Proof. Tempered distributions and finite field-label expansions. For every fixed ordered list of homogeneous original fields \(A_i\in\mathcal F_{\Delta_i}\), its already given vacuum distribution is tempered. To see this, remove the common translation coordinate. In the remaining relative coordinates \(\xi\), dilation invariance of the vacuum gives \[W(\lambda\xi)=\lambda^{-\sum_i\Delta_i}W(\xi),\qquad \lambda>0,\] in the distributional pullback sense. A compact neighborhood of the origin is controlled by the local finite order of the given distribution. On a fixed compact annulus it is bounded by finitely many test-function seminorms. Dilating that annulus transfers the bound to every outer dyadic annulus with a fixed power determined by the dimensions and the seminorm order. Summation against a sufficiently high Schwartz seminorm proves temperedness. The zero- and one-point cases are immediate.

Adjoints are homogeneous at the same real grade. For any fixed list of original fields \(A_i=\sum_{\Delta\in S_i}A_{i,\Delta}\), multilinearity on \(\mathcal D_{\mathrm{fact}}\) first gives the component expansion on compact factor tests. Compact tensor-test density and continuity of the given joint distributions then give, for the same compact joint kernel \(\varphi\), \[ W(A_1,\ldots,A_n)[\varphi] =\sum_{(\Delta_1,\ldots,\Delta_n)\in S_1\times\cdots\times S_n} W(A_{1,\Delta_1},\ldots,A_{n,\Delta_n})[\varphi]. \tag{3}\] The finite sum of homogeneous tempered extensions defines the unique tempered extension of the left side, so the same identity holds for every joint Schwartz kernel. It agrees with any originally specified Schwartz distribution by its stated continuity and compact-test density. Its constants and seminorm orders may depend on the fixed list; no single total scaling dimension is assigned to a nonhomogeneous sum.

Hilbert-valued words and insertion actions. For a fixed old list \(\mathbf A\), write \(\Psi^0_{\mathbf A}(u)\) for the original vacuum vector of a finite sum \(u\) of compact factor tests. The doubled tempered distribution gives a continuous Schwartz seminorm \(p\) and a constant \(C\) such that \[\|\Psi^0_{\mathbf A}(u)-\Psi^0_{\mathbf A}(v)\|^2 =W(\mathbf A^*,\mathbf A)[(u-v)^*\otimes(u-v)] \le C p(u-v)^2.\] Here the star reverses and adjoints the field list and conjugates and reverses the kernel. Compact finite tensor tests are dense in the joint Schwartz space. The estimate therefore constructs the unique continuous Hilbert-valued extension \(\Psi^{\mathrm{old}}_{\mathbf A}\) used in the definition of \(\mathcal D_{\mathrm{old}}\); polarization passes all its Gram identities to the limit. Any originally specified Schwartz vacuum-word map agrees with this extension by its stated continuity and the same density. Consequently the original vacuum-generation assumption also makes \(\mathcal D_{\mathrm{fact}}\) Hilbert-dense. This argument concerns only original products whose existence is assumed; it supplies no product containing a newly constructed field.

For a fixed old word \(\mathbf A\) and a fixed finite old prefix \(\mathbf B\), the same doubled vacuum-distribution estimate applies to the longer word. Consequently, if \(h_j\to h\) in the joint Schwartz topology, then for a fixed prefix kernel \(g\), \[\Psi^{\mathrm{old}}_{\mathbf A}(h_j)\longrightarrow \Psi^{\mathrm{old}}_{\mathbf A}(h),\qquad \Psi^{\mathrm{old}}_{\mathbf B,\mathbf A}(g\otimes h_j) \longrightarrow \Psi^{\mathrm{old}}_{\mathbf B,\mathbf A}(g\otimes h)\] in Hilbert norm. The estimates are continuous jointly in \(g,h\) on their Schwartz spaces. Reversed adjoint kernels obey the same bounds. Compact joint kernels can be approximated, after a finite partition into product patches, by finite sums of factor tests with the prescribed local supports; Schwartz kernels admit finite tensor approximations in the Schwartz topology. Concatenation, reversal, and differentiation are continuous in these test spaces.

Left concatenation of old test words is well-defined on their vacuum vectors. Indeed, if a finite old test polynomial \(P\) represents zero, positivity and Cauchy–Schwarz applied to \(P\) and \(Q^*QP\) give \(W(P^*Q^*QP)=0\) for every finite old test polynomial \(Q\). This is the squared norm of the proposed vector for \(QP\). Thus the null vectors form a left ideal, including cancellations between different words, and the same vacuum pairing places adjoint insertion inside the Hilbert adjoint. Concatenation therefore makes \(\mathcal D_{\mathrm{old}}\) invariant under the old field and polynomial insertions. It is Hilbert-dense because it contains the original factorized vacuum-polynomial vectors that generate \(\mathcal H\).

Agreement with the retained realizations. Let \(T_B(f)\) be a chosen closed affiliated realization of an original field with compact test \(f\), and choose finite sums of compact tensor tests \(h_j\to h\) as above. On the vectors in \(\mathcal D_{\mathrm{fact}}\subset\mathcal E\) the original extension clause gives \[T_B(f)\Psi^{\mathrm{old}}_{\mathbf A}(h_j) =\Psi^{\mathrm{old}}_{B,\mathbf A}(f\otimes h_j).\] Both sides converge in Hilbert norm by the two displayed continuity statements. Closedness of \(T_B(f)\) therefore proves the same equality for \(h\). Applying the argument to the closed Hilbert adjoint \(T_B(f)^*\) gives its equality with the insertion of \(B^\dagger(\bar f)\) there. Iterating this one-field statement proves the corresponding actions of every finite factorized prefix and its reversed adjoint on \(\mathcal D_{\mathrm{old}}\). Joint polynomial insertions are the continuous concatenations already defined; this argument does not assign an affiliated closure to an arbitrary joint smear.

If an original Schwartz smearing \(B(f)\) and its closed affiliated realization were already supplied, the same conclusion holds for that realization. To identify its action on a fixed vector of \(\mathcal D_{\mathrm{fact}}\), approximate \(f\) by compact tests in Schwartz topology. The longer doubled correlations give Hilbert convergence of the inserted vectors, while the original Schwartz distributional continuity gives the same limiting matrix elements against \(\mathcal D_{\mathrm{fact}}\). Density identifies the vectors, also for the adjoint. The preceding \(h_j\) closed-graph argument then applies unchanged. This uses only an originally supplied Schwartz action and its stated continuity; it asserts no pre-existing realization for a smearing that was not originally supplied.

Identities and local Ward actions. The same paired vector and inserted-vector limits extend every finite field-label identity from \(\mathcal D_{\mathrm{fact}}\) to \(\mathcal D_{\mathrm{old}}\). The converse follows from \(\mathcal D_{\mathrm{fact}}\subset\mathcal D_{\mathrm{old}}\), so the two equality conventions for \(\mathcal F\) agree. The homogeneous dilation law and the adjoint and derivative grade laws extend in the same way, because the transformed, reversed, and differentiated kernels remain in the stated test spaces. Thus none of these identities on joint vectors was used to construct those vectors.

The original covariance and flat field identities pass from the factorized vectors to joint vectors by the same longer-word continuity, with transformed or differentiated kernels. Locality and compact-slab Ward identities pass under their stated support conditions. For a general Schwartz spectator vector, keep the localized slab and insertion tests fixed and approximate only the spectator kernel in Schwartz topology. If a localized insertion polynomial itself has a compact joint kernel, use the support-preserving compact approximants. Thus the operational identities used below hold on the dense invariant vacuum-generated domain \(\mathcal D_{\mathrm{old}}\), with the original vacuum correlations, Hilbert space, and bounded net unchanged. The factor approximations just used converge in the graph of each specified \(T_B(f)\) on these joint vectors; Hilbert density alone was not used for that conclusion. No claim is made that \(\mathcal D_{\mathrm{old}}\) is a graph core for a chosen realization, that the closure of its field restriction equals that realization, or that a newly constructed field acts on arbitrary additional vectors of \(\mathcal E\). ◻

Proof. We first prove that, for every \(r>0\), \[ e^{rK_2}j e^{-rK_1}\ \text{is everywhere defined and contractive}. \tag{8}\] Its exponential growth on a high target band will then exclude that band from the image of a low source band.

Set \(A=e^{-iK_2}je^{iK_1}\). It maps \(X_1\) into \(X_2\), so \(\mathfrak s_2A\xi=A\mathfrak s_1\xi\) on \(\mathop{\mathrm{Dom}}\mathfrak s_1=\mathop{\mathrm{Dom}}\Delta_1^{1/2}\). Equivalently, \[\Delta_2^{1/2}A\xi =J_2AJ_1\Delta_1^{1/2}\xi \qquad(\xi\in\mathop{\mathrm{Dom}}\Delta_1^{1/2}).\] Thus \(\Delta_2^{1/2}A\Delta_1^{-1/2}\), initially on \(\mathop{\mathrm{Dom}}\Delta_1^{-1/2}\), has the contractive bounded extension \(J_2AJ_1\). The operators \[F(z)=\Delta_2^{iz}A\Delta_1^{-iz},\qquad -\tfrac12<\operatorname{Im}z<0,\] have contractive boundary values: the real boundary multiplies \(A\) on either side by a unitary, and the other boundary is \(\Delta_2^{it}J_2AJ_1\Delta_1^{-it}\). The analytic-extension theorem (Araki and Zsidó 2005, Theorem 2.3), followed by three lines, gives the contractive analytic continuation to the strip. Equivalently, sandwich between bounded spectral intervals of the modular logarithms, apply three lines to their scalar matrix coefficients, and remove the intervals. The displayed Tomita relation supplies the full source domain needed at the second boundary. Write \(\widetilde F(\zeta)=F(z)\) for \(\zeta=e^{-2\pi z}\), which maps the strip bijectively onto the upper half-plane. Boost covariance and the intertwining property of \(j\) identify its positive real boundary value as \(e^{-i\zeta K_2}j e^{i\zeta K_1}\). To justify the interior expression despite its potentially unbounded first factor, take \(\xi\) and \(\eta\) in bounded spectral bands of \(K_1\) and \(K_2\), respectively. The scalar function \[\left\langle e^{i\overline\zeta K_2}\eta,j e^{i\zeta K_1}\xi\right\rangle\] is entire and agrees with \(\left\langle\eta,\widetilde F(\zeta)\xi\right\rangle\) on a positive real interval. Boundary uniqueness, or reflection of their zero difference across that interval, identifies them throughout the upper half-plane. At \(\zeta=ir\), \(r>0\), contractivity therefore gives \[\left|\left\langle e^{rK_2}\eta,j e^{-rK_1}\xi\right\rangle\right| \le\left\lVert\eta\right\rVert\,\left\lVert\xi\right\rVert.\] Bounded spectral vectors are a core for \(e^{rK_2}\). The adjoint-domain criterion implies \(j e^{-rK_1}\xi\in\mathop{\mathrm{Dom}}e^{rK_2}\) and identifies its image with \(\widetilde F(ir)\xi\). Finally approximate any source vector by its \(K_1\) spectral truncations. Boundedness of \(e^{-rK_1}\) and \(\widetilde F(ir)\), followed by closedness of \(e^{rK_2}\), proves that \(e^{rK_2}je^{-rK_1}\) is everywhere defined and contractive. If \(\xi=\mathbf1_{[0,R]}(K_1)\xi\), apply this extension to \(e^{rK_1}\xi\) to obtain \[\left\lVert e^{rK_2}j\xi\right\rVert\le e^{rR}\left\lVert\xi\right\rVert.\] Projection onto \(K_2\ge R+\epsilon\) and \(r\to\infty\) shows that this projection annihilates \(j\xi\). Taking \(\epsilon\downarrow0\) proves (7). The resulting tail inequality is \[\left\lVert\mathbf 1_{(a,\infty)}(K_2)j\xi\right\rVert \le\left\lVert\mathbf 1_{(a,\infty)}(K_1)\xi\right\rVert.\] Integrate its square against \(2s a^{2s-1}\mathrm da\). Truncation followed by monotone convergence proves both the domain assertion and the norm bound. ◻

Proof. Put \(H=H_{p_0}\). Completion of the square in (18) shows that \(W_{p_0/(2R)}\) is the portion, with \(u<0\), of the future cone with tip \((u,v,y)=(-R,0,0)\). The null rotations about this tip have generators \(G_i+RP_i\). They preserve the algebra and the vacuum, so commute with its modular group \(e^{i(1-e^s)H/(2R)}e^{isB}\). Set \(\tau=(1-e^s)/(2R)\). Boost conjugation of the rotation group then gives, at the level of Weyl elements, the transformation \[((1-2R\tau)\beta,R\beta)\longmapsto(\beta,R\beta)\] for coefficients of \((G,P_\perp)\); its central coefficient \(b\) is fixed. For a fixed sufficiently small nonzero \(\tau\), vary \(R\) over an open positive interval. Two distinct values give linearly independent coefficient pairs. Multiplying the corresponding Weyl elements, with the same central Weyl phase on both sides, yields the full shear \[ \mathop{\mathrm{Ad}}(e^{i\tau H})G_i=G_i,\qquad \mathop{\mathrm{Ad}}(e^{i\tau H})P_i=P_i+2\tau G_i. \tag{23}\] The group law extends it to every real \(\tau\).

Positive energy confines the nonzero momentum spectrum on \(\ker b\) to one null ray. A small spatial rotation takes this ray to a disjoint ray; a vector supported there would be orthogonal to its arbitrarily small rotations, contradicting strong continuity. Thus \(\ker b\) is precisely the zero-momentum subspace. On it \(P_i=0\), and (23) forces \(G_i=0\). Repeating in all null directions makes every connected Lorentz generator vanish there. In particular \(B=0\), and \(e^{isB}He^{-isB}=e^sH\) forces \(H=0\) on this subspace.

For \(b>0\), the null rotations and transverse translations form the Weyl pair \(G_i=bY_i\), \(P_i=-k_i\). Their regular representation is the Schrödinger representation with a multiplicity space; the boost action on \(\log b\) trivializes that multiplicity over \(b\). This gives the representation in (22). The operator \(b|Y|^2\) implements exactly the shear (23). Consequently \(e^{-itb|Y|^2}e^{itH}\) commutes with the transverse Weyl representation and with \(b\). It is a one-parameter unitary group on the multiplicity space, fiber by fiber. Boost homogeneity identifies its generator as \(bh\) with \(h\) independent of \(b\). Since \(H\ge0\), its expression \(b(|Y|^2+h)\) implies \(h\ge0\): localize the \(Y\) variable near zero and test an arbitrary spectral interval of \(h\).

Transverse translation covariance replaces \(Y\) by \(Y-d\), proving the formula for \(p_d\). The modular-conjugation relation for \(H\) and \(b\), together with geometric conjugation of the null rotations, gives the three statements involving \(J_W\). ◻

Proof. The formula (22) and constant shifts make all parabolic generators commute. Consequently \(\mathop{\mathrm{Ad}}(e^{itH_{p_d}})\mathcal M(W_q)=\mathcal M(W_{q+tp_d})\) for every positive-curvature circular parabola \(q\) and \(t\ge0\). To pass to a bounded \(g\) we need an algebra-generation argument. For \(x_0=(u_0,v_0,y_0)\), \(u_0<0\), its null shadow is \[q_{x_0}(z)=v_0+\frac{|z-y_0|^2}{-2u_0},\qquad W_{q_{x_0}}=I^+(x_0)\cap\{u<0\}.\] Given a double cone compactly contained in \(W_g\), choose \(x_0\) slightly timelike before its lower tip, still inside \(W_g\). Its shadow strictly dominates \(g\) and its future cone contains the double cone. Conversely, \(q\ge g\) implies \(W_q\subset W_g\). Inner generation therefore proves \[ \mathcal M(W_g)=\bigvee_{q>g}\mathcal M(W_q). \tag{24}\] For \(W_{g+tp_d}\) the interior-shadow construction supplies a generating subfamily of dominating parabolas of curvature strictly greater than \(t\). Indeed lower curvature cannot dominate \(g+tp_d\) at transverse infinity, while equality corresponds to the boundary value \(u_0=-1/(2t)\) and cannot be the shadow of an interior point. Thus the join defining \(\mathcal M(W_{g+tp_d})\) may be restricted to all dominating parabolas of curvature greater than \(t\). Subtraction of \(tp_d\) bijects this restricted family with the positive-curvature parabolas dominating \(g\). Arbitrary dominating parabolas may have curvature exactly \(t\); they are not needed for the generating family. Taking joins yields \[\mathop{\mathrm{Ad}}(e^{itH_{p_d}})\mathcal M(W_g)=\mathcal M(W_{g+tp_d}).\] Conjugate the modular group and use \(\mathop{\mathrm{Ad}}(e^{itH_{p_d}})B=B-tH_{p_d}\). Applying the same identity to \(\lambda g\) and \(\lambda t\), then dividing by \(\lambda\) by boost homogeneity, gives \[ H_{g+tp_d}=tH_{p_d} +e^{i\lambda tH_{p_d}}H_g e^{-i\lambda tH_{p_d}} \quad(\lambda>0). \tag{25}\] There is no need to differentiate an unspecified unbounded sum. On a band \(\epsilon\le b\le L\), \(H_g\) is bounded by \(L\left\lVert g\right\rVert_\infty\). The second term in (25) is independent of \(\lambda\), and its strong limit as \(\lambda\downarrow0\) is \(H_g\). It follows that \(H_g\) commutes with the full group of every \(H_{p_d}\). Exhausting the \(b\) bands proves strong commutation globally. In the joint calculus of the commuting parabolic family, \[H_{p_d}-H_{p_0}=b(-2d\cdot Y+|d|^2).\] These differences recover \(Y\) on \(b>0\), and \(H_{p_0}/b-|Y|^2\) recovers \(h\). Hence \(H_g\) commutes with \(Y,h\). Finally (21) gives the bounded operator \(h_g=H_g/b\), and boost homogeneity makes it independent of \(b\). ◻

Proof. For every transverse center \(z\), \[g(y)\le g(z)+\nabla g(z)\cdot(y-z)+C|y-z|^2.\] The right side is a signed positive-curvature parabola if \(C>0\). Inclusion, modular-log order, and (22) give \[h_g\le g(z)+\nabla g(z)\cdot(Y-z)+C|Y-z|^2+Ch.\] Applying the same argument to \(-g\) and then (26) gives the lower inequality with both occurrences of \(C\) negated. For this fixed \(g\), all operators involved commute strongly by Proposition 6. Impose the inequalities first for a countable dense set of \(z\) and then, on each joint spectral fiber, let \(z\) tend to the value of \(Y\). They become \(-Ch\le h_g-g(Y)\le Ch\) in a commuting spectral calculus, which proves both assertions. If \(C=0\), boundedness makes \(g\) constant and the result is (20). ◻

Proof. Let \(j_\nu\) embed the scalar comparison copy. The derivative-removal argument in Section 2.6 gives its real wave localization inside the physical cut space. By (17) and (19), \[j_\nu e^{icH_\nu}X_\nu(W) \subset e^{icH_{p_0}}X(W)\qquad(c>0).\] The embedding intertwines the flat wedge boosts, so Lemma 4 applies. It also intertwines the external variables \(b,Y\). To see the latter point independently of the internal operator \(h\), a null rotation \(N_\beta\) fixes \(n,b,M\) and sends \(k\) to \(k+b\beta\) and \(e_i\) to \(e_i+\beta_i n\). The adapted axes in (11) therefore satisfy \(N_\beta e_A^{(n)}(p)=e_A^{(n)}(N_\beta p)\) for \(A=z,1,2\). The constant labels and their fixed derivative phases transform accordingly. Thus \(j_\nu\) intertwines the null-rotation groups generated by \(G=bY\), as well as \(b\); disintegration on \(b>0\) gives the claimed \(Y\) intertwining. This uses the Poincaré action on the comparison waves and does not require \(h\) to preserve their individual channels. Both sides are decomposable in \(b,Y\), and \(j_\nu\) intertwines these variables. In the common spectral fibers their generators are \(b(|Y|^2+h_\nu)\) and \(b(|Y|^2+h)\). Subtracting the common scalar \(b|Y|^2\) and dividing by \(b>0\) in the band-inclusion conclusion gives the first assertion in (28). More formally, apply the whole-generator conclusion for rational band endpoints, localize \((b,Y)\) to shrinking measurable rectangles, and use monotone convergence of spectral projections. The power bound follows by integrating spectral tails as in Lemma 4.

If \(f\le g\), the cut inclusion is \(W_g\subset W_f\). Apply the same lemma with source \(g\), target \(f\), and the identity embedding. If needed add the same constant to make both profiles nonnegative. After disintegrating \(b\), removing this common constant gives (29). This is spectral order obtained from an analytic strip, rather than from ordinary quadratic-form order. ◻

Proof. The quadratic sup-convolution obeys \(\mathcal L_{d+\delta}f=\mathcal L_d(\mathcal L_\delta f)\). Completing the square in the Lipschitz estimate gives \[\mathcal L_\delta(\lambda g)(y) \le\lambda g(y)+\frac{\delta\lambda^2 C_1^2}{2}.\] Hence the cut with profile \(\lambda g+\delta\lambda^2C_1^2/2\), translated by \(u=-\delta\), lies inside \(W_{\lambda g}\). The translated boost is \(B-\delta a\). Modular-log order thus gives \[B-\delta a-\lambda H_g(-\delta) -\frac{\delta\lambda^2C_1^2}{2}b \le B-\lambda H_g.\] This is the upper sign of (30). Apply \(J_W\), which fixes \(a,b,H_g\) and reverses the translation parameter by antiunitarity, and then translate by \(-\delta\); this gives the lower sign. One may initially use vectors smooth for translations and the boost, then remove the boost smoothing in the resulting form bound.

Put \(E=1+a+b\) and \(Q_\lambda=a/\lambda+\lambda C_1^2 b/2\). Differentiating at \(\delta=0\) on momentum-smooth vectors defines the Hermitian form \[c_g(\phi,\psi)=i\bigl( \left\langle a\phi,H_g\psi\right\rangle-\left\langle H_g\phi,a\psi\right\rangle\bigr).\] Differentiation is legitimate using \(H_g=bh_g\), bounded \(h_g\), and translation smoothness of both vectors; it does not presume that \(H_g\) preserves \(\mathop{\mathrm{Dom}}a\). The two inequalities give \(|c_g(\psi,\psi)|\le\left\langle\psi,Q_\lambda\psi\right\rangle\). Polarization, or representation of this Hermitian form after \(Q_\lambda^{1/2}\), gives \[ |c_g(\phi,\psi)|\le \left\lVert Q_\lambda^{1/2}\phi\right\rVert\, \left\lVert Q_\lambda^{1/2}\psi\right\rVert. \tag{31}\] For fixed \(\lambda\) this proves continuity in the \(E^{1/2}\) form norm.

For the domain assertion set \(\chi=H_g\psi\). The \(b\) bound and strong \(b\) commutation imply \(\chi\in\mathop{\mathrm{Dom}}\sqrt{1+b}\). On a joint spectral core for \(a,b\), \[\left\langle a\phi,\chi\right\rangle =\left\langle H_g\phi,a\psi\right\rangle-i c_g(\phi,\psi),\qquad |\left\langle H_g\phi,a\psi\right\rangle| \le\left\lVert g\right\rVert_\infty\left\lVert\sqrt b\,\phi\right\rVert \left\lVert\sqrt b\,a\psi\right\rVert.\] Both terms are continuous in \(\left\lVert\sqrt E\,\phi\right\rVert\). Adding \(\left\langle(1+b)\phi,\chi\right\rangle\) consequently gives \[|\left\langle E\phi,\chi\right\rangle|\le C_\psi\left\lVert\sqrt E\,\phi\right\rVert.\] The spectral theorem characterizes this inequality as \(\chi\in\mathop{\mathrm{Dom}}\sqrt E\): test with spectral truncations of \(\chi\) and let their endpoints tend to infinity. Thus \(H_g\psi\in\mathop{\mathrm{Dom}}\sqrt a\), as required. ◻

Proof. Start with a Hermitian finite algebraic sum \(C_0\) of factorized original words compactly localized in the opposite wedge. Its closed affiliated realization was constructed in Section 2. The boost Ward identity applies to its original polynomial restriction; affiliation and opposite locality allow the compact current flux around \(C_0\) to pass the bounded operator \(A\).

