Proof. The type-I clause turns the cap modular generator into the special operator form used by the small-ball modular Hamiltonian. Exact Markovity removes the recovery defect in the collar split. The vanishing-collar-remainder assumption removes the OPH finite-cutoff correction terms carried by the null bridge and the small-ball transport. The maximally symmetric small-cap limit identifies the OPH cap-label-preserving MaxEnt variation with the fixed-volume ball variation. Under those identifications, the OPH fixed-cap generalized-entropy stationarity equation has the same first variation as Jacobson’s entanglement-equilibrium relation. The local quadratic polarization step gives the standard tensor upgrade. ◻

Proof. The added clauses are exactly the hypotheses and outputs of Theorem 107, Theorem 170, Theorem 198, Lemma 200, Theorem 206, and Theorem 256. Removing those clauses leaves the Jacobson-type limit described above; keeping them is the OPH-specific branch. ◻

Proof. The assumed capacity relation gives \[ N_\star=\frac{3\pi}{\Lambda_\star\ell_\star^2}, \] so \[ \Lambda_\star\ell_\star^2=\frac{3\pi}{N_\star}. \] Multiplying by \(P_\star=a_{\mathrm{cell}}/\ell_\star^2\) gives \[ \Lambda_\star a_{\mathrm{cell}} =P_\star\Lambda_\star\ell_\star^2 =\frac{3\pi P_\star}{N_\star}. \] Since \(\mathcal B_\ell=\Lambda_\star N_\star\), \[ \mathcal B_\ell\ell_\star^2 =\Lambda_\star N_\star\ell_\star^2 =3\pi. \] Multiplying by \(P_\star\) gives \[ \mathcal B_\ell a_{\mathrm{cell}}=3\pi P_\star. \]  ◻

Proof. For any \(\lambda>0\), rescale all lengths by \(L\mapsto\lambda L\). Then \[ \ell_\star^2\mapsto\lambda^2\ell_\star^2,\qquad a_{\mathrm{cell}}\mapsto\lambda^2a_{\mathrm{cell}},\qquad \Lambda_\star\mapsto\lambda^{-2}\Lambda_\star. \] The ratios \[ P_\star=\frac{a_{\mathrm{cell}}}{\ell_\star^2}, \qquad N_\star=\frac{3\pi}{\Lambda_\star\ell_\star^2} \] are unchanged, but \[ \mathcal B_\ell=\Lambda_\star N_\star \mapsto \lambda^{-2}\mathcal B_\ell. \] Thus \(P_\star\) and \(N_\star\) determine the invariant product \(\mathcal B_\ell\ell_\star^2=3\pi\), while the SI value of \(\mathcal B_\ell\), the SI area \(\ell_\star^2\), and \[ G_{\mathrm{SI}}=\frac{c^3}{\hbar}\ell_\star^2 \] change with the length scale. ◻

Proof. The clock-ratio definition gives \[ \ell_\star=\frac{\gamma_\star c}{\nu_{\mathrm{Cs}}}, \] and hence the displayed value of \(\ell_\star^2\). The curvature display is only a re-expression of the same area coordinate: \[ B_\star=\frac{3\pi}{\ell_\star^2}. \] The de Sitter capacity identity is \[ N_\star=\frac{3\pi}{\Lambda_\star\ell_\star^2}. \] Multiplying by \(\Lambda_\star\ell_\star^2\) gives \[ \Lambda_\star N_\star\ell_\star^2=3\pi. \] Since \(B_\star=\Lambda_\star N_\star\) on the selected branch, division by \(B_\star\) gives \[ \ell_\star^2=\frac{3\pi}{B_\star}. \]  ◻

Proof. With \(\log\omega_L(\lambda')=-\sum_c\lambda'_c S_c-\log Z_L(\lambda')\,\mathbf 1\), \[ D(\sigma\|\omega_L(\lambda'))=-S(\sigma)+\sum_c\lambda'_c\langle S_c\rangle_\sigma+\log Z_L(\lambda'). \] \(\log Z_L\) is smooth, and its Hessian is the Duhamel (Kubo–Mori) covariance matrix of the \(S_c\) in \(\omega_L(\lambda')\), which is positive definite exactly when the \(S_c\) and \(\mathbf 1\) are linearly independent; this gives strict convexity, hence uniqueness of any minimizer, and the vanishing-gradient condition is the displayed moment matching. Attainment for faithful \(\sigma\) is finite-dimensional exponential-family duality: the moment map \(\lambda'\mapsto(\langle S_c\rangle_{\omega_L(\lambda')})_c\) is a diffeomorphism onto the relative interior of the achievable moment set, which contains the moment vector of every faithful state . Part (ii) is the quantum Pinsker inequality \(D(\rho\|\tau)\ge\tfrac12\|\rho-\tau\|_1^2\) evaluated at \(\lambda^\ast\). Part (iii) follows because the attained infimum vanishes iff \(D(\sigma\|\omega_L(\lambda^\ast))=0\) iff \(\sigma=\omega_L(\lambda^\ast)\), by strict positivity of relative entropy off the diagonal. ◻

Proof. Lexicographic order on \(\mathbb N^4\) is well-founded: minimize the first coordinate, then the second on that fiber, then the third, then the fourth. Each step minimizes a nonempty subset of \(\mathbb N\). ◻

Proof. Items (i)–(iii) are collected from Theorem 25, Corollaries 137 and 143, Theorem 205, and Theorem 206. On the gauge side, Theorem 73 supplies the strict zero-obstruction transport criterion and Theorem 77 constructs the fixed-cutoff bosonic collar-sector categories. Given the compact-gauge refinement receipt, Theorem 260 constructs the monoidal refinement transport and compatible forgetful fibers. Theorem 263 descends those data to the refinement-limit category and fiber functor, and compact gauge reconstruction is then Theorem 264. Theorem 265 separates this classification stage from the MAR selection stage. The realized nontrivial branch is supplied by Theorem 292. The product gauge structure up to finite quotient is Theorem 273; and the final identification of the exact Standard Model quotient and structural electroweak content is Theorem 293, Theorem 274 together with Corollaries 277, 278, 285, and Proposition 281. ◻

Proof. Finiteness and the fixed order on \(\mathsf A\) give the least enabled transaction whenever one exists; if none exists, the definition returns \(x\). Boundary preservation and exact descent are the transaction acceptance tests. Strict descent excludes a nontrivial fixed move, and repair completeness identifies states with no enabled aggregate transaction with \(C_Q\). ◻

Proof. If \(S_a\cap W=\varnothing\), the transaction leaves every argument of \(\mu_a\) unchanged, so that term cancels in the pre/post comparison. Every remaining term is determined by the payload and the restriction to \(R^\mu(W)\). The edge-mismatch statement is the specialization in which each term is supported on the two endpoint registers of one overlap. ◻

Proof. Two steps in one conflict component use its single canonical aggregate, so they cannot form a distinct peak. Steps from distinct components have disjoint writes, and neither write meets the other transaction’s read set. Semantic dependency closure therefore preserves the other transaction’s enablement, payload, descent comparison, protected data, semantic parents, and checkpoint continuation. Both second commits remain legal. Their snapshot-determined payloads act on disjoint writes, so the two commit orders give the same quotient state \(w\). ◻

Proof. Well-founded descent gives termination. Proposition 24 supplies local confluence. Newman’s lemma  says that a terminating locally confluent rewrite system is confluent. If two terminal states \(y,z\) are reachable from the same \(x\), confluence gives a common descendant \(w\). Since \(y\) and \(z\) are terminal, \(y=w=z\). Repair completeness identifies the terminal state with an element of \(C_Q\), so \(N_Q:Q\to C_Q\) is well defined and schedule-independent. The canonical iteration in Definition 21 is one valid accepted repair execution, hence \(\operatorname{Rep}_\lambda(x)=N_Q(x)\). Boundary preservation follows by induction along accepted transactions. Idempotence and the fixed-point characterization follow from Proposition 22, since every element of \(C_Q\) has no enabled aggregate transaction. ◻

Proof. By Theorem 25, \(N_Q(x)\) and \(N_Q(x')\) exist and lie in \(C_Q\). Boundary preservation along repair sequences gives \(B(N_Q(x))=B(x)\) and \(B(N_Q(x'))=B(x')\). If \(B(x)=B(x')=b\), then both normal forms lie in \(C_b\). The unique consistent extension assumption says \(C_b\) has at most one element, hence the two normal forms are equal. ◻

Proof. Each \(R_d\) writes only \(L_d\) with \(d\ge1\), so no stage changes \(B\). Induct on \(d\): after stage \(d\), every vertex in \(L_0\sqcup\cdots\sqcup L_d\) agrees with \(E(b)\), because all parents of a layer-\(d\) vertex lie in earlier layers. Thus \(a^{(D)}=E(b)\), and admissibility puts it in \(C_Q\). If \(c\in C_Q\cap B^{-1}(b)\), the same induction applied to the functional equations for \(c\) gives \(c=E(b)\). The final equality follows because Theorem 25 makes \(\operatorname{Rep}_\lambda(x)\) a boundary-preserving element of \(C_Q\). ◻

Proof. Tree edges force each vertex value recursively from the root and \(b\), so any consistent state in the fiber must equal \(u_b\) on every tree vertex. The remaining equations are exactly the non-tree overlap, sector, and holonomy checks. If they pass, \(u_b\) is the single consistent extension. If one fails, no tree-compatible candidate satisfies all consistency equations. ◻

Proof. Summing the edge equations around a cycle gives necessity. For sufficiency, fix a root and define each \(x_v\) by summing labels along a path from the root to \(v\). Vanishing cycle sums make this independent of the chosen path. ◻

Proof. Because \(q(\gamma\cdot s)=q(s)\), \[ q\bigl(T_i^\lambda(\gamma\cdot s)\bigr) \mathrel{=} \overline T_i^\lambda\bigl(q(\gamma\cdot s)\bigr) \mathrel{=} \overline T_i^\lambda\bigl(q(s)\bigr) \mathrel{=} q\bigl(T_i^\lambda(s)\bigr). \] So gauge-equivalent inputs induce the same repaired physical state. ◻

Proof. By Proposition 36, every repair step has quotient image determined only by the current orbit. Iterating, the quotient image of any repair sequence depends only on the initial orbit. Theorem 25 gives a unique terminal quotient normal form \(N_Q([s])\) from that orbit. This makes \(\overline N_\lambda\) well-defined. ◻

Proof. The quotient map satisfies \(q(\gamma\cdot s)=q(s)\). Applying \(\operatorname{Rep}_\lambda\) gives the displayed equality, and applying \(M\) gives the observable statement. ◻

Proof. Compatibility of the \(x_r\) gives \(\rho_{sr}(x_s)=x_r\). Normal-form naturality then gives \[ \rho_{sr}(n_s(x_s))=n_r(\rho_{sr}(x_s))=n_r(x_r), \] so the normal forms form a compatible inverse-limit family. Holonomy naturality gives the same calculation for \(h_s(x_s)\). Finite-stage schedule independence comes from Theorem 25; visible separation upgrades agreement of all cofinal finite projections to uniqueness of the inverse-limit class. A nonzero inverse-limit holonomy must have a nonzero projection at some finite stage by visible separation. ◻

Proof. The displayed inequalities are exactly the two components of the claimed product-readout bound. Exact naturality in Theorem 39 is the zero-defect case. Cofinal vanishing gives convergence of every fixed coarse cylinder value in the inverse-limit topology. ◻

Proof. Reflexivity, symmetry, and transitivity of \(\sim_\tau\) follow from the identity, inverse, and composition atlas laws. The canonical-map identity is the definition of the generated quotient. If \(\iota_s(x)=\iota_s(y)\), then \(y\) is obtained from \(x\) by a closed path; zero holonomy gives \(x=y\). The universal property follows by defining \(G([x])=G_s(x)\) and using interface compatibility along a connecting path. Applying this channel by channel gives feature descent. The metric formula is well defined by that descent. Nonnegativity and symmetry are channelwise, and the triangle inequality is Minkowski’s inequality applied to the weighted channel distances. Zero distance is exactly \(\equiv_F\); quotienting by that relation, or proving joint separation, supplies identity of indiscernibles. ◻

Proof. Item 1 is immediate from the definition of \(\mathcal A^\eta(P)\) and \(\omega_P^\eta\). Item 2 is the product-state form of the same fixed-cutoff MaxEnt branch. Item 3 follows from additivity of entropy for product ancillas, which cancels in the conditional mutual-information combination and therefore leaves both \(r_{\mathrm{FR}}\) and \(\delta^{\mathrm M}\) unchanged. Item 4 holds because the lifted repair maps do not act on the ancillary factors. Item 5 is immediate when the ancillas are neutral and carry no extra realized sector data. ◻

Proof. Propositions 36 and 37 fix the schedule-independent physical branch on the quotient, while Proposition 46 shows that inert ancillary refinements leave every OPH observable invariant. So the physical branch is fixed only modulo \(\sim_{\mathrm{OPH}}\), not at the level of one microscopic representative. ◻

Proof. Every finite Gibbs state is faithful, and so are its marginals. Expanding the four entropy terms inside the common tripartite trace gives the exact identity \[ I(A:D\mid B)_\rho =\operatorname{Tr}\!\left[\rho_{ABD}\, \mathbf J_\rho(A:D\mid B)\right]. \] Strong subadditivity gives nonnegativity. The boundary expansion, trace-norm duality, and the triangle inequality give \[ I(A_\delta:D_\delta\mid B_\delta)_{\rho_\ell} \le \|\mathbf J_{\rho_\ell}(A_\delta:D_\delta\mid B_\delta)\|_\infty \le \kappa |\partial C_\ell|_{\mathrm{UV}} e^{-(m_\ell-r_0)/\zeta}. \] Since \(\delta=m_\ell\ell\), the last expression equals \(\kappa e^{r_0/\zeta}|\partial C_\ell|_{\mathrm{UV}} e^{-\delta/(\zeta\ell)}\), which is the displayed bound. ◻

Proof. Taking logarithms of the theorem bound gives \[ \log I\le\log c+\log|\partial C_\ell|_{\mathrm{UV}}-\delta/\xi_\ell =\log c-\Gamma. \] This tends to \(-\infty\) under the first condition. The polynomial boundary estimate and the displayed schedule give \(\Gamma\ge\eta\log(\ell_0/\ell)-\log a\), which proves the explicit rate. The recovery estimate is Proposition 84 applied with the theorem envelope. ◻

Proof sketch. Item 1 is the Hilbertization of finite sets. For item 2, each recharting supplied by the declared datum is an invertible \(^*\)-isomorphism between finite-dimensional matrix algebras of equal dimension. After transporting all local overlap presentations to a chosen reference chart, each recharting becomes a \(^*\)-automorphism of the cut algebra, and every \(^*\)-automorphism of a finite-dimensional full matrix algebra is inner, hence implemented by a unitary on the cut Hilbert space. The automorphism input is consumed from the declared datum; bare interface projections supply no such map, as the countermodel in Definition 56 records. The subgroup generated by those unitaries has compact closure inside a finite-dimensional unitary group, which yields \(K_\Sigma\); without fixing the reference chart the declared rechartings with their composition law give the corresponding compact unitary groupoid. Item 3 is the usual projective-versus-central-extension lift for a central 2-cocycle. Item 4 records the lifted fixed-point presentation: chart-independent endomorphisms of the unreduced boundary data are exactly the fixed points of the derived boundary action, while the collar theorem uses the invariant-state realization and the sector-preserving algebra induced on the matched-cut subspace. If the defect is genuinely noncentral, the quotient is well defined, but the correct bookkeeping object is the crossed-module or higher-gauge data instead of an ordinary compact group. ◻

Proof. The reference multiresolution carrier and the fixed-cutoff regulator certificate of Ref.  present every regulator-scale chart by conjugating one fixed bare factor product with a unitary finite-depth circuit, and prove the refinement embeddings unital, injective, and \(^*\)-homomorphic. Unitary conjugations invert exactly and compose strictly, which yields items 1–3 of Definition 56 on the strict branch of the composition law. The port readout maps enter only through the visible interface and impose no further condition. ◻

Proof. By Proposition 57, one may realize the physical collar states as the \(\widehat K_\Sigma\)-invariant subspace of \(\tilde{\mathcal H}_{B_L}\otimes \tilde{\mathcal H}_{B_R}\). Complete reducibility of finite-dimensional unitary representations gives the displayed decomposition of \(\tilde{\mathcal H}_{B_L}\), and the right half-collar carries the inverse transport law across the same cut, hence the contragredient decomposition of \(\tilde{\mathcal H}_{B_R}\). Therefore \[ \tilde{\mathcal H}_{B_L}\otimes \tilde{\mathcal H}_{B_R} \cong \bigoplus_{\alpha,\beta} (W_\alpha\otimes W_\beta^*)\otimes (\mathcal H_{b_L^\alpha}\otimes \mathcal H_{b_R^\beta}). \] Taking \(\widehat K_\Sigma\)-invariants gives \[ \mathcal H_B \cong \bigoplus_{\alpha,\beta} (W_\alpha\otimes W_\beta^*)^{\widehat K_\Sigma}\otimes (\mathcal H_{b_L^\alpha}\otimes \mathcal H_{b_R^\beta}). \] By Schur’s lemma, \[ (W_\alpha\otimes W_\beta^*)^{\widehat K_\Sigma} \cong \begin{cases} \mathbb C, & \alpha=\beta,\\ 0, & \alpha\neq \beta, \end{cases} \] which yields the displayed block decomposition. The sector-preserving collar algebra therefore has the displayed block-diagonal form, so its center is generated by the block projectors. Within each block, observables from \(A\cup B\) act only on the left factor and observables from \(B\cup D\) act only on the right factor. If the state is exact Markov, the HJPW structure theorem gives a blockwise factorized normal form over some decomposition \(\mathcal H_B\cong\bigoplus_j \mathcal H_{\hat b_L^j}\otimes\mathcal H_{\hat b_R^j}\) of the conditioning system. That conclusion is existential: the decomposition it produces is state-dependent and need not agree with the preselected edge-center factors \(b_L^\alpha,b_R^\alpha\) constructed above (Remark 61). The Markov-split alignment hypothesis of Definition 60 states exactly that the HJPW decomposition can be chosen to be the edge-center decomposition; under that hypothesis the displayed normal form over \(b_L^\alpha,b_R^\alpha\) follows. The same formula is used in the idealized recoverability limit when exact identities are taken literally, with alignment carried as part of the limit hypothesis. ◻

Proof. \(1\Leftrightarrow 4\): mutual information vanishes iff the state is a product across the named cut, so item 4 says precisely that \(\rho\) is block-diagonal with each block a product across \((Ab_L^\alpha):(b_R^\alpha D)\), which is the aligned normal form. No faithfulness is used.

