Proof. Positivity of the pair terms says that a zero vector has no pair relative-angular-momentum-one component. An antisymmetric polynomial of the same homogeneous degree in each of two spinors vanishes when the spinors coincide. Separate homogeneity extends that vanishing to proportional spinors, so the irreducible bracket \([z_1,z_2]\) divides it. Dividing by the bracket and then setting \(z_1=z_2=z\) gives an equivariant map to the degree-\((2q-2)\) polynomials in \(z\), the spin-\((q-1)\) representation. This map is nonzero on \(e_0\wedge e_1\) and hence is onto. Its kernel is exactly divisibility by the bracket cubed: the divided polynomial is symmetric and separately homogeneous, and if it vanishes on the diagonal it is divisible by the bracket; the resulting quotient is antisymmetric and therefore divisible by the bracket once more. By the two-spin decomposition, the map detects precisely the spin-\((q-1)\) summand. Thus absence of that component is equivalent to cubic divisibility.

It follows that every pair bracket cubed divides the polynomial. The distinct pair brackets are nonassociate irreducible polynomials; unique factorization makes their product divide it. Dividing by the cubic product changes antisymmetry to symmetry and leaves degree \(q-3(n-1)\) in each variable. This proves both directions of (13). A symmetric polynomial with individual degree \(h\) has the symmetrized monomial basis indexed by \(0\le j_1\le\cdots\le j_n\le h\), giving the binomial dimension. The remaining claims follow at once, with the vacuum treated separately. ◻

Proof. Choose the axial gauge and azimuth convention in which the coefficient of \(e_j\) in \(k_y\) is \(e^{ij\phi_y}p_{q,m}(j)^{1/2}\). Its modulus is \[p_{q,m}(j)^{1/2},\qquad p_{q,m}(j)=\binom qj(m/q)^j(1-m/q)^{q-j}.\] They obey, after extension by zero outside \([0,q]\), \[ |\Delta^l p_{q,m}(j)^{1/2}| \leq C_{D,l}w_q(m)^{-1/2-l} \left(1+\frac{|j-m|}{w_q(m)}\right)^{-D}. \tag{24}\] Here and below finite differences may shift the argument by a fixed number of indices, which only changes the constant. We include a verification to specify uniformity near the ends. First put \(p=m/q\) and \(\sigma^2=qp(1-p)\), so \(w_q(m)^2=1+q\sigma^2/(q+1)\asymp1+\sigma^2\). Fourier inversion of a binomial law of variance \(\sigma^2\) gives \[\sup_j p_{q,m}(j)\leq\frac1{2\pi}\int_{-\pi}^{\pi} \bigl(1-4p(1-p)\sin^2(t/2)\bigr)^{q/2}\,dt \leq \frac C{1+\sigma}.\] For \(0<p<1\), tilt the law by \(e^{\lambda j}\). Its new parameter is \(p_\lambda=pe^\lambda/(1-p+pe^\lambda)\), and its variance differs from \(\sigma^2\) by a factor between \(e^{-|\lambda|}\) and \(e^{|\lambda|}\). Therefore the centered log moment generating function satisfies \(\log\mathbb E e^{\lambda(X-m)}\leq\sigma^2\lambda^2\) for \(|\lambda|\leq1/2\). At \(z=j-m\), choose \(\lambda=z/(2(\sigma^2+|z|))\). The exact tilting identity and the preceding point-mass bound give \[ p_{q,m}(j)\leq\frac C{w_q(m)} \exp\left(-\frac{c|j-m|^2}{w_q(m)^2+|j-m|}\right). \tag{25}\] The cases \(p=0,1\) are direct. This proves every polynomial tail bound with the required height, uniformly also when \(w_q(m)\) is bounded.

For the derivative gain first assume \(w_q(m)^2\geq16(l+1)\), and let \(h=\min(m,q-m)\asymp w_q(m)^2\). Interpolate the square root of the binomial probability by gamma functions, writing the resulting positive function as \(F(x)\). Put \(\psi=(\log\Gamma)'\) and \(\psi_1=(\log\Gamma)''\). On \(|x-m|\leq h/2\), the explicit formulas \[\begin{aligned} (\log F)'(x)=\tfrac12\left(\log\frac p{1-p} -\psi(x+1)+\psi(q-x+1)\right),\\ (\log F)''(x)=-\tfrac12\bigl(\psi_1(x+1)+\psi_1(q-x+1)\bigr) \end{aligned}\] and the positive-real polygamma series give \(|(\log F)'(m)|\leq Cw^{-2}\), \((\log F)''(x)\leq-cw^{-2}\), and \(|(\log F)^{(k)}(x)|\leq C_kw^{-k}\) for \(k\geq2\). Stirling’s inequalities give \(F(m)\leq Cw^{-1/2}\). Differentiating \(F=e^{\log F}\) consequently proves \[|F^{(l)}(x)|\leq C_{D,l}w^{-1/2-l} (1+|x-m|/w)^{-D},\qquad |x-m|\leq h/2.\] If \(|j-m|\leq h/4\), integrate this derivative over the unit cube that represents \(\Delta^lF(j)\); the entire stencil stays in the displayed interval. Outside that region, each nonzero stencil entry has \(|j+a-m|\geq cw^2\), \(0\leq a\leq l\). Equation (25) then supplies an exponential in that distance, which pays both \(w^{-l}\) and every polynomial tail. This also handles stencils crossing an endpoint. Finally, if \(w^2<16(l+1)\), bound the finite difference by its \(l+1\) values and use (25); the factor \(w^{-l}\) is now a constant depending only on \(l\). This completes the proof of (24), including the bounded-width regime.

Insert (24) in (20). For windows whose radial centers are a bounded distance apart, their widths are comparable; summation by parts \(l\) times gives the factor \((1+w_a|\phi_x-\phi_y|)^{-l}\), with angular difference modulo \(2\pi\). The absolute sum before summation by parts is bounded. For separated radial windows, the tail in (24) first supplies arbitrary powers of radial distance. The ratios of the widths cost only polynomial powers by (18). Spending additional tail orders therefore gives the same angular estimate. Radial distance plus the shorter latitude arc bounds the spherical distance from above, proving (23).

The coherent resolution of the identity has density \((q+1)/(2\pi q)\) with respect to area. Inserting it between two frame atoms reduces the frame-to-frame bound to \[\int_{S_q}(1+d(x,z))^{-D-3}(1+d(z,y))^{-D-3}\,dz \leq C_D(1+d(x,y))^{-D}.\] The inequality follows by dividing the integral into the sets where \(d(x,z)\geq d(x,y)/2\) and where \(d(z,y)\geq d(x,y)/2\). Finally \[|\langle e_j(y),k_z\rangle|^2 =\binom qj\sin^{2j}(\vartheta/2)\cos^{2(q-j)}(\vartheta/2),\] where \(\vartheta\) is the angle between \(y\) and \(z\). For bounded \(j\), this is bounded by \(C_j(1+d(y,z))^{2j}e^{-c d(y,z)^2}\); increasing the constant covers bounded \(q\). Inserting the coherent resolution once more proves the last assertion. ◻

Proof. Use the windows and Fourier lengths at flux \(q\) for both spaces, and restrict their index sequences to \(0\leq j\leq t\) for the smaller space. Equation (21) still holds on those indices. Only a bounded number of end indices have been removed. The windows meeting them have bounded width, or have rapidly small tails in the estimates in which an endpoint extension occurs. The slot positions for the two sphere metrics differ by a bounded distance; the constants may depend on \(d_0\). For all sufficiently large \(q\), every window centered in the northern third has its full support below \(q/2\), because its width is \(O(\sqrt q)\). On these supports the construction uses only \(\eta(\sqrt{1+j}-k)\) and their fixed normalization, and their Fourier length is the fixed integer \(16(k+1)\). These complete slot vectors are therefore independent of \(q\) in index coordinates. The buffer between \(q/3\) and \(q/2\) permits changing the frame outside this region without changing its northern slots. Bounded fluxes may use pairwise padded frames; no long sequence of common slots is needed there. ◻

Proof. Take approximants at radii \(C_*2^\nu r\) and telescope, starting with the first approximant as the core. The core norm is \(Ca\); the other differences are bounded by the two adjoining errors in (29). Sum with any \(D>B+1\) to obtain (30). ◻

Proof. The product estimate follows from \[\|AB-A_rB_r\| \leq\|A-A_r\|\|B\|+\|A_r\|\|B-B_r\|.\] For the commutator use approximations of radius less than \(d(x,y)/3\); their graded commutator vanishes. For the weighted assertion expand both factors as in (28). Terms with disjoint balls vanish, while intersecting balls of radii \(R,S\) require \(d(x,y)\leq R+S\). The number or measure of their possible centers is \(O((1+R+S)^2)\), and (27) costs at most \(C(1+R+S)^\kappa\). To obtain an output moment of order \(D\), bound these costs by \(C(1+R+S)^{D+\kappa+2}\) and choose input decay order larger than \(D+\kappa+6\). The two geometric sums then converge. This argument also permits assigning each output term to the center of either input; that convention will be used for differences of interactions.

For (31), the first estimate follows by the disjoint-ball argument at exponent \(l\). For approximation radii \(r\leq3D\), use zero as an approximant to \(C_{xy}\), giving \(r^l\|C_{xy}\|\leq C\). For \(r>3D\), approximate \(A\) in \(B_{r/3}(x)\) and \(J_y\) in \(B_{r/3}(y)\); their commutator is supported in \(B_r(x)\) and has error \(Cr^{-l}\). Thus the second estimate holds. Apply the product estimate with the small norm \(\|C_{xy}\|\leq CD^{-l}\) on one factor to obtain the third estimate. Finally a ball \(B_r(x)\) contains \(O((1+r)^2)\) slots, and tempered ratios within it cost \(O((1+r)^\kappa)\). The same sums give the stated polynomial core costs. For tails beyond \(Cr\), the distance from the original ball is comparable to the tail radius, so increasing a tail order does not increase the core-volume exponent. ◻

Proof. Coefficient localization gives operator-norm localization directly. If \(E_R\) is the one-particle projection onto slots in \(B_R(x)\), the CAR identity \(\|c(f)\|=\|f\|_2\) gives \[\|c(\mathsf S_q f)-c(E_R\mathsf S_q f)\| =\|(I-E_R)\mathsf S_q f\|_2.\] For \(f=k_x\) or a bounded polar mode at \(x\), Lemma 4 and the slot count (22) make the right side at most \(C_D(1+R)^{-D}\) for every \(D\); the approximating annihilator is supported in \(B_R(x)\).

The coherent resolution gives \[d\Gamma(T_q(\varphi))=\frac{q+1}{2\pi q} \int_{S_q}\varphi(x)c^*(k_x)c(k_x)\,dA(x).\] For \(H_q\), the highest pair vector is \(v_0=e_0\wedge e_1\). Schur’s Lemma on the spin-\((q-1)\) pair space and its dimension \(2q-1\) give, with Haar mass one, \[ H_q=(2q-1)\int_{SU(2)}B(gv_0)^*B(gv_0)\,dg. \tag{32}\] The product \(B(gv_0)=c(ge_1)c(ge_0)\) is rapidly local at the rotated pole by Lemmas 4 and 7. Its phase along the stabilizer disappears in its square, so the center measure on the sphere has density \((2q-1)/(2\pi q)\). This and \((q+1)/(2\pi q)\) are uniformly bounded. The same two lemmas localize \(c^*(k_x)c(k_x)\). No derivative of \(\varphi\) enters either the representation or its bound. ◻

Proof. Use (28) to write \(K\) as a sum of terms on balls. Taking Hermitian and even parts makes each term Hermitian and even without changing the total or its estimates. First retain only terms of diameter at most \(\ell\). For this truncated interaction define \[M_{ab}=2\sum_{X\ni a,b}\|K_X\|.\] Uniform summability, including the number of sites in \(X\), gives \(\sup_a\sum_b M_{ab}e^{d(a,b)/\ell}\leq C\); indeed each retained distance is at most \(\ell\), and the ball-volume costs are polynomial. The iterated commutator inequality, with internal evolution removed unitarily at each step, therefore gives \[ \|[\tau_t^{K_{\leq\ell}}(A),B]_{\mathrm{gr}}\| \leq C\|A\|\|B\|\,|X|\,|Y| e^{C|t|-d(X,Y)/\ell} \tag{34}\] for \(A\in\mathfrak A(X)\) and \(B\in\mathfrak A(Y)\). For completeness, the \(n\)th iterated integral contributes \(|t|^n/n!\) times a chain of \(n\) matrix factors \(M\). Multiplication by \(e^{d(X,Y)/\ell}\) is bounded by the product of the exponential weights along the chain. Summing the intermediate sites gives \(C^n\), and summing \(n\) gives (34). Evenness of the generator makes the same calculation valid for odd \(A,B\) with the graded commutator.

Restore the removed terms using Duhamel’s formula. For each removed ball term use the smaller of the trivial commutator bound and (34). The centers within distance \(C\ell(1+|t|)\) of \(X\), together with the radius of that term, occupy at most a polynomial volume. Outside that region the exponential bound is summable. Consequently, for every \(s\), the restoration error is at most \[ C_s\|A\||X|(1+|t|) (1+r+\ell(1+|t|))^4\ell^{-s}, \tag{35}\] when \(X\subset B_r(x)\). To see the power allowance explicitly, a removed term of radius \(a\) contributes its norm times at most \(C(1+r+\ell(1+|t|)+a)^2\) possible centers and at most \(C(1+a)^2\) sites. Its arbitrarily high radius moment makes the sum over \(a>\ell/2\) bounded by the right side of (35), after increasing the input moment by \(s+6\).