Choose the partition for the flux around \(A\) once. Its time-zero causal shadow is compactly contained in the positive spatial halfspace, so a small slab neighborhood of that shadow lies compactly inside \(W\) and is spacelike to every opposite-wedge polynomial. Denote the resulting fixed compact current vacuum vector by \(\eta_A\). For each \(C_0\), its own time-zero causal shadow is a compact subset of the negative halfspace. Thin or deform the slab only near that shadow, leaving the \(A\) partition and the homogeneous shape at infinity unchanged. Conservation and (37), applied to each deformation, transfer the opposite flux to this same \(\eta_A\). The signs of the commutator give \[\left\langle BC_0\Omega,A\Omega\right\rangle =\left\langle C_0\Omega,BA\Omega\right\rangle =\mathcal C_\epsilon(\eta_A,C_0A\Omega).\] In particular the current side obeys \[|\mathcal C_\epsilon(\eta_A,C_0A\Omega)| \le2\left\lVert\eta_A\right\rVert\left\lVert A\right\rVert\left\lVert C_0\Omega\right\rVert,\] since \(C_0A\Omega=\epsilon AC_0\Omega\). No uniform bound on the changing flux near \(C_0\) is needed. The boost has been moved to the fixed vector \(A\Omega\) before taking any Hilbert limit.

Hermitian factorized polynomial vacuum vectors are real dense in the opposite wedge standard subspace by the Tomita-graph argument in Section 2. The displayed bounded functional therefore extends to \(C_s\Omega\). For the fixed family of truncated localizations, the compact spatial partition can be replaced by the negative half-slab: \(C_s\) is spacelike there, and the commutator vanishes outside the fixed causal intersection of \(A\). This gives the first identity in [N:undifferentiated-Ward]. Interchanging the fixed side gives the second. Its minus sign comes from placing the boost on the bra; the opposite flux contributes \(\epsilon\left\langle CA_s\Omega,j_+\Omega\right\rangle-\left\langle j_+\Omega,CA_s\Omega\right\rangle\). The operators \(A_s,C_s\) need only lie in their fixed truncated algebras; compact localization of these varied operators is not required. ◻

Proof. First let \(A\) be a Hermitian finite algebraic sum of factorized physical polynomial words on the invariant field domain \(\mathcal D_{\mathrm{old}}\). Sums and products of its affiliated single-field realizations, followed by closure, give a closed affiliated operator \(T\) such that \(T|_{\mathcal D_{\mathrm{old}}}=A\) and \(T^*|_{\mathcal D_{\mathrm{old}}}=A\). The product is closable because its adjoint contains the reversed adjoint product on the dense domain \(\mathcal D_{\mathrm{old}}\). Write \(T=U|T|\), let \(E_L=1_{[0,L]}(|T|)\), and set \[T_L=TE_L,\qquad A_L=\tfrac12(T_L+T_L^*).\] Then \(A_L\) is bounded Hermitian in the same local algebra, with \(\left\lVert A_L\right\rVert\leq L\). For \(\psi\in\mathcal D_{\mathrm{old}}\) and every nonnegative integer \(m\), the spectral-tail bounds give \[\begin{align*} \left\lVert(T-T_L)\psi\right\rVert &\leq L^{-2m}\left\lVert|T|^{2m+1}\psi\right\rVert,\\ \left\lVert(T^*-T_L^*)\psi\right\rVert &\leq L^{-2m}\left\lVert|T|^{2m}T^*\psi\right\rVert. \tag{55}\end{align*}\] Both right sides are finite polynomial-domain norms. Indeed \(\mathcal D_{\mathrm{old}}\) is invariant under \(T,T^*\), and powers of \(T^*T\) and \(TT^*\) there agree with the corresponding iterated physical polynomials. Hence \(A_L\psi-A\psi=O(L^{-N})\) for every prescribed \(N\). This proves the same assertion for any fixed finite family of polynomial states and moments needed in the argument. No essential self-adjointness is required.

One must not differentiate the sharp spectral projection \(E_L\). Instead first convolve \(A_L\) under the spacetime group, with a fixed compact smooth kernel \(f\). Group derivatives then fall on \(f\). The estimate [N:polar-tails] is uniform for the transformed polynomial tests on the compact support of \(f\), since its right sides are continuous norms of smeared polynomial vectors. Thus the convolved vacuum vectors converge in every fixed translation Sobolev norm. Apply the same construction to \(C\) separately. Their localization regions remain spacelike. For example, \[C_LA_L\Omega-CA\Omega =C_L(A_L-A)\Omega+(C_L-C)A\Omega,\] whose first term is bounded by \(L\) times a faster-than-any-power tail and whose second is another polynomial-state tail. The same argument after convolution gives convergence of products in the finite Sobolev norms used above. To make the product convergence explicit, let \(\alpha_r\) denote the Poincaré action and write \(A_L(r)=\alpha_r(A_L)\), \(A(r)=\alpha_r(A)\), with analogous notation for \(C\). Uniformly for \(r,s\) in fixed compact group sets, \[\left\lVert C_L(s)A_L(r)\Omega-C(s)A(r)\Omega\right\rVert \leq L C_NL^{-N}+C_N'L^{-N}.\] For the second term apply the polar-tail estimate to the polynomial state \(A(r)\Omega\); its iterated \(C(s)\)-polynomial norms are uniformly bounded on these compact sets. For a product of the two convolved operators, a common translation is left translation of both group variables. Integration by parts consequently places each prescribed translation derivative on the convolution kernels. The resulting finite sum of compact kernel integrals has the same tail estimate, with new constants. This proves the required Sobolev convergence without differentiating either polar projection.

For these factorized polynomials, the left side of (54) converges by its \(b^{1/2}\) form bound, and the right side by the inverse-energy estimates for the half-step wave. Finally remove the convolution on the original smooth field tests. All constants are finite test seminorms, so this last limit and compact-parameter uniformity follow from distributional continuity.

It remains to pass to the admitted compact joint kernels. For an old word \(A\), write \(\Psi_A(\varphi_A)\) for its vacuum vector smeared with a joint kernel \(\varphi_A\). First partition a compact support into finitely many product patches inside the required localization region. On each patch, finite sums of tensor tests are dense in the smooth test-function topology. Choose such kernels \(\varphi_{A,j}\to\varphi_A\) with their supports in those patches; their reversed adjoint kernels converge as well. Do the same for the kernel \(\varphi_C\) of \(C\). Polarization gives the factorized identity for arbitrary complex word tests, so it applies to these approximants. This is a test-space approximation, not a finite-rank requirement on the given kernels.

For every fixed nonnegative integer \(k\), the original joint-distribution bounds imply \[\bigl\|(1+P^0)^k\bigl(\Psi_A(\varphi_{A,j})-\Psi_A(\varphi_A)\bigr)\bigr\| \longrightarrow0,\] and analogously for \(C\) and for the concatenated old word with kernel \(\varphi_{C,j}\otimes\varphi_{A,j}\to\varphi_C\otimes\varphi_A\). Indeed, a collective time-translation derivative of a vacuum word is the same word with the corresponding derivative of its joint test. The doubled original vacuum distributions therefore bound each displayed graph norm by finitely many joint test seminorms. Positive translation spectrum makes these norms control the \(b^{1/2}\) graph norm and every fixed translation-Sobolev norm used above. The left side of (54) is continuous by its \(b^{1/2}\) form bound; the right side is continuous by the inverse-energy half-step bound applied to the concatenated old vacuum word. Taking this kernel limit after the polar and convolution limits proves the stated Ward identity for the joint kernels.

On a compact smooth parameter family the same finitely many seminorms are uniformly bounded. The tensor approximations can be chosen uniformly in those seminorms, using a finite cover of the compact family and a smooth partition in its parameter. The preceding graph and flux bounds then give the asserted compact-parameter uniformity. This final passage does not assert a closed affiliated operator for an arbitrary joint smear. ◻

Proof. The map \(g\mapsto h_g\) is positive and contractive in the supremum norm, and \(h_{\mathbf1}=\mathbf 1\). It commutes strongly with \(Y,h,b\) and is boost independent. In the \(Y\) representation write \(h_g=M_{F_g}\), where \(F_g\) is an essentially bounded internal-operator-valued function. For \(\tau_tg(z)=g(z+t)\), transverse covariance gives \[F_{\tau_tg}(y)=F_g(y+t)\quad\text{almost everywhere},\qquad \|F_g(\cdot+t)-F_g\|_{L^\infty} \le\|\tau_tg-g\|_\infty.\] The right side tends to zero. Weak-operator convolution with a smooth approximate identity therefore converges to \(F_g\) in essential-supremum norm. The convolutions are norm continuous; their pairwise essential-supremum bounds are consequently actual supremum bounds. Their uniform limit selects a unique norm-continuous representative \(\widetilde F_g\). Covariance now holds pointwise, so \[\mathcal L(g)=\widetilde F_g(0),\qquad \widetilde F_g(y)=\mathcal L(\tau_yg)\] defines one positive contractive internal functional. Linearity, positivity, and the commuting relations pass to these representatives. For inequalities involving the unbounded \(h\), first restrict to a bounded spectral band of \(h\), evaluate the resulting continuous inequality at zero, and then exhaust the bands.

The Riesz representation theorem applied to scalar matrix elements of \(\mathcal L\) on compact tests, and polarized, gives a positive operator-valued measure \(\mu\). Commutativity of all \(h_g\) and commutation with \(h\) pass to its Borel values by bounded monotone approximation. This proves (56) first for compactly supported smooth profiles.

It remains to rule out mass at infinity, which is not automatic from a representation on compact tests. Choose a radial \(\chi_0\in C_c^\infty(\mathbb R^2)\), \(0\leq\chi_0\leq1\), equal to one near zero and nonincreasing along rays. Set \(\chi_{0,R}(z)=\chi_0(z/R)\). The Taylor estimate at the \(Y=0\) fiber gives, on \(\mathop{\mathrm{Dom}}h^{1/2}\), \[ 0\leq\mathbf 1-\mu(\chi_{0,R}) \leq C R^{-2}h. \tag{58}\] For \(\mathcal E_R=\mathbf 1-\mu(\chi_{0,R})\), positivity gives \(0\leq \mathcal E_R\leq\mathbf 1\), hence \(\left\lVert\mathcal E_R\psi\right\rVert^2\leq\left\langle\psi,\mathcal E_R\psi\right\rangle\). Equation (58) therefore gives strong convergence to zero first on \(\mathop{\mathrm{Dom}}h^{1/2}\) and then, by density and the contraction bound, on the whole internal space. Monotone convergence and density imply \(\mu(\chi_{0,R})\uparrow\mathbf 1\) strongly. This proves normalization and excludes escaping mass. It also extends the representation from compact tests to bounded smooth profiles. Indeed positivity and linearity at the \(Y=0\) fiber give \[|\mathcal L(g)-\mu(g\chi_{0,R})| \leq\left\lVert g\right\rVert_\infty\bigl(\mathbf 1-\mu(\chi_{0,R})\bigr).\] The right side tends strongly to zero; it controls the omitted scalar matrix elements, and polarization completes the extension.

For the precise second-moment constant, put \(q_R(z)=|z|^2\chi_{0,R}(z)\). These bounded smooth profiles increase to \(|z|^2\) and satisfy \(q_R\leq p_0\). Direct cut inclusion and log order give \(h_{q_R}\leq |Y|^2+h\). At \(Y=0\) this is \[0\leq\mu(q_R)\leq h.\] Monotone convergence defines the densely defined closed second moment form \(\kappa\), dominated by \(h\). All its bounded spectral approximants commute with \(h\), so \(\kappa\) is affiliated with the same commutative algebra. This proves \(\kappa\leq h\) with coefficient one, rather than a Hessian-dependent constant.

Finally let \(e\in\mathbb R^2\) and \(a_R(z)=(e\cdot z)\chi_{0,R}(z)\). Then \(a_R(0)=0\) and \(\left\lVert a_R''\right\rVert_\infty\leq C|e|/R\), so \[|\mu(a_R)|\leq C|e|R^{-1}h.\] The second-moment bound makes the first moment integrable on the \(h\) form domain: for \(\psi\in\mathop{\mathrm{Dom}}h^{1/2}\), \[\int|z|\,\left\langle\psi,\mu(\mathrm dz)\psi\right\rangle \leq\left\lVert\psi\right\rVert\,\left\lVert h^{1/2}\psi\right\rVert.\] Dominated convergence and polarization now give \(\int e\cdot z\,\mu(\mathrm dz)=0\). This proves (57). Covariance of the construction under changes of transverse axes and dilations follows from covariance of the original cut generators; boost independence has already removed the choice of boost coordinate. ◻

Proof. Wightman wedge reflection gives the same polynomial vacuum data and the same physical stress-tensor flux on the right side of (54). Proposition 12 therefore identifies the two bounded-profile forms on opposite polynomial vacuum vectors. Their \(b^{1/2}\) graph density and the uniform form bound identify the operators. The compact-profile measure and its monotone moments are consequently identical. This argument identifies the bounded functionals and their quadratic moment; it does not identify the full parabolic modular generator with \(K_n\) at this stage. ◻

Proof. Translate the plane normally by \(u\), test \(u\) with a compact smooth function, and choose \(\widehat g\) smooth with compact support in a transverse annulus. Average \(n\) over a small smooth patch, taking a smooth transverse frame and a sphere normalization of \(n\). Opposite wedge tests can be chosen uniformly for these small changes. Insert a cutoff in the longitudinal coordinate \(v\) and split its matrix by the separating step used in (54). One term is that Ward formula; the remaining term is an all-plane vacuum matrix. The latter vanishes at spacelike transfer by the vacuum spectrum. The operator on the Ward side at normal coordinate \(u\) is \(U_a(u)H_gU_a(-u)\).

The removal of the \(v\) cutoff has an independent distributional meaning. In momentum space it inserts \(2\pi\delta(n\cdot k)\) after averaging over \(n\). On its support, \(k=a_xn+q\) with \(q\) in the chosen annulus. The derivative of \(n\cdot k\) in an angular direction of \(n\) has the nonzero transverse component \(q\). The coarea formula therefore turns the averaged delta insertion into a smooth multiplier. The Fourier transform of the normal test is Schwartz. Derivatives of the moving coordinates cost polynomial factors, absorbed by this decay and the compact annular test. Thus the cutoff multipliers converge in the Schwartz seminorms needed to pair with any of the tempered external field distributions. For an arbitrary fixed list of original fields, these are the finite sums of homogeneous-list distributions established in Section 2; no single scaling dimension is assigned to the whole list.

The identity on opposite wedge tests extends to arbitrary external translations by positive-energy tube uniqueness. On the operator side, \(|H_g|\leq\left\lVert g\right\rVert_\infty b\) supplies the polynomial energy bound: insert positive-energy exponential damping in the two external vectors, obtain the holomorphic matrix elements, and remove the damping as distributional boundary values. Equality on the open wedge patch then determines the boundary distribution everywhere. This establishes the identity jointly with the normal parameters. The fixed-normal pullback needed here is justified in Lemma 16. ◻

Proof. Retain compact timelike output tests throughout the construction. Testing against a \(j\) label amounts to testing with \(V\) and smooth transverse projectors on that compact set. Lorentz and scale covariance trivialize the open timelike orbit, leaving only the finite rest-rotation representation. They identify the resulting distributions in simultaneous moving coordinates. Fixed-\(p\) formulas below refer to this covariant disintegration; their estimates are uniform on compact positive output mass ranges.

Local commutativity supports the relative-position commutator in \(\overline V_+\cup\overline V_-\). Off the origin choose future and past cutoffs constant near these two cones on the unit sphere. Finite distribution order allows their products to be extended through the origin by subtracting a sufficiently long Taylor jet of the test function. This is the cone-supported distribution extension used in causal splitting; see (Epstein and Glaser 1973, sec. 5.1). Restore the total commutator by a distribution supported at the origin. This gives tempered cone-supported distributions whose difference is the commutator. Average the splitting over rest rotations and transport it along the timelike Lorentz and scale orbit. Its local ambiguity is a finite sum of derivatives of a delta function. With the Fourier convention \(D(p,x)=\int e^{ikx}D(p,k)\,\mathrm dk\), the resulting boundary values \(D^+,D^-\) are holomorphic in the upper and lower primitive tubes, respectively, with polynomial bounds in momentum and inverse distance to the tube boundary (Bros and Iagolnitzer 1973, sec. 1, pp. 155–156).

Here is a direct path from those tubes to the required half-planes. In the rest frame of \(p\), set \[ k^0=\frac{M_o^2-X+z}{2M_o},\qquad R^2=(k^0)^2+X,\qquad \boldsymbol k=R\omega, \quad\omega\in S^2\text{ real}. \tag{63}\] Write \(k^0=u+iv\), \(R=s+it\). Direct calculation gives \[ v^2-t^2= \frac{2Xv^2}{u^2+v^2+X+|(u+iv)^2+X|}>0, \qquad |v|-|t|\geq\frac{X|v|}{2(u^2+v^2+X)}. \tag{64}\] Thus (63) lies strictly in the relevant tube whenever \(\operatorname{Im}z\ne0\). On compact positive \(X,M_o\) ranges the distance is bounded below by a constant times \(|\operatorname{Im}z|/(1+|z|)^2\). The tube bounds therefore give polynomial bounds in \(|z|+|\operatorname{Im}z|^{-1}\).

The square root in (63) introduces no extra branch. A tensor with the fixed finite collection of external indices has only finitely many rotation harmonics. A harmonic of degree \(s\) is \(R^s\) times a Cartesian harmonic polynomial in \(\omega\); its remaining scalar coefficient is even and analytic in \(R\). At \(R=0\), where the path is still inside a tube, analyticity in the Cartesian momentum components proves regularity of that coefficient. Divide by \(R^s\) and reconstruct with Cartesian harmonic polynomials. The scalar coefficients are single-valued functions of the two invariants. They can consequently be evaluated at the actual moving light-front momentum in (62), even though that complex momentum itself has null imaginary part. The branch points \(k^0=\pm i\sqrt X\) are nonreal; none belongs to the real locus on which a distributional restriction is taken.

We also verify that the real boundary obtained this way is the required restriction. For fixed real \(\omega\), \(\mathrm dR/\mathrm dk^0=k^0/\sqrt{(k^0)^2+X}\) has absolute value less than one. The real tangent is strictly timelike. Integration by parts against a compact test interval therefore gives the uniform decay on the dual cone needed for the Fourier–Laplace boundary limit. The harmonic reconstruction gives the moving-direction restriction. There is a useful equivalent noncharacteristic check. The tube boundary distributions have ordinary wavefront covectors in the corresponding causal cones: compact localization in momentum convolves the cone-supported tempered inverse Fourier transform with a Schwartz function, giving rapid decay in directions outside that cone. For the full finite tensor family, rest-rotation covariance gives equations \(L_{ij}D=\rho_{ij}D\), where \(L_{ij}\) is the angular rotation vector field and \(\rho_{ij}\) a constant matrix on the tensor indices. The principal symbols of these equations force every singular covector to annihilate the angular orbit. Since real spacelike \(k\) has nonzero spatial part, its annihilator is exactly \(\mathop{\mathrm{span}}\{p,k\}\). If \(\xi=A p+B k\) annihilates the tangent \(n\) in (62), then \(A b=0\), so \(\xi=B k\). As \(k^2=-X<0\), no nonzero such covector is causal. The pullback along \(a_x\mapsto a_xn+q\) is therefore noncharacteristic. On compact positive \(X\) and timelike output ranges the separation of its conormal bundle from these wavefront cones persists in a small angular patch of \(n\). The restricted distribution thus depends smoothly on that patch, removing the angular average in Lemma 15. The normal displacement \(u\) is still tested, or Fourier transformed; no point evaluation of an arbitrary distribution in \(u\) is needed. This is a restriction statement for this covariant three-point matrix, not for arbitrary null-plane field distributions.

In the first Wightman order the vacuum leg has momentum \(p-k\); in the reverse order it has momentum \(k\). At \(X>0\) the reverse order vanishes by the spectrum condition, and the first can occur only for \(z\leq0\). The two holomorphic boundary values glue in the positive \(z\) gap. If instead \(z>0\) is fixed and \(X\) varies, upper-half-plane continuation corresponds to the lower original tube; its jump is minus the commutator, hence the reverse Wightman order with a positive sign. Both gap values are values of the same split commutator at two spacelike virtualities. This proves the comparison directly from causal tubes, without a crossing assumption. ◻

Proof. Choose a positive subtraction point \(z_0\) and retain the full finite covariant tensor family. Use one integer \(d\) larger than all its polynomial growth orders, perform the Cauchy subtraction on that family, and then contract with \(nn,\bar n\bar n\) and the output frame. The subtracted Cauchy kernel is \[\frac{(z-z_0)^d}{(z'-z_0)^d(z'-z)}.\] Using (65), its value at \(z=|r|^2\) is proportional to \[ \left[\mu_n(q)\chi^n(r,M) \left(\frac{|r|^2-z_0}{-M^2-z_0}\right)^d\right]_{j_A}. \tag{69}\] Subtracting this Cauchy transform leaves zero jump. Its boundary Fourier transform is supported in both closed half-lines, hence at zero; the remainder is a polynomial in \(z\).

Its dependence on the transverse parameter must also be controlled. A null rotation with parameter \(\beta\), fixing \(n\), changes \[r\longmapsto r+b\beta,\quad a\longmapsto a+\beta\cdot(q+r)+\tfrac12b|\beta|^2,\quad a_x\longmapsto a_x+\beta\cdot q.\] It fixes \(b,M_o,q,X,z\) and transports the output adapted frame. The remaining tensor contractions transform polynomially in \(\beta\), by a matrix independent of \(z\). This matrix commutes with the common-order Cauchy subtraction. Covariance of the split commutator therefore makes the coefficients of its subtraction polynomial polynomial in \(r\) at fixed \(q\). Evaluation at \(z=|r|^2\) preserves this property. The opposite calculation gives a polynomial in \(q\) at fixed \(r\). The degrees are uniform because the distribution order in a rest frame and the degrees of the finite tensor representation are finite. The opposite calculation may hold its own boost coordinate \(a\) fixed; the evaluated contracted matrix has net boost weight zero, so it can then be written at the same \(b\) as the first calculation.

For completeness the wave regularization can be made explicitly. Set \(R_2=|r|^2\), \(D=M^2+R_2\). The change of spatial frame gives \[ c_T^{(\bar n)}=l/3+d_Ad_Bs_{AB},\qquad d_z=\frac{R_2-M^2}{D},\qquad d_i=-\frac{2Mr_i}{D}. \tag{70}\] Write \(s_{ij}^{\rm TF}=s_{ij}+\delta_{ij}s_z/2\). Expansion of (70) gives \[ \chi^n=\frac D3l+ \left(R_2-5M^2+\frac{6M^4}{D}\right)s_z +\left(-4Mr_i+\frac{8M^3r_i}{D}\right)s_{zi} +\frac{4M^2r_ir_j}{D}s_{ij}^{\rm TF}. \tag{71}\] In particular the radial coefficient follows from \(s_{11}+s_{22}=-s_z\): \(((R_2-M^2)^2-2M^2R_2)/D=R_2-5M^2+6M^4/D\). The vector coefficient is \(2Dd_zd_i\).

Choose a smooth mass cutoff \(\zeta\) equal to zero below \(1\) and one above \(2\), and subtract \(\zeta(M)\) times \[\frac{M^2+R_2}{3}l+(M^2-5R_2)s_z +(4Mr_i-8r_iR_2/M)s_{zi}+4r_ir_js_{ij}^{\rm TF}.\] The remaining large-mass terms are \(O(M^{-2})\) or better on compact \(r\) sets, and hence belong to \(L^2(M\,\mathrm dM)\). At small mass, \(|d|=1\) in (70), so the norm is bounded by a constant times \(M^2+|r|^2\). Dominated convergence now proves continuity also at \(r=0\); the same formulas give polynomial growth in \(r\).