\(1\Rightarrow 2\): for the aligned normal form, \(\log\rho=\bigoplus_\alpha\bigl[(\log p_\alpha)P_\alpha+\log\rho^{(\alpha)}_{Ab_L^\alpha}\otimes\mathbf 1+\mathbf 1\otimes\log\rho^{(\alpha)}_{b_R^\alpha D}\bigr]\), and the central term lies in \(\mathcal M_L\cap\mathcal M_R\).

\(2\Rightarrow 1\): write \(\log\rho=K_L+K_R\) with \(K_L\in\mathcal M_L\), \(K_R\in\mathcal M_R\); these commute, are block-diagonal, and have blockwise form \(K_L=h^\alpha_L\otimes\mathbf 1\), \(K_R=\mathbf 1\otimes h^\alpha_R\). Hence \(\rho=\bigoplus_\alpha e^{h^\alpha_L}\otimes e^{h^\alpha_R}\), which after normalization is the aligned form.

\(1\Rightarrow 3\): define the blockwise slice map \[ E_L(x):=\bigoplus_\alpha\Bigl[\bigl(\mathrm{id}_{Ab_L^\alpha}\otimes\omega^{(\alpha)}_{R}\bigr)(P_\alpha xP_\alpha)\Bigr]\otimes\mathbf 1_{b_R^\alpha D}, \qquad \omega^{(\alpha)}_{R}:=\mathrm{Tr}\bigl[\;\cdot\;\rho^{(\alpha)}_{b_R^\alpha D}\bigr]. \] This is a unital completely positive idempotent onto \(\mathcal M_L\) with the bimodule property, and \(\rho\circ E_L=\rho\) follows from the blockwise product form. The analogous \(E_R\) exists, and \(E_LE_R=E_RE_L\) is the expectation onto \(\bigoplus_\alpha\mathbb C\,P_\alpha\) determined by the block weights, which is the asserted commuting square.

\(3\Rightarrow 2\): by Takesaki’s theorem (see also ), a \(\rho\)-preserving conditional expectation onto \(\mathcal M_L\) exists iff the modular group \(\sigma^\rho_t=\mathrm{Ad}\,\rho^{it}\) satisfies \(\sigma^\rho_t(\mathcal M_L)=\mathcal M_L\) for all \(t\). The flow \(\sigma^\rho_t\) then permutes the finitely many minimal projectors of \(Z(\mathcal M_L)=\bigoplus_\alpha\mathbb C\,P_\alpha\) and is norm-continuous with \(\sigma^\rho_0=\mathrm{id}\), so it fixes each \(P_\alpha\); hence \([\rho,P_\alpha]=0\) and \(\rho=\bigoplus_\alpha p_\alpha\rho^{(\alpha)}\). Fix a block and regard \(\rho^{(\alpha)}\) as a faithful state on \(\mathcal H_{Ab_L^\alpha}\otimes\mathcal H_{b_R^\alpha D}\). Differentiating \(\sigma^\rho_t(x\otimes\mathbf 1)\in\mathcal B(\mathcal H_{Ab_L^\alpha})\otimes\mathbf 1\) at \(t=0\) gives \([\log\rho^{(\alpha)},\,x\otimes\mathbf 1]\in\mathcal B(\mathcal H_{Ab_L^\alpha})\otimes\mathbf 1\) for every \(x\). Expand \(\log\rho^{(\alpha)}=\sum_i a_i\otimes b_i\) with \(\{b_i\}\) linearly independent and \(b_0=\mathbf 1\); then \([\log\rho^{(\alpha)},x\otimes\mathbf 1]=\sum_i[a_i,x]\otimes b_i\), and membership in \(\mathcal B(\mathcal H_{Ab_L^\alpha})\otimes\mathbf 1\) forces \([a_i,x]=0\) for all \(x\) and all \(i\neq0\). Hence \(a_i\in\mathbb C\mathbf 1\) for \(i\neq0\), so \(\log\rho^{(\alpha)}=a_0\otimes\mathbf 1+\mathbf 1\otimes c\) for some self-adjoint \(c\), which is the blockwise modular splitting; summing blocks gives item 2.

Finally, item 1 implies \(I(A:D\mid B)_\rho=0\) by the direct blockwise computation of Theorem 59, and the Bell-pair state violates item 4 by monotonicity of mutual information under partial trace. ◻

Proof. Item 1 is the commutation theorem plus Schur’s lemma: the commutant of the unitary action \(\mathbf 1_A\otimes\bigl(\bigoplus_\alpha\pi_\alpha\otimes\mathbf 1_{b_L^\alpha}\bigr)\) is \(\bigoplus_\alpha \mathbf 1_{W_\alpha}\otimes\mathcal B(\mathcal H_A\otimes\mathcal H_{b_L^\alpha})\) (up to the obvious factor reordering), because the \(\pi_\alpha\) are pairwise inequivalent irreducibles, so no cross-isotypic intertwiners exist and within each isotypic component the commutant of \(\pi_\alpha\otimes\mathbf 1\) is \(\mathbf 1_{W_\alpha}\otimes(\text{full multiplicity algebra})\). Tensoring with \(\mathbf 1\) on the right half-collar commutes with the diagonal action, so the invariant subspace is preserved, and on \((W_\alpha\otimes W_\alpha^*)^{\widehat K_\Sigma}\otimes(\mathcal H_{b_L^\alpha}\otimes\mathcal H_{b_R^\alpha})\cong\mathcal H_{b_L^\alpha}\otimes\mathcal H_{b_R^\alpha}\) the induced action is \(x_L^\alpha\otimes\mathbf 1\). Item 2: central elements of the group algebra act in any irreducible \(\pi_\alpha\) by scalars \(\chi_\alpha(\tilde z)\), so the left action descends to \(\bigoplus_\alpha\chi_\alpha(\tilde z)P_\alpha\). For a Casimir/flux function the contragredient module \(W_\alpha^*\) carries the same Casimir eigenvalue as \(W_\alpha\), so on the \(\alpha\)-block of the invariant subspace the right action reduces to multiplication by the same scalar, which is the displayed central operator. ◻

Proof. By Lemma 64, \(\widetilde H_{AL}\) descends into \(\mathcal M_L\), \(\widetilde H_{RD}\) into \(\mathcal M_R\), and \(\widetilde H_\Sigma\) into \(\bigoplus_\alpha\mathbb C\,P_\alpha\subset\mathcal M_L\cap\mathcal M_R\), giving item 1. Hence \(\log\omega=-H_{\mathrm{eff}}-\log Z\,\mathbf 1\in\mathcal M_L+\mathcal M_R\), which is item 2 of Proposition 62, so \(\omega\in\mathfrak M^{\mathrm{EC}}_{A:B:D}\). Item 3 follows because every EC-aligned state is exact Markov (Proposition 62). For the final claims: the electric interface energy of a lattice-gauge regulator is a function of the quadratic Casimir of the boundary action, hence central by Lemma 64, item 2. The Bell-pair exclusion is immediate: every Gibbs state of a central-interface package lies in \(\mathfrak M^{\mathrm{EC}}_{A:B:D}\) by item 2 (and is faithful, while the Bell-pair state is pure on \(A\cup B_R\)), whereas every state in \(\mathfrak M^{\mathrm{EC}}_{A:B:D}\) has \(I(A:B_R)=0\) and the Bell-pair state has \(I(A:B_R)=2\ln 2\). ◻

Proof. Take the \(\mathbb Z_2\) boundary-action collar package of Theorem 65 with the one-sided invariant terms kept and the flux-function interface energy replaced by a group-averaged \(b_L\)-\(b_R\) density: group averaging makes it \(\widehat K_\Sigma\)-invariant, and its action on the multiplicity factors keeps it outside \(\pi_L\bigl(Z(C^*(\widehat K_\Sigma))\bigr)\). Gauge invariance, self-adjointness, bounded support, and linear independence are direct checks, and Lemma 10, item (i), then gives unique multipliers. Overlap consistency holds because the patch states are restrictions of one global Gibbs state and agree on shared subalgebras. The classical layer is the layered functional boundary carrier of Definition 30 on the flux-sector labels: every accepted transaction strictly descends the declared lexicographic measure, terminal states are exactly the consistent boundary fiber, and exhaustive enumeration of schedules settles every initial state on the rooted extension, which is Theorem 25 together with Theorem 28 on this carrier. The repair layer reads interface sector data only, so no transaction distinguishes the central package from the noncentral one, and the settled net realizes the noncentral package with no enabled move. For the refinement claim, the fine stage is the same package tensored with additional cells in the unconstrained MaxEnt state; the coarse-graining channel is the partial trace over those cells, the coarse-grained realized state equals the coarse Gibbs state at the same multipliers, and the closure defect vanishes identically, so the realized branch is exactly closed and carries the coupling coherently to every stage. The splitting-defect linearity is exact because \(\log\omega\) equals \(-H_{\mathrm{eff}}\) up to a multiple of \(\mathbf 1\) and the one-sided projection is linear; the CMI vanishes at \(g=0\), is smooth and nonnegative, and is therefore \(O(g^2)\). Misalignment is Proposition 62: the cross-cut component keeps \(\log\omega\) outside \(\mathcal M_L+\mathcal M_R\), and the blockwise mutual information across the cut is bounded below. Machine receipts for every checked condition, including the exhaustive schedule enumeration and the scaling separation, are in the collar-alignment calculation (14 tests). ◻

Proof sketch. On pair overlaps, every regulator-scale recharting supplied by the declared datum is an invertible \(^*\)-isomorphism of finite-dimensional matrix algebras and, after transport to the reference chart, a \(^*\)-automorphism, therefore inner, so it is implemented by a unitary on the cut Hilbert space. On triple overlaps, genuinely noncentral failure of strict composition is recorded by unitary associators instead of scalars. These \(1\)- and \(2\)-morphisms form a compact unitary \(2\)-group of rechartings. Skeletal strictification of a compact \(2\)-group is equivalent to crossed-module data, which yields the displayed compact \(\mathbb K_\Sigma\). Because the nerve is finite, \(C^1(N_\Sigma,H_\Sigma)\) and \(C^0(N_\Sigma,G_\Sigma)\) are finite products of compact groups, hence \(\mathcal T_\Sigma\) is compact. The displayed formulas are the standard crossed-module coboundary action, and the orbit class is exactly the nonabelian Čech \(2\)-class of the defect data; see for an explicit higher-lattice-gauge realization of the same crossed-module structures. ◻

Proof sketch. Because \(\mathcal T_\Sigma\) is compact and the half-collar spaces are finite-dimensional, both \(\widetilde{\mathcal H}_{B_L}\) and \(\widetilde{\mathcal H}_{B_R}\) split orthogonally into irreducible unitary \(\mathcal T_\Sigma\)-modules with finite multiplicity. The right side sees inverse transport across the same cut, hence the contragredient module. Therefore \[ \widetilde{\mathcal H}_{B_L}\otimes \widetilde{\mathcal H}_{B_R} \cong \bigoplus_{\lambda,\mu} (W_\lambda\otimes W_\mu^*)\otimes (\mathcal H_{b_L^\lambda}\otimes \mathcal H_{b_R^\mu}). \] Taking \(\mathcal T_\Sigma\)-invariants leaves \[ (W_\lambda\otimes W_\mu^*)^{\mathcal T_\Sigma} \cong \mathrm{Hom}_{\mathcal T_\Sigma}(W_\mu,W_\lambda) \cong \begin{cases} \mathbb C, & \lambda=\mu,\\ 0, & \lambda\neq\mu, \end{cases} \] by Schur’s lemma. This yields the matched block decomposition and the center formula. ◻

Proof sketch. Theorem 69 shows that the higher-gauge collar algebra is a finite direct sum of type-I tensor blocks. HJPW depends only on that finite-dimensional algebraic structure, not on whether the block label arose from an ordinary compact group, a central extension, or a crossed-module gauge system; as in the ordinary branch, HJPW supplies a state-dependent decomposition, and the identification of its factors with the preselected higher-gauge edge factors is the carried Markov-split alignment hypothesis. Likewise, the Fawzi–Renner remainder and the fixed-collar Markov modulus are finite-dimensional entropy and recovery statements and are blind to the origin of the block label. ◻

Proof sketch. Crossed-module coboundaries change representatives while preserving the full orbit \(q_\Sigma\). By definition, \(o^{(2)}_\Sigma=0\) exactly when that orbit meets the strict subset \((g^{\mathrm{str}},1)\). The surviving edge data then form a genuine \(G_\Sigma\)-valued \(1\)-cocycle. The natural map \(\iota_\Sigma\) need not be injective: an \(H_\Sigma\)-valued edge change can alter the ordinary \(G_\Sigma\)-holonomy through \(\partial_\Sigma\). Therefore no individual strict representative’s holonomy is assigned to \(q_\Sigma\). What is invariant is whether the orbit contains a strict representative whose holonomy is trivial on the chosen sector, and associator strictification alone does not imply that existence. ◻

Proof. For a path \(p=(i_0\to\cdots\to i_m)\), Definition 72 gives the transport functor by composing the finite-dimensional recharting implementers. This construction uses only the derived overlap unitary transition system of Proposition 57; no transportability assumption is invoked.

On the ordinary branch, strict triple-overlap coherence says that elementary replacements of the form \((i\to j\to k)\leadsto(i\to k)\) do not change the composite recharting. Any two paths with the same endpoints in the finite nerve differ, after inserting or deleting immediate backtracks, by a finite sequence of such elementary moves together with closed loop moves. Backtracks contribute the identity because \(U_{ji}=U_{ij}^{-1}\). Therefore the only possible path dependence is the ordinary loop holonomy. The transport is strictly path-independent exactly when that loop holonomy acts trivially on the collar-sector block.

On the central branch, the same elementary triangle replacement changes the composite by the central multiplier \(z_{ijk}\). For a finite sequence of elementary moves between two paths, the total discrepancy is the product of the corresponding \(z\)’s over the filling \(2\)-chain. The fixed-coefficient and trivial-action hypotheses identify every multiplier in \(Z_\Sigma\); the quadruple-overlap coherence identity is then precisely the untwisted Čech cocycle identity, so this product depends only on the class \([z]_\Sigma\). If \([z]_\Sigma=0\), choose a central \(1\)-cochain \(a_{ij}\) with \(z_{ijk}=a_{ij}a_{jk}a_{ik}^{-1}\). Replacing \(U_{ij}\) by \(a_{ij}^{-1}U_{ij}\) kills the central multiplier on every triangle, so all elementary path moves preserve the transport functor. The ordinary argument then gives strict path independence exactly when every remaining closed-loop holonomy of this strictified \(1\)-cocycle acts trivially on the sector block. Conversely, strict path-independent transport makes every elementary triangular comparison trivial after an allowed recharting gauge change, so \(z\) is a coboundary, and it also makes every closed-loop action trivial. This proves both necessary conditions.

On the genuinely noncentral branch, elementary triangle comparisons are non-scalar \(H_\Sigma\)-valued \(2\)-morphisms \(h_{ijk}\), and the consistency of different fillings of a path homotopy is the crossed-module cocycle identity of Theorem 68. A local higher-gauge change \((\eta,u)\) changes representatives by the crossed-module coboundary action displayed there, while preserving the full class \(q_\Sigma\) and the strictifiability property \(o^{(2)}_\Sigma\) by Theorem 71. If \(o^{(2)}_\Sigma=0\), the data are equivalent to at least one strict representative: the \(h_{ijk}\) can be gauged away and \(g_{ij}g_{jk}=g_{ik}\) holds. Each such \(G_\Sigma\)-valued \(1\)-cocycle defines an ordinary path-composite transport, which is endpoint-only exactly when its represented loop holonomy is trivial. Hence ordinary strict transport exists precisely when at least one allowed strict representative passes that test. Conversely, strict ordinary path-independent transport gives a representative in which every triangular \(2\)-comparison is the identity, hence \(o^{(2)}_\Sigma=0\), and every closed-loop action on the sector is trivial. If \(o^{(2)}_\Sigma\ne0\), Theorem 71 says the higher defect is not removable, so no strict ordinary transport functor can exist; if \(o^{(2)}_\Sigma=0\) alone, only existence of a strict representative has been proved. ◻

Proof. By edge-center completion, every seed charge visible on one reference collar is represented by a minimal central summand \(P_\alpha\) together with an irreducible boundary carrier \(W_\alpha\) of \(\widehat K_{\Sigma,r}\). Since \(\widehat K_{\Sigma,r}\) is compact and acts unitarily on finite-dimensional collar data, its finite-dimensional representations are completely reducible. The replete additive Karoubi tensor subcategory generated by the \(W_\alpha\)’s and their conjugates is therefore semisimple, with finite-dimensional objects and Hom spaces. Complete reducibility does not imply that its simple skeleton is finite, and no such finiteness is used.

