so its powers agree up to scalars with the tuple solution and commute with \(P_F\). The reverse-end heights solve the same intrinsic laws at opposite positions. All constructions use countable determining labels and tests before taking Borel output cuts; strong continuity then supplies every fixed label. The normal-copy argument of Section 4, together with uniqueness of the solution lines, therefore permits these one-ended heights to be copied to the two-measured-leg system. Exact end-input identities are copied normally too. Later commutation comparisons use one common all-gap path in that system, rather than unrelated marginal limits. ◻

Proof. Apply (151) to a tuple path at constant frequency, and compress using (154). Cancel the independent start power and pull the sharp reverse tuple through \(J\). At \(Y=F^-\) this gives the left side of (155), multiplied by \(\chi(\theta)\), equal to the unweighted ordered product.

This equality can be compared at both signs despite the ordering. Each height power commutes with its own spectral slot. Multiply outside the left and right slots by their inverse powers and insert the nonvanishing smooth pairing phase on the transverse patch. Applying the opposite-sign identity gives \[\chi(\theta)\chi(-\theta)\,P_Y(dw)P_F(dv) =P_Y(dw)P_F(dv)\] on such patches. Nonzero ordered rectangles exist by Lemma 105; hence \(|\chi(\theta)|=1\). Since \(\chi\) is a probability characteristic function, it is the characteristic function of a point mass. Its scalar is a positive constant raised to the imaginary power. Simultaneous source transport and the pairing covariance give \(C(Y,F)=c k_S(y_{i^*},x_i)\).

For clarity about subsequent substitutions, write \(P,Q\) for the reverse and same-label tuple slots on a conjugate end input, and \(A_t,B_t\) for the corresponding conjugated height powers. The compressed equation reads \[\chi(t)A_t P B_t Q\,k_T^{\mathrm it}=PQ.\] Thus the substitution used on a full path is \(P B_tQ\,k_T^{\mathrm it}=C^{\mathrm it}A_t^{-1}PQ\), in exactly that order. The relevant total range sum may be multiplied on the left by the inverse reverse power and remains total. Smooth multipliers and separated expansions on the buffered patches justify every localized identity. The isolated case follows from the first version of (151) in the same way. ◻

Proof. Insert (155) into (151) on a full path. Pulling back the sharp reverse tuple first produces the ordered conjugate-end expression \(\bar P_{F^-}(dw)\bar b_{F,i}^{\mathrm i\theta}\bar P_F(dv)\). The ordered substitution in the proof of Proposition 109 gives (156) on transverse ordered end ranges acted on by the inverse reverse power. Those ranges are total. Letting frequencies tend to zero and applying the shrinking-frequency argument proves tightness on the full path too.

Apply this for every type using any tuples in the chamber. The determinant weights form bases, so the first equation in (157) determines an invertible \(A_*\). To identify its column at \(e_i\), choose a tuple prefix with \(i\) fastest, add its other selected types at successively slower scales, and then add the remaining chamber types one at a time. Dividing the centered law by that fastest time and using (152), with the subsequent translations handled by exact precomposition, gives \(A_*e_i=s_i e_{p(i)}\). The last equality follows from the calibrated \(2\rho\) values in (152). The output Cartan phases commute; (156) pulls their sharp law through the isometry, giving the asserted joint law of \(\Delta\). ◻

Proof. The weighted packing estimate. Count inverse lattice indices in a fixed-width box around \(A_*L\) whose image chamber lies in a continuity patch \(O\subset B_T\). Pack with fixed disjoint right neighborhoods of those inverse indices. In all-gap windows, right thickening perturbs the image chamber by \(o(1)\) uniformly, by highest-line convergence, and changes the Cartan vector by a bounded amount. Cartan integration therefore gives \[\limsup\frac{\#\{\text{such indices}\}}{\exp(2\rho_S(L))} \leq C|O|.\] One may first enlarge angular patches by fixed slack and then shrink it by compact regular approximation. The finite-place count includes all double cosets in a fixed enlargement of the box, with bounded multiplicity per mesh box, as in Proposition 107.