Replacing (69) by this regularization changes only an \(L^2\)-valued polynomial in \(r\). Indeed \[(R_2-z_0)^d-(-M^2-z_0)^d\] is divisible by \(D\), canceling the sole denominator in (71). Both regulated expressions are \(L^2\) waves; polynomial interpolation then shows that every coefficient of their polynomial difference is an \(L^2\) wave. Choose the same mass cutoff and subtraction prescription in the other horizon.

Finally the output frame rotation is \[j_z^{(n)}=\frac{|w|^2-M_o^2}{M_o^2+|w|^2}j_z^{(\bar n)} -\frac{2M_ow_i}{M_o^2+|w|^2}j_i^{(\bar n)}.\] The common gap value from Lemma 16 and the common jump normalization therefore give (68) off \(q=0\) and \(r=0\).

There is no extra distribution supported on those axes. The left side is continuous in \(q,r\), with values on compact mass tests. Take polynomial interpolation projections \(\mathcal P_q, \mathcal P_r\) of degrees larger than the two degree bounds, using nodes away from zero. On the off-axis patch, \((1-\mathcal P_q)(1-\mathcal P_r)\) annihilates the left side. By continuity it does so everywhere. Decomposing with these two projections extends its separated-polynomial form to all \(q,r\). ◻

Proof. The Bessel recurrence \(J_2(x)=2J_1(x)/x-J_0(x)\) and \(\int_0^v rJ_0(Mr)\,\mathrm dr=vJ_1(Mv)/M\) give \[ (\mathcal H_2\mathcal H_0F)(v) =\frac2{v^2}\int_0^v rF(r)\,\mathrm dr-F(v). \tag{76}\] The identity first follows on smooth compact tests by the unitary Hankel inversion and then in \(L^2(v\,\mathrm dv)\). If \(F\) has zero first radial moment and support below \(R\), the right side also has support below \(R\). The order-two comparison in (61) therefore puts \(c\mathcal H_0F\) in the \(\lambda\leq R\) subspace.

Consequently the restriction of the transfer to any interval strictly above \(R\) factors through the single functional \(F\mapsto\int vF(v)\,\mathrm dv\). Choose one annular test with this moment equal to one. Overlapping support bounds identify the resulting vector-valued coefficient at every \(\lambda>0\). In these coordinates internal dilatations act as \(f(M)\mapsto e^sf(e^sM)\) on the input and \(\psi(\lambda)\mapsto e^{-s}\psi(e^{-s}\lambda)\) on the output. Since the moment of the corresponding Hankel-side test scales by \(e^s\), its coefficient must be \(\lambda^{-2}d_c\). A reference test and its square-integrable tail ensure \(d_c\in\mathcal K\). This proves an exact tail, not only its asymptotic order.

The order-zero comparisons on \(c_V\) and \(c_T\) say that \(c_V\mathcal H_0F\) and \(c_T\mathcal H_0F\) have no spectrum above the support radius of \(F\). Linearity of the tail gives \(d_l+d_{j_z}=0\) and \(d_l/3+d_{s_z}=0\). ◻

Proof. The kernels computed after (73), with \(t=M|z'|\), give explicitly in our transverse Fourier convention \[\begin{align*} f_L(M)&=6\int_0^\infty t^3K_0(t) \eta_L(\log t-\log M)\,\mathrm dt,\tag{83}\\ u_L(M)&=-4i\int_0^\infty t^2K_1(t) (2\eta_L+\eta_L')(\log t-\log M)\,\mathrm dt. \tag{84}\end{align*}\] Changing the Fourier orientation changes the fixed phase in the second formula only. The first has \[c_*=6\int_0^\infty t^3K_0(t)\,\mathrm dt=24\] with the convention fixed above. In particular it cannot vanish. On the logarithmic variable these are convolutions with integrable kernels having finite first moment. At \(t=0\) their densities decay exponentially in \(|\log t|\), and at infinity they decay faster; the relevant logarithmic derivatives have the same integrability. Convolution of the interval plateau differs from its kernel integral times that plateau by a uniformly bounded \(L^2\) error: the difference is supported near the two edges up to integrable tails. Positive derivatives act on these two fixed-width edges. This proves (79)–(80). The functions themselves and their derivatives have integrable tails at zero and infinity for each fixed \(L\).

Direct differentiation gives \[h_1(Mu)=-M^{-1}(\mathscr D^2+2\mathscr D)u, \qquad \left\lVert h_1(Mu)\right\rVert^2= \int_0^\infty|\mathscr D^2u+2\mathscr Du|^2\frac{\mathrm dM}{M}.\] The order-one comparison proves the first estimate in (81); the second is (79).

To prove (82) without assigning a divergent form norm to an individual nonzero-moment test, fix a smooth annular \(F\) with \(\int vF(v)\,\mathrm dv=1\) and put \(G=\mathcal H_0F\). Replace \(f_L\) by \[c_*\mathcal H_0\bigl(F-L^{-2}F(\cdot/L)\bigr)(M) =c_*\bigl(G(M)-G(LM)\bigr).\] Since \(G(0)=1\) and \(G\) decreases rapidly at infinity, this difference has the same plateau and the same uniform edge estimates as \(f_L\). Its error \(e_L\) satisfies \[\left\lVert\sqrt{h_2}e_L\right\rVert^2 =\int_0^\infty (|\mathscr De_L|^2+4|e_L|^2)\frac{\mathrm dM}{M}=O(1).\] The comparison (61) gives an \(O(1)\) error after multiplication by \(\lambda\). Above a fixed radius, the \(G\) term has the exact tail of Lemma 18. The norm of \(G(LM)\) in \(L^2(M\,\mathrm dM)\) is \(L^{-1}\left\lVert G\right\rVert\); hence its \(\lambda\)-weighted norm below \(CL\) is \(O(1)\) for any fixed \(C\). Above both Hankel support radii the two exact tails cancel, because their moments are both one. The remaining fixed lower interval and fixed-ratio upper interval contribute \(O(1)\) to the weighted norm. This proves (82). The \(\lambda=0\) spectral subspace contributes nothing to that estimate.

Using the difference of the two dilated tests is essential here: their individual \(\lambda^{-2}\) tails need not have finite \(\lambda\)-weighted norm, but the exact cancellation above the larger support radius makes their difference admissible. No estimate has been assigned to either divergent weighted tail separately. ◻

Proof. On the positive \(\lambda\) spectrum define \[A_L(\lambda)=\lambda^{-2}\mu(g_L),\qquad B_{i,L}(\lambda)=\lambda^{-1}\mu(\partial_i g_L).\] The moments in (60) imply \(0\leq A_L\leq\mathbf 1\) and \(\left\lVert B_{i,L}\right\rVert\leq C\) uniformly: \(|\partial_i g_L(z')|\leq C|z'|\), and positive-measure Cauchy–Schwarz bounds its integral by \(C\sqrt\kappa\). Scale covariance identifies the law of \(z'/\lambda\) in each fiber with one fixed positive measure on \(\mathcal K\). Its mean is zero and its trace-second moment is \(\mathbf 1\). Hence, on every fixed multiplicity vector, \[ A_L(\lambda)\longrightarrow\mathbf 1,\qquad B_{i,L}(\lambda)\longrightarrow0 \quad\text{if }\lambda\longrightarrow\infty, \quad L/\lambda\longrightarrow\infty. \tag{86}\] For the first limit use the integrable weight \(|z'/\lambda|^2\); for the second use \(|z'/\lambda|\) and the zero first moment. The limiting derivative integrand is \(2z'_i/\lambda\). Bounded convergence for the positive measure proves strong convergence; no uniform rate is required.

These are statements about the fixed multiplicity vectors that enter the tail formula. Positivity and the second moment control the truncated integrands; projection-valuedness of \(\mu\) is unnecessary. On \(\ker\kappa\) the second moment is zero, so \(\mu\) is concentrated at \(z'=0\) there. The profiles \(g_L\) and \(\partial_i g_L\) vanish near that point, and hence this spectral subspace makes no contribution to the pairing.

Pair (73) with \(f_L(M_o)/M_o\) and use (78). This is allowed although the identity was first tested on compact mass intervals. Cut off this test smoothly at zero and infinity. The approximants converge both in its \(L^2(M_o\,\mathrm dM_o)\) norm and after multiplication by \(M_o\). For fixed \(L\), the profile operators are bounded, and (81) controls the derivative wave, so every pairing has a limit.

The left side becomes \[\begin{align*} \left\langle j_zf_L,\mu(g_L)s_zf_L\right\rangle &=\left\langle\lambda\mathcal U(j_zf_L),A_L\lambda\mathcal U(s_zf_L)\right\rangle\\ &=|c_*|^2\int_1^L \left\langle d_{j_z},A_L(\lambda)d_{s_z}\right\rangle\frac{\mathrm d\lambda}{\lambda} +O(\sqrt{\log L}). \end{align*}\] The exact normalization \(|c_*|^2\) is common to the bra and ket. Equation (86) and boundedness imply, by averaging on the logarithmic interval, that this is \[ |c_*|^2\log L\,\left\langle d_{j_z},d_{s_z}\right\rangle+o(\log L). \tag{87}\] For example, discard two logarithmic edge intervals of fixed width, apply the strong bulk convergence on the remaining interval, and then let the discarded width grow. This proves the asserted Cesaro limit directly.

For the first term on the right of (73), (86), (82) and uniform boundedness give \[\left\lVert\mu(\partial_i g_L)s_zf_L\right\rVert =\left\lVert B_{i,L}\lambda\mathcal U(s_zf_L)\right\rVert =o(\sqrt{\log L}).\] The output test \(j_if_L/M_o\) has norm \(O(\sqrt{\log L})\). Their pairing is therefore \(o(\log L)\). For the last term, \(0\leq\mu(g_L)\leq\kappa\) and commutativity give \[\left\lVert\mu(g_L)\Phi(\partial_i g_L)\right\rVert \leq\left\lVert\kappa\Phi(\partial_i g_L)\right\rVert=O(1).\] Its pairing is \(O(\sqrt{\log L})\). Comparison with (87) forces \[0=\left\langle d_{j_z},d_{s_z}\right\rangle=\tfrac13\left\lVert d_l\right\rVert^2,\] by (75). Thus \(d_l=0\). If either of the spin labels used above is zero, the same conclusion already follows from (75).

Lemma 18 now says that \(l\mathcal H_0F\) has no \(\lambda\) spectrum above \(R\) whenever \(F\) is supported below \(R\). Smooth compact tests in \((0,R)\) are dense in \(L^2((0,R),v\,\mathrm dv)\). Hankel unitarity and closedness of spectral subspaces give (85) with \(s=R^2\); at \(s=0\) the regular \(h_0\) copy has no zero eigenvector. The result concerns the moment \(\kappa\). It does not identify the full parabolic operator \(h\) with that moment. ◻

Proof. The scalar wedge conjugation is the real scalar conjugation; it is identified already on the dense waves generated by \(M^2l\). To prove (88), cancel the common external factors and consider \(e^{-icb\kappa}\iota e^{icbh_0}\). Let \(E_i\) be the actual \(\kappa\) step projections and \(P_j\) the reference \(h_0\) step projections. Clip both spectral variables at one common finite bound, then approximate the clipped \(\kappa\) from below and the clipped \(h_0\) from above. If their step values are \(a_i,c_j\), spectral inclusion gives \(E_i\iota P_j=0\) whenever \(a_i>c_j\). Each surviving ordered term is therefore \(E_i\iota P_j e^{icb(c_j-a_i)}\), with nonnegative translation parameter; no joint spectral resolution is used. The coefficients intertwine the boost and wedge conjugation, so they map the reference real wedge space into the actual one. The positive translations preserve that actual space. At the common clipping endpoint both step values coincide, which preserves the same sign. Strong convergence follows from the separate spectral calculi as the mesh tends to zero and the clipping bound tends to infinity. Closedness of \(X(W)\) proves the inclusion. This uses the ordered products of two separate spectral resolutions, not a joint resolution of \(\kappa\) and \(h_0\).

For the second transport, positivity, normalization, and the zero first moment of \(\mu\) give \[H_g=b\int g(Y+z)\mu(\mathrm dz) \le b\{c(|Y-d|^2+\kappa)+c_0\}=K_p.\] This inequality is in the common commuting calculus. Its difference is \(b\) times a nonnegative boost-independent operator commuting with wedge conjugation. Thus \(e^{i(K_p-H_g)}\) is a direct integral of inward null translations and preserves \(X(W)\). Apply it to (88) to obtain membership in \(X(W_g)\).

Finally the actual parabolic generator is \(H_p=b\{c(|Y-d|^2+h)+c_0\}\). Since \(h\) and \(\kappa\) commute and \(\kappa\le h\), the same argument applies to \(H_p-K_p=bc(h-\kappa)\ge0\) and proves (89). The operators vanish on \(b=0\). Covariance and reflected agreement give the remaining placements. No equality of \(h\) with \(\kappa\) is used. ◻

Proof. Normalizing \(n=(1,\omega)\) and taking \(\omega=\pm e_a\) shows that \(K_y\) is compact, uniformly on compact spectator sets. We first remove its part outside \(C_y\), then recover the coefficients of the remaining wave solutions with bounds that preserve the unshifted spectator spectra.

We give the support argument explicitly. Choose an open spectator patch around distinct equal-time spatial extremal vertices, so that each \(y_i\) is a strict maximum and a strict minimum for open sets of future null directions. If \(x\in K_y\cap J^+(y_i)\), the maximizing directions give \(0\le n\cdot(x-y_i)\le0\). A causal vector orthogonal to an open set of null directions is zero. The minimizing directions treat \(J^-(y_i)\). Hence \[ K_y\cap C_y=\{y_1,\ldots,y_m\} \tag{93}\] on a smaller patch. Fix a test neighborhood \(U\) spacelike to all the vertices, and choose small vertex neighborhoods \(N_i\) whose closures remain spacelike to \(U\). There is a compactly supported smooth \(\chi(x)\) equal to one near \(K_y\) whose derivatives meet \(C_y\) only in \(N=\bigcup_iN_i\). Positive separation margins make one such cutoff work uniformly after shrinking the spectator patch. A compact spectator cutoff causes no change to \(\Box_x\). Thus \(v=\chi H\) has compact \(x\)-support and \[\Box_xv=[\Box_x,\chi]H=:f,\qquad \mathop{\mathrm{supp}}_xf\subset N.\] For compactly supported distributions the exact identity \(v=E_{\rm ret}*\Box_xv\) follows by moving \(\Box\) through convolution. Its right side vanishes on \(U\), since \(J^+(N)\cap U=\varnothing\). Near \(K_y\) one has \(v=H\); elsewhere the original support statement already suffices. This removes \(K_y\setminus C_y\) from (91), without any decomposition of distributions on touching closed sets.

In a common thin time slab the cone components are disjoint. A spatial partition separates them with derivatives off their supports. Each component still solves the wave equation there. In \(r=x-y_i\) coordinates its displacement and normal-derivative data on \(r^0=0\) are supported at \(\boldsymbol r=0\). Distributional solutions of the wave equation have distributional Cauchy data on this non-characteristic slice. The point-support structure theorem therefore writes both data as finite sums of spatial delta derivatives with spectator-distribution coefficients. On a smaller relatively compact patch their orders are uniformly bounded. With \(\Delta_0(0,\boldsymbol r)=0\) and \(\partial_0\Delta_0(0,\boldsymbol r)=\delta(\boldsymbol r)\), their evolutions are derivatives of \(\Delta_0\). Cauchy uniqueness first in the slab and then globally in \(x\) gives \[ H(x,y)=\sum_{i,P}C_{iP}(y)P(\partial_x)\Delta_0(x-y_i). \tag{94}\] Choose the finitely many polynomials \(P\) independently modulo \(k^2\). Enlarging the finite family if necessary, use homogeneous bases \(P_{q\alpha}\) of the quotient by \(k^2\) in each degree \(q=0,\ldots,N\).

We next justify continuation of the coefficients, including its growth bound. This is needed to recover the unshifted spectator spectra. Smear the Fourier transform of \(P(\partial_x)\Delta_0(x-y_i)\) against compact \(k\)-tests. Its column is, up to constant factors, \[\big\langle P(k)\widehat\Delta_0(k)e^{ik y_i},\varphi(k)\big\rangle.\] All augmented minors expressing that the corresponding tests of \(H\) lie in the span of these columns vanish on the initial patch. They are holomorphic in the ordered spectator tube; boundary uniqueness makes them vanish throughout that tube. The columns have full rank at every interior point, with a quantitative bound as follows. Choose finitely many fixed positive null vectors \(n_r=(1,\omega_r)\) such that, for every \(q\le N\), the evaluation matrix \[(E_q)_{r\alpha}=P_{q\alpha}(n_r)\] is injective. Such a choice exists: a homogeneous polynomial vanishing on every positive null ray is zero modulo \(k^2\), and finite dimensionality permits a finite separating set of evaluations. Taking the union of these sets for \(q\le N\) gives one collection of rays. Fix a left inverse for each \(E_q\).

On the \(r\)th ray the polynomial-exponential column sum is \[\sum_{i=1}^m\sum_{q=0}^N a_{iqr}s^qe^{is n_r\cdot y_i}, \qquad a_{iqr}=\sum_\alpha C_{i,q\alpha}(y)P_{q\alpha}(n_r).\] Let \(d=m(N+1)\), \(\varepsilon=c/(1+|y|)\), and use the \(d\) samples \(s=t\varepsilon\), \(t=1,\ldots,d\). The matrix from the coefficients \(a_{iqr}\) to these samples is \[(V_r)_{t,(i,q)}=(t\varepsilon)^qz_{ir}^{\,t}, \qquad z_{ir}=e^{i\varepsilon n_r\cdot y_i}.\] Its determinant is the confluent Vandermonde determinant \[ \det V_r=c_{m,N}\varepsilon^{mN(N+1)/2} \prod_i z_{ir}^{(N+1)(N+2)/2} \prod_{i<j}(z_{jr}-z_{ir})^{(N+1)^2}, \qquad c_{m,N}=\Bigl(\prod_{q=0}^Nq!\Bigr)^m. \tag{95}\] For example this follows by replacing powers of \(t\) by falling factorials and using the derivative columns of the ordinary Vandermonde matrix. Thus the ray inversion distinguishes the degree \(q\) and center \(i\) before the fixed left inverses of \(E_q\) recover each coefficient \(C_{i,q\alpha}\). This describes a left inverse for the full rectangular multi-ray matrix; independence on each ray alone would not suffice.

The scale of the samples is also significant. Keeping them at fixed nonzero momenta would preserve the exponential type of the shifted spectator cones. Taking \(s=O((1+|y|)^{-1})\) instead keeps that exponential bounded. The determinant estimate ensures that this shrinking choice costs only polynomial powers, which are compatible with tempered spectral boundary values.

For the tube distance \(\delta(y)\) in the statement, the moduli of the \(z_{ir}\) are bounded above and below, and for \(i\ne j\) \[|z_{ir}-z_{jr}|\ge\big||z_{ir}|-|z_{jr}|\big| \ge c'\varepsilon\delta(y)\quad(i\ne j),\] because the imaginary gaps are strictly ordered and the directions are fixed. The determinant formula and cofactor bounds therefore give the full left inverse polynomial norm in \(\varepsilon^{-1}\) and \(1+\delta(y)^{-1}\). To turn these point samples of the known columns into legitimate compact \(k\)-tests, replace each sample \(t\varepsilon n_r\) by a smooth bump of width \(w\), normalized to have integral one on the positive null shell. The centers are a distance comparable to \(\varepsilon\) from its vertex. For \(w\le c_1\varepsilon\), its shell Jacobian and the prescribed test derivatives therefore give \[p_N(\varphi)\le C\varepsilon^{-a}w^{-b}\] for fixed finite \(a,b\). Derivatives of the column functions \(P(k)e^{ik\cdot y}\) cost polynomial powers of \(1+|y|\) and \(\varepsilon^{-1}\), while their exponential is bounded on these supports. If \(A_0\) is the point-sample matrix and \(L_0A_0=1\), choose \[w=c_2\varepsilon(1+|y|+\delta(y)^{-1})^{-J}\] with fixed \(J\) sufficiently large and \(c_2\) sufficiently small. The resulting test-column matrix \(A\) then satisfies \(\|L_0(A-A_0)\|<1/2\). Its left inverse \((L_0A)^{-1}L_0\) has the same polynomial bound, and the displayed seminorm cost remains polynomial.

This proves full rank using genuine test columns. On a neighborhood of any fixed interior point, keep a full-rank finite set of tests fixed; its inverse and the vanishing augmented minors define holomorphic coefficients. Uniqueness identifies them on overlaps. The parameter-dependent test choices above serve only to estimate these already defined holomorphic functions. Their supports satisfy \(|k|=O(\varepsilon)\) and their seminorms have polynomial cost. Applying (92) with \(K=O(\varepsilon)\) therefore costs only polynomial powers: the remaining exponential is bounded by \(\exp(O(\varepsilon|\operatorname{Im}y|))=O(1)\). The left inverse consequently gives \[ |C_{iP}(y)|\le C(1+|y|)^N(1+\delta(y)^{-1})^N \tag{96}\] for some \(C,N\) on the tube. The coefficients have ordinary tempered ordered spectral boundaries. Testing all compact \(k\)-sets, including a neighborhood of zero, proves (94) globally. Translation covariance also shows that the coefficients are invariant under common spectator translation. ◻

Proof. Causal source solutions. The empty physical product has only the specified zero vacuum expectation, since \(l\) has no zero-momentum component. Suppose \(m\ge1\) and fix an order of homogeneous original entries \(A_i\in\mathcal F_{\Delta_i}\), \(i=1,\ldots,m\). Use the Fourier convention \(e^{-ikx-i\sum_i p_i y_i}\), and put \(q_h=\sum_{i\le h}p_i\), with \(q_0=0\). For an insertion just after \(A_j\), the required spectral conditions are \[ q_h\in\overline V_+\quad(1\le h\le j), \qquad k+q_h\in\overline V_+\quad(j\le h\le m-1), \qquad k+q_m=0. \tag{97}\] Redundant zero partial sums at the endpoints may be included. Write \(E_R,E_L\) for the known extreme values. For each \(i\), take the difference of the physical distributions with \(\Theta\) immediately before and immediately after \(A_i\). Its support, in \(r=x-y_i\), is causal. Split it into forward and backward parts by a finite-order Taylor subtraction in \(r\) alone, and apply respectively the retarded and advanced wave fundamental solutions. This gives a tempered \(B_i\) with the same causal support and the prescribed wave source. The convolutions are proper on the respective cones: two future causal vectors, or two past causal vectors, are bounded on the inverse image of a bounded set under addition. The same cone geometry gives the polynomial bounds needed for tempered convolution.

This construction preserves precisely the common spectral cones \[ q_h\in\overline V_+\ (h<i),\qquad k+q_h\in\overline V_+\ (h\ge i). \tag{98}\] Indeed in the coordinates \((r,y_1,\ldots,y_m)\) the spectator momenta are \(p_j\) for \(j\ne i\) and \(k+p_i\) for \(j=i\). All the conditions in (98) involve only those spectator momenta. Multiplication, subtraction, and convolution in \(r\) do not alter them. This is why solving the wave equation separately in the external momentum before doing causal splitting would be insufficient.

Removing the homogeneous defect. Set \[R_j=E_R+\sum_{i>j}B_i,\qquad L_j=E_L-\sum_{i\le j}B_i.\] Both have the wave source appropriate to the \(j\)th position. The first has all the initial cones in (97); the second has all its final cones. Their difference \(H_*=R_j-L_j\) is independent of \(j\) and satisfies \(\Box_xH_*=0\). After a compact Fourier test in \(k\), its spectator partial sums lie in \(\overline V_+\) up to bounded shifts of size \(O(\sup|k|)\). It consequently has an analytic continuation to the ordinary ordered spectator tube, with polynomial bounds multiplied by an exponential of type \(O(\sup|k|)\). Equation (90) and the causal support of the \(B_i\) give the support condition \(\mathop{\mathrm{supp}}_xH_*\subset C_y\cup K_y\) of (91).