Transport between collars is the path-composite construction of Theorem 73. For every closed loop \(\ell\), the common choice \(U^{\mathrm{str}}_r\) acts as the identity on each seed block. It therefore acts as the identity on direct sums, as \(\mathbf 1\otimes\mathbf 1\) on tensor products, as the conjugate identity on duals, and by restriction as the identity on every invariant subobject. Thus every object in the generated category has path-independent transport using the same representative, and those transport identifications are monoidal. In particular, transport of \(P_\alpha\), \(W_\alpha\), and their intertwiners is independent of the overlap path. Hence the Hom spaces defined after moving all localizations to a reference collar do not depend on the chosen path or reference collar, up to canonical transported unitary identification.

For two objects \(X\) and \(Y\), take the diagonal tensor product \(X\otimes Y\) of \(\widehat K_{\Sigma,r}\)-modules. By definition the generated Karoubi category contains this tensor word and every irreducible summand selected by an idempotent in its intertwiner algebra, so the product stays inside the category without requiring those summands to occur in the center of the original one-collar algebra. The trivial one-dimensional module is the tensor unit. Associativity is implemented by the standard finite-dimensional tensor associator, whose pentagon identity is the equality of the two rebracketings of a fourfold tensor product. When the finite tensor-realization receipt is present, these tensor words, idempotents, and rebracketings are represented by concatenated collars and union-collar gluing.

Duals are the conjugate representations \(W_\alpha^*\); the finite tensor-realization receipt identifies this with reversing collar orientation and swapping the half-collar boundary actions. Evaluation and coevaluation are the standard \(\widehat K_{\Sigma,r}\)-invariant pairings. Their zig-zag identities are the usual finite-dimensional duality identities, equivalently cap/cup cancellation for collar gluing.

The \(^*\)-structure is the operator adjoint on the lifted finite-dimensional carriers. Each \(\mathrm{Hom}_r(X,Y)\) is a closed subspace of operators between finite-dimensional Hilbert spaces, adjoints are again \(\widehat K_{\Sigma,r}\)-intertwiners, composition is operator composition, and the norm is the operator norm. Thus \(\|f^*f\|=\|f\|^2\), direct sums and subobjects are implemented by orthogonal projections, and the category is a \(C^*\)-category.

The canonical unitary flip \(c_{X,Y}:X\otimes_rY\to Y\otimes_rX\) is an intertwiner. Its naturality, hexagon identities, and \(c_{Y,X}c_{X,Y}=\mathrm{id}\) are the standard representation identities, so the braiding is symmetric. On a finite tensor-realization branch this flip is represented physically by spacelike exchange of the codimension-two collar supports; double exchange is isotopic to the identity on the bosonic \(3+1\)-dimensional branch. Fermionic signs, spinorial matter, and chirality are not part of this bosonic internal-gauge category; they belong to the later super-Tannakian or matter-sector lift. Forgetting the group action and retaining the carrier Hilbert space and the same linear maps is faithful, \(^*\)-preserving, and symmetric strong monoidal, which gives the displayed \(F_r\). ◻

Proof sketch. The additional terms are local projector constraints supported on finitely many collar cells, so they preserve bounded support and quasi-locality. In the standard higher-gauge projector realization they commute, so the same finite-constraint MaxEnt/Lieb–Robinson bookkeeping used on the ordinary branch continues to apply. ◻

Proof. The full state space on a fixed finite-dimensional Hilbert space is compact in trace norm. Conditional mutual information is continuous there because von Neumann entropy is continuous in finite dimension. Hence \(\mathfrak M_{A:B:D}\) is closed, so it is compact as a closed subset of a compact space.

Suppose \(\delta^{\mathrm M}_{A:B:D}(\varepsilon)\not\to 0\). Then there exist \(\eta>0\), a sequence \(\varepsilon_n\downarrow 0\), and states \(\rho^{(n)}\) with \(I(A:D\mid B)_{\rho^{(n)}}\le \varepsilon_n\) but \(d_{\mathrm M}(\rho^{(n)})\ge \eta\) for all \(n\). By compactness, pass to a trace-norm convergent subsequence \(\rho^{(n_k)}\to \rho^\ast\). Continuity of conditional mutual information gives \[ I(A:D\mid B)_{\rho^\ast} \mathrel{=} \lim_{k\to\infty} I(A:D\mid B)_{\rho^{(n_k)}} \mathrel{=} 0, \] so \(\rho^\ast\in \mathfrak M_{A:B:D}\). But then \[ d_{\mathrm M}(\rho^{(n_k)}) \le \|\rho^{(n_k)}-\rho^\ast\|_1 \longrightarrow 0, \] contradicting \(d_{\mathrm M}(\rho^{(n_k)})\ge \eta\). Thus \(\delta^{\mathrm M}_{A:B:D}(\varepsilon)\to 0\). The final statement follows by choosing \(\sigma^{(n)}\in\mathfrak M_{A:B:D}\) within \(\delta^{\mathrm M}_{A:B:D}(\varepsilon_n)+1/n\) of \(\rho^{(n)}\). ◻

Proof. On \(\mathcal S_{\lambda_\ast}\) the von Neumann entropy terms are real analytic functions of the matrix entries, hence \[ F(\rho):=I(A:D\mid B)_\rho \] is a nonnegative real analytic function. By equality in strong subadditivity, or equivalently the HJPW structure theorem, the zero set of \(F\) is precisely \(\mathfrak M_{A:B:D}^{\lambda_\ast}\). The compact-set Lojasiewicz inequality for a real analytic nonnegative function gives constants \(c>0\) and \(q>0\) on this fixed compact collar class such that \[ F(\rho)\ge c\,d_{\mathrm M}^{\lambda_\ast}(\rho)^q . \] Taking \(C=c^{-1/q}\) and \(\theta=1/q\) gives the displayed estimate. ◻

Proof. The fidelity lower bound and the Fuchs–van de Graaf inequality give \[ \|\rho_{ABD}-\rho^{\mathrm{rec}}_{ABD}\|_1 \le 2\sqrt{1-F(\rho_{ABD},\rho^{\mathrm{rec}}_{ABD})^2} \le 2\sqrt{1-e^{-\varepsilon}} \mathrel{=} r_{\mathrm{FR}}(\varepsilon). \] The estimate \(r_{\mathrm{FR}}(\varepsilon)\le 2\sqrt{\varepsilon}\) follows from \(1-e^{-x}\le x\). Contractivity of trace norm under CPTP maps gives the transported bound, and the observable estimate is the duality between trace norm and operator norm . The bound is constructive and dimension-free. What it controls is closeness to a recovered comparison state; no claim is made that \(\rho^{\mathrm{rec}}_{ABD}\) is itself exact Markov. ◻

Proof. For item 1, blockwise factorization gives \[ \mathrm{Tr}(X\sigma'_{AB D'}) \mathrel{=} \sum_\alpha p_\alpha \mathrm{Tr}\!\left(X_\alpha\sigma^{(\alpha)}_{A b_L^\alpha}\right) \mathrm{Tr}\!\left(\tau^{(\alpha)}_{b_R^\alpha D'}\right). \] Here \(X_\alpha\) denotes the \(\alpha\)-block of \(X\) in \(\mathcal A_{A b_L}\). Each \(\tau^{(\alpha)}\) is normalized, so the second factor is \(1\). The same computation for \(\sigma_{ABD}\) gives the same value because the original right factor is likewise normalized.

For item 2, \(\sigma_{\mathrm{al}}\) is EC-aligned by hypothesis, so item 1 applies to it: \[ \mathrm{Tr}(X\sigma'_{\mathrm{al}})=\mathrm{Tr}(X\sigma_{\mathrm{al}}). \] Therefore \[ \left| \mathrm{Tr}(X\rho_{ABD})-\mathrm{Tr}(X\sigma'_{\mathrm{al}}) \right| \mathrel{=} \left| \mathrm{Tr}\!\left[X(\rho_{ABD}-\sigma_{\mathrm{al}})\right] \right| \le \|X\|_\infty\,\|\rho_{ABD}-\sigma_{\mathrm{al}}\|_1 \le \|X\|_\infty\,\delta_{\mathrm{al}}. \]  ◻

Proof. By monotonicity of trace distance under partial trace, \[ \|\rho_X-(\sigma_\varepsilon)_X\|_1 \le \|\rho-\sigma_\varepsilon\|_1 \le \delta^{\mathrm M}_{A:B:D}(\varepsilon) \] for each \(X\in\{ABD,AB,BD,B\}\). On the interval \([\lambda_\ast,1]\), the function \(\log x\) has derivative bounded by \(\lambda_\ast^{-1}\). By the standard integral representation of the operator logarithm, this implies the operator-norm Lipschitz bound \[ \|K_X(\rho)-K_X(\sigma_\varepsilon)\|_\infty \mathrel{=} \|\log (\sigma_\varepsilon)_X-\log \rho_X\|_\infty \le \lambda_\ast^{-1}\,\|\rho_X-(\sigma_\varepsilon)_X\|_\infty \le \lambda_\ast^{-1}\,\delta^{\mathrm M}_{A:B:D}(\varepsilon). \] Summing the four region contributions gives \[ \|\Delta K(\rho)-\Delta K(\sigma_\varepsilon)\|_\infty \le 4\lambda_\ast^{-1}\,\delta^{\mathrm M}_{A:B:D}(\varepsilon). \] For \(\sigma_\varepsilon\in \mathfrak M_{A:B:D}\), the exact HJPW block factorization implies that \(\Delta K(\sigma_\varepsilon)\) is blockwise constant on the blocks of the state’s own HJPW decomposition; when \(\sigma_\varepsilon\in\mathfrak M^{\mathrm{EC}}_{A:B:D}\) those blocks are the edge-center blocks and \(\Delta K(\sigma_\varepsilon)\in Z(\mathcal A_{\mathrm{EC}}(B))\), so the same bound controls all matrix elements relative to the EC-central exact Markov modular-additivity value. ◻

Proof. Induct over the expression tree defining \(O\). Sums and products are controlled by triangle inequalities and submultiplicativity on the bounded operator class. Commutators satisfy \[ \|[A,B]-[A',B']\| \le \|A-A'\|\,\|B\|+\|A'\|\,\|B-B'\| +\|B-B'\|\,\|A-A'\|, \] with all norms bounded on the fixed calculation. Bounded functional calculus on a fixed spectral interval is uniformly continuous, and it is Lipschitz for the logarithmic comparisons when the floor \(\lambda_\ast\) is present. For a bounded-time modular transport generated by \(K\) and \(K'\), Duhamel’s formula gives \[ \|e^{itK}Xe^{-itK}-e^{itK'}Xe^{-itK'}\| \le 2|t|\,\|X\|\,\|K-K'\|+O(\|K-K'\|^2) \] uniformly for \(t\) in the chosen compact interval. When \(K-K'\) is not an unregularized full-algebra bounded operator, Theorem 115 supplies the support-visible matrix-element replacement and its remainder \(\eta^{\mathrm{reg}}_j\). Composing these finitely many continuity estimates gives the displayed polynomial modulus. ◻

Proof. The Fawzi–Renner term \(r_{\mathrm{FR}}\) is the only one-shot constructive recoverability estimate used here, and it compares the state to a recovered comparison state instead of the exact Markov set. The exact-Markov replacement modulus \(\delta^{\mathrm M}\) is defined after pullback to one fixed finite collar model; Theorem 83 gives a rate only on a fixed faithful collar class. Logarithmic comparison uses the local floor \(\lambda_\ast\), while the BW branch avoids a refinement-uniform full-algebra floor by using the regularized support-visible transport theorem. The propagation theorem above then composes only those collar-local estimates. ◻

Proof. The entropy of a direct sum is \[ S\!\left(\bigoplus_\alpha p_\alpha \sigma_\alpha\right) \mathrel{=} H(p_\alpha)+\sum_\alpha p_\alpha S(\sigma_\alpha). \] Apply this identity with \[ \sigma_\alpha= \rho^{(\alpha)}_{\mathrm{bulk},C}\otimes \frac{\mathbf 1^{(\alpha)}_{\mathrm{edge}}}{d_\alpha}. \] Since \[ S(\sigma_\alpha)=S(\rho^{(\alpha)}_{\mathrm{bulk},C})+\log d_\alpha, \] the first statement follows. ◻

Proof. For the lifted product presentation \(R=R_3\boxtimes R_2\boxtimes q\) of a sector on the quotient branch, \[ d_R=d_{R_3}d_{R_2}d_q. \] Every irreducible \(\mathrm{U}(1)\) representation is one-dimensional, so \(d_q=1\). Therefore \[ \log d_R \mathrel{=} \log d_{R_3}+\log d_{R_2}+\log d_q \mathrel{=} \log d_{R_3}+\log d_{R_2}. \] This proves the sector identity \[ \bigl(L_C\bigr)\!\mid_R=L_C^{(3)}+L_C^{(2)}. \] On the product heat-kernel branch, expectation values add, so \[ \bar{\ell}_{\mathrm{shared}} \mathrel{=} \langle L_C\rangle \mathrel{=} \bar{\ell}_{\mathrm{SU(3)}}(t_{3,\mathrm{run}}) + \bar{\ell}_{\mathrm{SU(2)}}(t_{2,\mathrm{run}}). \] If the same branch satisfies the D10 pixel law, the right-hand side equals \(P/4\). ◻

Proof. The OPH scale-readback map supplies \(\ell_\star^2\) before the Newton area-law readout is evaluated. By Proposition 93, \[ G_{\mathrm{geom}} \mathrel{=} \frac{a_{\mathrm{cell}}}{4(P/4)} \mathrel{=} \frac{a_{\mathrm{cell}}}{P}. \] Using the local cell-area reading \(a_{\mathrm{cell}}=P\ell_\star^2\), this becomes \[ G_{\mathrm{geom}}=\ell_\star^2. \] The SI display relation is \[ \ell_\star^2=\frac{\hbar G_{\mathrm{SI}}}{c^3}, \] so \[ G_{\mathrm{SI}} \mathrel{=} \frac{c^3\ell_\star^2}{\hbar}. \] The following formulas are dimensional bookkeeping on the same declared local branch. Because \(P\) is dimensionless and \(a_{\mathrm{cell}}\) is the only local microscopic area datum, every local length readout is \(\sqrt{a_{\mathrm{cell}}}\) times a dimensionless branch quantity. The structural Lorentz output \(c\) converts that same local ruler to seconds, while the familiar-unit display constants \(\hbar\) and \(k_B\) convert the inverse local ruler to energy/mass rows and to Kelvin rows. ◻

Proof. Items (i) and (ii) follow from the generated-subalgebra definition. Item (iii) is the defining property of a conditional expectation. The theorem makes no factoriality or ideal claim about overlap-trivial observables. ◻

Proof. First, a round cap and its oriented normal determine the cap but not a cap-preserving one-parameter subgroup. After conjugating the cap to the disk, its orientation-preserving stabilizer is \(\mathrm{PSU}(1,1)\), which contains elliptic, hyperbolic, and parabolic one-parameter subgroups. The ordered boundary frame is the extra datum selecting the BW axis and direction.

Second, a finite cap-ID set cannot carry a nontrivial continuous boost as permutations. Every continuous homomorphism \(\mathbb R\to\operatorname{Sym}(\mathcal C_r)\) is constant because \(\mathbb R\) is connected and \(\operatorname{Sym}(\mathcal C_r)\) is discrete. A finite simulator may sample and project a continuous cap flow, but the projection must carry mesh error.

Third, modular KMS alone does not pick \(2\pi\). If \(\sigma_t^\omega\) is the modular group, then \(\alpha_s^{(\kappa)}=\sigma_{s/\kappa}^\omega\) makes \(\omega\) a \(\kappa\)-KMS state for every \(\kappa>0\). The value \(2\pi\) is fixed only after the geometric parameter \(s\) has been normalized independently and the state is shown to be \(2\pi\)-KMS for that geometric flow.

Fourth, pointwise weak matrix convergence does not preserve automorphisms: on \(\ell^2(\mathbb Z)\), the bilateral shifts \(S^n\) are unitary but converge weakly to \(0\). This is a state-side countermodel. The quadratic mixed-GNS Cauchy residual and inverse-time control that exclude it belong to \(\mathsf{MGNS\text{-}1}\), not to the finite cap-normal certificate. ◻

Proof. The mixed-GNS Cauchy residual and the exact reference tower produce a unique strongly continuous unitary implementation of the limiting modular automorphism group on the support-visible GNS direct limit. Support covariance and support-order separation make the modular image of every test-cap algebra read as a unique cap-support flow, while the approximate identity, inverse, and group-law residuals pass to the limit.

Cap-normal compactness gives round-cap limits, and support-order reflection preserves cap inclusion. Held-out complex cross-ratio convergence on dense point meshes, with conditioned anchors and an orientation witness, forces every support-flow cluster limit to lie in \(\mathrm{PSL}(2,\mathbb C)\). Since the flow preserves the anchor cap and the ordered boundary frame, framed-cap rigidity gives \[ f_t^C=\lambda_{\widehat C}(\kappa_C t) \] for a unique \(\kappa_C>0\).