Integrate both compact rotations and split shifts \(\eta\) in a bounded box, using the frame inputs \(\bar b_0\otimes f_l\) of Lemma 106. The coefficient bound and the Cartan density, normalized by \(\exp(2\rho_S(L))\), show that a constant times \(|O|\) bounds the weighted integral of limiting probabilities of \[\Delta+A_*\eta\ \text{in the fixed box}, \qquad\text{image chamber in }O,\] with weight \(\exp(2\rho_S(\eta))\). Begin with smaller interiors and continuous tests. Exact input translations and the compact-net argument permit the bounded parameter integral before each limit. Equation (156) identifies these translated tests with the same sharp law, shifted by \(A_*\eta\) about the old center; compact rotations introduce no extra line scalings.

Now exhaust the whole split space, using a sum on finite-place meshes and ordinary integration on real coordinates. Choose the fixed counting box sufficiently wide. For every \(\Delta\), the set of permissible offsets for which \(\Delta+A_*\eta\) lies in a smaller box has uniformly positive sum-volume near \(-A_*^{-1}\Delta\). Since \(A_*\) is invertible and \(2\rho_TA_*=2\rho_S\), this yields \[\int_{\{\eta:\,\Delta+A_*\eta\ \text{in the box}\}} e^{2\rho_S(\eta)}\,d\eta \geq c e^{-2\rho_T(\Delta)}.\] Here and below this notation includes the indicated mesh sums. A full lattice has a bounded covering radius, which proves the assertion at finite places; the real subbox has a fixed positive volume. Only bounded offset windows are used before a path limit, so all chambers remain deep in the prescribed iterated order.

Positive integration thus bounds the expectation of \(e^{-2\rho_T(\Delta)}\) with the angular cut. By (158), independence of the two compact rotations, and the product input \(\bar b_0\otimes f_l\), that positive form factors into an opposite factor and a forward factor, up to a fixed constant. No exponential moments have been presumed. Perform this step with positive spectral truncations first; any strictly positive bounded truncation of the averaged opposite factor is enough to divide. It follows that \[ \sum_l\int_{B_S} \langle f_l,D_Z^{-1}P_Z(O)f_l\rangle\,dZ \leq C'|O|. \tag{160}\] Regular approximation extends the bound from continuity patches to all positive measurable tests.

The positive density map. For bounded \(f\geq0\) define the extended positive form \[T(f)=\int_{B_S}D_Z^{-1}P_Z(f)\,dZ.\] The height commutes with its angular slot, so this integral is positive. Its source covariance and the chamber Jacobian place it in the extended positive part of \((\mathcal A_j)^\sigma\). Indeed the inverse line factor is canceled by the change of angular probability under the source transport. After undoing induction a \(\sigma\)-fixed field is constant with value in the complementary commutant. Evaluation on the finite Parseval frame, summed over \(l\), is precisely its faithful finite trace. Thus (160) makes \(T\) a bounded positive map \[L^1(B_T)\longrightarrow L^1((\mathcal A_j)^\sigma).\] It intertwines transport of chamber densities with the trace-preserving target action. This also justifies the claimed \(L^1\) values of the unbounded positive form integral.

Why this map is scalar. Induce the finite algebra \((\mathcal A_j)^\sigma\) along the target lattice inclusion, with its induced finite trace. If \(\mathcal D_g\) denotes transport of chamber densities, the induced equivariant map is represented by fields \[g\longmapsto T(\mathcal D_{g^{-1}}f).\] Their equivariance under the right lattice action follows from that of \(T\). They are integrable over the quotient: \(\mathcal D_{g^{-1}}\) is a positive \(L^1\) isometry, and the norm of \(T\) gives a uniform bound on their \(L^1\) norms. Let \(h_0\) be the induced image of the constant density \(1\), and let \(\alpha\) be the induced group action.