Its shifted spectator cones also give (92): for a transfer test supported in \(|k|\le K\), every partial cone is shifted by at most a fixed multiple of \(K\). Fourier–Laplace damping contributes \(\exp(CK|\operatorname{Im}y|)\); the finite-order tempered bound on the joint distribution controls the remaining terms by one fixed seminorm \(p_N(\varphi)\) and polynomial powers of \(|y|\) and the inverse tube distance. This estimate is uniform in the test, including when its support shrinks toward \(k=0\). Lemma 22 therefore gives the expansion (94) for \(H_*\).

Let \(H_i\) denote the summand centered at \(y_i\). In Fourier variables it shifts \(p_i\) by \(k\), and no other spectator momentum. Consequently it satisfies exactly the common cones (98). The required distributions are \[ W_j=R_j-\sum_{i>j}H_i =L_j+\sum_{i\le j}H_i. \tag{99}\] The first expression supplies the initial cones and the second the final cones; \(W_m=E_R\), \(W_0=E_L\), and \(W_j-W_{j-1}=-B_j+H_j\) is supported in \(J(y_j)\).

Locality and descent to polynomial vectors. For an adjacent interchange of physical fields in the suffix, smear all suffix variables with any compact test supported where that pair is spacelike. Keep this test fixed. The two orderings have the same partial spectral cones in the prefix and \(x\): \(q_h\in\overline V_+\) for \(h\le j\) and \(q_j+k\in\overline V_+\). Smearing the suffix does not change those cones. Their difference is therefore the boundary of a holomorphic function in the ordered tube of the \(j+1\) variables \(y_1,\ldots,y_j,x\), with the last imaginary difference measured from the real boundary. On the nonempty open set where \(x\) is spacelike to every prefix point, the already proved adjacent \(L\) interchanges move it to the extreme left. At that extreme, physical locality makes the fixed suffix swap zero, regardless of the prefix. Boundary uniqueness in this partial tube makes the difference zero for all prefix and \(x\) variables. Since the fixed suffix test was arbitrary on the spacelike-pair patch, this proves the required distributional locality. For a swap in the prefix, hold its test fixed instead and move \(L\) to the extreme right. The common final cones, equivalently the negative reversed partial sums by total momentum conservation, give the opposite ordered tube in \(x\) and the suffix variables. The same argument applies.

For a fixed homogeneous list, locality moves \(L\) to an extreme position on a spacelike open set. Ordered tube uniqueness therefore shows that the specified extreme values determine the form uniquely. In particular, the result is multilinear in the homogeneous labels and is covariant, independently of the auxiliary causal splittings. For arbitrary original entries write \[A_i=\sum_{\Delta\in S_i}A_{i,\Delta},\qquad |S_i|<\infty,\] and take the finite sum of the constructed forms over \(S_1\times\cdots\times S_m\), using the same joint test kernel in every term. The common ordered spectral cones, locality, wave equation, and extreme pairings survive this finite sum. Its dilation covariance is the termwise action on the labels, not homogeneity of one total degree. The distribution orders and constants may depend on this fixed finite list.

The vector-dependence assertion uses exactly the preceding choice of which variables to hold fixed, now for the whole finite summed form. If a fixed ket polynomial \(A\Omega\) is zero, even by cancellation between component terms or word lengths, vary only the opposite prefix and \(x\). Once the entire ket has been smeared and held fixed, the finite sum has the same prefix-and-\(x\) tube, regardless of the word lengths on that fixed side. On the open set where the \(L\) interchanges through that prefix are spacelike, the extreme-left value pairs \(l\) with \(C^*A\Omega=0\). The partial-tube argument just given makes the insertion zero everywhere. It never requires a null polynomial vector to remain zero under independent translations of its individual factors. If the bra polynomial is zero, hold it fixed and use the extreme right and the opposite tube. For a permitted joint kernel, distributional continuity first supplies the summed form and its common partial-tube boundary for that fixed whole kernel. The zero extreme pairing is then used for the whole kernel, not for its approximants. Hermiticity and the stated covariance follow from uniqueness of the whole form. This proves a statement about forms with one \(L\), without a positivity assertion for repeated insertions.

In particular, the construction may now be differentiated in the \(L\) slot and paired with arbitrary physical polynomial vectors. These are precisely the operations needed to form the conserved currents below; none requires composing two \(L\) insertions. ◻

Proof. The mass projection commutes with the wedge modular data and hence preserves all wedge real spaces containing \(Z\). Wedge separation therefore makes the self-commutator function of \(E_IZ\) vanish at bounded times and sufficiently large spatial separation, with the spatial bound independent of \(I\). Its time derivative at zero has compact spatial support and is bounded in absolute value by \(2\langle Z,P^0E_IZ\rangle\). If \(\mu_I\) is the positive translation spectral measure of \(E_IZ\), the spatial Fourier transform of this derivative, after division by its fixed imaginary factor, is the sum of the positive marginal measures of \(p^0d\mu_I(p)\) at \(\boldsymbol p\) and \(-\boldsymbol p\). Fourier transformation of the compactly supported bounded function bounds each of these positive measures by \(C_Z\langle Z,P^0E_IZ\rangle d^3\boldsymbol p\). On the positive-energy cone \(p^0\ge|\boldsymbol p|\). Hence \[\int_{p^0<\Lambda}d\mu_I \le C_Z\langle Z,P^0E_IZ\rangle \int_{|\boldsymbol p|<\Lambda}\frac{d^3\boldsymbol p}{|\boldsymbol p|} \le C'_Z\Lambda^2\langle Z,P^0E_IZ\rangle.\] The integrable singularity at \(\boldsymbol p=0\) causes no problem; the positive mass band has no atom there. Taking square roots proves the estimate. ◻

Proof. Choose \(\chi\in C_c^\infty(\mathbb R)\) equal to one on \([0,1]\), put \(\chi_j(\tau)=\chi(\tau-j)\), and write \(L=e^j\), \(j\ge0\). At Fourier frequencies \(\zeta<-\delta L\), split the convolution \(\widehat\chi_j*\widehat g\) at source frequency \(\nu=-\delta L/2\). The Tomita-domain assumption gives \[\|\mathbf1_{\nu<-\delta L/2}\widehat g\| \le e^{-\pi\delta L/2}\|e^{-\pi B}g\|.\] For the remaining source frequencies, \(|\zeta-\nu|>\delta L/2\). The Schwartz \(L^1\) tail of \(\widehat\chi_j\) is \(O_N(L^{-N})\), uniformly in \(j\), since translation changes only its phase. Young’s inequality therefore gives \[\|\mathbf1_{\zeta<-\delta L}\widehat{\chi_jg}\| \le C_NL^{-N}\bigl(\|g\|+\|e^{-\pi B}g\|\bigr).\] On \(\zeta\ge-\delta L\), take \(\delta\) smaller than half the minimum of \(ce^{\tau-j}\) on \(\mathop{\mathrm{supp}}\chi_j\). The phase \(-ce^\tau-\zeta\tau\) then has derivative of absolute value at least \(C(L+|\zeta|)\). Repeated integration by parts in the Hilbert-valued integral gives \[\|\widehat{\chi_jg}(\zeta)\| \le C_N(L+|\zeta|)^{-N}\sum_{k\le N}\|f^{(k)}\|_2.\] Its squared integral on this half-line is \(O(L^{-2N+1})\) times the displayed Sobolev bound squared. Since \(\|\chi_jf\|=\|\chi_jg\|\), these two estimates give arbitrary inverse powers of \(e^j\) for the localized norm. Choose \(N>r+1\) and sum over \(j\); on \(\tau<0\) the weight is at most one. This proves (107) and the stated uniformity. ◻

Proof. On the lower side choose \(c>0,d\in\mathbb R\) so that all supports lie in \(W_{cp_0+d}\). Tomita affiliation, after undoing \(U_H(c)U_b(d)\), gives the domain condition for \(e^{-\pi B}\). For a compact joint kernel, the same condition is the closed Tomita-graph limit of factorized tests from Section 2. Collective boost derivatives act on the smooth joint kernel, so their norms are controlled by the old joint-distribution seminorms. No affiliated joint-smear operator is needed for this vector statement. Remove the fixed constant translation \(U_b(d)\) first; it commutes with \(H\) and preserves boost smoothness of these field vectors. On the positive part of the affine representation use \(H=e^\tau\), \(B=-i\partial_\tau\). The resulting vector \(f\) satisfies \[e^{-ice^\tau}f\in\mathop{\mathrm{Dom}}e^{-\pi B},\] where the original vector \(f\) has Sobolev derivatives of every order in \(\tau\). Lemma 25 gives \(e^{r\tau}f\in L^2\) for every \(r>0\). The \(H=0\) part is harmless. Since \(K_n\le H\) strongly and the two commute, this proves the assertion for the lower placement. Reflection and equality of the reflected moments prove the upper placement. The choices have positive geometric margins, giving locally uniform bounds.

For \(U(x+i\eta)A\Omega\), \(\eta\in V_+\), consider the bounded Hilbert-valued holomorphic function \[1_{(L,\infty)}(K_n)U(x+s\eta)A\Omega, \qquad \operatorname{Im}s>0.\] A real interval of \(s\) places the entire translated support on one side of the plane. On that interval its norm is \(O(L^{-N})\) for every \(N\), locally uniformly; elsewhere it is bounded by the original vector norm. The subharmonic logarithmic norm estimate with harmonic measure of that interval yields any prescribed inverse power at every fixed interior point, by starting with a sufficiently large \(N\). Spectral-tail integration gives all the power domains, locally uniformly. Closedness of \(K_n^r\) and uniform convergence of spectral truncations give strong holomorphy and the stated boundaries. ◻

Proof. We first treat the scalar labels \(l,j_z,s_z\). The comparison (85), together with \(c_V=l+j_z\) and \(c_T=l/3+s_z\), gives order-zero comparison on each of them. For smooth compact waves with \(b,M>0\) compare the compression of \(K_n\) with \(K_{0,n}=b(|Y|^2+h_0)\). Their nonnegative expectation difference, evaluated on \(U_a(-s)\psi\), has zero second derivative in \(s\). Indeed the second-difference identity from (109) transfers to these waves by symmetry against \(\mathscr D\); the domains follow from the spectral comparison. Direct differentiation gives the same second difference for \(K_{0,n}\). A nonnegative affine function on the whole real line is constant.

Testing with separated compact multipliers in \(b,k\) now shows that the positive radial form \(h_0-\kappa_{\rm compr}\) is invariant under multiplication by \(e^{itM^2}\). Its distribution kernel is annihilated by \(M^2-M'^2\). On \(M,M'>0\) this forces a diagonal kernel; positivity makes it a positive measure \(\nu(dM)\). It is bounded by the \(h_0\) form. For any compact interval \(I\Subset(0,\infty)\), choose logarithmic cutoffs \(f_L=1\) on \(I\), vanishing outside successively larger compact intervals, with \[\int_0^\infty M|f_L'(M)|^2\,dM\longrightarrow0.\] For example piecewise linear ramps in \(\log M\) have energy \(O(L^{-1})\), and smoothing preserves that bound. Consequently \(\nu(I)\le\int|f_L|^2d\nu\to0\). The compression forms agree.

This is also equality of operators on a reducing copy. The spectral comparison gives \[\|1_{(s,\infty)}(\kappa)\psi\| \le\|1_{(s,\infty)}(h_0)\psi\|.\] Equality of the first moments makes the squared tail norms equal almost everywhere. The support inclusion underlying this inequality and equality in the projection norm inequality give equality of the projected vectors. Spectral continuity gives it at all continuity points, hence equality of all spectral measures and reduction. Comparing the odd translation differences with the canonical scalar formula shows that \(Q\mathscr D\) is orthogonal to these labels. Varying the null direction spans all polarizations of \(\Pi\), proving the last assertion of the Theorem.

For clarity, the projection argument above does not require the two spectral measures to commute. Write \(\iota\) for the scalar-copy embedding, \(E_s=\mathbf1_{(s,\infty)}(\kappa)\) and \(P_s=\mathbf1_{(s,\infty)}(h_0)\). Spectral comparison gives \(E_s\iota=E_s\iota P_s\). Equality of the first moments is equality of the integrals of \(\|E_s\iota\psi\|^2\) and \(\|P_s\psi\|^2\); their pointwise difference is nonnegative, so it vanishes almost everywhere. Equality in the contraction estimate for the orthogonal projection \(E_s\) therefore gives \(E_s\iota\psi=\iota P_s\psi\). One first uses a countable dense form core and then spectral continuity to obtain the operator identity for every threshold. This proves reduction rather than merely equality of expectations.

We recall explicitly the reference generator on the remaining polarizations. On transverse helicity of absolute weight \(d\) it is \[ K_{c,n}=b(|Y|^2+h_d). \tag{110}\] For a rank-\(r\) primary of dimension \(r+2\), \(r=1,2\), contract \(r-d\) slots with \(n\) and use \(\partial_vO_i-\partial_iO_v\) in each of the other slots, discarding joint transverse traces. The wave is a constant multiple of \(b^rM^d\) times its helicity label. Under the null conformal map of Section 2, the transverse Jacobian is \(\gamma(e_i+2cy_i n)\). The curl cancels the \(y_i n\) terms; derivatives of those terms contribute only the discarded traces. The transformed wave thus has the effective scalar weight \(2+d\), giving (110) by the scalar computation. This uses only the positive primary two-point kernels and also acts on their descendant combinations \(V,T\). Its origin translation formula contains \(D\), without \(Q\).

On smooth compact timelike \(\Pi\) waves the actual domains follow from (28). Put \(X_n=K_n-K_{c,n}\) there. Symmetry-testing (109) on \(\mathscr D\), using the orthogonality just proved, shows that \(X_n\) intertwines translations and is linear in \(n\). The same is true on the energy-damped compact \(V,T\) waves in \(\mathscr D\): symmetry now gives \(Q=0\) as a vector on these waves, and their reference domains follow from the primary and descendant tube action. The translation formula gives polynomial graph bounds. Integrating translates against Schwartz functions therefore commutes \(X_n\) through compact timelike momentum multipliers. Such multiples of the damped \(V,T\) waves span the smooth compact timelike tests of \(\Pi\).

It follows that \(X_n\) is a measurable fiber coefficient map. Linearity in \(n\) supplies a covector index \(\lambda\). The coefficient tensors \[X_\lambda V_\mu(p)=X_\lambda J_\mu(p),\qquad X_\lambda T_{\mu\nu}(p)=X_\lambda t_{\mu\nu}(p)\] are locally square integrable up to the nonzero light cone from inside: this follows first with the damping and smooth test multipliers, and then locally without them by choosing those multipliers nonvanishing. Their dimensions are respectively \(d'=2,3\), and they are covariant tensors. A rest-frame boost component of weight \(\omega\) has norm proportional to \(M^{d'-2-\omega}\) when boosted to a fixed nonzero energy and \(M\downarrow0\). The transverse mass measure is proportional to \(M\,dM\). Local integrability therefore excludes every weight \(\omega\ge d'-1\).

For the first tensor, contracting the \(\mu\) slot with a null vector leaves the \(\lambda\) covector parallel to its lowered null vector: its transverse and same-null contractions have the forbidden weights. At rest conservation makes the \(\mu\) slot purely spatial. Reversing its spatial direction changes the contracted tensor by a sign, but would require it to be parallel to a different null covector. It is zero. For the second tensor, contract both \(\mu,\nu\) slots with the same null vector. The forbidden weights again leave only that parallel covector. Both slots are spatial at rest, so reversing direction leaves the contraction unchanged and again forces it to be zero. Varying the direction and using symmetry recovers every spatial tensor coefficient. Hence both tensors vanish and \(K_n=K_{c,n}\) on compact timelike tests.

For \(d\ge1\), conjugation by \(M^{1/2}\) turns the radial operator into \(-\partial_M^2+(d^2-1/4)M^{-2}\), which is limit-point at both endpoints. Compact smooth radial tests, combined with smooth transverse cores and compact \(b\) cutoffs, are thus graph cores for (110). The \(d=0\) part was already reducing with its specified Hankel extension. The closed graph of the actual self-adjoint \(K_n\) contains the graph of the reference self-adjoint operator on each copy. Resolvent uniqueness gives reduction of their sum \(\Pi\) and equality there, as claimed. ◻

Proof. Average the spectral measures of \(T_R^\pm\) over \(S<\log R<2S\), with normalized Lebesgue measure, and take common weak limit points as \(S\to\infty\). One may first test continuous functions on the compactified real line. The uniform second moments on dense vectors remove mass at infinity. The limits are normalized positive operator-valued measures, not assumed projection-valued. Dilatations shift \(\log R\) by a fixed amount, whose boundary contribution in the averages is \(O(S^{-1})\). Thus the limits commute with \(D\). They have joint positive measures \(E^\pm(dd,ds)\) with the spectral measure of \(D\). Uniform second moments also give uniform integrability of first moments, including polarized cross moments on the vectors just described.

Only these first moments will be compared with \(D+Q\). The logarithmic averaging is used to obtain commutation with the known self-adjoint operator \(D\), while the uniform second moments prevent loss of first moments in the weak limit. Neither operation turns the limiting positive operator-valued measures into spectral measures.

Let \(O\) be a Hermitian scaling component of real dimension \(\alpha\) and \(\xi=O(f)\Omega\), with \(f\) real, smooth and compact in the future cone. The scaling orbit continues for \(0<\operatorname{Im}z<\pi\) because \(e^zx\) has forward timelike imaginary part there. Compact interior-cone support and integration by parts give Hilbert boundary values at both ends; real scale covariance gives uniform norm bounds along the strip. Consequently \[ e^{-\pi D}\xi=e^{i\pi\alpha}O(f^-)\Omega, \qquad f^-(x)=f(-x). \tag{112}\] We need this strip identity also for complex linear combinations of real dilation translates. Write \(\xi_s=U_D(s)\xi\) and let \(\xi_s^-\) be its reflected-test wave. Hermitian conjugation of the two-point function, with both test coordinates reflected, gives \[\langle\xi_s^-,\xi_t^-\rangle=\langle\xi_t,\xi_s\rangle.\] For \(p(D)=\sum_jc_jU_D(s_j)\) put \(p^\#(D)=\sum_j\overline{c_j}U_D(s_j)\); the real parameters \(s_j\) are unchanged. With \(\omega_\alpha=e^{i\pi\alpha}\) and \(\eta=e^{-\pi D}\xi=\omega_\alpha\xi^-\), the reflected Gram identity yields \[ \|p(D)e^{-\pi D}\xi\|=\|p^\#(D)\xi\|, \qquad \|p^\#(D)\eta\|=\|p(D)\xi\|. \tag{113}\] For all sufficiently large truncations, the comparison Tomita partner of \(p(D)\xi\) is \(p^\#(D)\xi\), and that of \(p(D)\eta\) on the past side is \(\omega_\alpha^{-2}p^\#(D)\eta\). Their norms are therefore exactly the two norms in (113). The finite set of dilated supports remains compactly in the respective cone. In particular, we do not require a complex filter to preserve its real space.

Real-space containment and these Tomita norm bounds, followed by the spectral-measure limits in the future and past cones, give \[\begin{align*} \int |p(d)|^2e^{-2\pi s}\,\mathrm d\langle E^+\rangle_\xi &\le\|p(D)e^{-\pi D}\xi\|^2,\tag{114}\\ \int |p(d)|^2e^{-2\pi d}e^{2\pi s} \,\mathrm d\langle E^-\rangle_\xi &\le\|p(D)\xi\|^2. \tag{115}\end{align*}\] Truncation followed by monotone convergence justifies the unbounded exponential tests in the limiting measures. The first moments in \(s\), with every such filter, agree for the two signs by (111) and the uniform moment bounds. Exponential polynomials determine finite measures: polarization recovers every exponential \(e^{itd}\), and Fourier uniqueness then recovers the measure. Thus these equalities and inequalities hold conditionally on the common \(d\) marginal \(\mu_\xi\).

To spell out the first-moment assertion, the finite filters \(p(D)\) preserve the class of smooth cone-supported single-field vectors; they also commute with \(e^{-\pi D}\). Thus both signs admit every same filter. Their limiting first-moment measures in \(d\) are finite by the uniform second-moment bounds. Taking \(p(d)=1+z e^{itd}\) and polarizing in \(z\in\mathbb C\) shows that these two measures have identical Fourier transforms, hence agree. This is equality of conditional means, rather than just equality of their integrals over \(d\).

Disintegrate the two scalar joint measures as \(\mu_\xi(dd)\nu_d^\pm(ds)\). Their conditional first moments \(m_\pm(d)\) agree almost everywhere, while (114)–(115) say \[\int e^{-2\pi s}\nu_d^+(ds)\le e^{-2\pi d}, \qquad \int e^{2\pi s}\nu_d^-(ds)\le e^{2\pi d}.\] Jensen gives \(m_+(d)\ge d\ge m_-(d)\). Equality of the means forces both to equal \(d\); strict convexity then forces both conditional laws to be \(\delta_d\). This argument requires only positivity and normalization of the measures, not multiplicativity of the operator-valued limits. Positivity also gives Cauchy–Schwarz for each measurable set, so cross measures against \(\xi\) are supported on \(s=d\). Their first moments, paired with \(\phi\in\mathscr D\), now give the following identity. Indeed, if a measurable set is disjoint from \(s=d\), its positive operator has zero quadratic value on \(\xi\), and Cauchy–Schwarz makes every mixed value against \(\xi\) zero as well. The mixed measure is therefore concentrated on the diagonal; its \(d\) marginal is the mixed spectral measure of \(D\). Its first moment is consequently that of \(D\), while the uniformly integrable first-moment limits from (111) give the form of \(D+Q\). Subtracting the two gives \[ \langle Q\phi,\xi\rangle=0. \tag{116}\]

The strip and Jensen argument just completed used a Hermitian homogeneous field of dimension \(\alpha\) with a real test. Splitting a complex test into its real and imaginary parts extends (116) to all tests. A complex homogeneous field is the sum of its Hermitian real and imaginary parts, \[O_\Delta=\frac{O_\Delta+O_\Delta^\dagger}{2} +i\,\frac{O_\Delta-O_\Delta^\dagger}{2i},\] which have the same grade because adjunction preserves real scaling dimensions. Linearity therefore extends (116) to every homogeneous original field. For \(O=\sum_{\Delta\in S_O}O_\Delta\), its finite scaling support then extends the conclusion to \(O\) with the same smearing test. This step sums conclusions; it does not assign the nonhomogeneous field one strip phase or apply Jensen’s inequality to cross-grade terms.

It remains to pass from single fields to products. Define the insertion \(\delta_QO(y)\) by the commutator with \(J(y+r)\) integrated against a compact flux cutoff in \(r\). Conservation and locality shrink the flux as needed. This insertion is relatively local to physical fields and is tempered with the ordered spectra: in this cutoff operation the adjacent momenta are first combined, so the spectator spectral cones are unchanged. Its extreme vacuum values against every physical polynomial vanish by (116) and the charge identity. To justify that passage, damp both collective translations first, remove the ket damping on the smooth future-cone single-field test, and remove the bra damping distributionally. Translation covariance removes the future-cone support restriction. Move the insertion to the extreme on a spacelike open set; ordered tube uniqueness makes the zero equality global. Finally the commutator flux obeys the derivation rule over every finite list of product slots. This rule is the telescoping of the adjacent single-insertion commutators just constructed, not a new operator axiom for \(J\) or \(L\). For a joint test kernel the telescoping identity is first an identity of joint distributions and is then tested with that same kernel. Each summand is zero, proving the Proposition. ◻

Proof. By Proposition 28, \(T_R^\pm\psi\to D\psi\) on \(\mathscr D\). This is a core for \(D\): it is dense and invariant under its unitary group, and smoothing by compactly supported smooth functions of the group parameter supplies a graph core. The self-adjoint core convergence criterion therefore gives \(T_R^\pm\to D\) in strong resolvent sense, and their real unitary groups converge strongly.