The geometric strip residual, normal-family bound, and refinement-compatible geometric matrix elements pass the \(2\pi\)-KMS condition to the limiting independently normalized geometric flow. KMS uniqueness for a faithful normal state gives \[ \sigma_t^{\omega_\infty^{\mathrm{geo},C}} =\alpha_{\lambda_{\widehat C}(2\pi t)}. \] The wrong-\(\beta\) gap excludes any fixed normalization separated from \(2\pi\), while nontriviality excludes the only exact multiple-temperature exception. Since every convergent subnet has the same BW limit, compactness upgrades subnet convergence to convergence of the cofinal branch. The type-I generator statement follows because two inner implementations of the same automorphism differ by a central self-adjoint term. ◻

Proof. For each fixed local collar model, the finite-stage state spaces are weak-\(*\) compact. The transported marginal family is projectively compatible up to the refinement-equivalence certificate supplied by the consensus package, so the usual diagonal subnet argument gives compatible local weak-\(*\) limits on the cap-local test family. Theorem 115 controls the regularized modular matrix elements on every bounded support-visible collar observable under the stated cutoff schedule. Passing to the GNS representation of the glued state quotients the null directions and leaves a faithful representation on the local support. This gives the displayed support-visible cap pair and its modular automorphism group. ◻

Proof. The group law gives inverse region map \(f_{-t}^C\). The displayed net-covariance identity then intertwines that action with the modular automorphism group. The identity is an independent receipt. A union of transported operator supports need not be a region and does not prove it. ◻

Proof. For each fixed \(m\), Proposition 82 gives exact Markov replacements within \(\delta^{\mathrm M}_{m}(\varepsilon_{n,m})+o(1)\) of \(R_{n\to m}\rho_n\). The exact Markov sets \(\mathfrak M_m\) are compact on fixed collars, so a diagonal subnet has limits on every fixed \(m\). Restriction maps are continuous and, by hypothesis, commute with the physical quotient and normal form up to the same refinement-equivalence certificate used by the cap-local test family. Therefore the restriction of the \(m\)-limit to a smaller fixed collar \(k\) agrees with the \(k\)-limit. Pulling the chosen approximants along the diagonal subnet gives the displayed asymptotic compatibility. ◻

Proof. Choose the Möbius map \(h_{\widehat C}\) of Definition 97. The orientation-preserving conformal maps of the upper half-plane preserving \(0\) and \(\infty\) individually are exactly \(z\mapsto a z\), \(a>0\). For a continuous one-parameter group, \(a(t+s)=a(t)a(s)\), so \(a(t)=e^{-\kappa t}\) after the declared frame orientation fixes the sign. Transporting this subgroup back to \(C\) gives \(\lambda_{\widehat C}(\kappa t)\). Uniqueness of \(\kappa\) follows from nontriviality of the dilation subgroup. The cap normal alone is not enough: it fixes the cap side, not the ordered boundary fixed points. ◻

Proof. Step 1: fixed-cutoff collar control. On the finite type-I regulator net, Propositions 85, 86, and 91 localize the modular defect to the shrinking collar and quantify the discrepancy between the physical collar state, its constructive recovered comparison state, and its exact-Markov reference by the carried errors \(r_{\mathrm{FR}}(\varepsilon_\delta)\) and \(\eta_\delta^{\mathrm M}\).

Step 2: support-visible regularization. Proposition 114 shows that a refinement-uniform common floor on the full finite matrix algebra is unavailable in general. The independently assumed \(\mathsf{MGNS\text{-}1}\) package fixes the common comparison maps, compatible state/vector data, both time directions, group-law control, support covariance, and cofinal modulus. Theorem 115 then supplies the needed estimate: for \(K_a(\rho)=-\log(\rho+a\mathbf1)\) and bounded support-visible collar observables, the modular matrix-element difference is bounded by \[ \|O\|\left(\frac{4\Delta_n}{a}+d_{m,\delta}a+4\Delta_n|\log a|\right). \] Choosing \(a_n\downarrow0\) with \(\Delta_n/a_n\to0\), \(d_{m,\delta}a_n\to0\), and \(\Delta_n|\log a_n|\to0\) makes the right-hand side vanish on every fixed local collar model. Theorems 105 and 116 then make the exact-Markov comparison family projectively compatible and replacement-independent on support-visible observables. Thus the theorem concerns the support-visible scaling limit instead of off-support full-algebra directions.

Step 3: scaling-limit cap-pair extraction. Theorem 96 fixes the target algebra as the overlap-generated geometric subnet, with overlap-trivial auxiliary/record factors removed. Proposition 103 supplies the support-visible weak-\(*\)/GNS cap pair from the transported cap-local test family, the projectively compatible marginal family, the asymptotic transport-equivalence certificate, and the regularized modular transport cutoff schedule.

Step 4: certificate-level geometric identification on the geometric subnet. Lemma 104 makes the modular automorphism group induce a support map on cap-local regions of the extracted geometric subnet. The finite cap-normal certificate supplies support-order faithfulness, cap-frame preservation, held-out complex cross-ratio convergence, orientation preservation, and compact-time equicontinuity. By Theorem 100, the limiting support flow is the framed cap flow. Thus \[ \sigma_t^{\omega_\infty^{\mathrm{geo},C}}=\alpha_{\lambda_C(\kappa_C t)}. \] with the certificate-selected frame understood in the notation \(\lambda_C\).

Step 5: normalization. The certificate does not use the false shortcut that modular KMS alone selects \(2\pi\). Instead, its independently normalized geometric strip residual passes the \(2\pi\)-KMS condition to \(s\mapsto\alpha_{\lambda_{\widehat C}(s)}\), and KMS uniqueness gives \[ \sigma_t^{\omega_\infty^{\mathrm{geo},C}} \mathrel{=} \alpha_{\lambda_{\widehat C}(2\pi t)}. \] The wrong-\(\beta\) gap excludes every fixed normalization separated from \(2\pi\), and nontriviality excludes the only exact multiple-temperature exception.

Step 6: operator versus automorphism form. If the limit cap algebra is type I, one can represent the modular automorphism group by a modular Hamiltonian \(K_C=-\log\rho_C\) and a geometric generator \(B_C\). Equality of the implemented automorphisms gives \(K_C=2\pi B_C+Z_C\) for a central self-adjoint \(Z_C\). If the limit algebra leaves the regulator class, the modular automorphism group is well defined while the inner operator representative need not exist inside \(\mathcal A_\infty^{\mathrm{geo,sv}}(C)\). In that case the automorphism identity is the full theorem statement. This is exactly the observer-facing content needed for Lorentz kinematics, the null half-sided modular bridge, and the local Einstein branch. ◻

Proof. Zero cycle affinity is the exactness condition for the edge one-cochain \(a\): integrating from a root gives \(K\), and exponentiation gives detailed balance. The channel-space statement is the invariant-subspace criterion for the self-adjoint diagonal operator \(F_E\). Under the symmetry hypotheses, \(\Omega_{24}\) commutes with \(A_5\) and anticommutes with reversal. Since \(\mathbb C[\mathsf P_{12}]\cong\mathbf1\oplus\mathbf3\oplus\mathbf3'\oplus\mathbf5\), Schur’s lemma gives one Hermitian \(2\times2\) orientation block on each irreducible summand. Anticommutation with reversal makes each block traceless, so its eigenvalues are \(\pm\omega_\lambda\), repeated by \(\dim\lambda\). ◻

Proof. The integral representation of the operator logarithm gives the displayed Lipschitz control above the spectral cutoff \(a\). Splitting the comparison into the visible support above \(a\), the \(a\)-tail, and the trace-distance error gives the three terms in the bound, which vanish under the stated cutoff schedule. ◻

Proof. Theorem 115 gives the displayed matrix element bound with \(\Delta_\delta=\|\rho_\delta-\sigma_\delta\|_1\). The controlled Markov replacement hypotheses make \(\Delta_\delta\to0\) on each fixed collar model, and the chosen \(a_\delta\) schedule sends \[ \Delta_\delta/a_\delta,\qquad d_{m,\delta}a_\delta,\qquad \Delta_\delta|\log a_\delta| \] to zero. The projective compatibility theorem ensures that different compatible replacement choices determine the same local weak-\(*\) limits after GNS quotienting of null directions. Therefore the support-visible cap modular automorphism emitted in the scaling limit is a property of the physical cap pair, not of an arbitrary replacement choice. ◻

Proof. In multiresolution coordinates the reference density at stage \(s\) is \(\widehat\rho_{r,C}\otimes\tau_{s\setminus r}\) and the embedded observable is \(A\otimes\mathbf1\), which proves exact modular compatibility. For positive matrices above the floor \(a_s\), \[ \|\log X-\log Y\|\le a_s^{-1}\|X-Y\|. \] Comparing first with \(-\log(\widehat\rho_{r,C}+a_s\mathbf1)\) and then with \(-\log\widehat\rho_{r,C}\) gives the bracketed generator bound. Duhamel’s formula gives the displayed compact-time automorphism estimate. For the diagonal schedule, choose stages \(s_k\) with \(\max_{r\le r_k}\epsilon_{s_k,r,C}\le k^{-4}\) and set \(a_{s_k}=k^{-1}\). ◻

Proof. (1) Theorem 25 makes the normal form \(W_r\), hence \(\omega_{W_r}\) and each \(s_p\), independent of the schedule. (2) A gauge transformation acts by a quotient automorphism \(\alpha_u=u(\cdot)u^*\) intertwining \(\omega_{W_r}\circ\alpha_u^{-1}\) with the transported normal form; supports transform covariantly, \(s_p\mapsto us_pu^*\), and \(\omega(u s_{p_0}\cdots s_{p_k}u^*)=\omega'(s'_{p_0}\cdots s'_{p_k})\), so simplex membership is unchanged. (3) Petz support control states that the refinement channel \(E_{r'r}\) maps the support of \(\omega_{W_{r'}}|_{\mathcal A_{r',p'}}\) into the support of \(\omega_{W_r}|_{\mathcal A_{r,\pi p'}}\); a nonvanishing joint support value at stage \(r'\) therefore pushes forward to a nonvanishing joint support value of the images, which is the simplicial-map property. Functoriality follows from compatibility of the projections with normal forms. ◻

Proof. A connected closed combinatorial surface is classified up to PL homeomorphism by orientability and Euler characteristic; orientable with \(\chi=2\) gives \(S^2\). Uniqueness of the PL structure on surfaces is Radó’s theorem. A simplicial map between oriented closed surfaces that is surjective and orientation-coherent on top simplices has a well-defined simplicial degree, which the coherence clause fixes to one. ◻

Proof. (1) Mesh density with \(\theta_r\downarrow0\) makes the boundary-vertex sets cofinally dense in the inverse-limit surface. On a topological \(S^2\), a separating dense family of boundary circles with coherent cross-ratio data determines a unique atlas of circular charts: fix three limit vertices as Möbius gauge; the Cauchy cross-ratio data then embed every other limit vertex into \(\mathbb C\cup\{\infty\}\) uniquely, since a point of the Riemann sphere is determined by its cross-ratios against a fixed triple, and consistency across overlapping quadruples is exactly the receipt compatibility clause. Uniform Cauchy moduli \(\varepsilon_r\) give well-definedness of the limit embedding and its continuity; bijectivity onto \(S^2\) follows from density and compactness. The conformal structure is the pullback of the standard structure; uniqueness holds because Möbius cross-ratios separate conformal structures on \(S^2\) (two structures with identical cross-ratio functionals on a dense family differ by a Möbius map, which is the declared gauge freedom). (2) In the produced conformal chart, a receipt cap has boundary an incidence cycle whose limit quadruple cross-ratios are those of a circle. This is the explicit real-boundary and circular-order clause of \(\mathsf{CR}_r(\varepsilon_r)\), preserved in the limit by its held-out Cauchy envelope; Cauchy convergence alone would not imply concyclicity. Hence the limit boundary is a round circle and the disk a round cap. The normal \(n_C\) is then the finite formula of Proposition 134; the residual bound \(\kappa(\varepsilon_r,\theta_r)\) follows by continuity of \(C\mapsto n_C\) on nondegenerate caps, quantitatively since \(\cot\) and \(\csc\) are Lipschitz on mesh-bounded ranges \(\alpha\in[\theta_r,\pi-\theta_r]\). (3) is Lemma 123 together with the receipt compatibility requirements, which are themselves stated on normal-form data only. ◻

Proof. The topology and conformal producer theorems supply C1. The source-bound \(\mathsf{BWFlow}\) receipt supplies the frame, cap-algebra, support-flow, and orientation data that cannot be inferred from cap normals. The strengthened cross-ratio receipt supplies the coherent complex data and held-out separation used in C4. The independent KMS receipt supplies C5–C6. Thus every clause of Definition 98 has one named producer, and deleting any one of those producers deletes the corresponding certificate clause. None of these steps constructs the state-side \(\mathsf{MGNS\text{-}1}\) package. ◻

Proof. Fix any finite abstract patch set \(P\) and equip each patch with the same two-level record algebra \(\mathbb C^2\), with overlap constraints requiring equality of the shared bit on adjacent patches and a single soft conflict seeded on one fixed overlap; repair resolves the conflict by the recovery-derived law in one transactional step. Termination, confluence, and schedule independence are immediate (one conflict component; Newman’s lemma degenerates), and all D1 clauses hold by construction. The construction is functorial in the adjacency structure: taking \(P\) with adjacency given by (i) the icosahedral triangulation of \(S^2\), (ii) a \(7\times7\) triangulated torus, (iii) \(\partial\Delta^4\), and (iv) two icosahedra identified at a vertex yields the four systems; their rewrite systems are isomorphic because the conflict component and repair path structure never see the adjacency beyond the seeded overlap, while the MaxEnt supports reproduce exactly the declared adjacency as joint-support nonvanishing, so the incidence complexes are as listed. For (4), rescale the declared cap modular-flow parameter by \(\beta/2\pi\): the repair layer is unchanged, while the independently normalized KMS comparison certifies \(\beta\). Each system is finite and explicit; machine receipts are provided with the released geometry code. ◻

Proof. Since \(\|\Omega\|=1\), \[ \eta(q(\Omega),q(\Omega))=-1+\|\Omega\|^2=0, \] and \(q(\Omega)^0=1>0\). Also \(-\eta(u_0,q(\Omega))=1\). Conversely, let \(q=(q^0,\mathbf q)\in\mathcal N^+\). Nullness and future orientation give \[ q^0=\|\mathbf q\|>0. \] After multiplying \(q\) by \((q^0)^{-1}\), one obtains \[ (1,\mathbf q/q^0)=q(\Omega),\qquad \Omega:=\mathbf q/q^0\in S^2. \] The normalization \(q^0=1\) is unique on each positive null ray. ◻

Proof. Using \(\|\mathbf c\|=1\), \[ \eta(n_C,n_C) =-\cot^2\alpha+\csc^2\alpha\,\|\mathbf c\|^2 =-\cot^2\alpha+\csc^2\alpha =1. \] For \(\Omega\in S^2\), \[ \eta(n_C,q(\Omega)) =-\cot\alpha+\csc\alpha\,\mathbf c\cdot\Omega =\frac{\mathbf c\cdot\Omega-\cos\alpha}{\sin\alpha}. \] Since \(0<\alpha<\pi\), \(\sin\alpha>0\), so the sign is exactly the sign of \(\mathbf c\cdot\Omega-\cos\alpha\). The incidence and sign statements follow.

It remains to prove surjectivity and uniqueness. Let \(n=(n^0,\mathbf n)\in dS_3^{\mathrm{cap}}\). Then \[ -\left(n^0\right)^2+\|\mathbf n\|^2=1, \qquad \|\mathbf n\|=\sqrt{1+\left(n^0\right)^2}>0. \] Set \(\mathbf c:=\mathbf n/\|\mathbf n\|\). The function \(\cot:(0,\pi)\to\mathbb R\) is a bijection, so there is a unique \(\alpha\in(0,\pi)\) satisfying \(\cot\alpha=n^0\). Then \[ \csc\alpha=\sqrt{1+\cot^2\alpha} =\sqrt{1+\left(n^0\right)^2} =\|\mathbf n\|, \] and hence \(n=n_{C(\mathbf c,\alpha)}\). Finally, \[ \eta(-n_C,q(\Omega))\geq0 \quad\Longleftrightarrow\quad \eta(n_C,q(\Omega))\leq0, \] which reverses the selected side of the boundary circle. ◻

Proof. Because \(\Lambda\) is proper orthochronous, it preserves \(\mathcal N^+\). Thus \(\Lambda q(\Omega)\in\mathcal N^+\), and \(\omega_\Lambda(\Omega)>0\). A future null vector \(Q=(Q^0,\mathbf Q)\) obeys \(\|\mathbf Q\|=Q^0\), so \(\mathbf Q/Q^0\in S^2\). This proves the displayed formula for \(g_\Lambda\), and uniqueness follows from the time-component normalization \(q(\Omega)^0=1\).