Now let \(R\geq C(r+(1+|t|)^4)\) and choose \(\ell=R^{1/3}\). The truncated evolution differs from the evolution generated by its terms contained in \(B_R(x)\) by \(C\|A\||X|\operatorname{poly}(R)e^{C|t|-cR^{2/3}}\). This is again Duhamel’s formula and (34), summed over the boundary terms. Both restoration errors are bounded by \(C_D\|A\||X|R^{-D}\), choosing \(s\) sufficiently large in (35). This supplies an approximation in \(\mathfrak A(B_R(x))\). At smaller radii use the norm bound \(\|\tau_t(A)\|=\|A\|\). Since \(|X|\leq C(1+r)^2\), this proves the assertion for supported \(A\) and then, by (28), (33). The proof uses only uniform bounds on the interaction at each time, so also proves the time-dependent assertion. ◻

Proof. Integrate (33). The approximation norm is a seminorm up to its operator-norm summand, so the integral of local approximants gives the required estimate. The remaining assertions follow directly by commuting the evolution with gauge transformations, the physical algebra, and adjoints. ◻

Proof. Use the exact formula \[\tau_t^K(A)-\tau_t^{K'}(A) =i\int_0^t\tau_s^K ([K-K',\tau_{t-s}^{K'}(A)])\,ds.\] Apply Lemma 9 to the inside observable, Lemma 7(ii) to the commutator, and Lemma 9 once more to the outside evolution. The tempered inequality (27) is precisely what allows the insertion at \(z\) to be charged to \(a(x)\). All time costs are polynomial and therefore integrable against the specified filters. The time-dependent version follows from its identical Duhamel formula. Linearity handles a changed input.

We give the additional radius bookkeeping for (38)–(39). For a ball decomposition about \(x\), use the weighted sum \(\sum_R(1+R)^p\|A_R\|\). Lemma 9 bounds its evolved value by \(C_p(1+|t|)^{4p}\) times its initial value. Indeed, for a radius-\(r\) input, begin a dyadic decomposition at \(Q=C(1+r+(1+|t|)^4)\). The core contributes \(C\|A\|Q^p\); the subsequent shells contribute at most \(C_M\|A\|(1+r)^2\sum_{R\geq Q}R^{p-M}\). Choosing \(M>p+2\) bounds this by \(C_p\|A\|Q^p\). The weighted insertion estimate in the proof of Lemma 7 costs \(s=\kappa+2\) powers of the input radius. Duhamel at \(p=0\) proves (38).

For the tail first take \(|t|\leq cR^{1/4}\) in the Duhamel integrand, with \(c>0\) a sufficiently small fixed constant; bounded \(R\) are covered by (38). Fix also a small \(\varepsilon>0\). Choose \(\varepsilon,c\) and then \(C_*\) so that the propagation lemma applies both from radius \(r\) to \(r+\varepsilon R\) and from radius \(r+3\varepsilon R\) to \(r+R\). This requires only fixed constants: if \(C\) is the constant in that lemma, take \(\varepsilon<(100C)^{-1}\), \(c^4<\varepsilon/(10C)\), and \(C_*>100C/\varepsilon\). Cut the inner evolved observable at radius \(r+\varepsilon R\). Its remaining shells have weighted sum of order \(s\) at most \(C_M\|A\|(1+r)^2R^{s-M}\), by the short-time bound in Lemma 9; their commutator with the insertion has the same bound times \(Ca(x)\). In the retained part, discard insertion terms of radius greater than \(\varepsilon R\). Their arbitrary radius moments give an error \(C_Ma(x)\|A\|R^{-M}\). The remaining commutator is supported in \(B_{r+3\varepsilon R}(x)\) and has norm at most \(Ca(x)\|A\|R^s\). Evolving it in the outside factor of Duhamel and using the short-time propagation estimate gives an approximation on \(B_{r+R}(x)\) with error \(C_Ma(x)\|A\|R^{s+2-M}\). Choose \(M\) in each of these three bounds larger than the requested order plus \(s+4\), and integrate over an interval of length at most \(R^{1/4}\). This gives the stated tail bound for short times. For \(|t|>cR^{1/4}\), use (38) and \((1+|t|)^{4D}(1+R)^{-D}\geq c_D\).

Finally suppose an insertion ball of radius \(a\) can occur only if its center \(z\) satisfies \(d(x,z)+a\geq d\). If that ball meets an input shell of radius \(R\), then \(d\leq R+2a\). Inserting \((1+d)^{-M}(1+R+2a)^M\) into the shell summation proves the last assertion, after increasing the input moments. This includes changes supported outside a common comparison region. ◻

Proof. Put two slot spaces side by side. If \(S,T\) are their analysis isometries, the partial isometry \(TS^*\) has rapidly decaying matrix entries by Lemma 4. On their direct sum, the anti-self-adjoint one-particle operator \[\begin{pmatrix}0&-ST^*\\TS^*&0\end{pmatrix}\] rotates the two physical subspaces into each other through angle \(\pi/2\) and acts as zero on their complements. Its second quantization is a bounded-strength, rapidly local quadratic interaction: absolute row sums and every distance moment are bounded by (22). Lemma 9 applies over this fixed angle. Conjugate \(A\otimes I\) by this rotation and take the vacuum expectation in the entire first slot copy, retaining the entire second copy. On a CAR monomial, the physical components have moved from \(S U_q\) to \(T U_q\), while the complementary components in the first copy are evaluated in their vacuum. The resulting map is exactly (40), including for odd monomials; linearity gives the identity for every operator. Vacuum expectation is contractive and preserves support in the doubled slot geometry. This proves the assertion, and \(S=T\) gives re-embedded physical compression. In particular no projection onto the output complementary vacuum is left in the resulting operator.

For the functional-calculus assertion extend \(f\) smoothly to a compactly supported function on \(\mathbb R\). If \(A_r\) approximates \(A\) locally, its Hermitian part does so with the same error. The Fourier formula and Duhamel’s identity give \[\|f(A)-f(A_r)\| \leq\left(\int_{\mathbb R}|t|\,|\widehat f(t)|\,dt\right) \|A-A_r\|.\] Since \(f(A_r)\) belongs to the same local algebra, the approximation bounds follow. A constant value of \(f\) contributes only a scalar identity, which is local in every ball. ◻

Proof. Vandermonde’s identity identifies \(\binom qi\binom q{b-i}/\binom{2q}{b}\) as a hypergeometric law of mean \(b/2\) and variance \(b(2q-b)/(4(2q-1))\). Multiplying its second centered moment by four proves (41). The moment formula itself also follows directly by applying \(i\binom qi=q\binom{q-1}{i-1}\) twice and Vandermonde’s identity.

Here are symbol estimates that include the dependence on the moving row center. Set \[\ell_q(x)=\log\Gamma(q+1)-\log\Gamma(x+1) -\log\Gamma(q-x+1),\quad t=i-m,\] and \[E(m,t)=\tfrac12\{\ell_q(m+t)+\ell_q(m-t)\}-\ell_q(m), \qquad T(m)=S_{2m}e^{-2\ell_q(m)},\] where the gamma formula defines the smooth extension. Then \(a_b(i)=-2t e^{E(m,t)}/\sqrt{T(m)}\). The gamma duplication formula gives the exact expression \[ T(m)=\frac{4m(q-m)}{2q-1}\sqrt\pi\, \frac{R(m)R(q-m)}{R(q)}, \qquad R(x)=\frac{\Gamma(x+1)}{\Gamma(x+1/2)}. \tag{44}\] For \(m,q-m\geq1/2\), Stirling’s inequalities yield \(T(m)\asymp w_q(m)^3\). Its logarithmic derivatives of order \(r\geq1\) are bounded by \(C_r(u^{-r}+v^{-r})\). These derivative assertions can be obtained directly by differentiating Euler’s gamma product: for \(k\geq1\), \[\frac{d^{k+1}}{dx^{k+1}}\log\Gamma(x) =(-1)^{k+1}k!\sum_{n=0}^{\infty}(x+n)^{-k-1}.\] In the region \(|t|\leq c\min(u,v)\), the second derivative \(\ell_q''\) is negative and comparable in magnitude to \(u^{-1}+v^{-1}\asymp w^{-2}\). Thus \(E(m,t)\leq-ct^2/w^2\). Differentiating \(E\) in \(i\) and \(b\) gives polynomials in \(t/w\), with one factor \(w^{-1}\) per derivative; (44) has the same property. The discrete product rule, or integration of these derivative bounds over unit cubes, proves (42) in this region. There is also a global Gaussian bound, including the ends: convexity of the trigamma function \(\psi_1=(\log\Gamma)''\) and \(\psi_1(x)\geq1/x\) give \[E_{tt}(m,t)=-\tfrac12\bigl\{ \psi_1(m+t+1)+\psi_1(m-t+1) +\psi_1(q-m+t+1)+\psi_1(q-m-t+1)\bigr\} \leq-\frac1u-\frac1v\leq-\frac c{w^2}.\] Since \(E(m,0)=E_t(m,0)=0\), we have \(E(m,t)\leq-ct^2/w^2\) on the entire allowed interval. Outside the central region \(|t|/w\geq cw\) when \(w\) is large, so increasing the tail order pays all derivative and endpoint-extension factors. The rows with bounded \(w\) require only bounded finite differences. This proves the full uniform estimate without differentiating a lattice normalization sum.

Choose smooth windows \(\zeta_\nu(b)\) with \(\sum_\nu\zeta_\nu(b)^2=1\), supported on intervals of length \(O(W_\nu)\) around \(b=2M_\nu\), where \(W_\nu=w_q(M_\nu)\). The construction (19), with \(m=b/2\), supplies them. Define \[ L_{\nu,\phi}=W_\nu^{-1/2} \sum_b\zeta_\nu(b)e^{-ib\phi}Q_b, \qquad d\mu_\nu(\phi)=W_\nu\frac{d\phi}{2\pi}. \tag{45}\] Fourier orthogonality proves (43) exactly. The radial centers have bounded density in physical distance, while their circle lengths are comparable to \(W_\nu\) outside the polar balls. Hence \(\mu\) has bounded mass on unit balls.

After removing its linear phase \(e^{-i(i+j)\phi}\), the two-index coefficient of \(L_{\nu,\phi}\) has height \(O(W_\nu^{-1})\), and every mixed finite difference of orders \(r,s\) costs \(W_\nu^{-r-s}\). Its tails are arbitrary powers of \(1+|i-M_\nu|/W_\nu+|j-M_\nu|/W_\nu\). To express it in slot fermions, multiply in each index by the window coefficients from (20) and sum. On comparable windows, repeated summation by parts in both indices gives, for arbitrary \(D\), coefficients bounded by \[C_D(1+d(x,x_{\nu,\phi}))^{-D} (1+d(y,x_{\nu,\phi}))^{-D}\] for the slot monomial \(c_xc_y\). The height is bounded because the two summation lengths are \(O(W_\nu)\), the coefficient height is \(O(W_\nu^{-1})\), and the two Fourier normalizations contribute \(O(W_\nu^{-1})\). Noncomparable windows have radial separation; their scale ratios cost polynomial powers by (18), paid by increasing the radial tail orders. This proves the displayed bound for all slots. Its absolute sum, including every distance moment, is bounded by (22). Since \(\|c_xc_y\|\leq1\), truncating that sum proves rapid locality in operator norm. Products give locality of \(L_{\nu,\phi}^*L_{\nu,\phi}\). Each operation used was linear in the row coefficients before the final square; keeping any extra tempered factor proves the last assertion. ◻

Proof. Insert the analysis map on both sides of the diagonal one-particle matrix. The slot matrix elements are \[(L_{a_1}L_{a_2})^{-1/2}\sum_j a(j)\xi_{a_1}(j)\xi_{a_2}(j) e^{ij(\phi_2-\phi_1)}.\] The two radial windows must overlap, so their widths are comparable and their radial physical separation is bounded. Taking \(K\) finite differences of the summand and summing by parts gives the bound \(C_K A(m_1)(1+d(x_1,x_2))^{-K}\). The normalization cancels the length of the summation interval. At bounded-width end windows the same statement follows from finite sums; all involved slots are inside a bounded polar ball.

Assign the quadratic monomial to \(x_1\), and pair it with its adjoint to obtain Hermitian terms. Vacuum-compress each assigned term using (40). Compression fixes the physical total \(D_q(a)\), so the resulting decomposition is still exact and now each term is physical. A monomial joining points at distance \(R\) is supported in \(B_{1+R}(x_1)\); the compression lemma bounds its approximation norm of order \(D\) by \(C_D(1+R)^{D+2}\). The slot count (22), the tempered ratios, and \(K>D+\kappa+6\) therefore make the assigned absolute sum converge with the required \(D\)th moment. The stated value of \(K_D\) more than covers these losses and the fixed finite differences of the windows. The grade and reality are preserved by compression. Every step is linear before pairing adjoints, which also proves the difference assertion. ◻

Proof. Fix an allowed unperturbed Fock gap \(\gamma>0\). We compare columns of annihilators, rather than Hamiltonians. This avoids a factor equal to the number of particles.

Choose a smooth function \(\ell_B(x)\) on \(x\ge1\), constant for \(x\le B\), equal to \(\log x\) for \(x\ge2B\), and with \(|\ell_B^{(k)}(x)|\le C_k(B+x)^{-k}\) for \(k\ge1\). Replace each logarithm \(\log(q-j-r)\) in Equation (49) by \(\ell_B(q-j-r)\), retaining its constant denominator term, and denote the result by \(\bar g\). Let \(\bar B_b(s)\) be the corresponding normalized rows. At \(m=b/2\), the pair-row moment bounds of Lemma 13 imply \[ \|\bar B_b(s)-B_{t,b}\| \le C\sqrt w\,\frac{w^2}{(v+B)^2}. \tag{54}\] Here is the normalization detail in this application. Subtract the affine Taylor polynomial of \(\bar g\) at \(m\) from each of its two arguments. The subtracted contribution is constant on \(i+j=b\) and cancels in row normalization. Write \(z=|i-m|/w\). Taylor’s formula, the derivative bounds on \(\ell_B\), and \(w^2\le v\) give a bound \(Cw^2(v+B)^{-2}(1+z)^K\) for the remaining exponent; exponentiating costs at most another fixed power of \(1+z\). To check this also outside \(|i-m|\le(v+B)/2\), observe that this exterior region has \(z\ge c\sqrt v\) whenever \(v\ge B\), whereas logarithmic differences are bounded by \(C\log(1+v/B)\). If \(v<B\), all the logarithms have bounded scaled derivatives on the whole row. Thus the elementary inequality \(|e^x-1|\le |x|e^{|x|}\) and the row moment estimates give both an \(\ell^2\) coefficient error \(Cw^2/(v+B)^2\) and an \(\ell^1\) error \(C\sqrt w\,w^2/(v+B)^2\). The normalization factor differs from one by the first of these errors. Since \(w^2/(v+B)^2\le C/B\), it is bounded away from zero when \(B\) is large. Finally \(\|\sum e_{ij}c_jc_i\|\le\sum|e_{ij}|\) proves Equation (54).