For every target group element \(s\), averaging \(\mathcal D_k\mathcal D_s1\) over the special compact group gives \(1\), because that compact group is transitive on chambers and the density has total mass one. Equivariance therefore gives \[h_0=\int_K\alpha_k\alpha_s(h_0)\,dk.\] Every integrand has the same spectral distribution as \(h_0\), since the induced action preserves the finite trace. For a threshold \(t\) which is not an atom of that distribution, the projection \(p_t=1_{(t,\infty)}(h_0)\) uniquely maximizes \(\tau(p(h_0-t))\) among projections. The maximum is the same for each integrand. Testing the displayed average against \(p_t\) gives equality in the average of these upper bounds, so \(p_t\) is the maximizing spectral projection for almost every integrand. Take a countable dense set of atom-free thresholds; their spectral projections determine the whole positive \(L^1\) element. Hence \(\alpha_k\alpha_s(h_0)=h_0\) for almost every \(k\). Strong \(L^1\) continuity extends this to all \(k\), and in particular \(\alpha_s(h_0)=h_0\).

The span of the group transports of \(1\) is dense in \(L^1(B_T)\). To see this directly, take a regular split contraction. It pushes chamber probability to a point mass away from its null repelling set. Convolution of these concentrating densities with continuous compact-rotation weights approximates continuous chamber densities uniformly, by lifting to the compact group modulo the chamber stabilizer. Continuous densities are dense in \(L^1\). The induced map therefore depends only on total integral. Take a common conull section value for a countable dense set of density tests. Because density transport is an onto \(L^1\) isometry, this rank-one conclusion descends to the original map. Its value is fixed by both source and target actions, hence scalar. We obtain \[T(f)=c_0\int_{B_T} f(v)\,dv.\] The trace bound makes \(c_0\) finite, and positivity together with the injectivity of the heights makes it strictly positive.

Finally, for bounded angular tests, commutation of \(D_Z\) with \(P_Z\) gives the norm identity in (159) from this formula for \(T\). Polarization handles finite sums of elementary tensors, and completion gives the claimed isometry. Positive truncation and separability give the measurable \(L^2\) representative in that equation, without imposing an operator bound on \(D_Z^{-1/2}\). ◻

Proof. For any fixed spectral input value the nontransverse ordinary chamber set is null. Fubini therefore gives transversality on the full product of ordinary chamber probability and the input spectral measure. The powers in the two brackets have modulus one and commute with their own angular slots, proving unitarity.

Localize in the ordinary chamber variable and in the input \(P_F\) coordinates on buffered transverse rectangles; these give total domains. Before the factor \(D_Z^{-1/2}\) in \(V\), substitute at \(P_Z\) using (155). Chamber tests may be placed on the left because the \(Y(Z)\) height commutes with them. Use separated smooth expansions, or finite clopen partitions at finite places, for the pairing tests, and truncate the \(D_Z\) height to bounded intervals.

The scalar Weyl relations obtained from (156) show that commuting \(b_{Y(Z),i^*}^{-\mathrm i\theta}\) through \(D_Z^{-1/2}\) costs at most a positive scalar, while its height truncations are only dilated. This scalar is constant on the generic chamber orbit: simultaneous source transport preserves the scalar commutator, and the tuple choices inside a chamber have fixed type sets. Remove the truncations using the isometry (159). We obtain (161) initially up to that positive scalar. Both bracketed operators are unitary and \(V\) is an isometry, so comparison of norms forces the scalar to equal one. For every real frequency this makes the Weyl coefficient vanish; hence the corresponding unitary groups, and therefore the positive affiliated heights, commute strongly. The same argument proves (161) exactly on the completed domains. ◻