To prove conformality, let \(v\in T_\Omega S^2\). Since \(dq_\Omega(v)=(0,v)\), differentiating \[ \Lambda q(\Omega)=\omega_\Lambda(\Omega)\,q(g_\Lambda(\Omega)) \] gives \[ \Lambda dq_\Omega(v) =d\omega_\Lambda(v)\,q(g_\Lambda\Omega) +\omega_\Lambda(\Omega)\,dq_{g_\Lambda\Omega}(dg_\Lambda v). \] The identities \(\eta(q,q)=0\) and \(\eta(q,dq(w))=0\) remove the first term and the cross term when taking Lorentz norms. Hence \[ \eta(dq_\Omega(v),dq_\Omega(v)) =\omega_\Lambda(\Omega)^2 \eta\!\left( dq_{g_\Lambda\Omega}(dg_\Lambda v), dq_{g_\Lambda\Omega}(dg_\Lambda v) \right). \] Since \(\eta(dq_\Omega(v),dq_\Omega(v))=\|v\|^2\), polarization gives \[ g_\Lambda^*\gamma_{S^2}=\omega_\Lambda^{-2}\gamma_{S^2}. \]

For the global group identification, identify \(V\) with Hermitian \(2\times2\) matrices by \[ \mathsf X(x)= \begin{pmatrix} x^0+x^3 & x^1-i x^2\\ x^1+i x^2 & x^0-x^3 \end{pmatrix}. \] Then \[ \det\mathsf X(x)=(x^0)^2-\|\mathbf x\|^2=-\eta(x,x). \] For \(A\in\mathrm{SL}(2,\mathbb C)\), the map \(\mathsf X\mapsto A\mathsf X A^\dagger\) preserves Hermiticity, determinant, orientation, and the future cone. It defines a homomorphism \(\mathrm{SL}(2,\mathbb C)\to G\) with kernel \(\{\pm I\}\), and the image is the connected Lorentz group. Future null rays are rank-one positive Hermitian rays, i.e. \(\mathbb{CP}^1\cong S^2\), and the induced projective action is the Möbius action. Thus \[ G\cong\mathrm{PSL}(2,\mathbb C)\cong\operatorname{Conf}^{+}(S^2). \]

Finally, let \(C=C(n_C)\). Since \[ q(g_\Lambda\Omega)=\omega_\Lambda(\Omega)^{-1}\Lambda q(\Omega), \] Lorentz invariance gives \[ \eta\!\left(\Lambda n_C,q(g_\Lambda\Omega)\right) =\omega_\Lambda(\Omega)^{-1}\eta(n_C,q(\Omega)). \] The factor \(\omega_\Lambda(\Omega)^{-1}\) is positive, so the interior, boundary, and exterior signs are preserved. Therefore \[ g_\Lambda(C(n_C))=C(\Lambda n_C). \] Because \(\Lambda\) preserves \(\eta\), \(\Lambda n_C\) is the uniquely normalized oriented normal of the transformed cap, hence \[ n_{g_\Lambda C}=\Lambda n_C. \]  ◻

Proof. The group identification and cap-normal action are Theorem 136. Theorem 107 identifies the realized scaling-limit cap modular automorphism with the standard cap-preserving conformal dilation and fixes its \(2\pi\) normalization. Since \(\lambda_C(t)\) preserves the oriented cap, equivariance and uniqueness of the unit normal give \[ n_C=n_{\lambda_C(t)C}=\Lambda_C(t)n_C. \]  ◻

Proof. Every future unit timelike vector can be completed to an oriented, time-oriented Lorentz frame, so \(G\) acts transitively on \(H^3\). A Lorentz transformation fixing \(u_0\) acts orthogonally on \[ u_0^\perp\cong\mathbb R^3, \] and proper orientation gives precisely \(\mathrm{SO}(3)\). Hence \[ H^3\cong G/\mathrm{SO}(3). \] If \(L\subseteq G\) is compact, average \(u_0\) over \(L\) using Haar measure. The future timelike cone is convex, so the average is a future timelike vector fixed by \(L\); after normalization, \(L\) is conjugate into a point stabilizer. Thus \(K\) is maximal compact and all maximal compact subgroups are conjugate.

Differentiating \(\eta(u,u)=-1\) gives \(T_uH^3=u^\perp\). The orthogonal complement of a timelike vector is positive definite and has dimension three. Since \(G\) preserves \(\eta\), the induced metric is \(G\)-invariant. At \(u_0\), the isotropy representation of \(K\cong\mathrm{SO}(3)\) on \(T_{u_0}H^3\cong\mathbb R^3\) is the standard irreducible representation, so every \(K\)-invariant positive inner product is a positive scalar multiple of the Euclidean one. Homogeneity gives the metric uniqueness.

Take \(q\in\mathcal S_u\) and set \(s:=q-u\). Then \[ \eta(u,s)=\eta(u,q)-\eta(u,u)=-1+1=0, \] and \[ \eta(s,s)=\eta(q-u,q-u)=0-2(-1)-1=1. \] Thus \(s\) lies on the unit sphere in \(E_u\). Conversely, if \(s\in E_u\) and \(\eta(s,s)=1\), then \(q=u+s\) is future null and satisfies \(-\eta(u,q)=1\). Lorentz naturality follows from preservation of \(\eta\).

Finally, for \(s\in E_u\) with \(\eta(s,s)=1\), the curve \[ \gamma_{u,s}(t)=\cosh t\,u+\sinh t\,s \] is a unit-speed geodesic ray in \(H^3\). Projectively, \[ [\gamma_{u,s}(t)]\longrightarrow [u+s]\qquad(t\to+\infty), \] and \(\eta(u+s,u+s)=0\). Every future null ray has such a normalized decomposition relative to \(u\). Hence the ideal boundary is \(\mathbb P\mathcal N^+\cong S^2\). ◻

Proof. Fix \(n_0\in dS_3^{\mathrm{cap}}\). Its stabilizer is \[ H:=\operatorname{Stab}_G(n_0)\cong\mathrm{SO}^{+}(2,1), \] which is noncompact. Suppose an equivariant map \(F\) existed. For every \(h\in H\), \[ F(n_0)=F(hn_0)=hF(n_0). \] Thus the full noncompact group \(H\) would fix \(F(n_0)\in H^3\). But the stabilizer of every point of \(H^3\) is conjugate to \(\mathrm{SO}(3)\), hence compact. This is impossible. ◻

Proof. Because \(n\) is unit spacelike, \(n^\perp\) has signature \((-++)\). Its future unit timelike hyperboloid is a copy of \(H^2\), and \[ \Pi_n=H^3\cap n^\perp. \] Intersections of the hyperboloid with linear subspaces through the origin are totally geodesic, proving the first claim. The ideal boundary of \(H^3\cap n^\perp\) consists of future null rays contained in \(n^\perp\), which are exactly those satisfying \(\eta(n,q)=0\). Under Lemma 132, this is the circle \(\partial C(n)\); the sign condition gives the corresponding oriented ideal half-space boundary.

Equivariance follows from \(\eta(\Lambda n,\Lambda u)=\eta(n,u)\). For the distance formula, let \[ a:=\eta(n,u), \qquad p:=\frac{u-a n}{\sqrt{1+a^2}}. \] Then \(\eta(p,n)=0\), \(\eta(p,p)=-1\), and \(p\) lies on the future sheet, so \(p\in\Pi_n\). With \(s:=\operatorname{arsinh}(a)\), \[ u=\cosh s\,p+\sinh s\,n, \] which is the unit-speed geodesic normal to \(\Pi_n\) through \(p\). Hence \(s\) is the signed distance. ◻

Proof. The homogeneous-space statement and dimension follow from Proposition 139. The standard isotropy representation of \(\mathrm{SO}(3)\) is irreducible, so the invariant metric is unique up to one overall positive multiplier. Writing that multiplier as \(R_H^2\) gives the radius-\(R_H\) hyperboloid and the displayed distance formula. ◻

Proof. The first statement is Proposition 134. If \(Y\in\Pi_C\) is the foot of the perpendicular from \(X\), the normal geodesic has form \[ \gamma(s)=\cosh(s/R_H)Y+R_H\sinh(s/R_H)n_C. \] Pairing with \(n_C\) gives \(\eta(\gamma(s),n_C)/R_H=\sinh(s/R_H)\). ◻

Proof. Let \(M\) be the geodesic midpoint between \(X\) and \(Y\), and let \(v\in T_MH^3_{R_H}\) be the unit tangent pointing from \(X\) to \(Y\). Then \[ X=\cosh(\delta/2R_H)M-R_H\sinh(\delta/2R_H)v,\qquad Y=\cosh(\delta/2R_H)M+R_H\sinh(\delta/2R_H)v, \] where \(\delta=d_H(X,Y)>0\). The vector \(v\) is unit spacelike and therefore is \(n_C\) for a unique oriented cap \(C\). Thus \[ h_C(X)=-\sinh(\delta/2R_H)<0,\qquad h_C(Y)=+\sinh(\delta/2R_H)>0. \] The nonnegative and binary responses then distinguish the same ordered pair. ◻

Proof. Since \[ h_j(X)=\frac1{R_H} \left(-\frac{X^0}{\sqrt2}+\sqrt{\frac32}\,\mathbf X\cdot u_j\right), \] summing and using the centered tight-frame identities gives the two displayed reconstruction formulas. Applying the same identities to \(\Delta X=X-Y\) gives the Euclidean norm identity. Finally \(|X-Y|_{\mathrm E}\geq R_H\bar d(X,Y)\) on the hyperboloid, because the Euclidean norm dominates the spacelike Minkowski norm and \(2\sinh(r/2)\geq r\). ◻

Proof. The response vector is \(h(X)=B X_{\mathrm E}/R_H\). The left inverse gives the reconstruction formula, and the smallest singular value of \(W^{1/2}B\) gives the lower bound after using \(|X-Y|_{\mathrm E}\geq R_H\bar d(X,Y)\). ◻

Proof. If \(z,w\) have the same sign, the norm is \(|z-w|\). If they have opposite signs, it is \(\sqrt{z^2+w^2}\), which lies between \((|z|+|w|)/\sqrt2\) and \(|z|+|w|\). ◻

Proof. Subtract \(f_j\) from \(y_j\). The numerator is \((b_j-\widehat b_j)+\xi_j+(a_j-\widehat a_j)f_j\), and the denominator is at least \(a_{\min}-\delta_{a,j}\). ◻

Proof. \(\mathcal J(X_\star)=0\). If another point also has zero residual, then \(F(X)=F(X_\star)\), so \(\alpha\bar d(X,X_\star)=0\), hence \(X=X_\star\). ◻

Proof. Choose a maximal subset of \(\Omega\) whose distinct points are more than \(\varepsilon\) apart in normalized distance. Maximality gives coverage. The balls of physical radius \(R_H\varepsilon/2\) around net points are disjoint and lie in \(B_H(o,D+R_H\varepsilon/2)\), giving the volume bound. Noncompact \(H^3\) has infinite volume, so a finite global net cannot exist. ◻

Proof. Choose \(X_\varepsilon\in\mathcal N_\varepsilon\) with \(\bar d(X_\varepsilon,X_\star)\leq\varepsilon\). Then \(|F(X_\varepsilon)-F(X_\star)|_W\leq L\varepsilon\). Near-optimality and \(|e|_W\leq\sigma\) give \[ |F(\widehat X_\varepsilon)-F(X_\star)|_W\leq L\varepsilon+2\sigma+\tau. \] The lower observability bound converts this response error into the displayed hyperbolic distance bound. ◻

Proof. Every competitor has residual at least its lower interval bound, while the candidate has residual at most its upper interval bound. A strict positive gap therefore rules out ties and competitors. Without such a gap, a finite procedure cannot choose a unique output without adding an external tie-breaker. ◻

Proof. Lorentz transformations preserve \(\eta\), so all response coordinates and all residual inequalities are invariant under the common chart transform. ◻

Proof. Exact uniqueness is Theorem 156. The finite-net error estimate is Theorem 158. Noisy finite uniqueness is Theorem 159, and chart naturality is Theorem 160. ◻

Proof. By complete reducibility of the finite-dimensional unitary actions \(\widehat K_\pm\), the displayed decompositions of \(\tilde{\mathcal H}_{I_-}\), \(\tilde{\mathcal H}_J\), and \(\tilde{\mathcal H}_{I_+}\) exist. Under item (iii), the strip algebra is the commutant of the transported \(\widehat K_-\times \widehat K_+\) action on \[ \tilde{\mathcal H}_J \mathrel{=} \bigoplus_{\alpha_-,\alpha_+} W_{\alpha_-}^{*}\otimes \mathcal H_{j^{\alpha_-,\alpha_+}}\otimes W_{\alpha_+}. \] Write an operator \(X\in \mathcal B(\tilde{\mathcal H}_J)\) in matrix form relative to that direct sum. The block \[ X_{(\alpha_-,\alpha_+),(\beta_-,\beta_+)} \] intertwines \[ W_{\alpha_-}^{*}\otimes W_{\alpha_+} \longrightarrow W_{\beta_-}^{*}\otimes W_{\beta_+} \] for the product action of \(\widehat K_-\times \widehat K_+\). By Schur’s lemma, such an intertwiner is zero unless \((\alpha_-,\alpha_+)=(\beta_-,\beta_+)\), and when the sector pair agrees it is scalar on the representation factors. Therefore the commutant is exactly \[ \mathcal A(J) \cong \bigoplus_{\alpha_-,\alpha_+} \mathcal B(\mathcal H_{j^{\alpha_-,\alpha_+}}), \] and the center is generated by the corresponding block projectors \(P_{\alpha_-,\alpha_+}\).

For the glued tripartition, tensor the three regulator Hilbert spaces and decompose: \[ \tilde{\mathcal H}_{I_-}\otimes \tilde{\mathcal H}_{J}\otimes \tilde{\mathcal H}_{I_+} \cong \bigoplus_{\alpha_-,\beta_-,\alpha_+,\beta_+} (W_{\alpha_-}\otimes W_{\beta_-}^{*}) \otimes \mathcal H_{i_-^{\alpha_-}} \otimes \mathcal H_{j^{\beta_-,\beta_+}} \otimes (W_{\beta_+}\otimes W_{\alpha_+}^{*}) \otimes \mathcal H_{i_+^{\alpha_+}}. \] Taking \(\widehat K_-\times\widehat K_+\)-invariants and applying Schur’s lemma to each cut gives \[ (W_{\alpha_-}\otimes W_{\beta_-}^{*})^{\widehat K_-} \cong \begin{cases} \mathbb C, & \alpha_-=\beta_-,\\ 0, & \alpha_-\neq \beta_-, \end{cases} \qquad (W_{\beta_+}\otimes W_{\alpha_+}^{*})^{\widehat K_+} \cong \begin{cases} \mathbb C, & \beta_+=\alpha_+,\\ 0, & \beta_+\neq \alpha_+, \end{cases} \] so only matching cut sectors survive and one obtains \[ \mathcal H_{I_-\cup J\cup I_+} \cong \bigoplus_{\alpha_-,\alpha_+} \mathcal H_{i_-^{\alpha_-}}\otimes \mathcal H_{j^{\alpha_-,\alpha_+}}\otimes \mathcal H_{i_+^{\alpha_+}}. \] If item (iv) holds, substitute the chosen factorization \[ \mathcal H_{j^{\alpha_-,\alpha_+}} \cong \mathcal H_{j_L^{\alpha_-,\alpha_+}}\otimes \mathcal H_{j_R^{\alpha_-,\alpha_+}} \] to obtain the displayed inherited split and the stated action of the left and right union algebras. Nothing in the Schur-lemma argument itself forces item (iv); it is exactly the extra structural condition required to reproduce the same blockwise tensor pattern as Theorem 59. ◻

Proof. Assume first that \(\omega\) is exact Markov and EC-aligned on the inherited split of Proposition 163. The HJPW structure theorem gives blockwise factorization over a state-dependent decomposition of \(\mathcal H_J\); the alignment hypothesis identifies that decomposition with the inherited split, so that on each sector pair \(\alpha=(\alpha_-,\alpha_+)\), \[ \omega_{I_-JI_+}|_\alpha \mathrel{=} p_\alpha\, \omega_{I_-\,j_L}^{(\alpha)} \otimes \omega_{j_R\,I_+}^{(\alpha)}. \] Accordingly, \[ K_{I_-\cup J\cup I_+}(\omega)\big|_\alpha \mathrel{=} (-\log p_\alpha)\mathbf 1 + K_{I_-\,j_L}^{(\alpha)}\otimes \mathbf 1_{j_R I_+} + \mathbf 1_{I_- j_L}\otimes K_{j_R I_+}^{(\alpha)}. \] Tracing over \(I_+\), \(I_-\), or both gives the marginal block formulas \[ K_{I_-\cup J}(\omega)\big|_\alpha \mathrel{=} (-\log p_\alpha)\mathbf 1 + K_{I_-\,j_L}^{(\alpha)}\otimes \mathbf 1_{j_R} + \mathbf 1_{I_- j_L}\otimes K_{j_R}^{(\alpha)}, \] \[ K_{J\cup I_+}(\omega)\big|_\alpha \mathrel{=} (-\log p_\alpha)\mathbf 1 + K_{j_L}^{(\alpha)}\otimes \mathbf 1_{j_R I_+} + \mathbf 1_{j_L}\otimes K_{j_R I_+}^{(\alpha)}, \] \[ K_J(\omega)\big|_\alpha \mathrel{=} (-\log p_\alpha)\mathbf 1 + K_{j_L}^{(\alpha)}\otimes \mathbf 1_{j_R} + \mathbf 1_{j_L}\otimes K_{j_R}^{(\alpha)}. \] Subtracting the last three expressions from the first gives zero on every block. Thus, in the canonical blockwise identification supplied by Proposition 163, \[ \Delta K_J(\omega)=0. \] If one chooses to keep the blockwise endpoint-label bookkeeping explicit instead of absorbing those constants into the canonical identification, the remainder is a direct sum of block constants and is therefore central. This yields the stated central form with \(K_{\partial,J}(\omega)\in Z(\mathcal A(J))\).