Take two bounded operations in a fixed compact subregion of the future cone. For all sufficiently large \(R\), their vacuum and adjoint-vacuum vectors belong to the comparison standard space and its complexification with the required Tomita domains. Writing the two operations as \(A,B\), the standard-space two-vector identity gives the scalar holomorphic continuation of \(\langle B^*\Omega,e^{itT_R^+}A\Omega\rangle\) to \(0\le\operatorname{Im}z\le2\pi\). Its values are defined by \[F_R(t+iy)= \big\langle e^{-yT_R^+/2}B^*\Omega, e^{itT_R^+}e^{-yT_R^+/2}A\Omega\big\rangle, \qquad 0\le y\le2\pi,\] with the opposite boundary \[F_R(t+2\pi i) =\langle e^{itT_R^+}A^*\Omega,B\Omega\rangle.\] One obtains the width \(2\pi\) by assigning up to \(\pi\) of imaginary evolution to each vector and using \(e^{-\pi T_R^+}=J_RS_R\); the comparison conjugation disappears from the displayed boundary formula. The split-vector pairing uses only \(\mathop{\mathrm{Dom}}e^{-\pi T_R^+}\); no vector \(e^{-2\pi T_R^+}A\Omega\) is required. The bounds depend only on the four vector norms. Normal-family convergence on the strip, together with the strong convergence on the real edge, gives the vacuum KMS identity for \(D\) on these actual operations. The identical argument in the past gives the identity for \(-D\). Bounded strong-star approximation extends the identity from compactly localized operations to the cone algebras.

Scaling already acts as an automorphism of each actual cone algebra. The vacuum is cyclic by Reeh–Schlieder. For clarity, faithfulness is part of the conclusion: the KMS identity implies that a bounded operation annihilating the vacuum has adjoint annihilating it. Apply this to \(BA\) for arbitrary \(B\) if \(A\Omega=0\). Then \(A^*B^*\Omega=0\); cyclicity gives \(A^*=0\). Thus the vacuum is separating. Uniqueness of the modular automorphism group of this faithful vector state identifies the generators as claimed. ◻

Proof. Choose two line parameters \(a<b\). Let \(A,B_t\) be the dilation generators centered at \(ae,be\), and let \(C\) be the modular generator of the physical diamond with those tips. Inclusion of the diamond into the future cone at \(ae\) and the past cone at \(be\) is half-sided modular by Theorem 29 and isotony. The common vacuum is cyclic and separating for both inclusions; no cyclicity of a relative commutant is needed. The half-sided inclusion theorem gives the affine relations corresponding to \[ A=(z-a)\partial_z,\qquad B_t=(z-b)\partial_z,\qquad C=\frac{(z-a)(b-z)}{b-a}\partial_z. \tag{117}\] More specifically, \(A-C\ge0\) scales with weight \(-1\) under \(A\), with the negative inward inclusion convention \(N=U_{A-C}(-1)MU_{A-C}(1)\); and \(-B_t-C\ge0\) scales with weight \(+1\) under \(-B_t\). The pair \(A,B_t\) already has the physical affine relation with \(P_e\ge0\).

These are relations of unitary groups, not just commutators on a tentative common domain. Ordered products of the three classical one-parameter groups are coordinates near the identity. The three affine relations give exact pair-switching identities, so reordering products proves a local unitary homomorphism. It integrates uniquely to the simply connected cover. This is the three-subgroup integration criterion of (Guido et al. 1998, Lemma 1.1); see also (Morinelli and Tanimoto 2019, Lemmas 2.3–2.4). In its circle formulation the intervals are \((a,b)\), \((b,\infty)\), and \((\infty,a)\), with stabilizer fields \(C,B_t,-A\). Their three pair relations are precisely the two cone–diamond relations above and the physical affine relation between the two cone groups. Its translation subgroup is the physical one, so the representation has positive energy.

Affine covariance transports \(C\) to the modular generator of every line diamond. A sufficiently small Möbius transformation relative to a fixed finite interval factors into an affine map of its tips and an element of its connected stabilizer, with the lift near the identity. Both factors have the asserted action on its actual algebra. This proves the local geometric action, without replacing the physical net by a larger net.

The construction is made for each chosen time direction using its actual cone and diamond algebras. Its local geometric statement is the one needed below: the connected interval stabilizer acts by that diamond’s modular group. Compatibility sufficient to integrate all spacetime conformal transformations on the full net has not been used in this argument. ◻

Proof. Let \(J\) be the actual future-cone modular conjugation. The positive-translation relation gives \(JP_eJ=P_e\). The half-sided cone–diamond relation gives the same formula for its positive special-conformal partner, while \(JDJ=-D\). These three generating groups show that \(J\) implements time reflection on the integrated Möbius representation.

In each lowest-weight ladder choose the canonical reflection conjugation, measurably in \(h\); its existence and multiplicity form are given in (Longo 2008, Theorem 1.6.3 and its universal-cover extension). The product of this conjugation with \(J\) commutes with the full representation. Inequivalent lowest weights cannot be mixed by that commutant, so \(J\) preserves every measurable weight band. In the common affine realization it is pointwise energy conjugation together with an energy-independent multiplicity conjugation \(j\), preserving the spectral projections of \(\mathsf h\). This argument distinguishes weights \(h\) and \(1-h\) even when their Casimir values coincide.

For a real compact half-line test \(f\), the strip continuation gives \[e^{-\pi D}\bigl(E^h\widehat f(E)v\bigr) =e^{i\pi h}E^h\widehat f(-E)v.\] Hence its Tomita fixed-point condition is \(v=e^{-i\pi h}jv\), the phased real structure in the statement. These tests are complex cyclic by Fourier uniqueness. Their real closure is invariant under dilatations and contained in the half-line real space. Gaussian smoothing in \(D\) gives graph density for Tomita, hence equality of the two real spaces. On compact positive weight bands these operations have uniform bounds, and exhaustion treats a bounded band approaching zero. The half-sided unitary sends the half-line space onto the finite-interval space and transports the tests by primary covariance. All these operations commute with the weight-band projections. Each such projection therefore commutes with the diamond conjugation, strongly commutes with its modular operator, and preserves the domain and action of its Tomita operator. ◻

Proof. The actual parabolic containment (89) of Lemma 21, followed by the reduction in Theorem 27, gives the desired projected membership for every parabolic diamond \(N\) containing \(\overline O\): actual localization is contained in the comparison localization, and the comparison generator acts canonically on \(\Pi\). The conjugation on \(\Pi\) is the primary reality operation by wedge Bisognano–Wichmann; the same statement holds in the reverse orientation. We explain why these containments determine the bounded diamond.

The point of the geometry is to recover a bounded diamond from the larger parabolic regions already controlled by the null-charge comparison. We carry out this recovery in the canonical real spaces, where the complementary diamond can be described explicitly.

Use the Einstein cylinder, with coordinates \((\tau,x)\in\mathbb R\times S^3\) and round distance \(d\). By a Poincaré transformation and a dilation, the physical patch and the chosen diamond can be written \[|\tau|+d(x,N_0)<\pi, \qquad O=\{\,|\tau|+d(x,N_0)<\alpha\,\},\quad 0<\alpha<\pi.\] On the canonical representations, \[ X_c(O)'=X_c(O^\perp),\qquad O^\perp=\{d(x,N_0)>\alpha+|\tau|\}, \tag{120}\] where prime denotes symplectic complement. Indeed, a connected conformal transformation carries the wedge and its opposite to this diamond pair. Transporting wedge modular data proves the equality. There is no assumption here about a physical algebra’s duality.

For completeness, these transports are well-defined across the required charts. The two-point kernels are rational functions with the finite Jacobian and weight factors of their integral bosonic primaries. On strict complex point waves in sufficiently small boxes the rational covariance identity preserves every inner product. Their span is dense, so it defines local unitaries; the same identity gives local composition and hence integration to the connected cover. On nonsingular real patches, the differentials preserve the future cone, so smaller strict imaginary approach cones have distributional boundary values after transport. On overlaps, both boundary prescriptions are continuations of the same rational complex kernel. In strict spacelike separation the two prescriptions agree. Additivity therefore transports the real smear spaces, even when the opposite diamond meets several charts.

Here is an explicit covering of the complement needed in (120). Let a target have \(\tau\geq0\) and \(\theta=d(x,N_0)>\alpha+\tau\). Choose \(\alpha<\beta<\theta-\tau\) and take the past tip of \(N\) to be \((-\beta,N_0)\). Its future tip is \((\pi-r,q)\), where \(d(q,N_0)=r<(\pi-\alpha)/2\); it lies on the future null boundary of the physical patch. This parabolic diamond contains \(\overline O\) with strict margins. The target is outside the causal future of its past tip because \(\theta>\beta+\tau\). To put it outside the causal past of the future tip, it suffices to choose \(q\) so that \[d(x,q)+r>\pi-\tau.\] At fixed \(r\), the maximum left side is \(\min(\theta+2r,2\pi-\theta)\). As \(r\) increases to \((\pi-\alpha)/2\), this tends to \(\min(\theta+\pi-\alpha,2\pi-\theta)\), which exceeds \(\pi-\tau\), except when \((\tau,x)=(0,-N_0)\). Time reversal treats \(\tau<0\). Consequently the spacelike shadows of these \(N\) cover \(O^\perp\) except possibly its central antipode.

If a vector symplectically annihilates all these shadows, its pairing with canonical real point waves vanishes on their union. The resulting distribution cannot be supported only at the omitted point: locally its singular covectors are causal, from the positive-energy tube and its conjugate, whereas a nonzero point-supported distribution has all nonzero covector directions in its wavefront set. It therefore annihilates \(X_c(O^\perp)\). Taking symplectic complements in (120) proves (119). ◻

Proof. Choose the line direction as the time axis and write the individual Fourier variables as \(k=(\omega,\boldsymbol k)\). On the ordered spectral support \(\Gamma_n\), positive energy of the partial sums gives \(\sum_i|\boldsymbol k_i|\leq C_n\sum_i|\omega_i|\). Choose a smooth cutoff \(\chi=1\) near \(\Gamma_n\), supported where \(|\boldsymbol k|\leq C'_n(1+|\omega|)\), with polynomially bounded derivatives. For a joint Schwartz temporal test \(h\), \(\chi(k)\widehat h(\omega)\) is therefore Schwartz in all variables, with each Schwartz seminorm bounded by finitely many temporal ones. Compact families of spatial translations and fixed transverse derivatives introduce only polynomial factors and obey the same bound.

For the tempered Fourier distribution \(\widehat W_n\), fix a finite Schwartz seminorm \(p_m\) controlling its action. Transverse mollifier multipliers are uniformly bounded in the required weighted derivative seminorms, and \[|\partial_{\boldsymbol k}^{\gamma} (\widehat\rho(\delta\boldsymbol k)-1)| \leq C_\gamma\delta(1+|\boldsymbol k|),\qquad 0<\delta\leq1.\] After multiplication by \(\chi\widehat h\), this gives an \(O(\delta)\) bound in \(p_m\), with finitely many additional temporal seminorms. It constructs the restriction and proves the stated distributional convergence. For the squared one-field vacuum error, the doubled correlation has two mollifier differences; its seminorm is \(O(\delta^2)\), giving vector error \(O(\delta)\). The adjoint and all fixed alternating source-field words are covered by the same original vacuum distributions. Thus every fixed moment required below is uniformly bounded. This argument tests tempered distributions; it assumes no ordinary Fourier density for an \(n\)-point function.

For a real line test of a Hermitian field, sufficiently thin compact transverse mollifications are supported in \(O\). Their vacuum vectors belong to its closed real space by affiliation. Their Hilbert limit has the same membership. Translation and Lorentz covariance give the assertion for all the lines under consideration. ◻

Proof. For real \(f\in C_c^\infty(I)\) the primary norm, up to a positive constant, is \[\int_0^\infty E^{2s-1}|\widehat f(E)|^2\,\mathrm dE.\] Reality determines negative frequencies. To evaluate \(\partial_z^k\widehat f(z)\), choose a compactly supported smooth test which agrees on \(I\) with \((it)^k e^{izt}\). Modify it outside \(I\) to annihilate its first \(m\) Fourier jets at zero, with \(m>s-1\). Such a modification exists: finitely many disjoint small bumps outside \(I\), with distinct centers, have a moment matrix arbitrarily close to a nonsingular Vandermonde matrix. Cauchy–Schwarz with dual weight \(|E|^{1-2s}\) now bounds the pairing. Its high-frequency integral converges by smoothness, and its low-frequency integral converges by the vanished jets. The construction and all bounds are uniform on compact sets of \(z\) and compact bands of \(s\). They extend to the real Hilbert closure. The extended evaluations satisfy the Cauchy–Riemann equations and give the asserted entire function.

The regularity at zero frequency now detects the allowed weights. In the common affine realization, changing the primary weight changes only a power of \(E\). Entire continuation permits precisely the nonnegative integral powers of that variable.

An intertwiner commuting with \(E\) and its rescalings is an energy-independent bounded multiplicity map \(T_s\). For a real multiplicity section \(v\) in a compact \(H\)-band, projection of the actual localized wave has quotient \[ \widehat f(E)T_s E^{H-s}v. \tag{121}\] The assumed real-space inclusion and the preceding continuity make this holomorphic at \(E=0\). Choose \(\widehat f(0)\ne0\). On a compact band \(T_s\exp((H-s)z)v\) is an entire function of \(z=\log E\). Single-valued continuation in \(E\) implies \[T_s\bigl(e^{2\pi i(H-s)}-\mathbf 1\bigr)v=0.\] Complexifying real sections and applying the spectral theorem shows that \(T_s\) is supported on \(h-s\in\mathbb Z\). Only finitely many such values occur on the band. Their finite Laurent expansion in (121) has no pole, so its negative powers vanish. Exhausting the positive weight axis proves the first assertion.

For the second, write a dimension-\(d\) point wave as \(E^d u\) in the actual affine representation. Project onto compact \(H\)-bands, which commute with interval Tomita data. In each fiber division by its primary power gives \(\widehat f(E)E^{d-h}u_h\). The same argument forces \(d-h\in\mathbb Z_{\geq0}\). There is no translation-invariant component of a positive-dimension point wave: its zero-momentum vacuum amplitude would have to transform by a nonunit-modulus dilation character. For original fields, the required timelike restrictions and real localization are supplied by Lemma 33. The same assertion for a subsequently constructed wave uses its stated Hilbert-valued restriction and physical real localization. ◻

Proof. There are three issues to settle: the timelike spectral densities, the possible massless component, and the action of the actual null generators on the resulting scalar. The projection identity (122) will link the first two issues: comparison on one energy slice fixes both the normalization and the dependence on the time direction.

We first prove a two-point claim: the timelike entries of the joint spectral measure of the family \(S_e\) and \(l\) are constant multiples of the ordinary measure \(\mathrm d^4p\) on the future timelike cone. No density is assumed in this claim. Write \(\mu_{ee'}\) for a joint spectral entry, including \(l\) as an additional label. The corresponding commutator distribution is \[C_{ee'}(x)=\int e^{ip\cdot x}\,\mathrm d\mu_{ee'}(p) -\int e^{-ip\cdot x}\,\mathrm d\mu_{e'e}(p).\] Every invariant-mass projection commutes with wedge Tomita data. Wedge separation and reality therefore give causal support for this commutator after projection to any compact mass interval. For \(l\), the real wedge data are the canonical scalar data already identified in Section 2; physical point-field membership is not needed.

Homogeneity disintegrates the timelike measure as \(M^3\mathrm dM\,\mathrm d\nu(p/M)\), with a polynomially bounded measure on the unit hyperboloid. Testing arbitrary mass intervals shows that the shell commutator has causal support for almost every mass; rescaling then gives the same statement for the reference shell. This is a disintegration of distributions, not an assumed field on a fixed mass shell. The shell commutator solves the Klein–Gordon equation. Its displacement and normal-derivative Cauchy data at zero time are supported at the spatial origin, and hence are finite sums of delta derivatives. Fourier solution of this Cauchy problem expresses its shell measure as the ordinary commutator measure times a polynomial in \(p/M\), modulo \(p^2=M^2\). In particular this proves absolute continuity rather than presupposing it.

For a self-pair the shell polynomial is even under \(p\mapsto-p\). Take its representative of least degree modulo the shell relation. If its highest degree is \(k\), that homogeneous term is nonzero on an open set of null directions; otherwise divisibility by \(p^2\) would reduce the degree. At fixed nonzero energy and \(M\downarrow0\) it contributes order \(M^{-k}\) to the density with respect to \(\mathrm d^4p\). The transverse mass integration is \(M\mathrm dM\). Positivity and local finiteness therefore require \(k<2\). Evenness leaves only a constant. Positivity of the matrix measure bounds each cross density by the geometric mean of the self densities. A polynomial on the unit hyperboloid with such a global bound is constant by the same highest homogeneous term argument. All timelike density entries are therefore constant, proving the two-point claim. Covariance makes the self constant of \(S_e\) independent of \(e\).

There is no zero-momentum part. We must also exclude a massless part; positivity alone would allow one. Let \(\rho=\sqrt{-W_\mu W^\mu}\) be the massless Pauli–Lubanski length and project the entire Lorentz family onto a compact band \(0<a\leq\rho\leq b<\infty\). This projection commutes with wedge modular data. Its commutators are causal massless solutions. The preceding point-supported Cauchy-data argument makes them derivatives of the massless commutator, hence supported on the light cone in four dimensions.

On the complex cyclic span of this projected family, let \(X_+\) and \(X_-\) be the real spaces generated in the future and past cones. Translation-spectrum uniqueness makes both spaces complex cyclic: a vector orthogonal to smearings on an open cone is orthogonal to all translates by tube uniqueness. Huygens support gives \(X_-\subset X_+'\), so \(X_+\) is standard. Lorentz transformations preserve it and hence commute with its modular group. Every future translation maps it into itself. The positive-generator modular translation relation therefore scales all \(P_\mu\) by one common nontrivial exponential under that modular group. Because the Lorentz generators commute with the group, \(\rho\) scales by the same factor. No nonzero spectrum contained in \([a,b]\) is invariant under all these rescalings. The projected space is zero. Exhausting positive bands excludes nonzero Pauli–Lubanski length.

The remaining massless representations have trivial little-group translations and decompose into helicities. A rotational scalar about \(e\) has only helicity zero, since a spatial rotation fixing both \(e\) and a massless momentum acts trivially on its wave. Use the scalar massless invariant measure. The possible internal part of dilations is a momentum-independent unitary group \(W\) on scalar multiplicities. Lorentz covariance and dimension two give the amplitude \[ S_e(p)=(p\cdot e)W(\log(p\cdot e))\beta. \tag{124}\] For two directions the commutator argument makes the cross density a homogeneous quadratic shell polynomial. A quadratic scalar invariant under rotations fixing \(e,e'\) is, on the null shell, a linear combination of \((p\cdot e)^2\), \((p\cdot e)(p\cdot e')\) and \((p\cdot e')^2\). It follows that \[\varphi(u)=\left\langle\beta,W(\log u)\beta\right\rangle,\qquad u=\frac{p\cdot e'}{p\cdot e},\] is a linear combination of \(u^{-1},1,u\) on the attainable interval. As the relative rapidity grows these intervals exhaust \((0,\infty)\); the coefficients agree on overlaps. Since \(|\varphi(u)|\leq\|\beta\|^2\), both nonconstant coefficients vanish. Thus the massless cross density is \(c(p\cdot e)(p\cdot e')\), with \(c=\|\beta\|^2\geq0\).

The only massless possibility left is the scalar density just displayed. Its growth with the relative rapidity of \(e\) and \(e'\) is incompatible with the bounded timelike cross densities. The next energy-slice calculation makes this comparison with its exact normalization.

Now apply the defining projection in (122) to \(F=S_{e'}\). A dimension-two wave with physical real localization has actual weights at most two; \(u_l\) has actual weights \(2,3,\ldots\). In the multiplicity fiber at \(P_e=1\), \[ \left\langle u_F,u_{S_e}\right\rangle=\left\langle u_F,u_l\right\rangle. \tag{125}\] Let \(B(e',e)\) and \(D(e')\) denote the constant timelike density entries for \((S_{e'},S_e)\) and \((S_{e'},l)\), respectively. Write \(b=B(e,e)\), which is independent of \(e\) by covariance, and let \(\rho_{ll}\) be the self density of \(l\). With timelike measure \(\mathrm d^4p\) and massless measure \(\theta(p^0)\delta(p^2)\mathrm d^4p\), (125) reads explicitly \[ \frac{4\pi}{3}B(e',e)+2\pi c(e\cdot e') =\frac{4\pi}{3}D(e'). \tag{126}\] Indeed the timelike slice \(p\cdot e=1\) is the unit spatial ball in the rest frame of \(e\), of volume \(4\pi/3\). Lorentz invariance of \(\mathrm d^4p\) makes this normalization independent of the slicing direction. On the massless slice the angular integral is \(\frac c2\int_{S^2}(e'^0-\boldsymbol e'\cdot\boldsymbol n)\mathrm d\Omega =2\pi c(e\cdot e')\). Positivity gives \(|B(e',e)|\leq b\) and \(|D(e')|\leq\sqrt{b\rho_{ll}}\) uniformly. Arbitrarily large relative rapidity therefore forces \(c=0\). Setting \(e=e'\) in (126) now gives \(D(e')=b\), and hence \(B(e',e)=b\) for all pairs. Thus \(\|S_e(f)-S_{e'}(f)\|^2=0\) for every test \(f\). It also shows that the constant two-point density of \(S\) with \(l-S\) is zero. This is orthogonality of the scalar wave channels, including all their spacetime translates; it does not assume that the actual time-weight projection commutes with spatial translations.

Finally, physical real localization supplies the scalar order-zero parabolic comparison for \(S\). We can also apply the proof of Lemma 26 to its smooth compact vacuum smearings in the same strictly one-sided placements, and then to all their strictly energy-damped translates. That proof requires real localization, giving the Tomita bound after parabolic transport, and boost smoothness of every order. Both are available here: localization was obtained from (123), and the scalar covariance just proved identifies every iterated boost derivative with another compact smooth \(S\)-smearing. Its Tomita and Hardy estimates therefore give the asserted \(K_n\) domains without treating \(S\) as an operator. Symmetry against the same dense core transfers the even second difference of (109), \(K_n(s)+K_n(-s)-2K_n=4s^2a\), to these vectors as a form identity. The \(D'\) term cancels, so no \(D'\) operator domain for \(S\) is assumed. The second-difference and zero-capacity argument of Theorem 27 therefore applies: the nonnegative comparison defect is translation-diagonal in \(M^2\), and is bounded by \(\int_0^\infty M|f'(M)|^2\mathrm dM\). Logarithmic cutoffs equal to one on a prescribed compact mass interval make this bound tend to zero. Spectral comparison then gives reduction and the exact canonical null action. Its wedge reality and two-point kernel are scalar, so the proof of Lemma 32 includes this enlarged canonical space as well. ◻

Proof. The multiplicity label \(u_\ell\) has actual weights \(2+j\), \(j\geq1\). First take a Hermitian polynomial localized in a diamond on the line. Its vacuum vector \(\psi\) is in the corresponding real space. The scalar quotient \[F(E)=E^{-2}\left\langle u_\ell,\psi(E)\right\rangle\] is regular at zero, with continuous evaluation, by Lemmas 32 and 34. For compact actual weight-band projections of \(\psi\), its overlap with \(u_\ell\) has only positive integral powers \(E^{h-2}\), so \(F(0)=0\). These projections preserve the interval real space and converge in Hilbert norm; continuity of evaluation proves the same equality without the band cutoff. With the common \(\mathrm dE/E\) normalization, \[\left\langle\ell(iRe),\psi\right\rangle=\int_0^\infty E^3e^{-RE}F(E)\,\mathrm dE.\] The contribution near zero is \(o(R^{-4})\), and the complementary contribution is exponentially small by Cauchy–Schwarz. Every compact polynomial is a complex linear combination of Hermitian ones, and its support fits in a line diamond. For a compact joint polynomial kernel, the same real-space statement is the closed Tomita-graph limit of factorized tests, using the old joint-distribution continuity from Section 2. This proves the claim. ◻

Proof. For an ordinary field smearing with closed affiliated realization \(A=v|A|\), take its polar cutoff \(A_R=v|A|\mathbf1_{[0,R]}(|A|)\). It has norm at most \(R\). The tails on the vacuum and adjoint vacuum satisfy, for every positive integer \(m\), \[\begin{align*} \|(A-A_R)\Omega\|&\leq R^{-m}\||A|^{m+1}\Omega\|,\\ \|(A^*-A_R^*)\Omega\|&\leq R^{-m}\||A^*|^{m+1}\Omega\|. \end{align*}\] Here the absolute-value moments follow from the original realization hypothesis together with the generated-domain action check in Section 2. Both \(A\) and \(A^*\) preserve the dense invariant vacuum-generated domain \(\mathcal D_{\mathrm{old}}\) and agree there with the physical field and its polynomial adjoint. Induction therefore gives \[\mathcal D_{\mathrm{old}}\subset \mathop{\mathrm{Dom}}(A^*A)^m\cap\mathop{\mathrm{Dom}}(AA^*)^m \qquad(m=1,2,\ldots),\] and these powers on that domain are the corresponding alternating polynomial words. Their vacuum norms are finite and are bounded by finitely many seminorms of the source tests, by the joint Wightman distributions. Since \(|A|^{2m}=(A^*A)^m\) and \(|A^*|^{2m}=(AA^*)^m\), spectral calculus also gives every lower integer absolute-value moment used in the tail bounds. This uses the adjoint-extension clause of Assumption 2 and does not require essential self-adjointness.