Proof. We spell out the finite-dimensional representation used here. For \(\operatorname{Sp}(n,1)\), use the irreducible representation of absolute highest weight \(e_1+e_2\) in the root system \(C_{n+1}\). Normalize the split generator so that its evaluations on the first two standard coordinates are both one. The highest restricted weight is then two, is attained on a single extremal weight, and has multiplicity one. The lowest restricted weight is minus two. For \(F_{4(-20)}\), use the representation of highest short root. In the customary coordinates with short roots \[\{\pm e_j\}\ \cup\ \left\{\tfrac12(\pm e_1\pm e_2\pm e_3\pm e_4)\right\},\] the split direction has \(e_1(H)=2\) and all other \(e_j(H)=0\). Again the extremal evaluations two and minus two are unique. The absolute vanishing root system is \(B_3\). These weight assertions also follow directly by maximizing the displayed linear functional on the Weyl orbit and its convex hull.

Here are also the determinant weights, which will identify the kernel exactly. In type \(C_{n+1}\) the roots of split value two are \(2e_1,2e_2,e_1+e_2\), and those of split value one are \(e_i\pm e_j\), \(i=1,2\), \(j=3,\ldots,n+1\). Thus \[(m_\alpha,m_{2\alpha},Q)=(4n-4,3,4n+2),\qquad \sum_{\beta(H)>0}\beta=(2n+1)(e_1+e_2).\] For \(F_{4(-20)}\) the roots of split value two are \(e_1\) and \(e_1\pm e_j\), \(j=2,3,4\), and those of split value one are \(\tfrac12(e_1\pm e_2\pm e_3\pm e_4)\). Hence \[(m_\alpha,m_{2\alpha},Q)=(8,7,22),\qquad \sum_{\beta(H)>0}\beta=11e_1.\] In either case the determinant of the positive nilradical has absolute highest weight \((Q/2)\lambda\), where \(\lambda\) is the highest weight chosen above. The quaternionic groups in the present class have \(n\ge2\); thus \(Q\ge10\) in that case and \(Q=22\) in the Cayley case.

We justify the required real representation. Let \(\kappa\) be split Chevalley conjugation on the complex Lie algebra and \(\sigma\) its given real-form conjugation. The types \(C_{n+1}\) and \(F_4\) have no diagram automorphisms, so the complex automorphism \(\sigma\kappa\) is inner, say \(\mathop{\mathrm{Ad}}(h)\). The highest-weight module has a Chevalley real form, with antilinear conjugation \(C\) intertwining \(\kappa\). The map \(J=\pi(h)C\) therefore commutes with the given real Lie algebra. Schur’s lemma gives \(J^2=a\mathop{\mathrm{id}}\). Since \(J\) preserves the unique complex top space for the real split generator, its square on that line is a positive real scalar. Thus \(a>0\), and rescaling \(J\) gives \(J^2=\mathop{\mathrm{id}}\). Its fixed space is the required real module \(E\). These highest weights lie in the root lattice, so the representations descend to the adjoint groups used here.

The longest Weyl element is \(-1\) in both types. Self-duality therefore gives a nondegenerate invariant complex bilinear form. Its conjugate by \(J\) is a scalar multiple of itself; rescaling the form makes it \(J\)-real, so it restricts to a nondegenerate real pairing on \(E\). Split-weight orthogonality says that the top line pairs only with the bottom line; nondegeneracy makes that pairing nonzero. The stabilizer of the top line contains the boundary parabolic and is proper. In rank one the Bruhat decomposition has only two cells, so this stabilizer is exactly that parabolic. Its projective orbit is the smoothly embedded boundary.

Choose a maximal-compact-invariant Euclidean norm on \(E\). The boundary is a sphere, \(S^{4n-1}\) or \(S^{15}\), and is simply connected. Its real line bundle consequently has a smooth global unit lift \(n\). Connectedness of the group implies that all its action scalars on this lift are positive. Distinct ordered boundary pairs form one group orbit, so the pairing is nonzero with constant sign off the diagonal. Change its sign and scale to make it positive there and one at standard compact opposites. By irreducibility its transpose is a scalar multiple of itself, with scalar \(1\) or \(-1\); positivity on both orders excludes \(-1\). The real pairing is therefore symmetric.