Let \(\mathcal C_0\psi=(B_{t,b}\psi)_b\) and \(\bar{\mathcal C}_s\psi=(\bar B_b(s)\psi)_b\). These are operators from \(\mathcal F_t\) to a direct sum of copies of \(\mathcal F_t\). Equation  (54) gives \[ \|\bar{\mathcal C}_s-\mathcal C_0\|^2 \le C\sum_b\frac{w_b^5}{(v_b+B)^4} \le C\int_0^\infty\frac{(1+x)^{5/2}}{(B+1+x)^4}\,dx \le C B^{-1/2}. \tag{55}\] The middle inequality uses \(w_b^2\le v_b\) and the spacing \(1/2\) of \(v_b\). Consequently the operator-column error is at most \(CB^{-1/4}\). Choose \(B\), independent of \(q\) and \(s\), so that this is at most \(\sqrt\gamma/2\). Equation (53), also valid for \(\bar g\), shows that the two columns have the same nullity in each number sector. The min–max principle for singular values and the supplied Fock gap therefore give \[\|\bar{\mathcal C}_s\psi\| \ge\frac{\sqrt\gamma}{2} \operatorname{dist}(\psi,\ker\bar{\mathcal C}_s).\]

It remains to remove the rounding. Put \(\delta=g_{q,d}-\bar g\) and \(S_s=e^{sD_t(\delta)}\). At most \(2B+d\) orbital entries of \(\delta\) are nonzero, and each is bounded in modulus by \(C_d\log(2B)\). Thus \[\|S_s\|\|S_s^{-1}\| \le\exp\Bigl(\sum_j|\delta(j)|\Bigr)=:K_B<\infty\] uniformly on the entire Fock space. If \(\mathcal C_s\) is the column of the unrounded normalized rows, row normalization gives \[\mathcal C_s=R_s(\mathbb 1\otimes S_s) \bar{\mathcal C}_s S_s^{-1}, \qquad \|R_s\|+\|R_s^{-1}\|\le C_B,\] where \(R_s\) is diagonal in the row index. Since \(\ker\mathcal C_s=S_s\ker\bar{\mathcal C}_s\), the preceding distance estimate implies \[\|\mathcal C_s\psi\| \ge \frac{\sqrt\gamma}{2C_BK_B} \operatorname{dist}(\psi,\ker\mathcal C_s).\] Squaring proves the assertion. The flux threshold only has to ensure \(t\ge q_\gamma\) and \(q\ge4B+d\). ◻

Proof. We first prove uniform row regularity and its flux difference, then estimate the affine-subtracted commutator that enters the spectral filter.

Row regularity. The unnormalized coefficients in Equation (52) are \[(j-i)\Bigl[ \binom ti^{1-s}\binom qi^s \binom tj^{1-s}\binom qj^s\Bigr]^{1/2}.\] They obey the pair-row bounds in Lemma 13 uniformly for \(0\le s\le1\). We give the normalization argument because the normalization is now a sum with moving endpoints. Write \(z=i-m\). Relative to the undeformed row, after canceling its midpoint value, the additional factor is exactly \[ E_s(m,z)=\prod_{r=0}^{d-1} \left(1-\frac{z^2}{(q-m-r)^2}\right)^{-s/2}. \tag{57}\] When \(v\) is bounded, all allowed \(z\) and \(w\) are bounded and the estimates follow from finite differences. When \(v\) is large and \(|z|\le c v\), the logarithm of this factor and its derivatives in scale-\(w\) variables are bounded by \(C(w^2/v^2)\) times fixed polynomials in \(|z|/w\). The positive exponential factor can be absorbed into half the Gaussian decay of the undeformed row, since its exponent is \(O(z^2/v^2)\) while that decay has exponent \(-c z^2/w^2\) and \(w^2\le v\). For \(|z|>c v\), the factor in Equation (57) is at most \(C v^{d/2}\), whereas \(|z|/w\ge c\sqrt v\). The Gaussian tail therefore pays this factor and every required inverse power of \(w\).

To control normalization differences, first retain a smooth central cutoff supported in \(|z|\le c\min(u,v)\) and equal to one on a smaller such interval. Use the smooth log-gamma expression for the undeformed row before normalization, and sum its square times \(E_s^2\) and this cutoff over all fixed integers \(i\). This is a smooth function of real \(m\), bounded below by \(c w^3\) after extracting the common midpoint binomial factor: the \(O(w)\) indices with \(|i-m|\asymp w\) contribute \(\asymp w^2\) each. Its derivative of order \(k\) is at most \(C_k w^{3-k}\), by the Gaussian moment bounds. The discarded tail is \(O_D(w^{3-D})\) for every \(D\), also after any fixed finite difference: bound that difference by its finitely many values at adjacent centers, where the scales are comparable. The chain rule for the retained positive normalization, followed by this tail estimate, proves the full finite-difference estimates for the exactly normalized rows. For bounded \(w\) the normalization is bounded above and below directly. Thus no derivative of a moving-endpoint sum is being assumed. Lemma 13 now proves that \(H_{q,d}(s)\) is a rapid interaction of bounded strength.

Flux differences. Use common windows for fluxes a bounded distance apart, as in Lemma 5. With orbital labels fixed from the north, their Hamiltonian interactions differ by size \(C/v\). Indeed \[\log\binom{Q+1}{j}-\log\binom Qj =\log(Q+1)-\log(Q+1-j).\] For the path profiles, after canceling their common midpoint values, the corresponding ratio is exactly \[\left(1-\frac{z^2}{(t+1-m)^2}\right)^{-(1-s)/2} \left(1-\frac{z^2}{(q+1-m)^2}\right)^{-s/2}.\] Its difference from one has size \(Cw^2/v^2\le C/v\) in the row-symbol bounds. The preceding central-cutoff normalization argument, applied to the difference of the two profiles, retains this extra factor in every finite-difference bound. In using that argument, the factor \(w^2/v^2\) is retained throughout \(|z|\le cv\), including the part discarded by the smaller central cutoff. Only on \(|z|>cv\) is it recovered from tail powers, using \(v^2/w^2\le C(|z|/w)^2\). This distinction is necessary when \(v\) is much larger than \(u\). If an individual pair coefficient is present only at the larger flux, its index excursion satisfies \(|i-m|\ge v-O(1)\). For large \(v\) its Gaussian tail pays both the inverse of \(w^2/v^2\) and every required symbol seminorm; for bounded \(v\) it is a bounded-scale estimate. The entirely new rows lie in a bounded south zone, where \(1/v\) is bounded below. This proves the claimed common-frame interaction difference, including the changed endpoint supports. Repeating the one-flux identity handles any fixed bounded change in flux.

Affine subtraction. For a smooth real function \(a\) on the orbital interval, set \(X_a=[H_{q,d}(s),D_t(a)]\). Decompose the Hamiltonian into annuli by a sum-of-squares partition \(\sum_k\vartheta_k(b)^2=1\), of row-index width comparable to \(w_k\), and then use its Fourier wavepackets. If \(m_k\) is the annulus center, put \[a_k^\circ(j)=a(j)-a(m_k)-a'(m_k)(j-m_k).\] The affine contribution commutes with the annular sum, because every row has a fixed pair index sum. More explicitly, if \(C_x\) is its Fourier wavepacket and \(C_x(a)\) is the same wavepacket with row coefficients multiplied by \(a_k^\circ(i)+a_k^\circ(j)\), then \[ X_a=\int X_{a,x}\,d\mu(x),\qquad X_{a,x}=C_x^*C_x(a)-C_x(a)^*C_x. \tag{58}\] The center measure has bounded mass in unit balls. The wavepacket normalization is \(w_k^{-1/2}\) and its circle measure is \(w_k\,d\phi/(2\pi)\), so Fourier orthogonality proves Equation (58) exactly. The weighted row bounds give size \(Cw^2|a''(m)|\) whenever Equation (47) holds with \(\beta=w^2|a''(m)|\). For \(a=g_{q,d}\) this size is \(C\alpha_q(m)\): for every \(k\ge2\), \[w(j)^k|g_{q,d}^{(k)}(j)| \le C_k\frac{w(j)^2}{v(j)^2} \left(\frac{w(j)}{v(j)}\right)^{k-2} \le C_k\alpha_q(j),\] and \(\alpha_q\) has tempered ratio exponent four, independently of \(k\). Upon changing \(q\) by a bounded amount, the source interaction changes by size \(C\alpha_q/v\): differences of \(g''\) gain \(v^{-1}\), and the row wavepackets change by \(C/v\). These assertions include every fixed rapid localization order, since the estimates just used also apply after the finite differences used in Fourier localization.

Spectral filter. We use the spectral-filter construction of quasi-adiabatic continuation [6, 2], with locality supplied by Section 3. Choose a smooth odd real function \(\eta\) that is zero on \([-\gamma_{\mathrm{pin}}/2,\gamma_{\mathrm{pin}}/2]\) and equals \(\operatorname{sgn}(\omega)\) for \(|\omega|\ge\gamma_{\mathrm{pin}}\). Let \(r(\omega)=\eta(\omega)/\omega\), with value zero near zero, and use the spectral calculus of \(\operatorname{ad}_{H}(A)=[H,A]\) to set \[Y_{q,d}(s)=r(\operatorname{ad}_{H_{q,d}(s)})X_g.\] The function \(r\) and each of its derivatives belong to \(L^2(\mathbb R)\). Its inverse Fourier transform therefore has every absolute polynomial moment: use Cauchy–Schwarz on \(|t|\le1\), and the \(L^2\) norm of a sufficiently high derivative of \(r\) on \(|t|>1\). Lemma 10 and Equation (58) give rapid locality with size \(\alpha_q\).

The spectral multiplier of \(D_t(g)\) is \(\eta(\omega)\), so \(Y^*=-Y\). If \(P=P_{q,d}(s)\) and \(Q=1-P\), the gap gives \[YP=QD_t(g)P,\qquad PY=-PD_t(g)Q,\qquad PYP=0.\] Differentiating Equation (53) gives \(\partial_sP=QD_t(g)P+PD_t(g)Q=[Y,P]\). The unitary solution therefore transports \(P\). Both \(H_{q,d}(s)\) and \(D_t(g)\) commute with number and with \(\sum_jj c_j^*c_j\), proving the two symmetries.

Finally compare two fluxes in common frames. The change of \(X_g\) has size \(\alpha_q/v\). The Hamiltonian change has size \(1/v\). Inserting the latter into the time evolution in the filter by Duhamel’s formula costs \(\alpha_q/v\), by Lemma 11. All filters are the same for every flux and every path parameter. This proves the difference assertion. ◻

Proof. At the center the cutoff gives \(v\asymp q\), so \(\alpha_q(m)\le C(1+m)/q^2\) and \(\alpha_q(m)/v(m)\le C(1+m)/q^3\). Use the supported-observable bounds in Lemma 11, first for the self-adjoint path generators \(iY_{q,d}(s)\) and \(0\). Since \(\alpha_q=u/((q+1)v)\), its tempered ratio exponent can be taken to be four: each of \(u\) and \(v^{-1}\) costs at most two powers of distance. Equations (38)–(39) give a core factor \(r^6\) and every tail order beyond radius \(Cr\). Apply the same bounds directly to \(iY_{q,d}(s)\) and \(iY_{q',d}(s)\) for the comparison. Their difference has size \(\alpha_q/v=u/((q+1)v^2)\), whose ratio exponent is at most six, and therefore costs \(r^8\). Thus \(c=8\) suffices for both assertions. Choose \(b\) large enough that the initial ball uses identical northern slots. The common-frame generator comparison already includes the padded south modes, so this same bound covers the endpoint changes. Take the ball approximations by the normalized partial trace onto the corresponding slot algebra. This contraction has error at most twice the best approximation error and preserves number grade. Differences at radii \(C_*2^\nu r\) give one decomposition satisfying Equation (61) for all tail orders, after increasing \(C_*\) by one fixed factor. Choose the tail order greater than \(B+1\) and sum the resulting geometric series to obtain Equation (62). ◻

Proof. Retain the annular decomposition of \(Y\) used in its construction. Each annular total commutes with number and axial angular momentum: this is true of its source in Equation (58) and remains true under the time filter. We may consequently replace \(f(j)\), in its commutator with this annular total, by \(f(j)-f(m_k)-f'(m_k)(j-m_k)\). Lemma 14 bounds the resulting one-body interaction near the annulus by \(C\beta_q\); its bounds away from the center grow only polynomially with distance. The weighted commutator calculus thus bounds \([D_t(f),Y(s)]\) by \(C\alpha_q\beta_q\). Integrating \[K_{q,d}(f)=\int_0^1 U_{q,d}(s)^*[D_t(f),Y(s)]U_{q,d}(s)\,ds\] proves the first assertion. For the difference, use the same padded \(D(f)\) on both systems: its additional end-orbital number operators commute with the smaller system’s generator. Thus no uncompensated end value of \(f\) occurs. The source and generator differences supply the factor \(v^{-1}\) proved above; the difference of the transporting evolutions supplies the same factor by Duhamel’s formula. The weights \(A_q\) and \(v^{-1}\) have uniformly polynomial ratios, so Lemma 11 applies also to their product. All decompositions start with an annular source and expand its evolved operator in shells about that source. This gives the last assertion. ◻

Proof. Fix \(s\) and abbreviate \(H=H_{q,d}(s)\) and \(P=P_{q,d}(s)\). The derivative in the transported zero space is the compression of \[ D_t(g)(1-P)D_t(f)+D_t(f)(1-P)D_t(g). \tag{65}\] We estimate this operator before integrating in \(s\).