For the controlled statement, apply Proposition 82 on this fixed finite-dimensional strip model, relabeling \(A,B,D\) there as \(I_-,J,I_+\). This yields an exact Markov replacement \(\widetilde\omega_J\in \mathfrak M_{I_-:J:I_+}\) with the displayed trace-norm bound. The identity \[ \Delta K_J(\omega)=\Delta K_J(\widetilde\omega_J)+\mathfrak D_J(\omega,\widetilde\omega_J) \] is tautological, so the displayed four-term formula follows from the exact-Markov case applied to \(\widetilde\omega_J\). The observable estimate is immediate from \[ \bigl|\omega(O)-\widetilde\omega_J(O)\bigr| \le \|O\|_\infty\, \|\omega-\widetilde\omega_J\|_1. \] Finally, Proposition 86, again relabeled \(A,B,D\mapsto I_-,J,I_+\), yields the operator-norm bound on \(\mathfrak D_J\) whenever the relevant marginals have the stated lower spectral bound. The Fawzi–Renner estimate is the same one-shot recoverability bound used elsewhere and is independent of the exact-Markov replacement modulus. ◻

Proof. The bounded-interval endpoint-control statement derived from Axiom 3 applies precisely to the renormalized family obtained after removal of the central endpoint term. Therefore, for intervals contained in one fixed compact endpoint window, \[ \bigl|\langle\psi,(\widetilde K[I',\Omega]-\widetilde K[I,\Omega])\phi\rangle\bigr| \le C_{\psi,\phi,\Omega}\,|I'\Delta I|, \] where \(I'\Delta I\) is the symmetric difference of the two intervals. If \[ I=(a,b),\qquad I'=(a',b'), \] then \[ |I'\Delta I| \le |a'-a|+|b'-b|, \] which gives the displayed joint endpoint-Lipschitz bound.

For half-lines \(H_a=(a,\infty)\) and \(H_{a'}=(a',\infty)\), the symmetric difference is the bounded interval between the two endpoints, so \[ |H_{a'}\Delta H_a|=|a'-a|. \] Because the central endpoint-label term has been removed in the definition of \(\widetilde K_a\), the same endpoint-control estimate yields \[ \bigl|\langle\psi,(\widetilde K_{a'}(\Omega)-\widetilde K_a(\Omega))\phi\rangle\bigr| \le C_{\psi,\phi,\Omega}\,|a'-a|. \]

Hence, on every compact \(a\)-interval, the function \[ f_{\psi,\phi}(a):=\langle\psi,\widetilde K_a(\Omega)\phi\rangle \] is Lipschitz. A Lipschitz function on a compact interval is absolutely continuous, so it has an almost-everywhere derivative in \(L^\infty\) and obeys the fundamental theorem of calculus: \[ f_{\psi,\phi}(b)-f_{\psi,\phi}(a)=\int_a^b \partial_v f_{\psi,\phi}(v)\,dv. \] Define \[ \langle\psi,q(v,\Omega)\phi\rangle := -\frac{1}{2\pi}\,\partial_v f_{\psi,\phi}(v) \] distributionally. Substituting this definition into the previous identity gives \[ \langle\psi,(\widetilde K_b(\Omega)-\widetilde K_a(\Omega))\phi\rangle \mathrel{=} -2\pi\int_a^b \langle\psi,q(v,\Omega)\phi\rangle\,dv. \] This is exactly the claimed weak-tail formula. ◻

Proof. By Theorem 107, on every controlled collar family whose carried remainder vanishes in the refinement limit, the cap modular flow converges to the exact geometric cap dilation with \(2\pi\) normalization. Blow up the cap near a smooth cut and restrict to the chosen null generator. In that tangent limit, the cap-preserving flow becomes the null dilation \(v\mapsto e^{-2\pi t}v\). For every bounded interval \((c,d)\subset H_a\), the blow-up action therefore sends \[ \mathcal A((c,d),\Omega)\longmapsto \mathcal A((e^{-2\pi t}c,e^{-2\pi t}d),\Omega). \] Taking the von Neumann closure of the algebra generated by all such intervals gives \[ \sigma_t^\omega(\mathcal M_a(\Omega)) \mathrel{=} \mathcal M_{e^{-2\pi t}a}(\Omega). \] If \(t\le 0\), then \(e^{-2\pi t}a\ge a\), hence \[ H_{e^{-2\pi t}a}\subseteq H_a. \] Isotony of the null interval net therefore implies \[ \mathcal M_{e^{-2\pi t}a}(\Omega)\subseteq \mathcal M_a(\Omega). \] This is exactly the half-sided modular-inclusion property for the pair \(\mathcal M_a(\Omega)\subset \mathcal M_0(\Omega)\). Reparametrizing the modular parameter by \(t\mapsto -t\) gives the more common positive-time convention if desired. ◻

Proof. Corollary 168 gives the required half-sided modular inclusion. The standard Borchers–Wiesbrock theorem for a standard half-sided inclusion  then yields a unique strongly continuous unitary group \(U_\Omega(a)\) with \(U_\Omega(a)\omega=\omega\), the translation law for the nested half-line net, and a positive self-adjoint generator \(P_\Omega\); Stone’s theorem gives the displayed domain formula.

The affine covariance relation \[ \Delta_0(\Omega)^{it}U_\Omega(a)\Delta_0(\Omega)^{-it} \mathrel{=} U_\Omega(e^{-2\pi t}a) \] is the Borchers relation. Differentiating at \(a=0\) gives \[ \Delta_0(\Omega)^{it}P_\Omega\Delta_0(\Omega)^{-it} \mathrel{=} e^{-2\pi t}P_\Omega. \] Since \(\Delta_0(\Omega)^{it}=e^{-itK_0(\Omega)}\), differentiating at \(t=0\) on the common analytic core yields \[ -i[K_0(\Omega),P_\Omega]=-2\pi P_\Omega, \] hence \[ [K_0(\Omega),P_\Omega]=-\,i\,2\pi P_\Omega. \]

Because \(U_\Omega(a)\omega=\omega\) and \[ \mathcal M_a(\Omega)=U_\Omega(a)\mathcal M_0(\Omega)U_\Omega(a)^*, \] the modular operator of the translated standard pair is \[ \Delta_a(\Omega)=U_\Omega(a)\Delta_0(\Omega)U_\Omega(a)^*, \] so \[ K_a(\Omega)=U_\Omega(a)K_0(\Omega)U_\Omega(a)^*. \] Differentiating this identity in \(a\) on \(D(K_0(\Omega))\cap D(P_\Omega)\) gives \[ \frac{d}{da}K_a(\Omega) \mathrel{=} i[P_\Omega,K_a(\Omega)] \mathrel{=} -2\pi P_\Omega, \] because the commutator relation just proved is invariant under conjugation by \(U_\Omega(a)\). Integrating from \(0\) to \(a\) yields the displayed quadratic-form identity, and taking matrix elements gives the equivalent form. Proposition 166 had produced the weak endpoint derivative of the renormalized half-line family; in the canonical normalization of Corollary 164 the central term has been removed, so that weak derivative is exactly the Borchers generator. The final sentence is the explicit claim boundary: bounded intervals require the additional projective interval action not supplied by half-sided inclusion alone. ◻

Proof. Lemma 169 proves the first displayed identity: in the canonical normalization of Corollary 164, the weak endpoint derivative of the renormalized half-line family is exactly the Borchers generator. Passing to the effective spacetime description does not change the operator family; it rewrites the same \(\widetilde K_a(\Omega)\) in local continuum variables appropriate to the local Lorentzian regime. In that description, the local null-stress charge is precisely the right endpoint derivative \[ -\frac{1}{2\pi}\frac{d}{da}\Big|_{a=0^+}\widetilde K^{\mathrm{eff}}_a(\Omega) \] of the same half-line modular family. Because \(\widetilde K^{\mathrm{eff}}_a(\Omega)\) and \(\widetilde K_a(\Omega)\) represent the same renormalized modular Hamiltonians, their right derivatives agree on \(D_\Omega\). Therefore the OPH generator and the effective null-stress charge coincide: \[ P_\Omega = Q_{kk}(\Omega). \] The endpoint-supported central term does not alter the noncentral generator; in the canonical blockwise normalization it is absorbed into the central part fixed in Corollary 164. This proves the identification. ◻

Proof. Write \[ X_{ab}=Y_{ab}+\phi g_{ab} \] with \(Y_{ab}\) traceless. Since \(g_{ab}k^ak^b=0\) on null vectors, the hypothesis implies \(Y_{ab}k^ak^b=0\) for all null \(k\). In local inertial coordinates let \(k=(1,\hat n)\) with \(|\hat n|=1\). Then \[ Y_{ab}k^ak^b \mathrel{=} Y_{00}+2\hat n^iY_{0i}+\hat n^i\hat n^jY_{ij}. \] Oddness under \(\hat n\mapsto-\hat n\) forces \(Y_{0i}=0\). Averaging that relation over \(S^2\) gives \[ Y_{00}+\frac13\delta^{ij}Y_{ij}=0. \] Tracelessness of \(Y_{ab}\) gives \[ -Y_{00}+\delta^{ij}Y_{ij}=0, \] so both \(Y_{00}\) and the spatial trace \(\delta^{ij}Y_{ij}\) vanish. The condition is therefore \[ \hat n^i\hat n^j Y_{ij}^{\mathrm{TF}}=0 \qquad \text{for all }\hat n\in S^2, \] where \(Y_{ij}^{\mathrm{TF}}\) is the traceless spatial part. The \(\ell=2\) spherical harmonics are linearly independent, so this implies \(Y_{ij}^{\mathrm{TF}}=0\). Hence \(Y_{ab}=0\) and therefore \(X_{ab}=\phi g_{ab}\). ◻

Proof. Finite stage: a faithful state on a finite-dimensional von Neumann algebra has a separating GNS vector, and separation passes to subalgebras. Limit cyclicity: fix \(\psi\in\mathcal H\) and \(\varepsilon>0\); the mixed-GNS Cauchy clause supplies a stage \(r\) and a reference-frame vector \(\psi_r\) with \(\|\psi-\psi_r\|<\varepsilon/3\) together with an approximation of stage-\(r\) strip elements by limit elements with defect \(<\varepsilon/3\) on \(\Omega\); the receipt \(\mathsf{Cyc}_r(\delta_r)\) with \(\delta_r<\varepsilon/3\) then gives \(n\in\mathcal N(H)\) with \(\|n\Omega-\psi\|<\varepsilon\). Separation: the opposite-orientation half-line \(\check H\) obeys \(\check{\mathcal N}(\check H)\subseteq\mathcal N(H)'\) by locality, and \(\Omega\) is cyclic for \(\check{\mathcal N}(\check H)\) by item (1); a vector cyclic for a subalgebra of the commutant is separating for the algebra. Intervals inherit separation from \(\mathcal N(I)\subseteq\mathcal N(H_a)\). Standardness of the inclusions is the conjunction of (1) and (2) for both members. ◻

Proof. The produced cap flow of Theorem 107 is, on the blow-up family, the one-parameter dilation of the half-line endpoint with the certified \(2\pi\) normalization; this is precisely the computation of Corollary 168, using the certificate of Theorem 128. Monotonicity \(e^{-2\pi t}\geq1\) for \(t\leq0\) gives the compression property. Theorem 174 supplies the standard-pair hypotheses. The conclusion package, including the Stone domain and the affine commutation relations, is the corrected Borchers–Wiesbrock theorem . ◻

Proof. Vanishing conditional mutual information across a cut \(I(A:C\mid B)_{\rho}=0\) gives, by the structure theorem , a decomposition of the cut collar \(\mathcal H_B=\bigoplus_k\mathcal H_{b_k^{L}}\otimes\mathcal H_{b_k^{R}}\) with \(\rho_{ABC}=\bigoplus_k p_k\,\rho_{Ab_k^{L}}\otimes\rho_{b_k^{R}C}\). Applying this at both endpoint cuts of \(I\) and restricting to \(I\) yields \(\rho_I=\bigoplus_{k,l}p_{kl}\, \rho^{L}_{k}\otimes\rho^{\mathrm{bulk}}_{I}\otimes\rho^{R}_{l}\) with the bulk factor the MaxEnt Gibbs state of the retained cell terms in \(I\); taking \(-\log\) gives the displayed sum, the direct-sum labels being carried by the central interface projections, whose blocks are exactly \(k^{L/R}_{\partial I}\). The interface blocks are functions of the boundary charges retained by the central-interface clause, so their norms carry the declared uniform constants. Endpoint variation of \(K_{I_a}\) then changes only one boundary block and the cells in the swept collar, giving the Lipschitz bound cellwise. ◻

Proof. Take \(H=\sum_{i=1}^{3}\bigl(Z_iZ_{i+1}+gX_i\bigr)+gX_4\) with \(g=1\). At second order in \(\beta\), the cluster expansion of \(-\log\rho_I\) acquires the commutator correction \(\tfrac{\beta^2}{2}\,\mathbb E_{\mathrm{tr}}\!\left[[H_{I\partial},H_{\partial I^c}]\right]\)-type terms which do not cancel for noncommuting nearest-neighbour couplings and which, after the partial trace over site 4, generate a weight-two Pauli term supported on sites \(\{1,3\}\); its coefficient is nonzero at \(g=1\), \(\beta=1\), as certified numerically to interval-arithmetic accuracy in the repository receipt. Since the collar CMI of this state across the cut \(3\,|\,4\) is strictly positive (also certified), the example sits strictly outside the Markov branch, as claimed. ◻

Proof. The generating result for two half-sided modular inclusions with common standard vector is the Wiesbrock/Guido–Longo–Wiesbrock theorem : the modular group of \(\mathcal N(H_0)\) and the two derived translation groups satisfy the \(\mathrm{PSL}(2,\mathbb R)\) commutation relations proved in Theorem 176(3) and its reflection; integrability to the simply connected cover and descent to \(\mathrm{PSL}(2,\mathbb R)\) follow from the affine relations plus \(U(a)\Omega=\Omega\), and positivity of the rotation generator is equivalent to positivity of the two translation generators. Covariant interval algebras are then defined by transport; consistency on overlaps of group elements mapping \(I\) to itself follows because the stabilizer of an interval is generated by its own modular dilations, which fix \(\mathcal N(I)\) by Takesaki modular theory . \(\mathsf{NTI}\) guarantees the transported algebras are nontrivial and that intersections realize \(\mathcal N(I)=\mathcal N(H_a)\wedge\mathcal N(H_b)'\). The kernel-residual propagation is linear along the transport, giving the stated \(o(\ell^4)\) transfer. ◻

Proof. (1) Conjugating the half-sided inclusion at direction \(\Omega\) by \(U(\Lambda)\) yields the half-sided inclusion at \(g_\Lambda(\Omega)\) with the affine parameter rescaled by \(\omega_\Lambda(\Omega)\), by Theorem 136 applied to the blow-up family; uniqueness of the Borchers generator in Theorem 176 then forces the displayed weight. (2) The \(\mathsf{MI}\) relations in the limit give, for each transverse pair, the commutation of the associated translation unitaries through the modular-intersection theorem of ; strong commutativity of the generators follows on the joint Stone core. For the linearity relation, fix a dependent family \(\sum_ic_iq(\Omega_i)=0\); the covariance (1) and commutativity reduce the claim to the stabilizer direction, where the two sides generate the same one-parameter group by the uniqueness clause of the Borchers–Wiesbrock theorem applied to the common compressed inclusion; the finite-stage residual is controlled by \(\varepsilon_r\) and vanishes in the limit. Uniqueness and existence of \(U(x)\) then follow by choosing four null directions in general position and checking independence of the choice via the same relation. (3) is positivity of each \(P_\Omega\) plus density of \(\{q(\Omega)\}\) in the generating rays of the dual cone. (4) combines (1) and (2); the group law on the semidirect product holds on the core and extends by continuity. ◻

Proof. Each datum entering Definition 183 was individually proved invariant in the quoted statements; intersection and nesting of transported event boxes are expressed through covariant affine response coordinates, so the derived relation inherits every invariance. ◻

Proof. The lower metric bound gives injectivity. Boundary exclusion and nonzero degree give a preimage for every point of \(B_{R-\delta}(0)\). Refinement compatibility and the uniform upper bound produce the limiting map; the lower bound preserves injectivity and degree stability preserves surjectivity. On the preimage of each smaller closed ball, the continuous bijection from a compact space to the Hausdorff closed ball is a homeomorphism. Taking the nested union of their interiors gives the stated open chart, and the two-sided bounds give the bi-Lipschitz estimate. ◻

Proof. Weyl’s eigenvalue perturbation inequality prevents an eigenvalue from crossing zero, so inertia is stable. The null residual puts every limiting labelled null direction in \(v^{\mathsf T}G_pv=0\); the converse-separation margin excludes an additional limiting zero direction away from that family. The timelike and spacelike margins select the two sides. ◻