The determinant highest line computed above generates the Cartan component of \(V_\lambda^{\otimes Q/2}\). Its lift is the tensor power \(n(z)^{\otimes Q/2}\). Compact transitivity makes its unit normalization agree with that of the determinant line, and its invariant pairing is \([n(w),n(z)]^{Q/2}\). Normalizing at the same opposite pair removes the single constant. Consequently \[k(w,z)=[n(w),n(z)]^{Q/2},\qquad K(w,z)=[n(w),n(z)].\] This uses the entire highest-line representation, not only its split character.

In a chart based at the lowest line, the unnormalized lift is \(\exp(X)n_-\), where \(X\) ranges over the graded positive nilradical. A term raising restricted weight from minus two to at most two has total graded degree at most four. Exponentiation is thus a polynomial of that weighted degree. Passing to a unit lift divides by its smooth positive norm. Write \(q(X)=\exp(X)n_-\) for the unnormalized lift, and let \(\delta_t\) multiply degree-one coordinates by \(t\) and degree-two coordinates by \(t^2\). Since the bottom weight is \(-2\), \[q(\delta_t X)=t^2\pi(a_{\log t})q(X).\] Invariance of the pairing therefore makes its two-point kernel homogeneous of degree four. Its invariance under simultaneous left translation in the nilradical reduces it to the difference coordinates. It is positive away from zero. Compactness of the homogeneous unit sphere gives comparison with the fourth power of a homogeneous gauge; finitely many charts give the asserted compact distance comparison.

The highest-line orbit is an embedded projective orbit. The unit lift and its positive radial parameter consequently give the claimed cone embedding. Its span is a nonzero invariant subspace, hence all of \(E\). Compact transitivity makes the three displayed integrals independent of \(u\). The first two are finite and positive. For the last one, a ball of radius \(r\) has measure bounded by \(Cr^Q\) and the integrand is bounded by a constant times \(d(v,u)^{-8}\). Dyadic annuli give \[\int_{d(v,u)<1}K(v,u)^{-2}\,dv \le C\sum_{j\ge0}2^{8j}2^{-Qj}<\infty.\] Here \(Q=4n+2\ge10\) for \(\operatorname{Sp}(n,1)\) and \(Q=22\) for \(F_{4(-20)}\). ◻

Proof. The range of \(V\) reduces each unitary group \(C_z^{\mathrm i t}\): Equation (162) applies to both \(t\) and \(-t\). The \(C_z\) strongly commute, since at each ordinary source chamber they are scalar multiples of the same operator \(b_{y(Z)}^{-\vartheta}\). The spectral theorem transfers their resolvents to the target representation. Hence \(A_z(v)\) and \(A_w(v)\) strongly commute for almost every \(v\). For fixed \(z\), bounded resolvents of \(A_z(v)\) depend strongly continuously on \(v\), by bounded convergence in the \(Q_z\) calculus. Products of their bounded resolvents have the same continuity. Their commutation therefore holds for every \(v\).

Positive spectral truncation in Equation (162) gives, including the value \(+\infty\), \[ a_2^T\|B_z\xi\|^2 =c_i^{2\vartheta}\int_{B_S^{\mathrm{ch}}} K_S(y(Z),z)^2 \big\|b_{y(Z)}^{-\vartheta}(VI\xi)(Z)\big\|^2\,dZ. \tag{164}\] Here \(B_S^{\mathrm{ch}}\) denotes the full chamber variable occurring in \(V\); its indicated vertex marginal is the ordinary probability on \(B_S\). The left-hand side follows by integrating \(v\) and using Lemma 113. Integrating the right-hand side in \(z\) replaces the kernel square by the constant \(a_2^S\). This identifies the graph norm with a closed weighted norm of \(VI\xi\), and proves completeness. Boundedness of \(K_S\) gives the first estimate. For the second, the triangle inequality implies that at least one of \(d(y,z),d(y,w)\) is at least \(d(z,w)/2\). The sum of the two kernel squares therefore dominates \(c\,d(z,w)^8\), uniformly in \(y\), giving Equation (163).