Choose a smooth \(\chi_+\) equal to zero for \(\omega\le\gamma_{\mathrm{pin}}/2\) and to one for \(\omega\ge\gamma_{\mathrm{pin}}\). Apply the filter \(r_+(\omega)=\chi_+(\omega)/\omega\) separately to the pieces in Equation (58), and set \[G_x=r_+(\operatorname{ad}_H)X_{g,x},\qquad F_x=r_+(\operatorname{ad}_H)X_{f,x}.\] As in the preceding filter argument, its time kernel has all absolute polynomial moments. Every piece has strictly positive spectral frequencies. Thus \[ G_x^*P=F_x^*P=0, \qquad \left(\int G_x\,d\mu(x)\right)P=(1-P)D_t(g)P, \qquad \left(\int F_x\,d\mu(x)\right)P=(1-P)D_t(f)P. \tag{66}\]

At a center of latitude index \(m\), put \(\rho_x=f''(m)/(-g''(m))>0\). The common wavepacket decomposition gives \[ F_x=-\rho_xG_x+E_x, \tag{67}\] where \(G_x\) has size \(C\alpha_q(m)\) and \(E_x\) has size \(C\beta_q(m)/w(m)\), at every fixed rapid order. We verify the improved error explicitly. The function \(a=f+\rho_x g\) has \(a''(m)=0\). Subtract its affine Taylor polynomial, as in Equation (58). In a row window of width \(w\), the third-order remainder is bounded, with all scaled derivatives and polynomially weighted tails, by \[Cw^3\left(\frac{f''(m)}{L+m}+ \frac{\rho_x(-g''(m))}{v}\right) \le C\frac{\beta_q(m)}{w}.\] The last inequality uses \(w^2\le u,v\) and \(u\le L+m\). For windows outside this central region, derivative ratios have polynomial growth and the row tails absorb those powers. The weighted row estimate applied to \(C_x(a)\) proves the claimed source error; the same time filter preserves it. This proves Equation (67) without making any assertion about the sign of individual unfiltered row pieces.

Write \(R_g=\int G_x\,d\mu(x)\), \(R_f=\int F_x\,d\mu(x)\) and \(T=\int\sqrt{\rho_x}G_x\,d\mu(x)\). Each integral is finite at fixed flux. From Equation (66), a reversed product with a raising operator on the left vanishes in the compression, giving \(PG_x^*G_yP=P[G_x^*,G_y]P\). Expanding Equation (67) therefore gives the exact identity \[\begin{align*} P(R_g^*R_f+R_f^*R_g)P ={}&-2PT^*TP \\ &-\iint(\sqrt{\rho_x}-\sqrt{\rho_y})^2 P[G_x^*,G_y]P\,d\mu(x)d\mu(y) \\ &+\iint P\bigl([G_x^*,E_y]+[E_x^*,G_y]\bigr)P \,d\mu(x)d\mu(y). \tag{68}\end{align*}\]

We give the norm bounds for the last two integrals, since their spatial summability is essential. If \(D=\operatorname{dist}(x,y)\), rapid localization and the even parity of the pieces imply, for every fixed \(M\), \[\|[G_x^*,G_y]\|\le C_M\alpha_x\alpha_y(1+D)^{-M}, \qquad \|[G_x^*,E_y]\|\le C_M\alpha_x\frac{\beta_y}{w_y}(1+D)^{-M}.\] Here subscripts denote evaluation at the corresponding center. The logarithmic derivative of \(\rho\) is bounded by \(C((L+m)^{-1}+v^{-1})\). Using radial resolution \(w\) and the polynomial ratio bounds of Section 3 consequently gives a fixed exponent \(k\) such that \[|\sqrt{\rho_x}-\sqrt{\rho_y}| \le C\sqrt{\rho_x}\,\frac{D}{w_x}(1+D)^k.\] This bound follows by integrating the logarithmic derivative along the meridian between the two latitudes when the distance is at most a small multiple of \(w_x\); outside that region its right side dominates the polynomial ratio bound after increasing \(k\). Since \(\rho_x\alpha_x^2\asymp A_x\), choose \(M\) above all these polynomial exponents plus the two-dimensional volume exponent. Summing first in \(y\) bounds the second line of Equation (68) by \(C\int A_x/w_x^2\,d\mu(x)\), and its third line by \(C\int A_x/w_x\,d\mu(x)\). Since \(w\ge1\) and \(-2PT^*TP\le0\), Equation (65) is at most \(C\int_0^t A_q(m)/w(m)\,dm\) in quadratic form. Conjugate back by \(U_{q,d}(s)\) and integrate over \(0\le s\le1\) to obtain Equation (64). ◻

Proof. The first assertion is the integrated interaction difference in Proposition 18; vacant extra modes are compressed after comparison in the common frame. For one growth step, differentiate the expectation of the interaction along \(U_{Q,1}(s)\). The generator has size \(\alpha_Q\le C/v\), while the observable interaction has size \(A_q\). The weighted commutator calculus bounds the derivative in operator norm by \(C\int A_q/v\). Finite-parameter conjugation preserves this bound. There are at most two growth steps, and all size functions for their fluxes are comparable. Applying this argument to each selected source piece before summing proves the last assertion, including comparisons with the directly embedded lower-flux reference on common northern slots. ◻

Proof. The absence of orbital \(q\) is equivalent to divisibility by the north spinor coordinate in every variable, which is precisely one pin at the south pole. Evaluation of one electron at that pole leaves the three pair zeros attached to every remaining electron. This proves that the range of \(c_q\) lies in \(Z_{n-1,q}^{(3)}\).

To prove surjectivity, use the affine coordinate in which the south-pole coefficient is the coefficient of the highest power. After factoring the Laughlin polynomial, the map on symmetric polynomials extracts the coefficient of \(z_n^h\). A basis of its target consists of monomial symmetric functions indexed by partitions with at most \(n-1\) parts, all at most \(h\). Append a part equal to \(h\) and take the corresponding monomial symmetric function in \(n\) variables, with each distinct monomial included once. Extraction returns the original monomial symmetric function. This proves surjectivity, including \(h=0\). Restriction to the orthogonal complement of the kernel is therefore bijective, and its polar part is unitary onto the range. Composing with the endpoint transports gives the stated isometries. The assertions about \(h\) follow directly from (70). ◻

Proof. We first prove the singular-value bound, keeping track of the restriction of pair rows. The normalized rows are \[B_{q,b}=\sum_{\substack{0\leq i<j\leq q\\i+j=b}} a_{q,b}(i,j)c_jc_i,\qquad a_{q,b}(i,j)= \frac{(j-i)\sqrt{\binom qi\binom qj}} {\sqrt{q\binom{2q-2}{b-1}}},\quad 1\leq b\leq2q-1.\] For completeness, their normalization follows by taking coefficients in \[\sum_{i<j}(j-i)^2\binom qi\binom qj x^{i+j} =(1+x)^q(x\partial_x)^2(1+x)^q -\bigl((x\partial_x)(1+x)^q\bigr)^2 =qx(1+x)^{2q-2}.\] If a row contains orbital \(q\), put \(k=2q-b\). Its removed coefficient mass is \[p_{q,k}=\frac{k\binom{q-1}{k-1}}{\binom{2q-2}{k-1}}.\] Here \(p_{q,1}=p_{q,2}=1\), and \[\frac{p_{q,k+1}}{p_{q,k}} =\frac{(k+1)(q-k)}{k(2q-k-1)}\leq1.\] It follows that every row with a retained pair has retained mass \[ r_{q,b}=\sum_{\substack{i<j<q\\i+j=b}}|a_{q,b}(i,j)|^2 \geq\frac{q}{4q-6}\geq\frac14. \tag{74}\] Rows with no coefficient at \(q\) have mass one. The two top rows retain no pair and are omitted.

For \(d=1\), the pinning weights obey \(g_j=\tfrac12\log((q-j)/q)\). Multiplication of its coefficients by \(e^{-g_i-g_j}\), followed by normalization, turns a flux-\((q-1)\) row into the normalized retained flux-\(q\) row. More precisely, if \(B_{q,b}^{\mathrm{ret}}\) denotes the row with the coefficient at orbital \(q\) deleted, then \(B_{q,b}^{\mathrm{ret}}=\sqrt{r_{q,b}}B_{q,1,b}(1)\), in the notation of Equation (52). Let \(\delta>0\) be the uniform endpoint gap from Lemma 15. By (74), the Hamiltonian compressed to the south-empty sector has a gap at least \(\delta/4\) above \(\bigoplus_n\mathcal F_{n,q}\).

The CAR give the exact identity \[\sum_b[B_{q,b},n_q]^*[B_{q,b},n_q] =n_q\sum_{k=1}^{q}p_{q,k}n_{q-k}.\] The coefficient sum is \[\sum_{k=1}^{q}p_{q,k}=\frac{4q-2}{q+1}<4.\] One way to see this is to write \(p_{q,k}=k\binom{2q-k-1}{q-1}/\binom{2q-2}{q-1}\) and apply the hockey-stick identity twice. For \(\psi\in Z_q\), the two top singleton rows also imply \(c_{q-1}c_q\psi=c_{q-2}c_q\psi=0\). Hence \[ \sum_b\|[B_{q,b},n_q]\psi\|^2 \leq2\|c_q\psi\|^2. \tag{75}\] Let \(Q=1-n_q\) and let \(P_F\) project onto \(\bigoplus_n\mathcal F_{n,q}\). Since \(B_{q,b}\psi=0\), the restricted gap gives \[\frac\delta4\|Q\psi-P_F\psi\|^2 \leq\sum_b\|B_{q,b}Q\psi\|^2 \leq2\|c_q\psi\|^2.\] The occupied and vacant components are orthogonal, so \[\|\psi-P_F\psi\|^2 \leq(1+8/\delta)\|c_q\psi\|^2.\] Thus (73) holds with \(\varepsilon=\delta/(\delta+8)\).

To construct local representatives, take a real smooth frequency cutoff \(\chi\) equal to one near zero and supported strictly inside the uniform unperturbed gap. Apply its time filter to \(c_q\): \[A_q=\int_{\mathbb R}\widehat\chi(t) e^{itH_q}c_qe^{-itH_q}\,dt,\] with the Fourier normalization incorporated into \(\widehat\chi\). Contraction preserves zero modes. The cutoff therefore gives \(A_qP_q=c_qP_q\) and \([A_q,P_q]=0\). The local calculus makes \(A_q\) rapidly localized at the south pole, with uniform constants. Choose a smooth function \(\eta\) on a fixed interval containing the spectrum of \(A_q^*A_q\), equal to zero near zero and to \(x^{-1/2}\) for \(x\geq\varepsilon\). Then \[\widetilde{\mathsf T}_q=A_q\eta(A_q^*A_q)\] is rapidly localized by smooth functional calculus and agrees with \(\mathsf T_q\) on \(Z_q\). Its products \(\widetilde{\mathsf T}_q^*\widetilde{\mathsf T}_q\) and \(1-\widetilde{\mathsf T}_q^*\widetilde{\mathsf T}_q\) represent the branch projections on \(Z_q\). The construction preserves number and index grades because \(H_q\) preserves both and \(A_q^*A_q\) has grade zero. It also preserves \(P_q\) in both directions. No claim about a spectral gap of \(A_q^*A_q\) outside \(Z_q\) is needed: \(\eta\) is smooth on that entire spectral interval. ◻

Proof. The \(F\) block is the definition of \(K_{q,1}(f)\). For an affine sequence, the grades of \(\mathsf T_q\) give \[D_q(f)\mathsf T_q=\mathsf T_qD_q(f)-f(q)\mathsf T_q\] on the zero spaces, and both branch projections commute with \(D_q(f)\). The endpoint transports commute with the number and index operators. This proves the affine assertion, including the scalar \(f(q)\) in \(E\).

For a general \(f\), subtract its affine Taylor polynomial at \(q\). The remainder \(a(j)=f(j)-f(q)-f'(q)(j-q)\) satisfies \[|a(j)|\leq C_p\frac{f(q)}{(L+q)^2} (1+q-j)^{p+4},\qquad 0\leq j\leq q.\] The same polynomial bounds hold for the finite differences needed for localization. In the south-pole frame this says that \(D_q(a)\) has one-body interaction size at most \(C_pf(q)/(L+q)^2\), times a fixed polynomial in distance from the pole. Commuting it with any of the rapidly localized representatives in Lemma 22 thus has norm at most that size: decompose the representative into ball shells and sum its arbitrarily high inverse-power tails against this polynomial. In the \(E\) block use \(\mathsf T_q[D_q(a),\mathsf T_q^*]\); for the off-diagonal blocks use the commutator with a branch projection. Compression and the endpoint isometries do not increase these norms. This proves (76). ◻

Proof. For \(F\), this is Corollary 17. Its norm and tail estimates are Equations (59) and (60); enlarge the fixed core constant to write the outer ball radius as \(R\) instead of \(r+R\). For \(E\), commute \(O\) through the local representative of \(\mathsf T_q\) before applying that corollary with \(d=3\). The support of \(O\) is a distance comparable with \(\sqrt q\) from the south pole when \(b_0\) is increased if necessary. Rapid locality therefore makes the polar commutator and the off-branch blocks \(O(q^{-D})\) for every \(D\). The same estimate handles the finitely many trimmed orbitals. The difference assertion for the remaining changes is the same-corollary comparison of the pinning transports at bounded flux difference. ◻

Proof. Expand a slot observable into normally ordered CAR monomials. Its expectation in an embedded state is a linear combination of the state’s normally ordered orbital correlations. The coefficients use only the analysis-frame entries of the slots occurring in the monomial. Entries at orbital labels above \(s\) contribute zero, because those orbitals are vacant. At labels at most \(s\), the common northern frame entries are identical. Thus every monomial has the same expectation in both realizations. This also proves the assertion after physical compression of the slot observable, which leaves these matrix elements unchanged. ◻

Proof. The proof keeps \(O\) unchanged while descending through common northern slots. Only the small changes caused by each transport are estimated inductively. This distinction prevents an increasing radius from being assigned repeatedly to the original observable.