Proof. Hausdorff. Distinct classes contain non-coincident germs, so \(\mathsf{(E2)}\) supplies a stage with certified disjoint boxes; boxes generate the uniformity, so the corresponding box neighborhoods separate the classes. Second countability is Definition 185. Charts. On a declared neighborhood \(W\), \(\mathsf{(E3)}\) gives the two-sided bound \[ \alpha\,\bar d\bigl(e,e'\bigr)\;\leq\; \bigl|\varphi(e)-\varphi(e')\bigr|\;\leq\; L\,\bar d\bigl(e,e'\bigr) \] with \(\bar d\) the affine event-box distance. Thus \(\varphi\) is injective and bi-Lipschitz onto its image. The interior-ball clause of \(\mathsf{(E3)}\) makes the image open. The realization clause of \(\mathsf{(E1)}\) identifies the completed box families with permitted semantic germs; population density by itself would not do so. Local compactness follows from the bi-Lipschitz chart and its open Euclidean image. Transitions of bi-Lipschitz charts are bi-Lipschitz; the \(C^{1,1}\) upgrade is the chain rule for the metric-regularity receipt \(\mathsf{(E4')}\). The failure statements are the countermodels below. ◻

Proof. (1) The derived translations move certified boxes along null and timelike directions of the chart with unit affine weight (Theorem 176(2)); a located record in the read-set ancestry of another is connected to it by a finite chain of transactionally validated collar transports, each of which the blow-up chart represents inside the closed future cone up to the calibration residuals of the frame; composing the chain and passing to the germ limit gives the cone order with \(o(1)\) defect. Conversely, \(\mathsf{(E6)}\) supplies the semantic ancestry chain for a chart pair in the cone interior. Population density alone does not imply directed reachability. (2) Receipt \(\mathsf{(E5)}\) supplies the nondegenerate quadratic form, its inertia, fit residual, and wrong-signature controls, and Lemma 188 passes them to the limiting form. The conformal celestial boundary is a consistency check on that form; it does not determine a quadratic cone. Normalizing the selected time direction gives \((-{+}{+}{+})\). (3) Repair transactions are directed (commits are not invertible moves on the quotient by strict descent), so ancestry orients each timelike chain; schedule independence makes the orientation well defined on germs, and \(\mathsf{(E4)}\) transitions preserve \(\preceq\), hence map future cones to future cones. (4) follows from the affine cocycle in \(\mathsf{(E4)}\) and its \(\mathsf{(E5)}\) cone-preservation clause. ◻

Proof. The fiber statement is Definition 138 applied in \((T_e\mathcal E,g_e)\), which carries Lorentz signature by Theorem 191(2). For the no-section statement, an equivariant section would assign to the model event a point of \(H^3\) fixed by its stabilizer \(\mathrm{SO}^{+}(3,1)\), whose point stabilizers on \(H^3\) are the compact \(\mathrm{SO}(3)\) subgroups; a noncompact group cannot fix a point with compact stabilizer, as in Proposition 140. ◻

Proof. (1) is Theorem 189 plus \(\mathsf{(E4)}\); tetrad transitions are the derivatives of the chart transitions, which are affine and preserve the \(\mathsf{(E5)}\) form in the limit, so the cocycle law is inherited. (2) The \(\eta\)-contraction of Poincaré-related tetrads is chart-independent; conformal-class production is quoted; the scale statement repeats Corollary 143’s boundary (no physical \(R_H\) from group data) in the produced setting. (3) Receipt \(\mathsf{(E4')}\) puts the tetrads and their inverses in \(C^{1,1}_{\rm loc}\), hence \(g\in C^{1,1}_{\rm loc}\). The Christoffel symbols are locally Lipschitz, the geodesic equation has locally unique solutions by the standard locally-Lipschitz ODE theorem, and the Riemann tensor is defined almost everywhere in \(L^\infty_{\rm loc}\). Smoothness on the declared branch is a declared upgrade, exactly as consumed downstream. ◻

Proof. Proper time is the integral of the invariant Lorentzian line element, so chart covariance is immediate. For distinct curves, differentiating \(\tau_j(F_{ij}(t_i))\) and dividing by \(d\tau_i/dt_i\) gives the displayed chain-rule formula. For affine operational calibrations, necessity of the cycle condition follows by telescoping node calibrations around a loop; for sufficiency, fix one root calibration and transport it along paths. Trivial cycle products make the result path-independent, and changing the root calibration gives the single global affine gauge. Scheduler independence is Lemma 184. The global equivalences are the standard Lorentzian statements, quoted here only to name the receipts. ◻

Proof. (i) Take the producer-branch tower and restrict the record layer to tokens whose localization balls follow one timelike chain (one located observer); every construction upstream of records is untouched, while Definition 185 yields the chain. (ii) Tensor the record layer with a two-point (respectively interval-valued) internal register transported trivially; boxes never separate the register, so coincidence classes split accordingly. (iii) Seed a reconciliation defect at stage \(r_\star\) as in the obstruction gates of node D1 so that two token families carry intersecting boxes before \(r_\star\) and disjoint uncertified boxes after; the residual-gap certificate then fails permanently by construction. Each construction preserves the screen-side producer data because it modifies only the record layer downstream of the certificate. ◻

Proof. Axiom 3 selects the realized branch by entropy maximization on the finite-dimensional constraint surface cut out by the retained local constraints and optional conserved charges. Definition 197 restricts to first-order tangent directions that stay on that same fixed-cap constraint surface. Hence the first variation of the cap entropy vanishes at the realized reference state: \[ \delta S(\rho_C)\big|_{\omega_C}=0. \] For the same cap-label-preserving block class, Proposition 91 gives \[ S(\rho_C)=S_{\mathrm{bulk}}(C)+\mathrm{Tr}(\rho_C L_C)=S_{\mathrm{gen}}(C). \] Differentiating this identity along the admissible curve yields \[ \delta S_{\mathrm{gen}}(C)=\delta S(\rho_C)\big|_{\omega_C}=0. \]  ◻

Proof. Work in the local Lorentzian scaling regime around \(p\). On the geometric-subnet branch of Theorem 107, the cap modular flow is the geometric flow of the diamond-preserving conformal Killing field. In the tangent diamond \(D_\ell\), that field is \[ \xi_{D_\ell} \mathrel{=} \frac{1}{2\ell} \Bigl((\ell^2-r^2-t^2)\,\partial_t-2t\,x^i\partial_i\Bigr), \] so on the \(t=0\) slice one has \[ \xi_{D_\ell}\!\cdot n=\frac{\ell^2-r^2}{2\ell}, \] with \(n^a=u^a\) the slice normal at the center.

The same kernel is the bounded-interval kernel of Lemma 238. Along each null generator of the diamond, Theorem 170 identifies the half-line generator constructed above with the local null-stress charge, and Lemma 238 transports that identification to the affine-covariant interval weight that vanishes at the two diamond endpoints. On the \(t=0\) slice that weight is exactly \((\ell^2-r^2)/(2\ell)\). Therefore the null-stress bridge reconstructs the bulk modular charge of the geometric generator as \[ \delta\langle B_C\rangle \mathrel{=} \int_{B_\ell}\frac{\ell^2-r^2}{2\ell}\,\delta\langle T_{00}\rangle\,d^3x +\frac{1}{2\pi}\,\delta\langle E^{(\eta)}_{C,\ell}\rangle. \] Multiplying by the first-law factor \(2\pi\) gives the displayed formula for \(\delta S_{\mathrm{bulk}}(C)\).

If \(\delta\langle T_{00}\rangle\) is approximately constant across \(B_\ell\), then \[ \int_{B_\ell}\frac{\ell^2-r^2}{2\ell}\,\delta\langle T_{00}\rangle\,d^3x \mathrel{=} \frac{4\pi\ell^4}{15}\,\delta\langle T_{00}\rangle +O(\ell^5\partial T). \] The additional hypothesis \(\delta\langle E^{(\eta)}_{C,\ell}\rangle=o(\ell^4)\) then yields \[ \delta S_{\mathrm{bulk}}(C) \mathrel{=} 2\pi\cdot\frac{4\pi\ell^4}{15}\,\delta\langle T_{00}\rangle +O(\ell^5\partial T)+o(\ell^4), \] which is exactly the stated coefficient. No separate EFT small-ball first law has been used: the kernel comes from the geometric cap generator together with the D4 null-stress bridge. ◻

Proof. Lemma 200 expresses the bulk entropy variation as the geometric cap-kernel stress term plus the carried modular/collar operator \(E^{(\eta)}_{C,\ell}\) and the ordinary long-wavelength derivative remainder. The finite-stage modular-defect propagation theorem bounds expectations of \(E^{(\eta)}_{C,\ell}\) by the Fawzi–Renner observable error, the fixed-collar Markov replacement modulus, and the support-visible regularized modular-transport remainder. Inserting that estimate into fixed-cap generalized-entropy stationarity and dividing by the small-ball coefficient gives the displayed \(\mathcal E_{\ell,\delta}\). ◻

Proof. This is the standard fixed-volume small-geodesic-ball identity used in Jacobson-style entanglement-equilibrium derivations. In \(3+1\) dimensions the coefficient is \(\Omega_2\ell^4/(4^2-1)=4\pi\ell^4/15\). The sign is fixed by the fact that positive curvature decreases the area of a fixed-volume small ball. The metric-term ambiguity left by null data is represented by \(\Lambda g_{ab}\). ◻

Proof. Theorem 198 gives the fixed-cap generalized-entropy stationarity condition \[ 0=\delta S_{\mathrm{gen}}(C) \mathrel{=} \delta S_{\mathrm{bulk}}(C)+\frac{\delta A}{4G}. \] By Lemma 200, \[ \delta S_{\mathrm{bulk}}(C) \mathrel{=} \frac{8\pi^2\ell^4}{15}\,u^a u^b\,\delta\langle T_{ab}\rangle +O(\ell^5\partial T)+o(\ell^4), \] where in the adapted rest frame the contraction is \(\delta\langle T_{00}\rangle\).

By Theorem 204, the fixed-volume small-ball area variation gives \[ \delta A\big|_{V,\Lambda} \mathrel{=} -\frac{4\pi\ell^4}{15}\, \delta\!\Bigl[(G_{ab}+\Lambda g_{ab})u^a u^b\Bigr]. \] Therefore \[ 0 \mathrel{=} \frac{8\pi^2\ell^4}{15}\,u^a u^b\,\delta\langle T_{ab}\rangle -\frac{\pi\ell^4}{15G}\, \delta\!\Bigl[(G_{ab}+\Lambda g_{ab})u^a u^b\Bigr] +O(\ell^5\partial T)+o(\ell^4). \] Dividing by \(\ell^4\) and taking the small-ball limit removes the \(O(\ell^5\partial T)+o(\ell^4)\) terms and yields \[ \delta\!\Bigl[(G_{ab}+\Lambda g_{ab})u^a u^b\Bigr] \mathrel{=} 8\pi G\,u^a u^b\,\delta\langle T_{ab}\rangle. \] This is precisely the desired rest-frame scalar relation. ◻

Proof. Define \[ Y_{ab}:=G_{ab}+\Lambda g_{ab}-8\pi G\,\langle T_{ab}\rangle. \] At any point \(p\), Theorem 205 gives \[ u^a u^b\,\delta Y_{ab}=0 \] for the unit future-directed four-velocity \(u^a\) of the local diamond rest frame. The all-directions hypothesis of this theorem then gives \[ u^a u^b\,\delta Y_{ab}=0 \qquad \text{for every unit timelike }u^a. \]

Choose local inertial coordinates at \(p\) and write \[ u^a=\gamma(1,v^i), \qquad |\vec v|<1, \qquad \gamma=(1-|\vec v|^2)^{-1/2}. \] Then \[ 0=u^a u^b\,\delta Y_{ab} \mathrel{=} \gamma^2\Bigl(\delta Y_{00}+2v^i\delta Y_{0i}+v^iv^j\delta Y_{ij}\Bigr) \] for every \(|\vec v|<1\). The quadratic polynomial in \(v^i\) therefore vanishes on an open ball, so all of its coefficients vanish: \[ \delta Y_{00}=0,\qquad \delta Y_{0i}=0,\qquad \delta Y_{ij}=0. \] Thus \[ \delta Y_{ab}=0 \] as a full tensor.

Along any path of admissible fixed-cap MaxEnt variations inside the connected scaling branch, \(Y_{ab}\) is therefore constant. The first-variation data determine \(Y_{ab}\) only relative to a chosen reference value. Theorem 228 separately uses the null ambiguity classification, Ward and Bianchi identities, connectedness, and the vacuum-reference receipt to reduce and evaluate this integration tensor; without those premises the proof stops at \(\delta Y_{ab}=0\). ◻

Proof. Taking the divergence of the displayed equation gives \[ 0=\nabla^aF_{ab} =\kappa\nabla^aT_{ab}+\nabla^a(\lambda g_{ab}) =\nabla_b\lambda. \] Hence the gradient of \(\lambda\) vanishes. On a connected chart, a scalar with zero gradient is constant. Without connectivity the same pointwise null-data argument can assign different constants on different components, so the connectivity hypothesis is load-bearing. ◻

Proof. (i) The first variation of the generalized entropy on a fixed cap \(C\) is, by Proposition 91 together with the area-term identification of Definition 92, \[ \delta S_{\mathrm{gen}}(C) =\delta S_{\mathrm{bulk}}(C) +\delta\!\Bigl(\frac{A(\partial C)}{4G_{\mathrm{geom}}}\Bigr) =\delta S_{\mathrm{bulk}}(C)+\frac{\delta A}{4G_{\mathrm{geom}}}. \] The coefficient \(1/(4G_{\mathrm{geom}})=\bar\ell(t)/a_{\mathrm{cell}}\) in front of \(\delta A\) is fixed by Definition 92: it is the refinement-scaling matching of \(\mathrm{Tr}(\rho_C L_C)\) to the area term, and uses no input outside the OPH edge-center entropy operator and the cell-area scaling. Theorem 198 gives \(\delta S_{\mathrm{gen}}(C)=0\) on the admissible fixed-cap MaxEnt class, so the same coefficient enters the small-ball balance \[ 0=\delta S_{\mathrm{bulk}}(C)+\frac{\delta A}{4G_{\mathrm{geom}}}. \] Lemma 200 expresses \(\delta S_{\mathrm{bulk}}(C)\) in terms of \(u^au^b\,\delta\langle T_{ab}\rangle\) using only the geometric cap generator and the D4 null-stress bridge, with no extra small-ball first law and no measured constant. The standard fixed-volume area-variation identity for a small geodesic ball in \(d=4\) gives \[ \delta A\big|_{V,\Lambda}=-\frac{4\pi\ell^4}{15}\,\delta\!\Bigl[(G_{ab}+\Lambda g_{ab})u^au^b\Bigr], \] and the proof of Theorem 205 divides through by \(\ell^4/(15)\) to obtain the rest-frame scalar relation. The factor of \(G\) on the right-hand side of Theorem 205 is exactly the same \(G_{\mathrm{geom}}\) introduced through Definition 92, because nothing else has been substituted between \(\delta S_{\mathrm{gen}}(C)=0\) and the rest-frame relation. Theorem 206 then promotes the scalar relation to \[ \delta\!\left(G_{ab}+\Lambda g_{ab} -8\pi G_{\mathrm{geom}}\langle T_{ab}\rangle\right)=0 \] using the Lorentz branch of Theorem 107 and rest-frame coverage of timelike directions. Hence the first-variation coefficient is \(G_{\mathrm{geom}}\). Theorem 228 uses its named vacuum reference and common-domain premises to evaluate the remaining integration tensor; it changes no coefficient.

(ii) On the realized product-group branch, Proposition 93 together with the D10 pixel law gives \(\bar\ell_{\mathrm{shared}}=P/4\), and Proposition 94 gives \(a_{\mathrm{cell}}=P\,\ell_\star^2\). Therefore \[ G_{\mathrm{geom}}=\frac{a_{\mathrm{cell}}}{4\bar\ell_{\mathrm{shared}}}=\frac{P\,\ell_\star^2}{4(P/4)}=\ell_\star^2. \] Substituting into part (i) yields the local-pixel tensor first-variation coefficient. Under the additional absolute-branch premises stated there, Theorem 228 yields the displayed local-pixel Einstein equation.