The invariant polynomial domain supplies these moments, uniformly on finite test-seminorm balls; subdivision supplies the requested local neighborhoods. Use a Hermitian basis or adjoint-paired cutoffs to preserve adjunction and grading.

For a nonhomogeneous original source, any comparison of shrinking patches by dilation is made on its finitely many homogeneous components and then summed. For this fixed field and requested accuracy, a maximum of the finitely many seminorm orders and powers is sufficient. This does not assign the source a single scaling dimension or require a bound uniform in the set of dimensions, and it does not change the fixed polynomial norm exponent in (128).

For \(S\) the order of construction matters. We truncate the ordinary source field before applying the actual modular unitary, and then add the resulting bounded operations. We will estimate the sum by its absolute quadrature weights, so no domain or moment estimate for an unbounded sum of modularly transformed fields enters.

For \(S\), use (123) on parallel lines. After undoing \(g\) in the test, a line smearing of \(S\) is a fixed linear combination of a line virial smearing and a modularly transformed line virial smearing, with smooth real coefficients. By Lemma 33, transverse approximation at width \(\delta_R\leq e^{-R}\) has vacuum error \(O(\delta_R)\), and every fixed source-field moment is bounded uniformly as \(\delta_R\downarrow0\). Its adjoint data obey the same estimates.

Polar truncate each such source field at \(R\) and apply the actual Möbius unitary. It fixes the vacuum and transports the bounded operation into the destination diamond, so both estimates in (129) survive. A transverse quadrature of mesh at most \(e^{-R}\) approximates a full spacetime smearing. Although its number of terms grows, the sum of absolute quadrature weights is bounded. Hence the sum of the operator norms is \(O(R)\), and the sum of the vacuum errors remains rapidly small. No moment bound for a sum of noncommuting transformed fields is being assumed.

All the operations just described use finitely many derivatives for each requested accuracy. Rescaling tests and using diamonds with margins proportional to \(\varepsilon\) therefore introduces only inverse powers of \(\varepsilon\). The fixed non-affine map can be chosen nonsingular on the entire compact parameter patch, with \(g'g''\) bounded away from zero. Take \(\delta_R\leq c\varepsilon e^{-R}\) and sufficiently fine quadrature to retain these margins. Subdivision completes the construction for an arbitrary compact source support. ◻

Proof. We first prove convergence where locality lets us estimate the whole product by vacuum errors. Holomorphic propagation then carries that convergence into the ordered tube. Finally, bounded-product continuity at a spacelike edge gives the common neighborhoods needed for ties; translation compatibility removes the source smearing.

At real translations making all approximation patches mutually spacelike, telescope two bounded products, whether at \(R,2R\) or from different approximation choices. Strong graded locality moves each difference to the vacuum. Its vacuum error is rapidly small by (129), while all other factors have polynomial norm growth by (128). The product difference is therefore \(O_N(R^{-N})\) for every \(N\), uniformly on smaller placement patches and appropriate test-seminorm balls. This proves independence, graded symmetry and multilinearity at these real placements.

We record the propagation estimate, since vacuum convergence alone would not justify the products elsewhere. For a holomorphic vector difference \(G_R\) bounded by \(CR^q\), restrict to a complex line \(z=x+\zeta\eta\) with \(\eta\) in the ordered cone. A real interval \(I\) can be chosen inside a good spacelike-placement patch. The subharmonic logarithm of its norm and harmonic measure give \[ \log\|G_R(x+\zeta\eta)\| \leq\omega(\zeta,I)(\log C_N-N\log R) +(1-\omega(\zeta,I))(\log C+q\log R). \tag{133}\] On compact subsets reached this way \(\omega\) is bounded below by a positive number. Interior three-circle chains propagate the same conclusion to every fixed compact subset of the connected tube. As \(N\) is arbitrary, convergence is rapid there. Above a smaller good real patch it is rapid uniformly up to the boundary, because \(\omega\) has a uniform positive lower bound for bounded approach directions. Summing the dyadic differences proves existence.

For ties, first keep the common translation of each block and its spectators strictly inside their allowed complex domains. The relative translations within a block have wedges of opposite imaginary-gap cones for its order and its reversal. At their common real edge the bounded products agree, with the graded sign, by strong locality. Each bounded product has a norm-continuous extension to that edge: the real translation groups and the positive-energy contraction semigroups are strongly continuous, and only finitely many bounded factors occur. On fixed smaller wedges and their common edge its norm is \(O(R^q)\). The continuous-edge form of the local edge-of-the-wedge theorem therefore applies directly; its continuation neighborhood depends only on the wedges and the edge patch, and its interior bounds are controlled by the supremum bounds on those wedges and their edge; see (Browder 1963, Theorems 1\('\) and 2).

To make uniformity explicit, apply the theorem after scalarization to the Hilbert-valued family \((R^{-q-1}F_R)_R\) in the square-summable direct sum, with dyadic \(R\). Its common edge is norm continuous: finitely many components are continuous and the remaining squared norms are bounded uniformly by a convergent tail of \(\sum_R R^{-2}\). Scalarization gives a bounded holomorphic Hilbert-valued continuation on one common neighborhood, and each unweighted component there has a bound \(O(R^{q+1})\). The same supremum-bound argument is uniform on the prescribed test-seminorm balls; no estimate from boundary distribution seminorms alone is being asserted. Interior propagation from the strict tube now gives rapid convergence on compact subsets of these neighborhoods. Adjacent permutations within a block are treated on overlapping smaller neighborhoods, where uniqueness glues the continuations.

By uniqueness from the spacelike real patch, multilinearity persists throughout these domains. The seminorm bounds and the test-function kernel theorem give joint distributions in the source variables, holomorphic in the translation variables. Translate a source test within its patch. At real spacelike placements this is the same as translating its insertion argument; the equality involves only the vacuum wave and hence survives all approximation choices. Analytic uniqueness extends this equality. If \(x\) denotes source variables, the kernel in coordinates \((x,\operatorname{Re}z+x,\operatorname{Im}z)\) is therefore independent of its first variables in the distributional sense. It satisfies the Cauchy–Riemann equations in the remaining variables. Thus it is a holomorphic Hilbert-valued function of \(x+z\), rather than merely a distribution in \(x\). This proves unsmearing. The same identity glues source patches and their translations.

For the split identity first keep all source tests smeared, and use the same bounded approximants on the two sides, taking the adjoints of the bra approximants in the scalar word. This is an admissible choice because both vacuum and adjoint-vacuum errors were controlled. Bounded adjunction gives (132) before taking limits. The displayed scalar word lies in its ordered scalar tube: its reversed bra gaps and ket gaps are future directed, and its central gap has imaginary part \(\operatorname{Im}w_1+\operatorname{Im}z_1\in V_+\). Local uniform convergence of the two vector factors and of the scalar word therefore proves the identity there. The preceding unsmearing then gives the point formula. Conversely, at any cut between two successive entries of an ordered scalar tuple, choose a common imaginary shift strictly between their imaginary parts. Subtracting that shift puts the reversed-adjoint prefix and the suffix in the two strict vector tubes of (132). Thus the formula supplies the analytic Hilbert splitting of every ordered scalar tuple at every cut. Finite linear combinations retain all cross terms. At mutually spacelike real placements, the same identity follows from the Hilbert-norm limits already proved on those patches; no unrestricted real-axis limit is needed.

For physical fields, polar truncations converge on the polynomial domain as well as on the vacuum. At mutually spacelike real placements their product limit is consequently the ordinary smeared product. For a joint source kernel, finite tensor-test approximation and the joint distribution estimates extend this equality; the kernel itself need not have finite tensor rank. Tube uniqueness identifies their point vectors everywhere. Finally the estimates above give distributional boundary convergence locally uniformly on the good real patches. In any closed strict subcone of approach, point bounds near such an edge are polynomial in inverse distance: test the real variables on balls at that scale and use the holomorphic mean-value estimate on smaller complex balls. ◻

Proof. Shrink the real patch and the approach cone so that the image of a small imaginary translate of size \(\delta\) is a graph at distance comparable to \(\delta\) inside one fixed strict convex subcone. Interpolate that graph to a straight imaginary translate there. Suppose the holomorphic function is \(O(\delta^{-m})\) on this interpolation. Extend a compact smooth test almost analytically to order \(N\): its \(\bar\partial\) is \(O(\delta^N)\), uniformly with the finitely many derivatives required on the patch. Stokes’ formula for the holomorphic volume form compares the two integrals. Their error is bounded by a constant times \(\delta^{N+1-m}\); take \(N>m+1\).

There is a minor test-function issue in this comparison. The desired coefficient on the moving graph is a fixed smooth real test, whereas an almost-analytic extension evaluated on that graph has nonzero imaginary Taylor terms. Include the holomorphic Jacobian, and choose the real test on the straight surface by inverting these Taylor terms through order \(N\). More explicitly, the graph evaluation operator on test jets is \(\mathbf 1+\delta B_1+\cdots+\delta^N B_N\), where each \(B_j\) is a differential operator with smooth coefficients. It has a unique inverse modulo \(\delta^{N+1}\), obtained successively in each power of \(\delta\). This is the negative-imaginary-displacement Taylor correction. Its corrected real tests converge in every smooth seminorm to the ordinary real change-of-variables test; the remaining graph error is \(O(\delta^{N+1-m})\). The boundary limit on the straight surface is therefore precisely the distributional real pullback. Pairing with arbitrary Hilbert vectors proves the weak statement, and the uniform estimates prove the asserted Hilbert-valued distributional convergence. ◻

Proof. Choose a future null direction \(m\) separated from the directions \(n\) and use \(m\) as the conjugate wedge axis. Work strictly below the null cut plane centered at \(o\). In the resulting coordinates put \(p(y)=|y|^2\). Each source patch fits, with a strict margin, inside a cut \(W_{\alpha p+c}\) with \(\alpha>0\) sufficiently small and \(c\) sufficiently negative. The same containment holds for the finitely many inverse coordinate images when the flow parameters are small enough.

Here is a two-sign bound uniform in quadratic truncation. Choose smooth bounded profiles \(0\leq q_R\leq p\), increasing to \(p\), and write \(U_R(t)=e^{itH_{q_R}}\). For \(0<t<\alpha\), cut inclusion and the constant-shift identity give \[\mathcal M(W_{\alpha p+c})\subset\mathcal M(W_{tq_R+c}) =\mathop{\mathrm{Ad}}(U_R(t))\mathcal M(W_c).\] Thus conjugation by \(U_R(-t)\) sends the source algebra into the fixed wedge algebra \(\mathcal M(W_c)\). For positive \(t\), the source algebra is already contained in \(\mathcal M(W_c)\), and \[\mathop{\mathrm{Ad}}(U_R(t))\mathcal M(W_c)=\mathcal M(W_{tq_R+c})\subset\mathcal M(W_c).\] Both signs therefore have the same containing wedge for \(|t|\leq t_0<\alpha\), independently of \(R\). This argument uses \(q_R\leq p\), not a bound on the diverging norms \(\|q_R\|_\infty\). The positive moment construction of Section 3 gives \(U_R(t)\to e^{itK_n}\) strongly. Weak closedness of the fixed wedge algebra now preserves the containment for each bounded conjugated operation in this limit. Translations give the centered version used here.

For finitely many choices, take \(m\cdot x\) sufficiently negative, then add a sufficiently large positive multiple of \(m\). This leaves \(m\cdot x\) unchanged and makes all the finitely many \(n\cdot x\) sufficiently positive. A small open double cone about such an \(x\) lies in the required opposite region. All inequalities are strict, hence stable under the small changes asserted in the statement. Local vacuum cyclicity in this double cone proves totality. No cyclicity of a relative commutant is used. ◻

Proof. Complete the polarizations to a generic basis and put \(z_j=b\cdot e_j\). A polynomial in the square-free monomials \(z_I\) that vanishes on an open part of the null cone vanishes on the full complex quadric and is divisible by its nondegenerate quadratic equation \(Q(z)\). This last assertion follows, for example, by solving the quadric for one variable on a nonsingular chart and applying polynomial division; the remainder vanishes on an open set and is zero. In a generic basis \(Q\) has a nonzero \(z_j^2\) coefficient for every \(j\). If \(Q R\) were nonzero and multi-affine, then \(\deg_{z_j}(QR)=2+\deg_{z_j}R\geq2\), a contradiction. Thus the square-free monomials are independent on every such cone patch. There is consequently a finite nonsingular evaluation matrix. Varying ray lengths separates its homogeneous degrees.

Scale all its sample vectors by \(\varepsilon>0\) and divide column \(I\) by \((4\varepsilon)^{|I|}\). As \(\varepsilon\downarrow0\), the matrix in (137) converges uniformly on compact source patches to the nonsingular monomial matrix. Its determinant remains nonzero for sufficiently small \(\varepsilon\). Matrix inversion gives the last assertion. Thus the interpolation involves multiplication of distributions by smooth functions, never pointwise inversion of a distribution. ◻

Proof. Proceed by induction on length, and at a fixed length by increasing number of \(t\) slots. Length one is the given one-vacuum construction. For accessible inputs use the localized bounded approximants of Proposition 37. Conjugate every factor of an approximating real product by the same \(U_g\). Lemma 40 allows the first factor to commute past \(C\). Vacuum-vector and adjoint-vacuum convergence for this factor, and convergence of the shorter tail, therefore yield the right side of (138). By the shorter induction assertions this is the multilinear expansion of (136), initially as a functional of \(C\Omega\).

Use \(T\) in positions intended eventually to contain \(t\). Terms with fewer \(t\) slots have representing vectors already and can be subtracted: replacing a transformed \(T\) by \(\mathcal P_gL\) lowers this induction index. Interpolation is then needed only for the remaining \(V/L\) choices. At the remaining \(V\) positions the unknown functionals are indexed by the subsets replaced by \(L\), with the coefficients (137). Lemma 41 expresses each unknown as a finite smooth linear combination of already existing Hilbert-valued distributions. They therefore have representing vectors. Generic polarization choices suffice: multilinearity, followed by a finite choice of generic bases, recovers all components. This proves (138) at the new length.

The same argument with any other sufficiently small \(g\) proves the expansion formula, because it proves equality against the total family \(C\Omega\) in a common opposite cone. Subdivision of patches or a change in all auxiliary choices gives the same vectors recursively by (138). Comparing a construction with a small translated, Lorentz-transformed or rescaled construction proves the similarity laws. Interpolating the symmetry of transformed accessible products proves spacelike permutation symmetry, again first subtracting terms with fewer \(t\) slots. Inserting a zero vacuum wave into (138), after permuting it to the first position, gives zero. Hence, in particular, \[ \Box L=\partial\cdot V=\Theta,\qquad \partial\cdot J=0,\qquad \partial^\mu t_{\mu\nu}=0,\qquad t^\mu{}_{\mu}=0 \tag{140}\] hold in these products. This argument concerns vector distributions, not operator-domain equalities.

For bounded approximants the vacuum expectation of a product splits at any position by Hilbert-space adjunction. The same is true for its common unitary conjugate. Expand the full scalar product and both factors by (136); real transformations respect adjunction. Subtract the terms already split by induction in the number of \(t\) slots. The identical finite interpolation now proves (139) for every remaining entry list, with the same adjoint and coordinate convention as (132). Thus short-length associativity and positivity come from actual bounded products.

It remains to continue the vectors, rather than merely their matrix coefficients. Fix a compact continuation path from a spacelike boundary patch to an interior tube point or one of the allowed ties. Choose the parameter scale in Lemma 41 small enough that every inverse image \(g^{-1}z_j\) stays in the accessible product domain along that path. Near a tie use its open holomorphic neighborhood. The analytic inverse matrices remain nonsingular on a small neighborhood of the compact path. Applying \(U_g\) to the accessible analytic vectors and performing the same successive subtractions and inversions constructs the desired holomorphic vectors. The boundary calculus of Lemma 39 identifies their distributional boundary with the real construction: the coordinate maps take a smaller approach wedge strictly into the original wedge.

There is no choice of branch. On overlapping connected compact continuation neighborhoods choose parameters sufficiently small for both constructions; their boundaries agree on a smaller original patch, and distributional edge-of-the-wedge uniqueness makes the holomorphic vectors agree. Exhaustion gives the full ordered tube. At a tie the accessible germs and previously constructed terms agree across the relevant permutations, so the interpolated germs agree as well. All leg identities and similarity laws continue by the same uniqueness. Finally, the two sides of each split identity have equal boundary distributions when bra and ket are separately smeared on the original spacelike patches and approached in strict subcones. Uniqueness in their common split tube proves the asserted continuation of the identity. Only a local distributional boundary and an open convex approach subcone are needed here; no global boundary estimate for arbitrary \(L\) products has been assumed. ◻

Proof. First take a finite sum of factorized compact physical bra words and translate it into a common opposite cone. Its affiliated bounded truncations can be used as \(C\) in (138). With \(L\) first, the adjoints of these truncations acting on an ordinary smeared \(V\Omega\) converge on the polynomial domain. Locality of the single-insertion forms therefore gives equality on an open real spacelike patch. For a compact joint bra kernel, approximate by factor tests in the permitted smooth topology, preserving the opposite-cone margin. Continuity of the vacuum and adjoint-vacuum graph vectors and of the single-insertion distributions extends this equality to that kernel. Give bra and pair strictly separated ordered imaginary translations. The known single-insertion spectral cones give analyticity in the two pair arguments and the common translation of the bra, with its internal spectator tests fixed. Analytic uniqueness extends the equality, and one may remove the separate bra damping while keeping the pair damped. The other pairs follow from their linear differential definitions. ◻

Proof. Fix the \(t\) and \(S\) positions. At the remaining \(m_0\leq4\) positions start with polarized vectors \(V(e_j)\), and let \(W_I\) be the function obtained by replacing the positions in \(I\) by \(L\). The canonical special-conformal Ward operator \(\widehat K_a\) acts diagonally by the primary differential actions and has off-diagonal part \[ (\widehat K_a W)_I\big|_{\mathrm{off}} =4\sum_{j\notin I}(a\cdot e_j)W_{I\cup\{j\}}. \tag{141}\] These operators commute: they are the canonical commuting special transformations of primaries and their derivatives. They may be centered at \(o\), since the difference from another center is an affine Ward operator, already zero on every row.

Unitarity and the expansion in Proposition 42 give \[ (e^{s\widehat K_n}W)_\varnothing=W_\varnothing \tag{142}\] for small \(s\) and the open family of null \(n\). Initially the tensor positions contain \(T\). Induct in their number, substitute \(T=t+\mathcal P_0L\), and subtract terms with fewer \(t\) positions, whose covariance is already established. This gives (142) with the indicated \(t\) positions.

Write \(X_I(a)=(\widehat K_aW)_I\). Differentiating (142) once and using the linear span of the open null family gives \(X_\varnothing(a)=0\) for every \(a\). Suppose that \(X_I=0\) when \(|I|<k\). Commuting two Ward operators on a row \(H\) of size \(k-1\) and differentiating (142) \(k+1\) times give, respectively, \[\begin{align*} \sum_{j\notin H}z_j(a)X_{H\cup\{j\}}(b) &=\sum_{j\notin H}z_j(b)X_{H\cup\{j\}}(a), \tag{143}\\ \sum_{|I|=k}z_I(n)X_I(n)&=0, \tag{144}\end{align*}\] where \(z_j(a)=a\cdot e_j\) and \(z_I=\prod_{j\in I}z_j\). Indeed all diagonal differentiations of already zero rows vanish; the first surviving paths through the triangular off-diagonal array are exactly the \(k!\) orders of adding the same \(k\) indices. The common nonzero factors \(4\) and \(k!\) have been canceled.

Complete a generic set of polarizations to a basis and write \(X_I(a)=\sum_{h=1}^4z_h(a)x_{Ih}\). Equation (143) says that, for each \(H\), the matrix with rows outside \(H\) equal to \(x_{H\cup\{j\},h}\) and its other rows zero is symmetric. If \(k\geq2\), choose \(H=I\setminus\{j\}\). For \(h\in I\) choose \(j\ne h\); its zero row kills \(x_{Ih}\). A completed-basis row \(h>m_0\) likewise kills that entry. For \(h\notin I\), symmetry identifies all the surviving entries with a coefficient \(c_{I\cup\{h\}}\), depending only on the union. Thus (144) becomes \[ (k+1)\sum_{|B|=k+1}c_Bz_B(n)=0. \tag{145}\] The square-free independence proved in Lemma 41 kills every coefficient, including the case where there is no such union.

For \(k=1\), (143) instead gives a symmetric matrix. Its quadratic form vanishes on the null cone, so it equals a scalar multiple of the metric quadratic. If \(m_0<4\), its zero completed-basis rows force that multiple to vanish. If \(m_0=4\), let \(G_{ij}=e_i\cdot e_j\). The coefficient matrix is \(\beta G^{-1}\), which says \[ X_{\{i\}}(e_j)=\beta\delta_{ij}. \tag{146}\] The function \(\beta\) is multilinear in all four polarizations: row \(i\) is independent of \(e_i\), and evaluation on \(e_i\) restores its linear factor. It vanishes when two polarizations coincide, by (146) and continuity from generic choices. It is therefore alternating. The sole possible exception is \[ X_{\{i\}}(a)= \lambda\,\epsilon(e_1,\ldots,e_{i-1},a,e_{i+1},\ldots,e_4). \tag{147}\] Here there are exactly four \(V/L\) positions and no other entries. Divergence in position \(i\) of \(X_\varnothing=0\) also gives \[ \Box_iX_{\{i\}}=0. \tag{148}\] For clarity, the canonical array preserves the relation \(\partial\cdot V=\Box L\): its difference is the divergence of the primary dimension-three vector \(J\), whose divergence transforms homogeneously. This is the intertwining identity used in deriving (148).

We eliminate (147) using its last position, \(i=4\). Smear the first three positions on a real mutually spacelike patch, keeping the last argument in the future tube. Split the scalar function into the triple bra and the one-wave ket. Its boundary in the first three positions is obtained from negative ordered bra dampings. Move the center of the generator sufficiently far to the future of these compact supports; affine covariance leaves \(X\) unchanged. The last-position canonical differential action on \(L\) is exactly the canonical generator applied to \(l(z_4)\). The ordinary physical triple \(VVV\) bra belongs to the domains of these centered actual \(K_n\) by Lemma 26. Moreover \(\Pi\) reduces these generators and their actions on \(\Pi\) are canonical by Theorem 27. Projecting the bra to \(\Pi\) thus allows the last-position generator to be moved to the fixed bra. Linear combinations of the available null parameters span arbitrary \(a\).

Every other term of the smeared \(X_{\{4\}}(a)\) pairs the unchanged last wave with a fixed constructed triple bra, possibly differentiated or with replacements. Consequently the whole expression is \(\left\langle\eta,l(z_4)\right\rangle\) for a fixed Hilbert vector \(\eta\), with no coordinate multiplier left on the last wave. Its translation spectrum is purely timelike, since that of \(l\) is. Equation (148) says that this spectrum is supported on \(p^2=0\); these disjoint supports force it to vanish. More explicitly, after any fixed future damping it is a finite spectral pairing, and multiplication by \(p^2\) kills it only if that pairing is zero off the null shell. There is no shell or zero-momentum component of \(l\). Boundary uniqueness in the first three arguments and generic polarization then give \(\lambda=0\).