We first describe the centered one-step identities under the northern condition of Lemma 24. For an \(E\) step put \(t=q-3\), so \(t^0=q^0-3\), and write \(R_{q,E}(O)\) for its uncentered polar remainder in Lemma 24. The filled \(E\) step at \(q^0\), together with compatibility of the northern references, gives the exact scalar identity \[\operatorname{ref}_q(O)-\operatorname{ref}_t(O) =\operatorname{ref}_{t^0}\bigl( \mathcal D_{q^0,3}(O)+R_{q^0,E}(O)\bigr).\] Thus the parent-to-child reference change is determined by a filled step. Define \[\begin{split} R_{q,E}'(O)&=R_{q,E}(O) -\operatorname{ref}_{t^0}(R_{q^0,E}(O))I,\\ s_q(O)&=\operatorname{ref}_t(\mathcal D_{q,3}(O)) -\operatorname{ref}_{t^0}(\mathcal D_{q^0,3}(O)). \end{split}\] The references in the second line use the same filled state. Lemma 25, first on common northern slots and then on rapid tails by norm approximation, therefore gives \[s_q(O)=\operatorname{ref}_{t^0}\bigl( \mathcal D_{q,3}(O)-\mathcal D_{q^0,3}(O)\bigr) +O(q^{-D}).\] The bounded-flux comparison in Lemma 24 supplies the additional factor \(q^{-1}\). Substituting these scalar identities into the branch comparison yields \[ \begin{split} J_{q,E}^*(O-\operatorname{ref}_q(O))J_{q,E} ={}&(O-\operatorname{ref}_t(O)) +\bigl(\mathcal D_{q,3}(O) -\operatorname{ref}_t(\mathcal D_{q,3}(O))\bigr)\\ &+s_q(O)I+R_{q,E}'(O),\\ |s_q(O)|\leq{}&Cr^c u/q^3+C_Dq^{-D},\qquad \|R_{q,E}'(O)\|\leq C_Dq^{-D}. \end{split} \tag{82}\] Here and below the symbols denote compressions to the relevant child zero space. When \(h=0\) the comparison is its own filled calibration, so all centered contributions vanish exactly.

For an \(F\) step, \(t=q-1\). The nearest smaller filled fluxes are either identical or differ by three. In the latter case apply the filled \(E\) comparison just used. Together with (77), this proves \[ J_{q,F}^*(O-\operatorname{ref}_q(O))J_{q,F} =O-\operatorname{ref}_t(O)+S_{q,F}(O),\qquad \|S_{q,F}(O)\|\leq Cr^cu/q^2+C_Dq^{-D}. \tag{83}\] The off-diagonal blocks have norm \(C_Dq^{-D}\). An \(F\) step already records a flag, so its entire error will be charged to that flag.

Here is a uniform closure argument, including the parameter choices. Let \(c,b_0\) be as in Lemma 24. Choose \(b>\max\{b_0,c\}\), then choose \(B>\max\{ab,c\}\) large enough to absorb the fixed polynomial costs of the local calculus. A constant \(M\) will be fixed after \(B\). Set \[\tau=Mr^b u.\] If \(q\leq\tau\) and \(h>0\), every flag has \(b'\leq q\), so \[W_m\geq(2Mr^b)^{-a}I.\] The norm bound \(\|O-\operatorname{ref}_q(O)\|\leq2\) proves (81) in this region with constant \(2(2M)^a\). When \(h=0\) the centered compression is zero. Increasing \(M\) includes all stopping cases in this argument.

Fix a finite \(Q\), and let \(K_Q\) be the least constant in (81) for fluxes at most \(Q\), with the chosen \(B\), all number sectors, and all allowed frames. This number is finite: for \(h>0\) use \(W_m\geq(1+Q)^{-a}\), while the filled compression is exactly centered. We prove \[ K_Q\leq C_M+\frac12K_Q \tag{84}\] with \(C_M\) independent of \(Q\).

Suppose \(q>\tau\). Expand the unchanged \(O\) down the branching tree, using (82) and (83), until the current flux is at most \(\tau\). The original northern slots, center, and radius are unchanged through this part of the descent. The terminal comparisons have just been bounded. In an \(E\) child, decompose \(\mathcal D_{q,3}(O)\) into its core and dyadic ball shells, all centered at \(m\). Equation (62), with a tail order larger than \(B+2\), gives \[\sum_\nu r_\nu^B\|\mathcal D_{q,3}(O)_\nu\| \leq C(B)r^{B+c}u/q^2.\] Applying the definition of \(K_Q\) to each smaller-flux shell, and using linearity of the reference, bounds its centered contribution by \[K_QC(B)r^{B+c}u/q^2\] times the remaining history weight in that child. Along any word, the flux strictly decreases; thus \[\sum_{q'>\tau}\frac{u}{(q')^2}\leq C\frac{u}{\tau} =\frac{C}{Mr^b}.\] Consequently all induction-dependent contributions are bounded by \[K_Q\frac{C(B)}M r^{B+c-b}W_m.\] Choose \(M\) so large that this is at most \(\tfrac12 K_Qr^BW_m\). This choice is possible because \(b>c\) and \(r\geq2\).

We verify that the other errors have a bound independent of \(K_Q\). At a node with positive remaining hole number, its identity is bounded by the sum of the identities assigned to its remaining flags. A scalar electron error may therefore be charged to every such flag. For a flag at \(\ell\), the total charge from ancestors in (82) is at most \[ C\frac{r^cu}{\max\{\tau,1+\ell\}^2}. \tag{85}\] The \(F\) error in (83) occurs at its own flag and obeys the same bound, up to a fixed constant. Write \(w_\ell=\min\{1,(u/(1+\ell))^a\}\). Since \(a<2\) and \(\tau=Mr^bu\), \[\frac{r^cu}{\max\{\tau,1+\ell\}^2} \leq C M^{a-2}r^{c+b(a-2)}w_\ell \leq C_M r^B w_\ell.\] The \(q^{-D}\) remainders, including the off-diagonal blocks, obey the same estimate after choosing \(D>a+2\). For a self-adjoint off-diagonal block, its norm bounds the full block matrix above and below by that norm times the identity at its node, so the same charge applies. Nodes with \(h=0\) contribute nothing, by the exact centered filled identity. Summation of these diagonal bounds along words proves (84).

It follows that \(K_Q\leq2C_M\) for every \(Q\), proving the uniform claim. Finally, the assertion for \(A=\sum_\nu A_\nu\) follows by linearity of the reference and by summing the form inequalities. ◻

Proof. Choose \(a\in(1,2)\) in Theorem 26. The symmetric-polynomial factor in (71) has degree one in each variable. It is the irreducible spin-\(n/2\) representation, so \(Z_{n,q}\) is irreducible and has dimension \(n+1\).

When \(h=1\), an \(F\) branch has \(h=0\) and hence dimension one. Before that branch all steps are \(E\), which lower the flux by three. Therefore at most one one-dimensional history has a flag at any specified nonterminal flux. The history that reaches the stopping region without an \(F\) step has bounded terminal dimension. It follows that, as an operator on \(Z_{n,q}\), \[ \operatorname{Tr}(W_0) \leq C+\sum_{b\geq0}(1+b)^{-a}\leq C_a. \tag{87}\] The trace is unchanged on conjugating back to the physical zero space. Write \(\widehat W_0=\mathcal R_{n,q}^*W_0\mathcal R_{n,q}\) for that physical operator.

Let \(V_q(g)\) denote the rotation representation, with normalized Haar measure \(dg\). Schur’s lemma gives \[\int_{\mathrm{SU}(2)}V_q(g)\widehat W_0V_q(g)^*\,dg =\frac{\operatorname{Tr}(\widehat W_0)}{n+1}I.\] For each rotation use the readout axis whose north pole is the image of the original north pole. Precisely, write \(x(g)=g\cdot\mathrm{north}\) and \(W^{g}=V_q(g)\widehat W_0V_q(g)^*\). Use the rotated frame and rotated branch construction at \(x(g)\). Theorem 26, applied to \(O_{x(g)}\), gives \[\bigl|\langle\psi,O_{x(g)}\psi\rangle -\langle\phi,O_{x(g)}\phi\rangle\bigr| \leq Cr^{B'}\bigl(\langle\psi,W^g\psi\rangle +\langle\phi,W^g\phi\rangle\bigr).\] The two reference expectations cancel in this inequality. Changes between local frames cost only a fixed polynomial in \(r\) by the local calculus. Integrating rotations and multiplying by the sphere area yields at most \(Cr^{B'}q\operatorname{Tr}(W_0)/(n+1)\), which is uniformly bounded because \(q=3n-2\). For any center measure with bounded mass in unit balls, spread the mass at each center over a ball of fixed radius. The resulting density relative to area is bounded uniformly. The operator originally assigned to \(x\) is supported in a ball of radius \(r+1\) about every point in that spreading ball, so the same pointwise bound applies before integration. This reduces the general center measure to the area estimate. Finally, summing ball shells proves the assertion for rapid families. ◻

Proof. The exact formula \[ \ell_q=f(q)-\left(1+\frac2{q+1}\right) \frac{f(q)+f(q-1)+f(q-2)}3 +\frac{2\sum_{j=0}^{q-3}f(j)}{(q+1)(q-2)} \tag{95}\] is obtained by separating the last three summands in \(C_q\). A useful positive form of this identity is obtained by putting \(d_j=f(j)-2f(j+1)+f(j+2)\) and \(Q_q=(q+1)(q-2)\): \[ \ell_q=\frac1{Q_q}\sum_{j=0}^{q-3}(j+1)(j+2)d_j +\frac{2q}{3(q+1)}d_{q-2}. \tag{96}\] Expanding the second differences verifies the formula coefficient by coefficient. In particular, \(\ell_q\ge0\), and constant and affine functions contribute zero.

Here are precise discrete-to-continuous error bounds: \[\begin{align*} |\ell_q-\ell(q)|+ \sup_{x\in[q-3,q]}|\ell(x)-\ell(q)| &\le C(L+q)^{-2},\tag{97}\\ |\ell_q-\ell_{q-r}| &\le \begin{cases} C L^{-2},&q\le L,\\ C(F(q)+f(q))q^{-2},&q>L, \end{cases} \tag{98}\end{align*}\] for \(r=0,1,2\). We give details because the sharper second estimate is needed for the summable weight. The identity \(d_j=\int (1-|t-j-1|)_+f''(t)\,dt\) expresses (96) as \(\int_0^q \kappa_q(t)f''(t)\,dt\). The function \(\kappa_q\) is linear between consecutive integers and has values \[\kappa_q(i)=\frac{i(i+1)}{Q_q}\ (0\le i\le q-2),\qquad \kappa_q(q-1)=\frac{2q}{3(q+1)},\qquad \kappa_q(q)=0.\] On \([0,q-2]\) its difference from \(t^2/q^2\) is at most \(C(t/q^2+t^2/q^3)\), and on \([q-2,q]\) that difference is bounded by a constant. The error in its integral against \(f''\) is consequently at most \[C\left(q^{-2}\int_0^q tf''(t)\,dt+ q^{-3}\int_0^q t^2f''(t)\,dt+ \int_{q-2}^q f''(t)\,dt\right).\] For \(q\le L\) use \(f''\le C/L^2\); for \(q>L\) use \(\int_0^q tf''\le1\), \(\int_0^q t^2f''\le2qF(q)\), and \(f''(t)\le C f(t)(L+t)^{-2}\). This proves the first term of (97). The continuous identity \(\ell(x)=x^{-2}\int_0^x t^2f''(t)\,dt\) implies \(\ell'(x)=f''(x)-2\ell(x)/x\), which proves the oscillation estimate.

For (98), difference the positive formula directly: \[\begin{split} \ell_q-\ell_{q-1} ={}&-\frac{2(q-1)}{Q_qQ_{q-1}} \sum_{j=0}^{q-4}(j+1)(j+2)d_j\\ &+\frac{(q-1)(q-2)}{3q(q+1)}d_{q-3} +\frac{2q}{3(q+1)}d_{q-2}. \end{split}\] The first sum is bounded by \(CqF(q)\), as follows by its integral representation and \(\int_0^q t^2f''\le2qF(q)\). This gives \(CF(q)/q^2\) when \(q>L\); the last two terms cost \(Cf(q)/q^2\). Below \(L\) use \(d_j\le C/L^2\) in the displayed formula. A bounded number of differences proves the claimed bound.

The exact F-step identity is \[C_{q-1}(f)-C_q(f)+\frac13 f(q) =\frac{2}{3q(q+1)}\sum_{j=0}^{q-1}\bigl(f(j)-f(q)\bigr).\] For \(q\le L\) its ratio to \(f(q)\) is \(O(L^{-1})\). For \(q>L\) use \[ F(q)\le \begin{cases} f(q)/(1-p),&p<1,\\ C_pL/q,&p>1. \end{cases} \tag{99}\] The resulting ratio is \(O(L^{-1})\) also when \(1<p<2\), since \(L^{1-p}q^{p-2}\le L^{-1}\) for \(q\ge L\). The case \(p=1\) follows from the explicit logarithmic integral and is not needed below. Dividing (98) by \(f(q)\) and summing gives \(O(L^{-1})\) below \(L\). Above \(L\) it gives \(O(L^{-1})\): the summands are \(O(q^{-2})\) for \(p<1\) and \(O(q^{-2}+L^{1-p}q^{p-3})\) for \(1<p<2\).

Twice integrating \(f''\) proves \[\ell(x)=\frac1{x^2}\int_0^x t^2f''(t)\,dt\ge0, \qquad \int_k^\infty\ell(x)\,dx=2F(k)-f(k).\] The E-step intervals \([q-3,q]\) have disjoint interiors along a history. Consequently their continuous contribution is at most \(\frac13\int_{(k-3)_+}^{\infty}\ell(x)\,dx\). For \(p<1\), (99) bounds this by \(\frac{1+p}{3(1-p)}f(k)\), up to \(O(L^{-1})f(k)\) from the bounded endpoint shift. Finally, the tail sum of (97) is bounded by \(C/(L+k)\), and \[\frac1{L+k}\le\frac1L f(k)\qquad(0<p<1).\] Thus it is \(O(L^{-1})f(k)\) uniformly in \(k\). This proves (94). ◻

Proof. For \(q\le L\), integrating \(u^2/L^2\) gives \(B_q\le C(q+1)L^{-2}\); for \(q>L\), dropping the factor \((u/(L+u))^2\) gives \(B_q\le C F(q)/q\). These estimates and (99) prove the first bound in (101). On the northern half of the sphere \(v\ge(q+1)/2\), whereas on the southern half \(A_q(m)\le C f(q)(L+q)^{-2}\). Therefore \[\frac{J_q}{f(q)}\le \frac{C B_q}{(q+1)f(q)}+ \frac{C\log(q+2)}{(L+q)^2}.\] The first term sums to \(O(L^{-1})\): above \(L\) it is bounded by \(Cq^{-2}\) for \(p<1\) and by \(CL^{1-p}q^{p-3}\) for \(p>1\). The second sums to \(O((1+\log L)/L)\).