If \(\sum_j a_j n_S(z_j)=0\), then \(\sum_j a_jK_S(y,z_j)=0\) for every \(y\). All the source operators in this finite linear combination are defined on \(VI\mathfrak D\) by Equation (164). Intertwining shows that \(\sum_j a_jA_{z_j}(v)\xi=0\) for almost every \(v\). Continuity in \(v\) on this domain extends the equality to every \(v\). Moreover, \[B_z\xi=(a_1^T)^{-1}\int_{B_T}A_z(v)\xi\,dv, \qquad \xi\in\mathfrak D,\] so the same relation holds for \(B_z\). Linearity in \(n_T(v)\) follows directly from the bilinear kernel representation. ◻

Proof. Density. Let \(M_l\) be multiplication on the source chamber-variable space by \(b_y^\vartheta1_{\{b_y\le l\}}\), and put \[T_l=I^*V^*M_lI_S.\] These are bounded operators. The constant-input formula in Equation (159) and the strong commutation of \(D_Z\) with its own-coordinate height give \[\langle\eta,T_l\eta\rangle =c_0^{-1/2}\int \left\langle D_Z^{-1/2}\eta, b_y^\vartheta1_{\{b_y\le l\}}\eta\right\rangle\,dZ\ge0.\] All pairings are integrable by the isometry and boundedness of \(M_l\). For nonzero \(\eta\), they cannot vanish for every \(l\): in the common \(D_Z,b_y\) calculus the union of the cuts \(b_y\le l\) is the identity, and both positive operators have zero kernel. Thus the positive operators \(T_l\) have zero common kernel and their ranges are total.

For \(\zeta=V^*M_lI_S\eta\), the reducing-range intertwining gives \[A_z\zeta =V^*C_z M_lI_S\eta =c_i^\vartheta V^* \big(K_S(y,z)1_{\{b_y\le l\}}I_S\eta\big).\] Its norm is bounded uniformly in \(z\). To estimate \(B_zI^*\zeta\), test against \(\eta'\in\operatorname{dom} B_z\) and write \[\langle I^*\zeta,B_z\eta'\rangle =\int_{B_T} \left\langle A_z(v)\zeta(v), K_T(v,P_z)^{-1}\eta'\right\rangle\,dv.\] This identity is first made with joint spectral truncations of \(Q_z\). The inverse-square bound in Lemma 113 permits their removal by Cauchy–Schwarz. It proves that \(I^*\zeta\in\operatorname{dom} B_z\) with a uniform bound. Integrating \(z\) proves \(\mathop{\mathrm{ran}}T_l\subset\mathfrak D\), and therefore density.

Smooth stability and graph cutoffs. The basic stability estimate concerns a bounded unary average \[R_F=\int_{B_T}F(v,A_z(v))\,dv.\] For \(\xi\in\mathfrak D\) and \(\eta'\in\operatorname{dom} B_w\), move \(A_w(v)\) through \(F(v,A_z(v))\), using Lemma 114, and pair against \(K_T(v,P_w)^{-1}\eta'\). Cauchy–Schwarz gives \[ \|B_w R_F\xi\| \le C\|F\|_\infty\|B_w\xi\|. \tag{165}\] The adjoint domain criterion used above justifies this computation before the left-hand side is known to be finite. Integrating \(w\) gives a bound on \(\mathfrak D\). Products of unary averages satisfy the iterated bound.

Choose \(v_1,\ldots,v_d\) such that the \(n_T(v_j)\) form a basis of \(E_T\). Allow each \(v_j\) to vary in a sufficiently small fixed neighborhood. The resulting inverse frame matrices are uniformly bounded. The evaluations \[\big(K_T(v_j,u)e^{\vartheta s}\big)_{j=1}^d\] then recover \(e^{\vartheta s}n_T(u)\) linearly. A compactly supported smooth function of \((u,s)\) extends smoothly to a compact set in the ambient vector space away from zero. Compose such an extension with the linear recovery, and integrate the independent \(v_j\) over their neighborhoods. On fixed compact coordinate boxes, its Fourier expansion in the separate variables has absolutely summable coefficients after finitely many derivatives. Each separated term is a product of unary averages. Equation (165) therefore proves the stated smooth stability estimate.