The induction on \(k\) is now complete, followed by the induction in the number of \(t\) positions. This proves every row of the canonical Ward system. ◻

Proof. We justify passage from the localized-vector result of Corollary 36 to these products; unrestricted timelike real boundaries for accessible products are unnecessary. For a real vector localized in a fixed line diamond, the projected Paley multiplier in that Proposition has zero value at energy zero. The same local Paley bound controls its first derivative there. Thus it is \(O(E)\left\lVert\xi\right\rVert\) at low energy. Integrating with the weight-two energy powers against \(e^{-RE}\) in the measure \(\mathrm dE/E\), and using exponential damping beyond a fixed energy, gives \[ R^4\left\lvert\left\langle\ell(iRe),\xi\right\rangle\right\rvert \leq C R^{-1}\left\lVert\xi\right\rVert,\qquad R\geq1. \tag{152}\] For complex localized operator data the right side is bounded by the sum of the vacuum and adjoint-vacuum norms. Consequently the estimate passes to mutually spacelike accessible products in that diamond by the two bounded-approximation limits of Proposition 38.

Choose tests on small patches about equal-time mutually spacelike points and continue their translations a short distance into strictly increasing imaginary time. Before using the improved real-edge estimate, the corresponding smeared pairing multiplied by \(R^4\) is \(O(R^2)\) uniformly up to a smaller good real patch: the one-wave norm is \(O(R^{-2})\) and the smeared product bounds are uniform there. Apply the subharmonic logarithmic estimate on a half-disk with this real edge. If its harmonic measure at an interior point is \(\omega\), the bound becomes \(O(R^{-\omega+2(1-\omega)})=O(R^{2-3\omega})\). There is an interior open set with \(\omega>2/3\), and hence convergence to zero on that set. The positive imaginary times can be fixed before further shrinking the source tests. On its strict Euclidean configurations (150) already converges locally holomorphically. Small real source offsets therefore commute with that limit. Unsmeared uniqueness gives (151) on an open set, and real analyticity continues it to every distinct configuration. Derivatives follow from the same locally holomorphic identity. ◻

Proof. For each length restrict its arguments to a relatively compact open box in its tube. These vectors span a dense subspace: a vector orthogonal to them gives a holomorphic scalar function zero on an open set, and hence zero throughout the tube. For all sufficiently small real conformal transformations, map these vectors by their primary transformation laws. The resulting inner products are unchanged by the split identities of Proposition 42 and the integrated scalar Ward identities of Proposition 44. Total length four is exactly what is needed for pair–pair inner products. The image is dense by the same open-box argument; thus the map extends to a unitary. On smaller boxes the local composition identity is exact. Holomorphic point dependence gives strong continuity. Continuing these local unitaries along paths integrates the action to the universal cover.

Uniqueness extends the local laws to all tube points and, for sufficiently small transformations, to Euclidean configurations in the positive-time half-space. The point vectors there are smooth group vectors. On translations the construction agrees with the actual translation action, whose spectrum is future directed. The time-Möbius subgroup conjugates its positive time-translation generator to \(K_e^{\mathrm{rep}}\), proving the final assertion. No conformal action on the full physical Hilbert space is asserted here. ◻

Proof. First take \(A=V\) and pair with an arbitrary ordinary physical polynomial bra. Lemma 43 identifies the analytic pairs with tempered single-insertion forms. Deform \(\Sigma\) to a cylinder whose relative Euclidean time caps are \(\pm\delta\) and whose spatial disk has radius \(a\). Give the whole relative time a small real Minkowski shift \(s\), with \(|s|<a\). The shifted contour remains in the holomorphic domains; where the imaginary relative time crosses zero on the side, strict spatial separation and tie gluing provide a common neighborhood.

Average smoothly and with integral one over \(a\) and \(s\), keeping the shift support strictly inside \(|s|<a\). The side contribution tends to zero as \(\delta\downarrow0\) because it lies on a compact nonsingular spacelike set and its time thickness tends to zero. The two caps now have smooth Cauchy-slab tests and converge to the opposite Lorentzian order boundaries at a common target damping. This use of the single-insertion boundary is legitimate even at a point in the common complex target translation: its total pair momentum is exponentially damped, while the smooth relative test decays rapidly in the relative momentum. The difference is the commutator charge with the time average and a spatial cutoff equal to one on its causal support. The \(J\) charge is zero by Proposition 28; the translation charge of \(t\) is the original translation charge by Section 5. They give (154)–(155) for \(V\).

The Wick time component contributes \(i\), and the tensor translation index contributes the pullback \(C_b^\nu\). With the order and charge convention in the statement, \(i[V,P_b]=-\partial_bV\), hence \(c=-1\). This calculation fixes one common normalization for all components and all subsequently transported Killing fields. Pairing with the dense polynomial bras proves equality of the Hilbert vectors.

A constant-current flux around a primary target extends to a holomorphic single-wave function of its Minkowski argument throughout the future tube: choose a relative sphere smaller than its common damping. Translation covariance makes it a translation wave; dilations give its flux dimension. Small Lorentz transformations act on the pair and its contour; homotopy of the transformed contour back to a small sphere gives the tensor transformation. Differentiate the identities for \(V\) in its target index and use \(\partial\cdot V=\Box L\) while keeping the enclosing contour fixed. One obtains the homogeneous wave equation for both \[ F_J(L),\qquad F^t_{\partial_b}(L)-c\,\partial_bL\Omega. \tag{157}\] These are, respectively, a scalar wave of dimension two and a vector wave of dimension three. Translation covariance and positive-energy holomorphy realize their values after any fixed future damping as vacuum spectral amplitudes; further damping agrees by uniqueness. Their wave equation confines their spectrum to \(p^2=0\). Scale covariance excludes a zero-momentum amplitude, and the light-cone boost-weight test of Section 2 excludes the nonzero shell: in these cases \(d-1>\max\omega\) for the finite Lorentz multiplet. Only local finiteness after exponential damping is needed for that test. Thus both defects vanish. Subtraction of \(\partial L\) from \(V\) proves the identities also for \(J\).

For \(A=L,J\), conjugate the translation identity by any sufficiently small real conformal transformation in Proposition 46. Since \(t\) is a conserved traceless primary of dimension four, its flux transforms exactly to the flux for the pushed-forward vector field: three surface powers, one vector-field power, and the two orthogonal index factors cancel its weight. Deform the transformed sphere back locally. The target primary factors are invertible, and the differential action on the right transforms by the same Lie algebra adjoint action. Translations and all their such adjoints span the conformal Lie algebra (successive conformal brackets give rotations, dilations and special transformations). Linearity and complexification give every Euclidean conformal Killing field and prove (156). ◻

Proof. We specify the smooth-density issue first. The element \(Z=e^{2\pi iH}\) is central, since its adjoint action on the conformal Lie algebra is the identity by (162). Its fixed subrepresentation contains every integer \(H\) eigenspace. The projection onto that subrepresentation commutes with the group, preserves smooth vectors and has dense smooth image. There the spectrum of \(H\) is supported on the integers, and \[ E_H(\{h\})v=\frac1{2\pi}\int_0^{2\pi}e^{it(H-h)}v\,\mathrm dt \tag{164}\] preserves smoothness. Averaging also over the compact rotations gives dense smooth vectors in each rotation type and dense smooth finite tensor-equivariant arrays. Rotations descend to \(SO(4)\), since their action on the dense radial kets is the ordinary tensor action. The computations below therefore extend by density to the stated eigenspaces.

At zero energy the expectations of both positive terms in (160) vanish. They annihilate a smooth zero-energy vector. The translation spectrum cone then makes every translation generator annihilate it, and the conjugate cone makes every special partner annihilate it. Their brackets annihilate it as well, proving triviality. If \(Hv=v\), lowering has energy zero and \(\left\langle w,p_a^-v\right\rangle=\left\langle p_a^+w,v\right\rangle=0\) for every zero-energy \(w\); hence \(p_a^-v=0\).

For any smooth lowest array of energy \(h\), raising norms give \[ \left\|\sum_a p_a^+v_a\right\|^2 =2\sum_{a,b}\left\langle v_a,(h\delta_{ab}+M_{ab})v_b\right\rangle\geq0. \tag{165}\] A two-by-two plane-rotation block gives \(|m|\leq h\) for each plane weight. An \(SO(4)\) tensor type has two \(SU(2)\) spins \(j_+,j_-\), their sum an integer, with maximal plane weight \(j_++j_-\). At \(h=1\) only scalars, vectors and the two chiral two-form types survive this test. The divergence arrays exclude the last two. In detail, the tensor index rotation rule gives \[\begin{align*} \sum_bM_{ab}|A_b\rangle&=-3|A_a\rangle,\tag{166}\\ \sum_cM_{ac}|F_{cb}\rangle &=-3|F_{ab}\rangle-|F_{ba}\rangle +\delta_{ab}\sum_c|F_{cc}\rangle. \tag{167}\end{align*}\] The matrix eigenvalues in (165) are therefore \(-3\) for the vector divergence, \(-4\) for a symmetric traceless rank-two divergence, and \(-2\) for an antisymmetric rank-two divergence. Their positivity bounds are \(h\geq3,4,2\), respectively. At \(h=1\) this leaves only scalars.

At \(h=2\) the types in two vectors are even under the central radial rotation \(y\mapsto-y\). Their lowered arrays are odd but would have to belong to the scalar-only energy-one space, so they are lowest arrays. The \(-4\) bound excludes the symmetric traceless type. The antisymmetric bound is saturated and makes the first norm in (163) zero. Apply the same argument to the Hodge dual, which intertwines rotations, to obtain its second identity. This proves exactly the needed low-energy statements without an expansion in a complete family of local operators. ◻

Proof. Let \(B_\nu(z)\) be the reflected ket of this exterior data; it is a smooth pair vector. Its exterior translation and similarity laws, including the full derivative reflection of \(V\), are \[ \partial_{z^a}B_\nu=-p_a^-B_\nu,\qquad (z\cdot\partial+1)B_\nu=-HB_\nu. \tag{168}\] Translation fixes the insertion at infinity; reflection changes its generator to the lowering generator with the indicated sign. For the second formula the exterior vector contributes weight \(-6\) and the reflected interior pair dimension five, giving total degree \(-1-H\). These statements need only similarity covariance of \(V\).

For a smooth Maxwell array, set \(Q_{\nu ab}(z)=\left\langle B_\nu(z),F_{ab}\right\rangle\). Equations (163) and (168) make \(Q\) homogeneous of degree \(-3\) and make it and its Hodge dual in \(a,b\) divergence-free. Rotation covariance and antisymmetry leave precisely \[ Q_{\nu ab}=A\,|z|^{-4}(\delta_{\nu a}z_b-\delta_{\nu b}z_a) +B\,|z|^{-4}\epsilon_{\nu abd}z_d. \tag{169}\] For \(n=z/|z|\), the divergence of the first tensor is \(|z|^{-4}(\delta_{\nu b}-4n_\nu n_b)\), while that of the second is zero. Hence \(A=0\). Hodge duality interchanges these tensors up to orientation sign, so dual divergence gives \(B=0\). Dense smooth arrays in the energy-two rotation types and continuity of the fixed exterior pairing prove the general assertion. ◻

Proof. Expand \(W(\ell_\infty V(y)T(z))=0\), which is a case of (151), using (149). The \(Jt\) term has just vanished; the terms with derivatives of \(L\) vanish by (158) and (180). What remains is \(\partial_yW(\ell_\infty L(y)t(z))=0\). Translation covariance and homogeneity of degree \(-4\) force that function to be zero. Its centered dilation flux around \(L\), using (156), now gives \[ 0=2c\,W(\ell_\infty L). \tag{185}\] The energy-slice calculation in Proposition 35 gave \(D(e)=B(e,e)=b\) after excluding the massless contribution. Thus the constant two-point density of \(S\) with \(l-S\) vanishes: the scalar wave channels are orthogonal, not merely their labels on one time slice. It follows that \(W(\ell_\infty L)\) equals the scalar two-point coefficient \(W(\ell_\infty\ell)\), which is the nonnegative radial norm \(\left\lVert|\ell\rangle\right\rVert^2\) and is strictly positive for a nonzero scalar wave. Since \(c\ne0\), (185) makes that norm zero. Therefore \(l=S\). ◻

Proof. A distributional wedge boundary has no analytic, hence no smooth, wavefront at the origin in a direction outside \(K\). This follows by shifting a localized Fourier test a small distance in a strict direction of \(C\) with negative frequency pairing. The usual distributional boundary bounds on smaller approach wedges justify the shift. The argument applies in Hilbert norm, or after scalarization uniformly on the unit ball.

Fix a smooth Fourier test supported in a compact subset of \(\mathbb R^N\setminus K\). Its pairing with \(\widehat u\) after dilation to frequency \(\Lambda\) decreases faster than every power of \(\Lambda\). To pass from the local wavefront estimate to this unlocalized pairing, insert a cutoff equal to one near the origin. The local part has the asserted rapid decay. Outside the cutoff, the rescaled Schwartz inverse Fourier test and all its derivatives decrease rapidly in every Schwartz seminorm, so temperedness controls the remainder. A finite covering treats all directions in the fixed frequency support.

The scaling defect has zero pairing with that test and with every one of its positive dilates, since \(K\) is a cone. The remaining scaling identity makes the norm of the dilated pairing a fixed power of \(\Lambda\) times the norm of the original pairing; \(V(\lambda)\) does not change that norm. Rapid decay forces the original pairing to vanish. Every compact Fourier test outside \(K\) is allowed, proving the assertion. ◻

Proof. First, the product has scalar-primary covariance under sufficiently small simultaneous special transformations. The geometry in Lemma 40 applies to every fixed finite family of source and destination patches: its proof uses only finitely many strict wedge inequalities. Choose the conjugate null axis \(m\) separated from the finitely many flow directions, take \(m\cdot x\) sufficiently negative, and then add a sufficiently large positive multiple of \(m\) to make all the required opposite inequalities strict. A small double cone \(O_-\) about that point is spacelike to every containing wedge and to the destination patches. Local vacuum cyclicity makes \(\{C\Omega:C\in\mathcal M(O_-)\}\) total.

For a fixed finite product length, reparametrize the source tests by a small null flow and its inverse primary factors. Write \(F_g^{(n)}\) for the limit of the resulting simultaneously conjugated bounded products and \(F^{(n)}\) for the accessible destination product. The former limit exists because the source product converges and the common conjugating unitary is bounded. For \(C\in\mathcal M(O_-)\), move its first bounded factor past \(C\) and take the vacuum and adjoint-vacuum limits. The canonical one-wave action and induction on the length give \[\begin{align*} \langle C\Omega,F_g^{(n)}\rangle &=\langle L(\bar f_1)\Omega,C^*F_g^{(n-1)}\rangle\tag{189}\\ &=\langle L(\bar f_1)\Omega,C^*F^{(n-1)}\rangle =\langle C\Omega,F^{(n)}\rangle. \end{align*}\] Every transformed one-vacuum wave is already the accessible \(L\) wave, so this induction uses no finite-length interpolation. Totality proves the covariance identity on the real spacelike patch, and boundary uniqueness proves it in the tube. Compositions of small null flows give small special transformations. For any fixed real compact set and strict approach subcone, choose the transformations small both there and on a finite interior path to the initial patch. Their unitary factors have norm one. Thus the identity is uniform near that compact set; localization under arbitrary large conformal transformations is not required.

Thus covariance has been obtained separately for every finite pure product length. The next step uses it to transport the known boundary estimates for products of \(\Theta\) into enough differential estimates to recover the original pure \(L\) product.

Here is an algebraic device that converts this covariance into a boundary estimate. On the ambient null cone use \[X=(X^+,X^-,X^\mu),\qquad q(X)=X^\mu X_\mu-X^+X^-, \qquad X=\lambda(1,x^2,x).\] Here \(F\) is the Hilbert-valued analytic vector product. Its lift is Hilbert-valued and homogeneous of degree \(w=-2\) in each slot; the differential identities below may be tested against arbitrary Hilbert vectors, with estimates uniform on their unit ball. On degree \(w\) homogeneous functions, define the flat Thomas derivative \[ \mathscr D_A(w)=(2+2w)\partial_A-X_A\partial^2_{\mathrm{amb}}. \tag{190}\] Successive derivatives use the new degree in their slot. The product rule, \(\partial^2_{\mathrm{amb}}q=12\), and Euler’s identity show that these derivatives preserve the quadratic ideal, commute successively, and have zero successive trace. They therefore act intrinsically on the cone. The same calculation gives \[ X^A\mathscr D_A(w)=w(2+2w)\quad\pmod{q}. \tag{191}\] These are the flat ambient identities underlying the Thomas operator; see also (Gover and Waldron 2009). None of the factors in (191) vanishes at \(w=-2,-3,\ldots\).

In particular, contraction can undo each successive derivative at the degrees encountered here. The weight in (190) must be lowered after each differentiation in that slot; using a fixed weight throughout would not give these intrinsic commuting operators.

Take the null direction \(a_\infty\) representing chart infinity, and extend the lifted function independently of \(X^-\). Then \(a_\infty\cdot\mathscr D\) is a nonzero smooth chart factor times \(\Box_x\). Consequently \[ G_aF:=\prod_{i=1}^n(a\cdot\mathscr D_i)F \tag{192}\] has polynomial boundary growth for \(a=a_\infty\), by the ordinary all-\(\Theta\) product and (186). Primary transport proves the same statement for \(a\) in an open real null-cone patch about \(a_\infty\). Finite derivatives preserve these bounds after shrinking the approach subcone, by Cauchy’s inequalities.

For this transport, write \(\widetilde F\) for the homogeneous lift. For each fixed sufficiently small transformation \(g\), its local covariance is \(\widetilde F(gX)=U_g\widetilde F(X)\), with the primary chart factors absorbed in the lift. The flat ambient chain rule gives \(G_{ga}\widetilde F(gX)=U_gG_a\widetilde F(X)\). The unitary \(U_g\) is fixed as the insertion coordinates vary, so the coordinate derivatives act only on \(\widetilde F\). Taking \(a=a_\infty\) transfers the known bound to the null direction \(ga_\infty\). The shifted homogeneous weights supply smooth chart factors bounded on the chosen compact set. Scalarization replaces a testing vector by its unitary image and therefore keeps the Hilbert norm estimate uniform.

We give the finite algebraic argument needed to recover \(F\). In constant complex variables \(\xi_{iA}\), let \(I\) be the ideal generated by \(\xi_i^2\) and by \(\prod_i(a\cdot\xi_i)\) for all \(a\) in that patch. If a common zero had every \(\xi_i\ne0\), the patch would be covered by the finitely many proper hyperplane sections \(a\cdot\xi_i=0\). The real null patch is Zariski dense in the irreducible complex null cone and spans the ambient space, so this is impossible. Thus every common zero has \(\xi_i=0\) for some \(i\).

For each of the finitely many monomials \(m=\prod_i\xi_{iA_i}\), the Nullstellensatz gives \(m^{e_m}\in I\). Let \(e\) dominate their exponents. Every monomial of degree at least \(6(e-1)+1\) in each slot is divisible by one of the \(m^e\) and belongs to \(I\). Noetherianity gives finite generator representations, and separate homogeneity in each slot gives multihomogeneous coefficient representations. Substitute the successively commuting operators (190). Trace terms vanish; the remaining terms are finite derivatives of (192). Hence sufficiently high derivative tensors of \(F\) have polynomial growth. Contracting their indices with the \(X_i\) and using (191) repeatedly recovers \(F\) with a nonzero constant factor. The chart factors are bounded on the compact set. This proves moderate growth of \(F\).

The algebraic recovery used only a finite number of null directions and derivatives for this fixed product length. Its constants may depend on that length; no estimate uniform over all lengths is needed for constructing the polynomial field domain.

The moderate-growth tube boundary theorem, applied weakly and then uniformly in unit Hilbert vectors, gives a Hilbert-valued distribution locally. Its exact similarity covariance globalizes this boundary and makes it tempered: rescaling dyadic annuli to a fixed compact set changes a finite distribution seminorm by only a power of the radius, and the dilation unitary leaves its norm unchanged. Lemma 52 gives its ordered vector spectrum. Fourier–Laplace uniqueness then identifies its continuation with the original product. ◻

Proof. The first assertion is the distribution case of the generalized FBI criterion of Hoepfner–Ragognette (Hoepfner and Ragognette 2021, Theorem 6.3 and Section 8.1). In their notation take the positive quadratic polynomial \(p(v)=|v|^2/(2r_*)\) and \(\xi=\Lambda\eta\), \(|\eta|=r_*\). Then \(|\xi|p(w-z)=\Lambda(w-z)^2/2\), so their transform is (194) up to an irrelevant fixed normalization. Their local inverse decomposition separates a distribution, obtained from the polynomially bounded part, from an analytic remainder. Finite covering of the frequency sphere makes the polynomial estimate uniform. The compact representative is immaterial because changing it away from \(w_0\) changes this estimate by an exponentially small term.

For the second assertion, use the fixed-height integration representation of a wedge boundary value (Hoepfner and Ragognette 2021, Remark 4.1), using a rounded box with smooth boundary. The difference is an analytic functional carried in an arbitrarily thin complex neighborhood of the boundary of the chosen real box. This is also obtained from the product Cauchy representation: entries that extend across one coordinate cancel under contour displacement. Axes strictly inside the wedge give the requisite product half-domain description. Let the distance between the smaller box of centers and that boundary be \(a>0\). For \(z=x+iy\), \(|\eta|\leq M\), the real exponent in the Gaussian is \[ \Lambda\left(y\cdot\eta+\frac{|y|^2}{2} -\frac{|w-x|^2}{2}\right). \tag{196}\] Choose all side contours with \(|y|\leq t\) and then the carrier neighborhood with width \(\gamma\) so small that \((t+\gamma)M+(t+\gamma)^2/2<a^2/4\). Equation (193) bounds their contributions by \(C e^{-c\Lambda}\). This is the localization estimate of (Hoepfner and Ragognette 2021, Proposition 4.5). Heights and carrier neighborhoods are fixed before \(\Lambda\to\infty\); no integrability of \(f\) down to the real edge has been assumed. ◻

Proof. We first work with \(|r^0|\leq C_0\varepsilon\) and bounded \(\mathbf r\). Shift \(\mathbf r\) outward by at most \(C_1\varepsilon\) so that the resulting \(r_*\) has spacelike clearance at least \(4c\varepsilon\). Put the real merged placements \(z_*\) sufficiently far apart on a common time slice. Their spacelike margins, including separation from both members of the pair, can be chosen of order one. The tests for the two nearby slots have diameter \(c\varepsilon\); all tests may be taken of that size. The bounds (188) hold on the resulting separated patch.

Here \(F_R\) denotes the scalar vacuum expectation of the bounded product (131). On the vector tube, vacuum invariance removes its initial factor \(U(z_1)\). The resulting expression depends only on consecutive gaps and extends holomorphically, with the same polynomial norm bound, to the scalar tube where those gaps are strict. Thus this scalar face estimate does not require the first-gap condition that remains necessary for vector products.

Write \(\eta=\operatorname{Im}z\). Consider the complex line \[ \zeta\longmapsto (r_*+\varepsilon\zeta v,\ z_*+\zeta\eta), \qquad \operatorname{Im}\zeta>0. \tag{198}\] For sufficiently small \(\varepsilon\) its direction is in the full target cone. There is a real interval \([-a,a]\) on which the approximant supports remain spacelike, with \(a>0\) independent of \(\varepsilon\): the relative displacement is of size \(\varepsilon\) and the corresponding clearance is also of that size, whereas all merged displacements and margins are of order one. The harmonic measure of this interval at \(\zeta=i\) is \(\alpha=2\arctan(a)/\pi>0\). The two-constants theorem applied to \(\log|F_{2R}-F_R|\) gives \[|F_{2R}-F_R| \leq C_k\varepsilon^{-\alpha B_k-(1-\alpha)A} R^{-\alpha k+(1-\alpha)Q}.\] Slightly reducing the clearances gives this estimate on a complex \(r\)-ball of radius \(c_1\varepsilon\) and a \(z\)-ball of fixed radius.