For the supremum in (102), use \(CL^{-2}\) when \(q\le L\) and \(C(q+1)^{-2}\) otherwise. The latter divided by \(f(q)\) is \(O(L^{-p}q^{p-2})\), whose sum is \(O(L^{-1})\) for \(p<1\). On the northern half, \(w\ge c\sqrt u\), so the last integral is at most \[\frac{C}{(q+1)^2}\int_0^{q/2} \frac{f(m)u^{3/2}}{(L+u)^2}\,dm +\frac{C\sqrt{q+1}\,f(q)}{(L+q)^2}.\] After division by \(f(q)\), both terms are bounded by \(C\sqrt{q+1}/L^2\) for \(q\le L\) and by \(Cq^{-3/2}\) for \(q>L\), provided \(p<1/2\). Their sums are \(O(L^{-1/2})\). ◻

Proof. In the filled E step at \(q^0\), both parent and child compressions of \(D(f)-C(f)\mathbf1\) vanish. Proposition 23 therefore gives \(|\ell_{q^0}+\langle K_{q^0}\rangle|\le C f(q^0)(L+q^0)^{-2}\). Corollary 20 compares this filled-step expectation first with the directly embedded lower-flux state and then with its at most two F-growth transports. Its bound is \(CJ_q\) for each comparison, proving (103) after adding \(|\ell_q-\ell_{q^0}|\).

For a northern source center, the linear reference functional of Theorem 26, applied at child flux \(q-3\), is the expectation in the same directly embedded filled state at \(q^0-3\), using the common slots. The last assertion of Corollary 20 applies to any selected collection of source pieces and bounds the sum of absolute comparison errors by \(CJ_q\). It therefore also permits the annulus-dependent choice of centering in the statement. ◻

Proof of Proposition 28 for \(p=1/8\). Apply Proposition 23 recursively to \(D_q(f)-C_q(f)\mathbf1\). The norm remainder at a node is at most \(C f(q)(L+q)^{-2}\), and its total cost is \(o_L(1)S_{L,p}\) by (104) and Lemma 30. F transport has norm at most \(CB_q\). Together with (92), its contribution is \[-\bigl(\tfrac13-o_L(1)\bigr)f(q)\] for the flag created at that step.

At an E step with one remaining hole, \(q\equiv1\pmod3\), so \(\chi_q\) also has one hole. Compare the child vector to \(\chi_q\) using Corollary 27. The difference of the \(K_q\) expectations is at most \(C\sup_m A_q(m)\). This is a two-sided form bound since it holds for every unit vector in the child space. The calibration error and all one-hole errors have total cost \(o_L(1)S_{L,p}\) by Lemmas 29 and 30.

At an E step with at least two remaining holes, Proposition 19 bounds the increase from \(K_q\) by \(C\int A_q(m)/w\,dm\). Its total cost is again \(o_L(1)S_{L,p}\). It remains to pay the positive part of \(\ell_q\). Divide that scalar equally among the remaining flags. A fixed flag then pays at most one half of each \((\ell_q)_+\) at such an ancestor. The E-step intervals are disjoint, so Lemma 29 bounds the total payment for a flag at \(k\) by \[\left(\frac{1+p}{6(1-p)}+o_L(1)\right)f(k) =\bigl(\tfrac3{14}+o_L(1)\bigr)f(k).\]

At terminal fluxes there are only finitely many spaces. Uniformly on them, \(f(j)=1+O(L^{-1})\), and hence \[D_q(f)-C_q(f)\mathbf1=-\frac h3\mathbf1+O(L^{-1}).\] For \(h=0\) the expression is exactly zero by rotational invariance; for \(h\ge1\) the error is \(O(L^{-1})h\), which is also \(O(L^{-1})\) times the terminal flag weight. Terminal flags therefore receive the same credit \(1/3-o_L(1)\). Combining all bounds yields \[\mathcal R_{n,q}(D_q(f)-C_q(f)\mathbf1)\mathcal R_{n,q}^* \le-\bigl(\tfrac13-\tfrac3{14}-o_L(1)\bigr)S_{L,p}.\] The strict margin is \(5/42\). Choose \(L=L_0\) so that the total error is less than \(5/84\) and take \(c=5/84\). ◻

Proof. To specify the rotation average on a physical zero space, let \(V_{n,q}(g)\) denote its rotation representation and define \[S_{L,p}^{g}=V_{n,q}(g)\mathcal R_{n,q}^*S_{L,p} \mathcal R_{n,q}V_{n,q}(g)^*.\] Thus \(S_{L,p}^{g}\) acts on \(Z_{n,q}\), whereas \(S_{L,p}\) itself acts in history coordinates. This is the history construction obtained by rotating a fixed readout frame by \(g\); its estimates are uniform in \(g\). Use this convention separately in each number sector.

For a local term centered at \(x(g)=g\cdot\mathrm{north}\), use Theorem 26 in the frame rotated by \(g\), with exponent \(a=7/4\). The local term itself need not belong to a rotation-covariant family; the estimate is pointwise in \(g\). Since \((1+b)^{-a}\le C f_{L_0,1/8}(b)\), the local comparison is bounded by the corresponding rotated \(S_{L_0,1/8}\). Its reference scalar cancels between the two vectors. By irreducibility of \(U_q\), normalized rotation averaging gives the one-particle identity \[ \int_{SU(2)} D_q(f)^{g}\,dg =\frac{\sum_{j=0}^q f(j)}{q+1}\,\mathcal N, \tag{106}\] where Haar measure has mass one. Proposition 28 for \(p=1/8\) consequently gives \[\int_{SU(2)}\langle S_{L_0,1/8}^{g}\rangle_\psi\,dg \le\frac{h}{3c(q+1)}\sum_{j=0}^q f_{L_0,1/8}(j).\] The physical area is \(2\pi q\), and the center density is bounded. Multiplying by that area and using \(\sum_{j=0}^q f_{L_0,1/8}(j)\le C(q+1)^{7/8}\) proves the result. The same argument applies to any center measure with bounded mass in unit balls: spread each center uniformly over a ball of fixed radius and regard its observable as centered at the new point. The resulting measure has bounded area density, and the support radius increases by only a fixed constant. Thus the preceding integral estimate applies to both discrete and continuous center measures. Finally, a rapid shell expansion multiplies the radius-\(r\) estimate by a summable series of shell norms times \((1+r)^B\). ◻

Proof of Proposition 28 for \(p=3/2\). First consider \(q\le T_M\). Center the annular interaction at \(\chi_q\). If the child has \(h>0\) holes, the calibration state has \(r\le h\) holes, so Corollary 32, applied at child flux \(q-3\), bounds the centered interaction in absolute quadratic form by \[ C(q+1)^{-2}f_M a_M(q+1)^{7/8}h\mathbf1. \tag{107}\] The child flag weight is at least \(hf(q)\). For \(M\le q+1\le T_M+1\), \(f_M/f(q)\le C M^{p\delta}\). Thus summing the relative cost (107) over these fluxes gives \[ C a_M M^{p\delta}\sum_{q+1\ge M}(q+1)^{-1-1/8} \le C a_M M^{-(1/8-p\delta)} =C a_M M^{-1/16}. \tag{108}\] The finitely many fluxes near the stopping range obey the same estimate after increasing its constant.

Now let \(q>T_M\). With \(C_0\) sufficiently large, the annulus lies in the common northern region. Center it by the linear reference functional in Theorem 26. Lemma 31 accounts for this change of centering with the already budgeted error \(CJ_q\). For a center in the annulus, each descendant history \(\omega\) satisfies \[ \begin{split} W_m(\omega) &\le C\sum_{k\in\mathcal B(\omega)} \min\left(1,\left(\frac{M}{1+k}\right)^a\right)\\ &\le\frac C{f_M}\sum_{k\in\mathcal B(\omega)}f(k) =\frac C{f_M}S_{L,3/2}(\omega). \end{split} \tag{109}\] This is also a diagonal operator inequality in the descendant history space. Indeed, for \(k\le M\), \(f(k)\ge f(M)\); for \(k>M\), use \(a=7/4>p\) and \(M/(1+k)\le C(L+M)/(L+k)\). The annulus has area \(O(M)\). The local readout estimate therefore bounds its centered interaction by \[C(q+1)^{-2} f_Ma_M\frac M{f_M}S_{L,3/2}.\] For rapid interactions this formula follows by applying the radius-\(r\) bound to each shell and summing its norm times \((1+r)^B\). The shell center stays in the same annulus even when its support extends beyond it; shells reaching the south are covered by the large-radius part of the readout estimate. Summing over \(q>T_M\) gives \[ C a_M M\sum_{q>T_M}(q+1)^{-2}S_{L,3/2} \le C a_M M^{-\delta}S_{L,3/2}. \tag{110}\]

Both resulting sums over dyadic \(M\) tend to zero with \(L\). Explicitly, for every \(0<\eta<2\), \[ \sum_{M\in\{1,2,4,\ldots\}} \left(\frac M{L+M}\right)^2M^{-\eta} \le C_\eta L^{-\eta}. \tag{111}\] For \(M\le L\) this follows by summing \(L^{-2}M^{2-\eta}\); for \(M>L\) it follows by summing \(M^{-\eta}\). Apply this with \(\eta=1/16\) and \(\eta=1/24\) to (108) and (110). The node summation rule after (104) now gives an error \(o_L(1)S_{L,3/2}\) for the entire centered E contribution.

There is no uncompensated electron scalar in this argument. Every F flag and every terminal flag contributes \(-(1/3-o_L(1))f(k)\), while all E contributions and all remainders cost \(o_L(1)S_{L,3/2}\). Choose a fixed \(L=L_1\), after \(L_0\) and the constant in Corollary 32, so that the combined error is at most \(1/6\). This proves (89) with \(c=1/6\). ◻

Proof. Use the local readout estimate with \(a=7/4\), now comparing \(W_0\) to \(S_{L_1,3/2}\). Since the flux is divisible by three, the reference is the filled Laughlin expectation. The rotation average (106) and the second part of Proposition 28 give \[\int_{SU(2)}\langle S_{L_1,3/2}^{g}\rangle_\psi\,dg \le\frac{h}{3c(q+1)}\sum_{j=0}^q f_{L_1,3/2}(j) \le\frac{C h}{q+1},\] because \(L_1\) is fixed and \(\sum_{j\ge0}f_{L_1,3/2}(j)<\infty\). Integration over centers costs area \(O(q+1)\), exactly as in the proof of Corollary 32. The result is \(C_Wh=3C_W(N-n)\); absorb the factor three into \(C_W\). Polynomial radius costs and rapid shell summability are preserved by that argument. Decomposing a density matrix into number sectors and then into pure states proves the last assertion. ◻

Proof. The following two elementary estimates specialize the general pair localization bounds to the cutoff used here. In particular, the exponential estimate on distant rows will let us preserve coefficient one on their energy. Both follow directly from the coefficients in (10). For \(b\le q/2\) and every fixed integer \(r\ge0\), \[ \sum_{\substack{i<j\\i+j=b}}|v_{b-1}(i,j)|\,|i-b/2|^r \le C_r b^{r/2+1/4}. \tag{121}\] For \(b>10M\) and \(q\ge30M\), \[ \sum_{\substack{i<j,\ i+j=b\\i<3M}}|v_{b-1}(i,j)| \le Cb^{3/2}e^{-c_1b}. \tag{122}\] Constants may change after decreasing \(c_1>0\). Here is a direct verification. Set \[p_{q,b}(i)=\frac{\binom qi\binom q{b-i}}{\binom{2q}b}.\] The exact pair formula gives \[ |v_{b-1}(i,b-i)|^2 =\frac{2(2q-1)(2i-b)^2}{b(2q-b)}p_{q,b}(i), \qquad \frac{p_{q,b}(i+1)}{p_{q,b}(i)} =\frac{(q-i)(b-i)}{(i+1)(q-b+i+1)}. \tag{123}\] When \(b\le q/2\), these ratios, their inverses, and symmetry about \(b/2\) show \[p_{q,b}(i)\le Cb^{-1/2} \exp\{-c_2(i-b/2)^2/b\}.\] For completeness, on the upper half \(i\ge(b-1)/2\), the first factor in the ratio is at most one and the second is at most \(\exp\{-(2i+1-b)/(b+1)\}\). Summing logarithms from a middle index gives the Gaussian upper bound relative to the middle value. In the range \(|i-b/2|\le\sqrt b\), the absolute value of that logarithm is at most \(C(1+|i-b/2|)/b\). Summing these lower bounds and using \(\sum_i p_{q,b}(i)=1\) bounds the middle value by \(C/\sqrt b\). Insertion in (123), followed by summing Gaussian moments, proves (121).

To prove (122), there is nothing to check if \(b>q+3M\), because the sum is empty. Otherwise \(b\le q+3M\le11q/10\), so \(2(2q-1)/[b(2q-b)]\le C/b\) in (123). For every allowed \(i\le2b/5\) and \(b\ge20\), the ratio in (123) is at least \(4/3\). Starting with \(i<3M<3b/10\) and taking at least \(b/10-O(1)\) steps toward \(2b/5\) gives \(p_{q,b}(i)\le Ce^{-c b}\), since each probability is at most one. The square root of (123) is therefore at most \(C\sqrt b\,e^{-c_1b}\), and there are at most \(b\) terms. This proves (122).