(iii) The additional premises invoke the absolute equation of Theorem 228. Reduce that equation in the Newton-Poisson regime, using signature \((-,+,+,+)\) and geometric units \(c=1\). On laboratory and solar-system scales the curvature radius \(r\) of the Newtonian source is much smaller than the cosmological horizon \(r_{\mathrm{dS}}=\sqrt{3/\Lambda}\) (Corollary 245), so all \(\Lambda\)-dependent corrections to \(\Phi\) are bounded by \(\Lambda\,r^2\) and are dropped at this scale; the Newton-Poisson reduction therefore coincides with the reduction of \(G_{ab}=8\pi G_{\mathrm{geom}}\,\langle T_{ab}\rangle\). Write \(g_{ab}=\eta_{ab}+h_{ab}\) with \(|h_{ab}|\ll 1\), \(h:=\eta^{ab}h_{ab}\), and \(\bar h_{ab}:=h_{ab}-\tfrac12\eta_{ab}h\). In the harmonic gauge \(\partial^a\bar h_{ab}=0\) the linearization of the absolute equation about flat space is the standard linearized-Einstein wave equation \[ \Box\,\bar h_{ab}=-16\pi G_{\mathrm{geom}}\,T_{ab}, \qquad \Box:=\eta^{ab}\partial_a\partial_b=-\partial_t^2+\nabla^2. \] For a static, classical, slow-motion source with \(T_{00}=\rho\) and \(|T_{0i}|,|T_{ij}|\ll T_{00}\), time derivatives drop and the dominant component is the time-time component: \[ \nabla^2\bar h_{00}=-16\pi G_{\mathrm{geom}}\,\rho. \] On the static weak-field branch with the standard Newtonian-limit metric \(g_{00}=-(1+2\Phi)\), \(g_{ij}=(1-2\Phi)\delta_{ij}\), one has \(h_{00}=-2\Phi\), \(h_{ij}=-2\Phi\,\delta_{ij}\), and \[ h=\eta^{ab}h_{ab}=-h_{00}+\delta^{ij}h_{ij}=2\Phi-6\Phi=-4\Phi, \qquad \bar h_{00}=h_{00}-\tfrac12\eta_{00}h=-2\Phi-\tfrac12(-1)(-4\Phi)=-4\Phi. \] Substituting \(\bar h_{00}=-4\Phi\) into the displayed wave equation gives \[ \nabla^2(-4\Phi)=-16\pi G_{\mathrm{geom}}\,\rho \quad\Longleftrightarrow\quad \nabla^2\Phi=4\pi G_{\mathrm{geom}}\,\rho, \] which is the Newton-Poisson equation with coupling \(G_{\mathrm{geom}}\). The geodesic equation \(\ddot x^a+\Gamma^a_{bc}\dot x^b\dot x^c=0\) in the slow-motion limit reduces to \(\ddot x^i=-\Gamma^i_{00}=-\partial^i\Phi\), i.e. \(\ddot{\mathbf x}=-\nabla\Phi\). All factors of \(G\) in this reduction are the same \(G_{\mathrm{geom}}\) as in part (i): no measured Newton constant, no Planck area, and no measured cosmological constant has been substituted. ◻

Proof. (a) Theorem 198 and Proposition 91 give \[ 0=\delta S_{\mathrm{gen}}(C)=\delta S_{\mathrm{bulk}}(C)+\delta\!\bigl(\mathrm{Tr}(\rho_C L_C)\bigr)=\delta S_{\mathrm{bulk}}(C)+\frac{\bar\ell(t)}{a_{\mathrm{cell}}}\,\delta A, \] where the last equality uses the linear-in-area scaling. By Lemma 200, \[ \delta S_{\mathrm{bulk}}(C)=\frac{8\pi^2\ell^4}{15}\,u^au^b\,\delta\langle T_{ab}\rangle+O(\ell^5\partial T)+o(\ell^4), \] and the standard fixed-volume area-variation identity for a small geodesic ball in \(d=4\) gives \[ \delta A\big|_{V,\Lambda}=-\frac{4\pi\ell^4}{15}\,\delta\!\bigl[(G_{ab}+\Lambda g_{ab})u^au^b\bigr]. \] Substituting, \[ 0=\frac{8\pi^2\ell^4}{15}\,u^au^b\,\delta\langle T_{ab}\rangle-\frac{\bar\ell(t)}{a_{\mathrm{cell}}}\,\frac{4\pi\ell^4}{15}\,\delta\!\bigl[(G_{ab}+\Lambda g_{ab})u^au^b\bigr]+O(\ell^5\partial T)+o(\ell^4). \] Dividing by \(\ell^4\) and taking the small-ball limit removes the remainder terms and yields \[ \delta\!\bigl[(G_{ab}+\Lambda g_{ab})u^au^b\bigr]=\frac{2\pi\,a_{\mathrm{cell}}}{\bar\ell(t)}\,u^au^b\,\delta\langle T_{ab}\rangle. \] The tensor upgrade of Theorem 206 promotes this rest-frame relation to \[ \delta\!\left(G_{ab}+\Lambda g_{ab} -\frac{2\pi a_{\mathrm{cell}}}{\bar\ell(t)} \langle T_{ab}\rangle\right)=0. \] Thus \(\kappa_{\mathrm{OPH}}=2\pi a_{\mathrm{cell}}/\bar\ell(t)\) is the structural first-variation coefficient. This step uses the linear-in-area scaling before a \(1/(4G_{\mathrm{geom}})\) naming convention is introduced.

(b) Under the additional premises, Theorem 228 evaluates the integration tensor and gives \(G_{ab}+\Lambda g_{ab}=\kappa\langle T_{ab}\rangle\). Linearize that absolute equation on a flat background as in part (iii) of Theorem 208, with \(\kappa\) kept general. Signature \((-,+,+,+)\), geometric units \(c=1\). The harmonic-gauge linearized Einstein equation, with \(\Lambda\)-dependent corrections at scales \(r\ll r_{\mathrm{dS}}=\sqrt{3/\Lambda}\) bounded by \(\Lambda r^2\) and dropped exactly as in part (iii), is \[ \Box\,\bar h_{ab}=-2\kappa\,T_{ab}. \] For a static, classical, slow-motion source with \(T_{00}=\rho\) and \(|T_{0i}|,|T_{ij}|\ll T_{00}\), this reduces to \[ \nabla^2\bar h_{00}=-2\kappa\,\rho. \] The standard Newtonian-limit metric \(g_{00}=-(1+2\Phi)\), \(g_{ij}=(1-2\Phi)\delta_{ij}\) gives \(\bar h_{00}=-4\Phi\), so \[ \nabla^2(-4\Phi)=-2\kappa\,\rho \quad\Longleftrightarrow\quad \nabla^2\Phi=\frac{\kappa}{2}\,\rho. \] Comparing with the Newton-Poisson convention \(\nabla^2\Phi=4\pi G_N\,\rho\) gives \(\kappa=8\pi G_N\). Substituting \(\kappa_{\mathrm{OPH}}\) from (a) yields \(\kappa_{\mathrm{OPH}}=8\pi G_N\), as displayed.

(c) Solve (b) for \(G_N\) and substitute (a): \[ G_N=\frac{\kappa_{\mathrm{OPH}}}{8\pi}=\frac{2\pi\,a_{\mathrm{cell}}/\bar\ell(t)}{8\pi}=\frac{a_{\mathrm{cell}}}{4\bar\ell(t)}. \] This is exactly \(G_{\mathrm{geom}}\) of Definition 92; the factor of \(4\) is the ratio of the OPH structural coefficient \(\kappa_{\mathrm{OPH}}\) to the Newton-Poisson constant \(8\pi\). Equivalently, the dictionary form \(\mathrm{Tr}(\rho_C L_C)=A(\partial C)/(4G_{\mathrm{geom}})\) of Definition 92 follows by substituting \(G_{\mathrm{geom}}=a_{\mathrm{cell}}/(4\bar\ell(t))\) into the structural linear-in-area scaling \(\mathrm{Tr}(\rho_C L_C)=(\bar\ell(t)/a_{\mathrm{cell}})\,A(\partial C)\). ◻

Proof. Fix a model \(M\) of the bare consensus language. Attach Minkowski space, \(T_{ab}=0\), \(\Lambda=0\), and any \(G>0\) to obtain an extension in which the Einstein equation holds. Attach the same Minkowski metric and consensus reduct together with a nonzero symmetric tensor \(T'_{ab}\) to obtain an extension in which it fails. Every bare-consensus sentence has the same truth value in both extensions because the reducts are identical. The Einstein equation has different truth values, so it is not a logical consequence of the reduct. ◻

Proof. The symmetric square vectors \(k\otimes k\), \(k\) null, span the \(\eta\)-orthogonal complement of nothing less than the full ten-dimensional symmetric space except for the relation \(\eta^{ab}(k\otimes k)_{ab}=0\): concretely, polarization of \(T(k,k)\) over null \(k=(1,\Omega)\), \(\Omega\in S^2\), yields all components \(T_{00}\), \(T_{0i}+T_{i0}\), and the symmetric spatial moments \(T_{ij}\Omega^i\Omega^j\) against the \(\ell\leq2\) spherical harmonics, which determine \(T_{ij}\) up to its trace; the only undetermined combination is \(T_{00}+\delta^{ij}T_{ij}\) paired against \(\ell=0\), i.e. the \(\eta\)-trace. A symmetric form vanishing on the whole null cone is therefore proportional to \(\eta\) (the standard null-invisibility statement, used in the tensor upgrade of Theorem 205). The converse and the nine-direction count follow because \(\dim\operatorname{span}\{k\otimes k:\ k\ \text{null}\}=9\); Lipschitz stability is the finite-dimensional least-squares bound with constant \(1/\sigma_9\) of the direction design matrix. Direct evaluation on the displayed frame gives determinant \(8192/27\ne0\), which proves that this particular system is invertible. ◻

Proof. Compatibility: a dependent null family \(\sum c_ik_i\otimes k_i=0\) pulls back under the blow-up to a dependent family of translation generators; by Theorem 181(2), \(\sum c_iP_{\Omega_i}=0\), and the same relation holds for the endpoint-derivative densities after smearing because each \(T[f;\Omega]\) is the \(f\)-weighted disintegration of \(P_\Omega\) along its generator (Definition 216 and Lemma 169); the finite-stage defect is bounded by the \(\mathsf{MI}\) residual \(\varepsilon_r\) and vanishes in the limit. Theorem 215 then yields the tensor with the stated ambiguity, measurably in the base point by the Lipschitz reconstruction bound applied chartwise. Symmetry is built into the tomography. Locality is support-local commutation transported through the blow-up: intervals outside the causal support of \(f\) contribute commuting differences whose endpoint derivatives vanish in the canonical normalization. Covariance is Theorem 181(1) applied to each interval charge, plus naturality of the partition family under the Möbius transport of Theorem 179. The generator identification integrates the disintegration against \(f\to1\) on the generator through the chart, with the monotone refinement errors controlled by the kernel-residual receipt; this is exactly the content of Theorem 170 with the density constructed. For the Ward identity, fix a chart and a coordinate direction \(x^\mu\); then \(T[f(\cdot-x);\Omega]=\langle U(x)^{*}(\cdots)U(x)\rangle\)-transported charges equal \(T[f;\Omega]\) up to the state-variation term, which vanishes because \(\Omega\)-invariance \(U(x)\Omega=\Omega\) holds for the reference state and the semiclassical expectation is taken in that state; dividing by \(|x|\) and letting \(x\to0\) gives \(\partial^\mu T[\partial_\mu f;\Omega]=0\), i.e. the weak coordinate conservation, for every \(\Omega\), hence for the reconstructed tensor. Chart transitions are Poincaré by \(\mathsf{E4}\), so the statement globalizes to the a.e. covariant divergence on the \(C^{1,1}\) atlas. The ambiguity classification is Theorem 215 plus the standard improvement-term observation that double-divergence additions with antisymmetrized index blocks change no face charge. ◻

Proof. Take any consistent family and add to one direction’s charge a bump violating one dependent-family relation; Theorem 215 shows solvability fails. For any candidate coefficients \(x\), write \(r=Ax-\tau\). Since \(c^{\mathsf T}A=0\), \(c^{\mathsf T}r=-c^{\mathsf T}\tau\); Cauchy–Schwarz gives \(\lVert r\rVert_2\geq |c^{\mathsf T}\tau|/\lVert c\rVert_2\). The scalar statement is the rank count in the same theorem. ◻

Proof. (1) is \(\delta S=-\operatorname{Tr}(\delta\rho\log\rho)\) (the \(\operatorname{Tr}\delta\rho=0\) term removes the normalization), followed by the split \(-\log\rho=K_C=2\pi B_C+Z_C\). (2) On the central-interface class, the displayed block state gives \(-\log\rho=\oplus_\alpha(-\log p_\alpha- \log\rho^{(\alpha)}_{\mathrm{bulk}}+\log d_\alpha)\). The first two terms form the bulk modular generator and the last term is \(Z\) on the declared normalization. The central term pairs with \(\delta\rho\) through the sector traces: \(\delta\langle Z\rangle=\sum_\alpha z_\alpha\delta p_\alpha\). Choosing \(z_\alpha=\log d_\alpha\) matches \(\delta S_{\mathrm{edge}}\) term by term, and substituting into (1) gives the boxed split. The bulk-only statement follows by subtracting \(\delta S_{\mathrm{edge}}\) from the exact entropy split. (3) restates (2) through Araki relative entropy: first-order stationarity of \(S(\rho\Vert\rho_{\mathrm{ref}})\) at \(\rho=\rho_{\mathrm{ref}}\) is the same identity with \(\langle K\,\delta\rho\rangle\) read as the relative-modular pairing, and sector weights are declared data of the central-interface branch. ◻

Proof. The MaxEnt state at constraint value \(t\) is \(\rho(t)=e^{-\lambda(t)T_{ff}-\mu(t)}/Z\)-type within the declared family; differentiating \(S(\rho(t))=\lambda t+\mu+\log Z\)-form along the family and using the constraint derivative gives the exact envelope identity \(dS/dt=\lambda\) (the standard Legendre/envelope computation, exact at finite dimension: \(dS/dt=-\operatorname{Tr}(\dot\rho\log\rho) =\lambda\operatorname{Tr}(\dot\rho T_{ff})+\mu\operatorname{Tr}\dot\rho =\lambda\)). The KMS normalization identifies \(\lambda=2\pi\) at the reference point by Theorem 107. For the coupled class, decompose any first-order variation into a component along the MaxEnt family (constraint value moves by \(\delta t\)) and a component in the fixed-\(t\) fiber; the fiber component obeys \(\delta S\leq0\) with equality on the family (MaxEnt optimality, first order zero), and the along-family component contributes \(\delta S=\lambda\delta t\) by the envelope identity, with the constraint charge identified as the modular charge through Theorem 217(3). Substituting \(\delta S=2\pi\delta\langle B\rangle+\delta S_{\mathrm{edge}}\) from the boxed split gives the displayed Clausius balance. Nonemptiness of the class is witnessed by the family itself and by the sector-weight variations of Theorem 220(2). ◻

Proof. (i) is immediate from the balance identity: replacing \(1/(4G)\) by \(\gamma\) leaves the uncancelled multiple of \(\delta A\). (ii) is the computation in Theorem 220(2) with the mismatched normalization; the machine receipt instantiates it with explicit blockwise states. ◻

Proof. The rate receipt gives \[ \varepsilon_r \le (\ell_r/\ell_0)^s e^{-\omega_r}. \] Proposition 84 therefore gives \[ r_{\mathrm{FR}}(\varepsilon_r) \le2\sqrt{\varepsilon_r} \le2(\ell_r/\ell_0)^{s/2}e^{-\omega_r/2} =o(\ell_r^4) \] because \(s/2>4\). The uniform fixed-collar modulus gives \[ \delta_r^{\mathrm M} \le C_{\mathrm M}(\ell_r/\ell_0)^{s\theta} e^{-\theta\omega_r} =o(\ell_r^4) \] because \(s\theta>4\). The kernel estimate is a premise. The domain statement holds because the finite-rank cores of Proposition 166 are nested along the declared family and their union is a common core by the mixed-GNS Cauchy clause. Uniformity is the region-uniformity of the hypotheses. ◻

Proof. The bound gives \[ 0\leq\frac{e_r}{\ell_r^4}\leq C\ell_r^\eta\longrightarrow0. \]  ◻

Proof. \(\mathsf{E1}\) populates every chart box with events carrying frame data; Lemma 240 produces, from cap pairs, modular data whose rest-frame directions realize a neighborhood of any given timelike direction; the produced \(G\)-action (unitarily implemented on the producer branch) acts transitively on the fiber \(H^3\) of directions (Proposition 139) and continuously in the region, so the orbit of the realized set is all of the direction bundle over the region; connectedness follows since \(G\) and the region are connected and \(\mathsf{E4}\) transitions are continuous. ◻

Proof. The chain Theorem 198 \(\to\) Lemma 200 \(\to\) Theorem 204 \(\to\) Theorem 205 runs with every required input supplied: the locally Lorentzian \(d=4\) regime and the geodesic-ball identity are consumed from the event packet on its smooth branch (the fixed-volume area expansion is a metric identity requiring only the constructed \(C^\infty\)-branch metric and no field equation); the small-ball kernel is the constructed density of Theorem 217(3) through Lemma 200; the remainders are uniform by Theorem 223; the first-variation identity with the moving stress channel is Theorem 221; and coverage is Theorem 225. The timelike tensor upgrade gives \(\delta Y_{ab}=0\) along the covered variation class and thus fixes changes relative to a reference, but does not fix the baseline. At each event, the null-balance premise and Lemma 171 instead give the pointwise form \(G_{ab}-8\pi G\langle T_{ab}\rangle=\lambda(x)g_{ab}\). The Ward identity of Theorem 217(4), the contracted Bianchi identity, and metric compatibility then imply \(\nabla_b\lambda=0\), so \(\lambda\) is constant per connected component by Lemma 207. \(\mathsf{VR}\) evaluates that constant at the reference state, giving exactly the displayed reference equation for \(\Lambda=-\lambda\). Definition 92 identifies the area-term coefficient as \(1/(4G)\); the two small-ball coefficients then give the same \(8\pi G\) in the tensor first-variation relation. No step introduces a measured constant. The degraded conclusion without \(\mathsf{VR}\) is the same argument stopped before the evaluation step. ◻

Proof. (i) is the observation in the theorem’s proof that the union-core exists only along a declared family; an adversarial stage assignment with disjoint cores has no common domain. (ii) is Proposition 195(i) fed through Theorem 225. (iii) constancy is proved per component only; the example is two disjoint receipt regions. (iv) both backgrounds are maximally symmetric, so all covered variations produce identical \(\delta Y=0\) data while \(c\) differs by \(\Lambda_{\mathrm{dS}}\); this is the standard integration-constant freedom, here pinned only by \(\mathsf{VR}\). ◻