For completeness, compact graph cutoffs can be constructed by this same procedure, without assuming a joint height law at different labels. Cut each of the basis evaluations smoothly from above at \(l\), and subtract the piece on which all evaluations are smaller than \(1/l\). Uniform frame inverses imply that the resulting function of \(e^{\vartheta s}n_T(u)\) is supported in a bounded radial interval bounded away from zero. It is consequently a compact smooth function of \((u,s)\). Each upper unary cutoff tends strongly to the identity. Each simultaneous-small piece tends strongly to zero. The same convergences hold in the graph norm: in the proof of Equation (165), apply dominated convergence to the vector \(A_w(v)\xi\), then integrate \(w\). The only possible zero of an evaluation at a finite positive height is \(P_z= v\); \(P_z(\{v\})=0\) for almost every ordinary \(v\), as follows by integrating against a faithful normal scalarization of \(P_z\) and using that the ordinary measure is nonatomic.

Individual cores. Let \(L\) be \(B_z\) or \(A_z(v)\). Cut a vector in \(\operatorname{dom} L\) by compact smooth functions of its own \(Q_z\) variable that tend to one and have bounded height support. These cuts converge in the graph norm of \(L\). For each fixed cutoff, approximate the uncut vector in Hilbert norm by vectors of the dense domain \(\mathfrak D\), and apply that cutoff. Smooth stability keeps the resulting vectors in \(\mathfrak D\), while boundedness of \(L\) on the cutoff gives convergence in its graph norm. A diagonal choice proves that \(\mathfrak D\) is a core. ◻

Proof. To specify the ordered assertion, let \(f\in C_c^\infty(B_T\times\mathbb R)\), let \(\rho\) equal one on its angular support, and let \(\chi'\) be supported in a patch transverse to \(\mathop{\mathrm{supp}}\rho\). For a smooth kernel on this transverse rectangle, write \[\mathcal M_{w,z}(F) =\int F(u',u)\,P_w(du')P_z(du).\] This denotes the operator-norm-convergent sum obtained from a summable separated smooth expansion of \(F\); it does not introduce joint spectral calculus for \(P_w,P_z\). We will prove the bounded operator identity \[ B_wP_w(\chi')f(Q_z) =c_i^\vartheta K_S(w,z)\, \mathcal M_{w,z}\!\left( \frac{\chi'(u')\rho(u)}{K_T(u',u)}\right) B_z^{-1}f(Q_z). \tag{166}\] In particular the entire range on the left belongs to the domain of \(B_w\). Initially insert a fixed rightmost smooth angular-height test \(f_0(Q_z)\) equal to one on \(\mathop{\mathrm{supp}}f\), with a slightly larger transverse angular support and a fixed compact height interval. Equation (155) then gives, for a Schwartz function \(a\) and by Fourier inversion, \[\begin{aligned} &a(\log b_w)P_w(\chi')f_0(Q_z)\\ &\quad=\int \chi'(u')\rho_0(u) a\!\left(\log\frac{c_i k_S(w,z)}{k_T(u',u)}-s\right) \,P_w(du')Q_z(du,ds)f_0(Q_z), \end{aligned}\] where \(\rho_0\) is a fixed enlarged angular cutoff. In this formula the height on the right belongs to the single \(Q_z\) calculus, and the remaining ordered kernel is defined by the same separated expansions. Every fixed number of derivatives of \(k_T(u',u)^{-\mathrm it}\) on the transverse rectangle is bounded polynomially in \(|t|\). The Schwartz Fourier transform therefore makes the integral and all required separated expansions absolutely summable.