The modified row vanishes when \(b\le M\). For \(M<b\le10M\), put \(x=b/(2M)\) and \(t=T(x)\). The smooth even function \(z\mapsto T(x+z)T(x-z)\) has zero first derivative at zero. By (117), every derivative of positive order there is bounded by \(C_r\sqrt{1-t}\). Taylor expansion through degree \(41\), with a bounded remainder of order \(42\), and (121) yield \[ \widetilde B_{b-1}=t^2 B_{b-1}+E_b, \qquad \|E_b\|\le CM^{-3/4}\sqrt{1-t}+CM^{-20}. \tag{124}\] We used \(\|c_jc_i\|\le1\). More explicitly, the degree-\(r\) contribution is bounded by \[C_rM^{-r}b^{r/2+1/4}\sqrt{1-t},\qquad 2\le r\le41,\] and the remainder is at most \(CM^{-42}b^{21+1/4}\).

Let \(\delta=M^{-8}\) and \(s=1-t+\delta\). For any vector \(\phi\), Cauchy’s inequality gives \[\|(t^2B_{b-1}+E_b)\phi\|^2 \le(t^4+s)\|B_{b-1}\phi\|^2 +(1+t^4/s)\|E_b\phi\|^2.\] Since \(t^4+s\le1+\delta\), (124) bounds the second term by \(CM^{-3/2}\|\phi\|^2\). Summing over the \(O(M)\) central rows costs \(CM^{-1/2}\|\phi\|^2\). The additional coefficient \(\delta\) also disappears in this error: (121) with \(r=0\) implies \(\sum_{M<b\le10M}\|B_{b-1}\|^2\le CM^{3/2}\).

Finally, for \(b>10M\) the coefficients of \(\widetilde B_{b-1}-B_{b-1}\) are supported where \(i<3M\) or \(j<3M\). The increasing ordering reduces this to \(i<3M\). Equation (122) bounds this difference in operator norm by \(Cb^{3/2}e^{-c_1b}\), while normalization of the pair vector gives \(\|B_{b-1}\|\le\sqrt b\). The norm of the difference of the two row squares is therefore summable over \(b>10M\), with sum at most \(Ce^{-c_3M}\). Combining the three ranges proves the assertion, with \(\varepsilon_M\le C(M^{-1/2}+M^{-13/2}+e^{-c_3M})\). ◻

Proof. The tensor product of two spin-\((q-1)\) representations is multiplicity free. Hence rotation invariance and \(0\le A_{q,M}\le I\) prove the first assertion. Put \(a=q-1\) and \(d_{q,\ell}=4q-3-2\ell\). If \(C_{q}(\ell,k)\) is the coupling coefficient of \(v_0\otimes v_k\) in total spin \(2a-\ell\) at total lowering degree \(k\), then \[ a_{q,M,\ell}=\frac{2q-1}{d_{q,\ell}} \sum_{k+1>M}|C_q(\ell,k)|^2. \tag{128}\] The unrestricted sum, including all \(k\), has coefficient exactly one.

We give the fixed-\(k\) limit, including its normalization. At a fixed total degree, the two spin lowering operators, divided by \(\sqrt{2a}\), converge to multiplication by two independent oscillator variables \(X,Y\). The highest vector of deficit \(\ell\) converges to \((X-Y)^\ell/\sqrt{2^\ell\ell!}\): applying the raising operator gives successive coefficient ratio \(-\sqrt{(\ell-p)/(p+1)}\) in the limit, which fixes this normalized vector. Normalized subsequent lowering multiplies by \((X+Y)/\sqrt{2(k-\ell)}\) at degree \(k\). Thus the degree-\(k\) coupled vector converges, in the orthonormal monomial basis, to \[\frac{(X-Y)^\ell(X+Y)^{k-\ell}} {2^{k/2}\sqrt{\ell!(k-\ell)!}}.\] Its coefficient of \(Y^k/\sqrt{k!}\) has squared modulus \(2^{-k}\binom{k}{\ell}\), and hence \[|C_q(\ell,k)|^2\longrightarrow2^{-k}\binom{k}{\ell}.\] There are only finitely many degrees below a fixed \(M\). Subtracting these from the exact unrestricted coefficient one gives \[\lim_{q\to\infty}a_{q,M,\ell} =1-\frac12\sum_{\ell\le k\le M-1}2^{-k}\binom{k}{\ell}.\] The identity \(\sum_{k=\ell}^{\infty}2^{-k}\binom{k}{\ell}=2\) proves (126). This finite-head argument requires no uniform convergence over an unbounded set of coupling coefficients.

For (127), choose a normalized highest vector \(z_{q,\ell}=\sum_{p=0}^{\ell}s_pv_p\otimes v_{\ell-p}\), with \(\sum_p|s_p|^2=1\). Its wedge annihilator is \(\sum_p\overline{s_p}B_{\ell-p}B_p\). Every \(B_{\ell-p}\) uses only the first \(\ell+2\) orbitals and has norm at most \(\sqrt{\ell+2}\). Therefore \[\|B(W_4z_{q,\ell})\psi\|^2 \le(\ell+2)\sum_{p=0}^{\ell}\|B_p\psi\|^2.\] Resolve \(P_{q,\ell}\) by the orbit of its highest vector and use (113). This yields \[\mathcal L_4(W_4P_{q,\ell}W_4^*) \le\frac{d_{q,\ell}}{2q-1}(\ell+2)(\ell+1)H_q,\] whose coefficient is bounded for fixed \(\ell\). ◻

Proof of Theorem 34. For the moment fix \(M\), and define \[J_{g,\alpha}=K_\alpha(g)B_0(g), \qquad d\nu_q(g,\alpha)=(2q-1)\,dg\] with counting measure in \(\alpha\). Completeness of the loss channel and (113) give \(\int J_y^*J_y\,d\nu_q=H_q\). Lemma 35, applied to \(B_0(g)\psi\) and then averaged, gives \[\begin{align*} \int J_y^*H_qJ_y\,d\nu_q(y) &=d_a\int B_0(g)^*\mathcal E_{M,g}^*(H_q)B_0(g)\,dg\\ &\le\mathcal L_4(W_4A_{q,M}W_4^*)+\varepsilon_MH_q. \tag{129}\end{align*}\] The equality between the averaged pair products and the lifted four-body operator uses ordered tensor factors. There is no factor of two: \(B(v\wedge w)=B(w)B(v)\) and the lift sums each ordered pair once.

By Lemma 36 and positivity of the lift, \[\mathcal L_4(W_4A_{q,M}W_4^*) \le\mathcal L_4(W_4R_q^{\mathrm{hi}}W_4^*) +\left(\sum_{\ell=0}^{22}C_\ell a_{q,M,\ell}\right)H_q.\] Fix \(c_0=1/50\) in (116). Choose \(M\) large enough, and then \(q\) large enough, that \(\varepsilon_M+\sum_{\ell=0}^{22}C_\ell a_{q,M,\ell}\le c_0/2\). Equations (116) and (129) prove (114) with \(c=c_0/2\).

Each \(K_\alpha\) is a product of \(3M\) operators of particle loss zero or one, so \(J_{g,\alpha}\) has loss between \(2\) and \(L=3M+2\). All modes involved in \(J_{g,\alpha}\) have index below \(3M\), apart from identity factors. To see operator-norm locality directly, use the frame isometry \(\mathsf S_q:U_q\to\ell^2(\Lambda_q)\) and write \(c(f)=B(f)\) on the slot Fock space. If \(P_R\) cuts off slots outside the radius-\(R\) ball around \(g\cdot\mathrm{north}\), the canonical anticommutation relations give \[\|c(\mathsf S_qe_j(g))-c(P_R\mathsf S_qe_j(g))\| =\|(I-P_R)\mathsf S_qe_j(g)\|_2\le C_{D,M}R^{-D}.\] The last estimate follows from Lemma 4 and the slot volume bound, with a larger decay order before summing squares. The same estimate for a product follows by telescoping its finitely many factors, or by Lemma 7. Thus the jumps are rapidly local around \(g\cdot\mathrm{north}\), uniformly in \(q\). There are only \(2^{3M}\) auxiliary labels, with \(M\) now fixed. The pushforward of \((2q-1)dg\) to the physical sphere has area density \((2q-1)/(2\pi q)\); the remaining rotation angle has mass one. Including all auxiliary labels therefore gives total center density \(2^{3M}(2q-1)/(2\pi q)\), uniformly bounded in \(q\) for this fixed \(M\). These observations prove all locality and density assertions. ◻

Proof. Theorem 34 gives \(\mathcal D_q(H_q)\le-cH_q\), proving (134). Since \([\mathcal N,J_y]=-r_yJ_y\), \[-\mathcal D_q(\mathcal N)=\int r_yJ_y^*J_y\,d\nu_q(y), \qquad 2H_q\le-\mathcal D_q(\mathcal N)\le LH_q.\] Integration proves (135) with \(C_{\mathrm{num}}=L/c\). Compactness of the density matrices in each fixed finite-dimensional Fock space gives limit points as \(t\to\infty\). By energy decay each is supported on \(\ker H_q\). Lemma 2 shows that this kernel contains no sector of particle number exceeding \(N\). Applying (135) to a state of definite particle number \(n\) and passing to a limit point gives \(n-N\le C_{\mathrm{num}}\langle H_q\rangle_0\). This proves (131) on each sector, and hence on Fock space.

Choose \(0<\mu\le\min(1,(2C_{\mathrm{num}})^{-1})\). In sectors \(n\le N\), \(G_q=H_q+\mu(N-n)\) dominates \(\mu(H_q+N-n)\). In sectors \(n>N\), (131) gives \(G_q\ge H_q/2\) and \(H_q+n-N\le(1+C_{\mathrm{num}})H_q\). This proves the first inequality in (133) with \(a=\min(\mu,[2(1+C_{\mathrm{num}})]^{-1})\). For \(n\ne N\), \(|N-n|\ge1\); for \(n=N\), the uniform gap and uniqueness in Theorem 3 give \(H_{N,q}\ge(1/25)(I-P_{\mathrm L,N})\). Thus the second inequality holds with \(g=a/25\).

Finally, energy decay and number control give \[\langle G_q\rangle_t \le\langle G_q\rangle_0+\mu C_{\mathrm{num}}\langle H_q\rangle_0 \le(1+\mu C_{\mathrm{num}}/a)\langle G_q\rangle_0.\] Take \(C_G=1+\mu C_{\mathrm{num}}/a\). Passing to a limit point, where \(H_q=0\), proves (137). ◻

Proof. Fix \(q\) for the moment and let \(K=K_q\) be the least constant in (139) for the stated class. It is finite: every admissible sum annihilates \(\Omega\) on both sides, its norm is bounded by the finite number of cells, and \(G\) has a positive gap on \(\Omega^\perp\). This observation gives no useful uniform bound; we obtain one from the loss evolution.

Let \(J_y\) be the jumps of Theorem 34, including their bounded auxiliary indices in the measure \(dy\). They obey \[\int J_y^*J_y\,dy=H, \qquad J_y\Omega=0\quad\text{for almost every }y.\] The second assertion follows by taking the expectation of the first in \(\Omega\). Put \(C_{xy}=[A_x,J_y]\) and \(D_{xy}=1+d(x,y)\). Since \(A_x\) is even, the ordinary commutator vanishes for disjointly supported approximants, including when \(J_y\) is odd. Lemma 7(iii) gives \[ \|C_{xy}\|\le C D_{xy}^{-\ell}, \qquad \|C_{xy}\|_{x,\ell}\le C, \qquad \|C_{xy}^*C_{xy}\|_{x,\ell}\le C D_{xy}^{-\ell}. \tag{140}\] The last estimate retains the same order \(\ell\) required of \(A_x\). Thus, after integration in \(y\), the positive operators \(C_{xy}^*C_{xy}\) remain in the class whose best relative-bound constant is \(K\). Fixing one polynomial norm makes this closure possible.

Every \(C_{xy}\) annihilates \(\Omega\), because both \(A_x\) and \(J_y\) do. Therefore \(C_{xy}^*C_{xy}\) is a physical, positive, neutral, centered operator to which the definition of \(K\) applies. Choose \[ 2<b<\ell-2. \tag{141}\] The two-dimensional volume bound and (140) give \[ \sum_x\int D_{xy}^{b}C_{xy}^*C_{xy}\,dy\le C K G. \tag{142}\] Indeed for each fixed \(x\) the integrated operator has localization norm at most \(C\int D_{xy}^{b-\ell}\,dy\le C\), and kills \(\Omega\). Collect it as a single admissible term at cell \(x\). This also explains the continuous-family extension in the statement.

Let \(\rho_t\) be the state evolution dual to the loss generator, starting at an arbitrary density matrix \(\rho_0\). All observables in the desired inequality preserve number, so replacing \(\rho_0\) by its number-diagonal part changes none of the relevant expectations. For \(A=\sum_xA_x\), the Lindblad identity reads \[\frac{d}{dt}\operatorname{Tr}(\rho_t A) =\operatorname{Re}\sum_x\int \operatorname{Tr}(\rho_t J_y^*C_{xy})\,dy.\] Apply Cauchy–Schwarz to this sum with weights \(D_{xy}^{-b}\) and \(D_{xy}^{b}\). Since \(\sum_xD_{xy}^{-b}\le C\) uniformly in \(y\), (142) yields \[ \left|\frac{d}{dt}\operatorname{Tr}(\rho_t A)\right| \le C\bigl(\operatorname{Tr}(\rho_t H)\bigr)^{1/2} \bigl(K\operatorname{Tr}(\rho_t G)\bigr)^{1/2}. \tag{143}\] The energy-descent estimates give constants \(c,C>0\) such that \[\operatorname{Tr}(\rho_t H)\le e^{-ct} \operatorname{Tr}(\rho_0 H),\qquad \operatorname{Tr}(\rho_t G)\le C\operatorname{Tr}(\rho_0 G), \qquad H\le CG.\] Integrating (143) therefore costs at most \(C\sqrt K\operatorname{Tr}(\rho_0 G)\).