Returning the spatial shift takes a bounded number of \(\varepsilon\)-balls. At fixed complex \(r\), moving the merged variables to any prescribed merged compact takes a bounded number of fixed-radius \(z\)-balls. Moving \(y\) moves \(x=y+r\) with it, so the small internal gap does not restrict those \(z\)-radii. More precisely, take the compact convex hull, inside the strict merged tube, of the initial placements and the prescribed compact set. Its gaps have a fixed positive margin; the two exterior gaps adjacent to the pair keep that margin when the internal displacement is \(O(\varepsilon)\). A finite cover therefore supplies radii and a number of balls independent of \(\varepsilon\). Take overlapping balls with radii \(\rho,2\rho,4\rho\) inside the tube. Each three-circle step retains exponent \(1/2\). Thus a fixed \(\sigma>0\), independent of \(\varepsilon\), gives \[ |F_{2R}-F_R| \leq C_k\varepsilon^{-E_k} R^{-\sigma k+(1-\sigma)Q}, \qquad E_k=\sigma\max(B_k,A)+(1-\sigma)A. \tag{199}\] Choose one finite \(k\) with \(\sigma k-(1-\sigma)Q>1\) and then fix its finite test seminorm. Summing dyadic differences gives a fixed inverse power of \(\varepsilon\). If localization starts at a dyadic \(R_0(\varepsilon)\) bounded by a power of \(\varepsilon^{-1}\), its initial norm contributes another finite power. The construction of Proposition 37 permits this choice. The accuracy order is never increased while \(\varepsilon\) tends to zero.

This order of choices is essential: the continuation geometry fixes \(\sigma\), then one chooses \(k\), and only then lets the relative gap shrink. The resulting \(E_k\) is a fixed finite exponent in the face bound.

For completeness, unsmearing this estimate does not involve point evaluation of a bounded approximant. The limit already has a holomorphic kernel by Proposition 38. Choose a smooth radial bump \(\rho\) on \(\mathbb C^4\) of integral one, supported in a sufficiently small ball. Holomorphic mean value in each of \(p\) slots gives, writing \(Z\) for the full slot coordinates, \[\begin{align*} W_2(Z) &=\int_{(\mathbb C^4)^p}\prod_i\varepsilon^{-8} \rho(w_i/\varepsilon)W_2(Z+w)\,\mathrm dw\tag{200}\\ &=\int \mathrm db\; W_2(f_{\varepsilon,b_1},\ldots,f_{\varepsilon,b_p}; Z+i\varepsilon b). \end{align*}\] Here \(f_{\varepsilon,b}(u)=\varepsilon^{-4}\rho(u/\varepsilon+ib)\), and \(b\) ranges over a fixed compact set and \(p_{K,m}(f_{\varepsilon,b})\leq C_m\varepsilon^{-4-m}\). The finite seminorm estimate just proved therefore bounds the last integral by a power. Cauchy’s inequalities in smaller \(\varepsilon\)-balls also bound all fixed derivatives. This proves (197) initially in the thin real-time strip.

Finally fix \(z\) and the imaginary displacement \(i\varepsilon v\). The function \(u_\varepsilon(t,\mathbf r)=W_2((t,\mathbf r)+i\varepsilon v;z)\) satisfies the ordinary inhomogeneous wave equation in the real variables. Its initial value, spatial derivatives, and first time derivative have the bounds just obtained. Its source is the Fourier–Laplace continuation of the inductively tempered \(S_2\), so it has a power bound of its own at this relative gap. Kirchhoff’s formula on \(|t|\leq T\), \(|\mathbf r|\leq B\) gives \[\begin{align*} \|u_\varepsilon\|_{T,B} &\leq \|u_\varepsilon(0)\|_{B+T} +T\|\nabla u_\varepsilon(0)\|_{B+T} +T\|\partial_tu_\varepsilon(0)\|_{B+T} +\tfrac12T^2\|S_2\|_{T,B+T}, \tag{201}\end{align*}\] with an immaterial fixed norm constant. This proves the bound on every real compact set. Derivatives obey the same estimate. Moderate-growth boundary values now give \(F_0\); the estimates uniform on \(z\)-compacts give holomorphy and locally uniform finite distribution order in \(r\). ◻

Proof. Fix a real merged center. Initially \(bC_j\) is only a hyperfunction. Choose a small real box about the center, a smaller box of centers \(w\), and a strict merged direction \(v_z\). Lemma 54 represents the Gaussian testing by a fixed translate \(it_0v_z\), up to exponentially small side errors.

Choose a strict relative direction \(v_r\) for the target ordering, small enough compared with the merged gaps that \((v_r,v_z)\) is a strict full target direction. Choose finitely many nonzero null covectors \(k_a\) on the wrong branch, so \(k_a\cdot v_r<0\). The sample points range over a radially open patch of that branch; they are not constrained to a fixed energy shell. The radius variation will be used when separating coefficient polynomials of different degrees; choosing directions only on one fixed shell would not give that independence argument. Then choose \(r_*>0\) sufficiently small that, uniformly for all samples and \(|\eta|=r_*\), \[ k_a\cdot v_r+\eta\cdot v_z\leq-c_0<0. \tag{204}\] Test (203) in \(r\) with \(\chi(r)e^{-i\Lambda r\cdot k_a-\Lambda r^2/2}\), where \(\chi=1\) near zero, and in \(z\) with the Gaussian (194) on the fixed translated box.

For the \(F_0\) term, deform the relative contour from zero to \(it_0v_r\), leaving \(z\) at height \(it_0v_z\). Lemma 55 justifies the initial endpoint. On the final central contour the real exponent, apart from negative spatial Gaussians, is at most \[\Lambda\left[-t_0c_0+ \tfrac12t_0^2(|v_r|^2+|v_z|^2)\right].\] Choose \(t_0\) so small that this is \(\leq-c\Lambda\). The continued \(W_2\) is bounded on that fixed interior compact, so this term is exponentially small.

We specify the cutoff error, since the relative contour starts at a distributional face. Let \(N\) be an inverse-height growth exponent from (197), increased to cover the derivatives used below. Replace \(\chi(x)\) on the moving contour by its almost-analytic Taylor lift \[\chi^{[M]}(x+iy)= \sum_{|\alpha|\leq M}(iy)^\alpha\partial^\alpha\chi(x)/\alpha!, \qquad M>N+1.\] Its antiholomorphic derivative is \(O(|y|^M)\), supported in the transition region. There \(|x|\) is separated from zero. By decreasing \(t_0\) before taking the large-parameter limit, (196) is \(\leq-c'\Lambda\) throughout the transition, even after all linear phase costs. Stokes errors are bounded by a polynomial in \(\Lambda\) times \(e^{-c'\Lambda}\int_0^{t_0}t^{M-N}\,\mathrm dt\). The same construction applies to any smooth central cutoff in \(z\). For the merely hyperfunctional coefficient boundary we instead use the fixed-height analytic-functional remainder of Lemma 54; we do not multiply that boundary by a smooth cutoff.

The \(T_0\) term is polynomially bounded in \(\Lambda\): move its merged contour back to the real boundary and use its tempered seminorm. Derivatives of the Gaussian and cutoffs cost only powers of \(\Lambda\). Its contour errors obey the same estimate, now with the known finite distribution order of \(T_0\).

It remains to check that these tests recover every coefficient. Put \(d_j=\deg P_j\) and \[M_{aj}(\Lambda)= \left\langle P_j(\partial)\Delta_0, \chi(r)e^{-i\Lambda r\cdot k_a-\Lambda r^2/2}\right\rangle.\] The Fourier transform of \(\Delta_0\) is a nonzero normalization times \(\operatorname{sgn}(p^0)\delta(p^2)\). Gaussian Fourier integration, with \(p=\Lambda k_a+\sqrt\Lambda q\), gives \[ \Lambda^{3/2-d_j}M_{aj}(\Lambda) \longrightarrow c(k_a)P_j(ik_a),\qquad c(k_a)\ne0. \tag{205}\] Indeed \(\mathrm dp\) contributes \(\Lambda^2\), the delta function \(\Lambda^{-3/2}\), and the four-dimensional Gaussian Fourier factor \(\Lambda^{-2}\). The remaining integral is a nonzero Gaussian integral on \(2k_a\cdot q=0\), with the fixed sign of the branch. The cutoff error is exponentially small by the temperedness of \(P_j(\partial)\Delta_0\). The nonzero factor \(c(k_a)\) can depend on the row.

The evaluation matrix \(P_j(ik_a)\) has full column rank for a suitable finite sample set. Otherwise a nonzero linear combination would vanish on an open real patch of one null branch, hence on the irreducible complex quadratic cone, contradicting independence modulo the wave quadratic. Radial openness matters for polynomials of different degrees. Selecting an invertible square submatrix, (205) bounds its inverse by powers of \(\Lambda\). The tested identity therefore gives \[|\mathcal T_\Lambda(bC_j)(w,\eta)|\leq C(1+\Lambda)^N \qquad(|\eta|=r_*),\] uniformly on the smaller box. Lemma 54 proves that every \(bC_j\) is an ordinary distribution there. This conclusion is local in the merged real variables. Growth at infinity and the merged spectral support still require the scaling argument below. ◻

Proof. Let \(\mathcal D_K\) be the span of \(C_1\cdots C_m\psi\), with \(\psi\in\mathcal D\) and each \(C_j\) in the commutant of an allowed neighborhood algebra, including \(m=0\). Every finite collection of these neighborhoods has a common smaller open neighborhood of \(K\). By isotony all its commutant coefficients belong to the commutant of this smaller algebra. Products and adjoints of the coefficients therefore obey the assumed exchange identities.

For a finite sum with product coefficients \(C_i\), prescribe \[\widetilde A\sum_iC_i\psi_i=\sum_iC_iA\psi_i, \qquad \widetilde A^\sharp\sum_iC_i\psi_i=\sum_iC_iA^\sharp\psi_i.\] If the input sum is zero, pairing the proposed first image with \(\phi\in\mathcal D\) gives \(\langle A^\sharp\phi,\sum_i C_i\psi_i\rangle=0\). Density proves well-definedness; the reverse exchange proves it for the second prescription. For two sums use exchange with \(D_j^*C_i\) in a common smaller algebra. This gives \[\widetilde A^\sharp\subset\widetilde A^*,\qquad \widetilde A\subset(\widetilde A^\sharp)^*.\] Both operators are thus closable. Their common domain is invariant under every unitary in every allowed commutant, and both operators commute with that unitary and its inverse. Their closed graphs retain these invariances, proving affiliation simultaneously with all the neighborhood algebras. The adjoint inclusions survive closure. In the symmetric case the two prescriptions coincide, so their closure is symmetric as well. ◻

Proof. We first prove exchange with every bounded operator of the actual commutant, not merely with opposite-region fields. If \(C\in\mathcal M(O)'\), then \[ \langle\phi,CL(f)\psi\rangle =\langle L(f)\phi,C\psi\rangle, \qquad \phi,\psi\in\mathcal D. \tag{211}\] First take \(\phi,\psi\) to be finite sums of words with factorized compactly supported tests, and by linearity treat one word on each side. Translate their slots into real mutually spacelike patches, all spacelike from \(\mathop{\mathrm{supp}}f\). Subdivide \(f\) into local patches and choose its self-adjoint bounded approximants in \(O\). The simultaneous approximating products converge there in Hilbert norm. Exchange holds at the bounded level because each approximant to \(L(f)\) commutes with \(C\). The norm limit proves (211) on that real open placement set.

Continue separately in the translated polynomial slots on either side. A fixed \(L(f)\) on the left of a polynomial leaves every suffix spectral cone of that polynomial intact; on a bra the corresponding opposite cones hold. The bounded operator \(C\) between the vectors does not affect these two separate vector analyticity statements. To see the common domain explicitly, use the positive vector tube of (130) for each polynomial before taking the bra adjoint. For a polynomial \(\psi\) of length \(n\), both \(\psi(b)\) and \(L(f)\psi(b)\) are holomorphic on that tube. In the latter product let \(q_1\) be its total momentum and \(q_2,\ldots,q_{n+1}\) its suffix momenta. Fixing the first, \(L\), coordinate real removes its imaginary damping, but the strict polynomial gaps still damp all \(q_j\) with \(j\geq2\). The real smearing contributes \(\widehat f(q_1-q_2)\), whose rapid decay controls the remaining momentum since \(|q_1|\leq|q_1-q_2|+|q_2|\). The tempered vector distribution therefore has this smeared face, with all finite derivative bounds. The same applies to \(\phi\) and \(L(f)\phi\). After conjugating the bra variables, the common scalar domain is the product of the negative bra tube and the positive ket tube. Boundedness of \(C\) suffices for these pairings; neither commutation with translations nor invariance of \(\mathcal D\) under \(C\) is used. Their scalar pairings therefore obey boundary uniqueness, which extends the identity to arbitrary placements and smearings. For a compact joint kernel, first use a finite partition into product patches and then approximate each piece by finite sums of factor tests in the required finite smooth seminorms. Theorem 51 gives Hilbert-norm convergence both for the polynomial vector and for the vector with the additional \(L(f)\) slot. Boundedness of \(C\) therefore passes the identity to the joint kernel. This test-topology limit neither asserts finite tensor rank nor uses invariance of \(\mathcal D\) under \(C\). Compact joint tests are dense in the joint Schwartz space, so the same Hilbert-norm continuity extends exchange to all joint Schwartz kernels in \(\mathcal D\).

Apply Lemma 57 with \(A=A^\sharp=L(f)\) and \(K=\mathop{\mathrm{supp}}f\). The exchange identity holds for every neighborhood, so the lemma gives one compatible closed symmetric affiliated extension. Essential self-adjointness is neither asserted nor required. ◻

Proof. Coordinates multiplying fields are held fixed when taking a Ward variation. Using (218), the conformal Killing equation, and conservation, direct differentiation gives \[ \delta_X j_{Y,\mu} =\partial^\lambda(X_\lambda j_{Y,\mu} -X_\mu j_{Y,\lambda}) -j_{[X,Y],\mu}. \tag{220}\] For example, expanding the divergence gives \(4\sigma_X j_{Y,\mu}+X\cdot\partial j_{Y,\mu} -(\partial^\lambda X_\mu)j_{Y,\lambda}\); subtracting \(j_{[X,Y]}\) cancels the derivative of \(Y\), and \(\partial_\lambda X_\mu+\partial_\mu X_\lambda =2\sigma_X\eta_{\lambda\mu}\) yields exactly the two index terms and the remaining \(2\sigma_X\) in (218).

Choose one compact flux \(Q_Y(F)\) implementing \(\delta_Y\) on an insertion \(A\) and its local \(X\) variation. Use a larger compact flux for \(\delta_X\) on \(Q_Y(F)\) and these insertions. The derivation rule on the joint domain gives \[[\delta_X,\delta_Y]A=i[\delta_X Q_Y(F),A].\] The first term on the right of (220) is an antisymmetric divergence. Its commutator flux about the compact insertion is zero: integration by parts leaves only the outer boundary, where locality permits the cutoff to be removed. The second term therefore gives \(-i[Q_{[X,Y]}(F),A]=-\delta_{[X,Y]}A\). All products and iterated insertions in this calculation are defined by compact joint kernels on \(\mathcal D\). Equation (216) and the affine identification were proved above. ◻

Proof. For a similarity \(g(z)=\lambda\Lambda z+a\), \(\lambda>0\), write \(U(g)\) for the already given Poincaré/dilation implementer. Changing variables in the smooth current test for the dimension-four \(t\) sends \(X\) to \(g_*X=\lambda\Lambda X\circ g^{-1}\) and an absolute normalized cap test \(h_0\) to \((h_0)_g=\lambda^{-1}\Lambda h_0\circ g^{-1}\), in the tensor convention of their contraction. For a relative cap, the translation cancels in this formula. Since \(g_*k_b^{[x]}=\lambda^{-1}k_{\Lambda b}^{[gx]}\), cap independence gives \[U(g)B_{b,A}(x)U(g)^{-1} =\lambda^{\Delta-1}\rho_A(\Lambda) B_{\Lambda b,A}(gx),\] where \(\rho_A\) is the Lorentz component matrix of \(A\). Thus \(B_{b,A}\) is a possibly zero translation-covariant homogeneous insertion of real grade \(\Delta-1\), in a finite Lorentz family with one additional vector index. Only the existing similarity covariance and the proved cap deformation have been used.

Let \(s_{b,c}\) denote the constant component matrix in the affine action \[\delta_{p_c}A=c\cdot\partial A,\qquad \delta_dA=(x\cdot\partial+\Delta)A,\qquad \delta_{m_{b,c}}A=(m_{b,c}(x)\cdot\partial+s_{b,c})A.\] Expanding \(k_b(z-x)=k_b(z)-2(b\cdot x)d(z)-2m_{b,x}(z)+p_{k_b(x)}(z)\) in the centered flux gives \[ \delta_{k_b}A(x) =\bigl(k_b(x)\cdot\partial+2\Delta(b\cdot x)+2s_{b,x}\bigr)A(x) +B_{b,A}(x). \tag{221}\] The polynomial-coordinate term is an insertion, in general not a translation-covariant point-field label; its coefficients multiply the target test after smearing. For example, (217) gives \(B_{b,L}=0\).

The local affine Ward actions on \(B\) also follow from the proved Lie relation. Indeed \([p_c,k_b]=2(b\cdot c)d+2m_{b,c}\) and \([d,k_b]=k_b\). At a fixed enclosing cap target differentiation commutes with the commutator, so (219) gives \[\begin{aligned} (\partial_c-\delta_{p_c})\delta_{k_b}A &=2(b\cdot c)\delta_dA+2\delta_{m_{b,c}}A,\\ \delta_d\delta_{k_b}A &=(x\cdot\partial+\Delta-1)\delta_{k_b}A . \end{aligned}\] The polynomial term in (221) has the same two defects. Subtraction gives \(\delta_{p_c}B=c\cdot\partial B\) and \(\delta_dB=(x\cdot\partial+\Delta-1)B\). The Lorentz bracket \([m_M,k_b]=-k_{Mb}\) for \(m_M(z)=Mz\) similarly gives the tensor-product component action. Thus these affine identifications are consequences, not an assumed conformal closure.

For any fixed finite collection of centered insertions, substituting their relative kernels into a joint product expands it into finitely many of the already constructed products. This is a continuous map of compact smooth or Schwartz test spaces: the relative supports are compact and the coefficients are polynomial. It supplies all joint products at every finite length on \(\mathcal D\), including adjoints and all mixed orderings. Merging each block retains the ordered suffix cones. For real \(b\) and \(h\), reversing the two products gives \(B_{b,A}^\dagger=B_{b,A^\dagger}\). Positivity holds in the original Hilbert space, and locality follows by shrinking the relative supports on spacelike product patches; a compact test on the spacelike configuration set is first partitioned into such patches. The preceding exchange argument supplies compatible original-net affiliated realizations and their adjoints for these finite homogeneous families.

At each depth, choose a shrinking representative of the previous depth inside an inner neighborhood, and its enclosing cap inside the prescribed target neighborhood. The previous depth’s locality makes its current commutator causally supported, so conservation gives the same normalized-cap deformation and shrinking property at the next depth. Every nested expression is still a finite joint polynomial in the original fields and \(L\). The joint continuity and full-commutant construction above therefore give all mutual products and compatible affiliation for any fixed finite collection of depths. The covariance calculation gives grade \(\Delta-r\) and at most \(4^rN_A\) components before field identities. Equation (221), the affine actions just proved, and finite Leibniz rules express every fixed finite Ward word as a finite sum of polynomial-coordinate multiples of finite derivatives of these families. For an original \(A=\sum_{\Delta\in S_A}A_\Delta\), do this separately for the finitely many components, using the same target joint kernel in every component term and assigning no single weight to their sum. Thus each finite Ward calculation uses homogeneous insertions in a jointly constructed finite family on \(\mathcal D\). This does not identify them with original labels in \(\mathcal F\), assert global closure under centered descendants, or construct a conformal action on the weakly closed bounded net. ◻

Proof. Regard the original fields and \(\mathcal G\) as the spectator fields in the boundary reconstruction of this section. Their joint products, common invariant realization, adjoints, and original-net affiliations are given by the admission criterion. Each fixed homogeneous list has tempered vacuum distributions by the scaling argument of Section 2; finite sums treat the original labels. Their affiliated polar truncations obey Proposition 37, since the required moments are alternating words in these already given joint products. Thus Proposition 38 supplies their analytic products with the same accessible \(L\) and the same reflected splittings. No current Ward identity on \(\mathcal G\) is needed for this step.

The pure \(L\) construction is unchanged. In the simultaneous mixed induction, replacing \(L\) by \(\Theta\) lowers the number of \(L\) entries and adds an original homogeneous spectator. The merged-face estimate, causal ambiguity, Gaussian coefficient recovery, and scaling argument therefore apply with \(\mathcal G\) among the fixed spectator labels. They prove Theorem 51 for every such enlarged list. Its positivity and full-commutant exchange then give the invariant joint domain \(\mathcal D_{\mathcal G}\) and the compatible affiliated realizations exactly as above. Products involving only the original fields and \(L\) agree with the previous ones: their analytic products are the same approximation-independent limits, and their tempered boundaries are unique. Consequently \(\mathcal D\subset\mathcal D_{\mathcal G}\) and all insertion actions agree there. Old-and-\(\mathcal G\) products likewise retain their given vacuum-generated restrictions, without requiring agreement of chosen closed realizations on an independently larger domain. The same reasoning proves agreement for nested compatible families. If derivatives of the added fields are included as labels, their dimensions lie in the finite union of progressions \(\Delta+\mathbb Z_{\ge0}\); each fixed grade receives only finitely many derivative components. This preserves finite scaling support and the stipulated locally finite, bounded-below grade convention.

Compact commutator fluxes now define \(\delta_X\) on the enlarged polynomial algebra. Conservation, locality, and joint continuity give cap deformation, the derivation rule, and preservation of the quotient by zero operators just as before. Its compact polynomial vectors have wedge reality by their affiliated realizations and the joint graph limits. Lemma 24 and its summable dyadic estimate therefore give vanishing complete vacuum flux and the Ward identity (216) on the enlarged domain.

We check the physical affine actions on the added fields, rather than assume them. Let \(X\) be a translation, Lorentz, or dilation generator, and let \(G_X\) be its given self-adjoint generator, with the convention that the physical variation is \(i[G_X,\cdot]\). For a compact smearing \(F\) of a member of \(\mathcal G\) and an old compact polynomial \(C\), the vacuum Ward identity gives \[\langle C\Omega,\delta_XF\,\Omega\rangle =-\langle(\delta_XC)\Omega,F\Omega\rangle =i\langle G_XC\Omega,F\Omega\rangle =\langle C\Omega,iG_XF\Omega\rangle.\] The middle identity uses the already proved affine action on old polynomials. Smooth compact smearing and the given similarity covariance place \(F\Omega\) and \(C\Omega\) in the generator domains; the vacuum is invariant. Density of old polynomial vectors thus identifies the vacuum action with the physical affine variation of \(F\), including its component and weight terms.

The discrepancy between these two variations is relatively local and closable on \(\mathcal D_{\mathcal G}\): it is a finite compact joint insertion with a densely defined reversed adjoint. It annihilates the vacuum. Choose an old double cone spacelike to its compact localization. Locality makes it annihilate every old polynomial vacuum vector from that cone, a dense set by Reeh–Schlieder. The closure of its graph therefore contains \(\mathcal H\times\{0\}\), and being an operator graph it can contain no nonzero output. The discrepancy vanishes on the whole joint domain. This proves the required affine identifications without assuming new current Ward identities.

The primary identities for \(L\) and \(t\) likewise persist on the enlarged domain: their relatively local discrepancies vanish on the original dense domain, and the same adjoint and closed-graph argument applies. The current-variation computation and compact nested flux proof of Proposition 59 consequently give the conformal Lie relation on every enlarged polynomial insertion. Finally the centered construction of Proposition 60 uses only these Ward identities, similarity covariance, cap shrinking, and finite joint products. Repeating it on \(\mathcal D_{\mathcal G}\) proves its conclusion for every fixed finite family of added targets and finite descendant depths. ◻