Along a sequence \(t_j\to\infty\), finite dimensionality gives a limit state \(\rho_\infty\) supported on \(Z_q\). Its sectors have \(n\le N\) by Lemma 2. The zero-mode bound and \(\langle\Omega,A\Omega\rangle=0\) show that \[|\operatorname{Tr}(\rho_\infty A)| \le C\operatorname{Tr}(\rho_\infty(N-\mathcal N)) =C\mu^{-1}\operatorname{Tr}(\rho_\infty G) \le C\operatorname{Tr}(\rho_0G).\] Combining the two bounds, and then taking the supremum over all admissible interactions and states, gives \[K\le C+C\sqrt K.\] Solving this quadratic inequality for \(\sqrt K\) proves a bound independent of \(q\), as required. ◻

Proof. Apply simultaneous lowering \(p\) times to \(-e_0\wedge e_1\) and divide by \(\sqrt{(2q-2)_p p!}\). Expand the lowering operator binomially. Combining the two contributions to each increasing pair gives (154). Taking the limit gives (155). The normalization can also be checked directly: the square of the coefficient in (154) is \[(i-j)^2\frac{\binom qi\binom qj}{q\binom{2q-2}{p}},\] and differentiating \((1+x)^q\) gives \[\sum_{i,j=0}^q(i-j)^2\binom qi\binom qj x^{i+j} =2qx(1+x)^{2q-2}.\] Take the coefficient of \(x^{p+1}\) and halve the sum to restrict to \(i<j\). ◻

Proof. Identify \(V_q\otimes U_q\) with tensors antisymmetric in their first two positions. Then \(W_3\) is \(\sqrt3\) times full antisymmetrization. Its Gram operator is the compression of \(I-P_{13}-P_{23}\), where \(P_{ij}\) swaps tensor positions. The two swap compressions coincide, since conjugation by \(P_{12}\) interchanges them and \(P_{12}=-I\) on the domain. By equivariance and multiplicity-freeness they act by the same scalar \(b_z\) on each spin block.

A highest vector of that block has coefficients \(s_p\) on \(v_p\otimes e_{z-p}\), \(0\le p\le z\), satisfying \[ \frac{s_{p+1}}{s_p}=-\sqrt{\frac{(z-p)(q-z+p+1)}{(p+1)(2q-2-p)}}. \tag{158}\] This follows by applying the raising operator. Using (154), the coefficient of the compressed \(P_{23}\) image on \(v_0\otimes e_z\) is \[s_0b_z=\frac12\left[ s_{z-1}\sqrt{\frac{z(q)_z}{q(2q-2)_{z-1}}} -s_z(z-1)\sqrt{\frac{(q)_z}{(2q-2)_z}}\right],\] with the first term absent for \(z=0\). Here the ordered tensor coefficients of a normalized wedge include a factor \(1/\sqrt2\). Iteration of (158) gives \[\frac{s_z}{s_0}=(-1)^z\sqrt{\frac{(q)_z}{(2q-2)_z}},\qquad \frac{s_{z-1}}{s_0}=(-1)^{z-1} \sqrt{\frac{z(q-1)_{z-1}}{(2q-2)_{z-1}}}\quad(z\ge1).\] Substitution gives \(q_z=-2b_z\) in the form (156). The three stated null sectors have \(q_z=-1\), and (157) follows directly. ◻

Proof. At \(T=z\), the raising equation gives the adjacent ratio \[-\sqrt{\frac{(z-p)(2j_2-z+p+1)}{(p+1)(2j_1-p)}}.\] With sign \((-1)^z\) at \(p=0\), normalization yields the coefficients of \((uX-vY)^z/\sqrt{z!}\) in the limit. The change of variables \((X,Y)\mapsto(uX-vY,vX+uY)\) is orthogonal, so these polynomials have norm one in the normalized-monomial inner product. Normalized lowering from \(T\) to \(T+1\) divides the sum of the two lowering operators by \[\sqrt{(T-z+1)\{2(j_1+j_2-z)-(T-z)\}}.\] In the limit this acts by multiplication by \((vX+uY)/\sqrt{T-z+1}\). Induction proves the statement for fixed descendants. ◻

Proof. The two positivity statements both assert that the quadratic form of \(C\) is nonnegative on \(\operatorname{ran} G=\operatorname{ran} W^\dagger\). For the second assertion, use \(\lVert c_kc_jB(v_p^*)\psi\rVert\le\lVert B(v_p^*)\psi\rVert\) and sum; one may take \(C_0\) to be the largest multiplicity of a pair label \(p\). ◻

Proof. An exact change of variables. Take \(q\ge24\). All annihilators in (175), including those in \(h_q\), involve only the modes \(0,\ldots,24\). First work on \(\mathcal F_{\mathrm{loc}}\). Define the positive diagonal operator \[\Delta_q\lvert A\rangle=\left(\prod_{x\in A}d_q(x)\right)\lvert A\rangle, \qquad d_q(x)=\sqrt{(q)_x/q^x},\] on each basis wedge \(\lvert A\rangle\). This local Fock space is fixed as \(q\) varies. Thus \(\Delta_q\) and \(\Delta_q^{-1}\) converge in norm to \(I\); also \(0<\Delta_q\le I\).

The pair coefficient formula (154) gives \[v_p(x,y)=r_p(q)d_q(x)d_q(y)v_p^*(x,y), \qquad r_p(q)=\sqrt{(2q)^p/(2q-2)_p}\ge1.\] Checking the factors attached to the removed orbital labels on a basis wedge gives the exact identities \[\begin{align*} B_p&=r_p(q)\Delta_q^{-1}B(v_p^*)\Delta_q, \tag{176}\\ B(v_p\wedge e_j\wedge e_k) &=\frac{r_p(q)}{d_q(j)d_q(k)}\, \Delta_q^{-1}B(v_p^*\wedge e_j\wedge e_k)\Delta_q. \tag{177}\end{align*}\] These are identities on the local Fock space, with the inverse diagonal acting on the output particle sector. Repeated removed labels make both sides zero. In particular, solving (176) for \(B(v_p^*)\Delta_q\) and using \(r_p(q)\ge1\) and \(\lVert \Delta_q\rVert\le1\) gives \[ \Delta_q h_*\Delta_q\le h_q. \tag{178}\]

The error is controlled by pair energy. Fix \(D\). Index a finite family by \(b=(p,j,k)\) with \(p+j+k=D\) and \(j<k\), and put \[A_b=B(v_p^*\wedge e_j\wedge e_k),\qquad \beta_{rb}^q=S_r^q(D,D;p,j,k) \frac{r_p(q)}{d_q(j)d_q(k)}.\] Equations (171) and (177) yield \[B(w_{Dr}^q)=\Delta_q^{-1} \left(\sum_b\beta_{rb}^qA_b\right)\Delta_q.\] The coefficients are real, and Lemma 43 gives \(\beta_{rb}^q\to\beta_{rb}^*:=S_r(D,D;p,j,k)\). Hence the finite real symmetric matrices \[M_q=(\beta^q)^TC_D^q\beta^q \quad\hbox{converge to}\quad M_*=(\beta^*)^T(2\chi-E_D+\varepsilon I)\beta^*.\] The quadratic expression in (175) is consequently \[\Delta_q\left(\sum_{b,c}(M_q)_{bc} A_b^\dagger\Delta_q^{-2}A_c\right)\Delta_q.\] Its limiting expression before the outer diagonals is \[ \sum_{b,c}(M_*)_{bc}A_b^\dagger A_c =\sum_{r,s}(2\chi-E_D+\varepsilon I)_{rs} B(w_{Dr}^*)^\dagger B(w_{Ds}^*)\ge0. \tag{179}\] Indeed, the certificate (173) and Lemma 44, applied to the map with columns \(w_{Dr}^*\), give positivity before lifting; the lift \(\mathcal L_4\) preserves positivity.

For clarity, the perturbation estimate does not require \(M_*\ge0\). Let \(R_q\) be the block matrix with entries \[(R_q)_{bc}=(M_q)_{bc}\Delta_q^{-2}-(M_*)_{bc}I.\] Its norm tends to zero, since both its index set and the local Fock space are fixed. For every vector \(\phi\), \[\left|\sum_{b,c}\langle A_b\phi,(R_q)_{bc}A_c\phi\rangle\right| \le\lVert R_q\rVert\sum_b\lVert A_b\phi\rVert^2 \le C_0\lVert R_q\rVert\langle \phi,h_*\phi\rangle.\] Combine this estimate with (179), take \(\phi=\Delta_q\psi\), and apply (178). This proves (175) on the local Fock space with a coefficient tending to zero. Taking a maximum over the fixed set \(1\le D\le23\) gives a common \(\rho_q\).

Additional particles are spectators. Order the 25 local modes before all other modes in the Fock factorization. Every operator in (175) acts on the local factor alone. Tensoring the local inequality with the identity proves it on \(\mathcal F_q\), with the same \(\rho_q\), regardless of the number of occupied modes outside this factor. ◻

Proof. Haar averaging the matrix unit \(\lvert D,D;r\rangle_q\langle D,D;s\rvert_q\) gives the corresponding copy matrix unit tensored with the spin identity, divided by \(d_D\). Therefore, on the blocks \(1\le D\le23\), the low-block target minus \(A_4\), with the \(\varepsilon I\) allowance added, is \[d_a\sum_{D=1}^{23}\int\sum_{r,s}(C_D^q)_{rs} B(w_{Dr}^q(g))^\dagger B(w_{Ds}^q(g))\,dg.\] Lemma 45, conjugated by each rotation, bounds this below by \(-o(1)H\), because Schur averaging gives \[d_a\int h_q(g)\,dg=24H.\] The number of blocks in the comparison is fixed, so this error remains uniform in particle number.

To bound the allowance, fix \(D\) and write \[\lvert D,D;r\rangle_q =\sum_{\substack{p+j+k=D\\j<k}}S_{rb}^q\, v_p\otimes(e_j\wedge e_k), \qquad b=(p,j,k).\] The uncoupled vectors on the right are orthonormal. So are the highest vectors on the left: distinct \(r\) belong to orthogonal spin summands of the second pair factor. Thus \(S^q(S^q)^T=I\). Apply the contraction \(S^q\otimes I\) to the Hilbert-space-valued vector \((c_kc_jB_p\psi)_b\). It follows that \[\sum_{r\in\mathcal R_D}\lVert B(w_{Dr}^q)\psi\rVert^2 \le\sum_{\substack{p+j+k=D\\j<k}}\lVert c_kc_jB_p\psi\rVert^2 \le\sum_{\substack{p+j+k=D\\j<k}}\lVert B_p\psi\rVert^2.\] This uses orthogonality before wedging; the vectors \(w_{Dr}^q\) themselves need not be orthogonal. The number of triples on the right is \[n_D=\sum_{n=1}^D\left\lceil\frac n2\right\rceil =\left\lfloor\frac{(D+1)^2}{4}\right\rfloor.\] Conjugate by rotations and integrate. Each pair norm contributes \(\langle \psi,H\psi\rangle/d_a\), so the total allowance is bounded by \[\varepsilon\sum_{D=1}^{23}n_DH =\varepsilon\left(\sum_{m=1}^{12}m^2+ \sum_{m=1}^{11}m(m+1)\right)H =1222\varepsilon H.\] Removing the allowance proves (180). No block of \(R_q^{\mathrm{hi}}\) has entered this estimate. ◻

Proof. For \(16\le z\le q\), the bracket in (156) is positive, since it is at least \(2z-2\). Thus only odd \(z\) contribute negatively. A normalized highest vector of \(R_{3,z}\) is \(\sum_{p=0}^zs_pv_p\otimes e_{z-p}\). Its orbit resolves that spin projector with factor \(d_{3,z}\). After wedging and lifting, its quadratic form is \[\lVert \sum_{p=0}^zs_pc_{z-p}(g)B_p(g)\psi\rVert^2 \le\sum_{p=0}^z\lVert B_p(g)\psi\rVert^2.\] Averaging each summand proves (181) with (182).

For \(q\ge4\), the ratios in \((q)_z/(2q-2)_z\) are nonincreasing, so \[\frac{(q)_z}{(2q-2)_z}\le\left(\frac q{2q-2}\right)^z \le(2/3)^z,\qquad \frac{d_{3,z}}{d_a}\le2.\] The summands, extended by zero for \(z>q\), are therefore dominated by \(6z(z+1)(2/3)^z\). Dominated convergence and (157) give \[\lim_{q\to\infty}\delta_q =\frac32\sum_{z=17,19,\ldots}(3z-1)(z+1)2^{-z} =\frac{61}{4096}.\] The last identity follows by writing \(z=17+2m\) and using the geometric-series identities for \(\sum t^m\), \(\sum mt^m\), and \(\sum m^2t^m\) at \(t=1/4\). ◻

Proof of Proposition 39. Write \(T_{\mathrm{lo}}=\mathcal L_4(W_4R_q^{\mathrm{lo}}W_4^\dagger)\) and \(T_{\mathrm{hi}}=\mathcal L_4(W_4R_q^{\mathrm{hi}}W_4^\dagger)\). The normal-square identity, the three-body comparison (168), and the tail estimate give \[H^2\ge (1-\delta_q)H+A_3+T_{\mathrm{lo}}+T_{\mathrm{hi}}.\] By (180), \(T_{\mathrm{lo}}\ge A_4-(1222\varepsilon+o(1))H\). Using \(\mathcal K_q=\eta H+A_3+A_4\ge0\) from (164), we obtain \[H^2\ge\mathcal K_q+ (1-\eta-\delta_q-1222\varepsilon-o(1))H+T_{\mathrm{hi}}.\] The coefficient of \(H\) tends to \[ 1-\frac{93527408868499}{10^{14}}-\frac{61}{4096} -1222\frac3{10^6} =\frac{4616733319001}{10^{14}}>\frac1{25}. \tag{183}\] For fixed \(0<c_0<1/25\), choose \(q\) large enough that this coefficient exceeds \(c_0\) and discard the positive operator \(\mathcal K_q\). This is (152). The matrix comparisons involved finitely many fixed orbital modes, the transfer estimates held on full Fock space, and the tail estimate was uniform in particle number. Therefore the same threshold applies simultaneously to every sector. ◻