Proof. This is the support-visible compact-gauge Yang–Mills form theorem of the compact OPH paper, where it appears as Theorem thm:oph-4d-euclidean-yang-mills-form . The proof spine is recalled here because it fixes the target Hamiltonian for the mass-gap step.

The present Yang–Mills branch takes compact simple \(G\) as branch data. When \(G\) is imported from OPH reconstruction, that import is conditional on the compact-gauge refinement receipt; on that tail the zero-obstruction transportable bosonic sector category and its faithful forgetful fiber functor reconstruct \(G\) . At fixed cutoff, the declared compact-gauge patch-carrier presentation gives support-visible link holonomies and plaquette holonomies. In the refinement limit, the zero-obstruction gluing law makes infinitesimal rectangle holonomies multiplicative and path-local. The Cauchy/regularity certificate in items (Y1)–(Y3) upgrades the generalized-holonomy cluster state to a local connection \(A\) on the four-dimensional scaling chart, and the infinitesimal plaquette defect is \[ U_{\mu\nu}(\varepsilon,x) \mathrel{=} \mathbf 1+\varepsilon^2F_{\mu\nu}(x)+O(\varepsilon^3), \qquad F=dA+A\wedge A. \]

The Euclideanized MaxEnt/local-Gibbs branch gives a local finite-range action density built from support-visible gauge-invariant collar data. Gauge quotienting permits only class functions of the curvature and its covariant derivatives. The four-dimensional scaling chart, Euclidean rotation invariance, locality, and reflection positivity leave one relevant dimension-four positive quadratic invariant in the pure gauge sector: \[ \langle F_{\mu\nu},F_{\mu\nu}\rangle. \] The possible topological density \(\langle F\wedge F\rangle\) is reflection odd and belongs to a separate topological-angle sector; it is absent on the ordinary reflection-positive zero-obstruction vacuum branch used here. Higher curvature powers and covariant-derivative terms are irrelevant operators under the declared continuum scaling; their disappearance in the extracted state is part of the uniform renormalized remainder control in (Y1)–(Y3). Normalizing the unique positive quadratic invariant defines the coupling \(g\) and gives (YM).

The finite-stage cylinder measures are the gauge-register / quantum-link Gibbs measures pushed to the support-visible quotient. Proposition 18 proves the compatible support-visible weak-\(*\) / GNS cylinder extraction. The generator and transfer-semigroup identification use the separate dynamic items (Y4)–(Y5) of Assumption 20; weak-* compactness alone does not provide them. ◻

Proof. The curvature \(F=dA+A\wedge A\) obeys the covariant Bianchi identity \(DF=0\). Varying the quadratic curvature action \[ S[A,J] \mathrel{=} -\frac{1}{2g^2}\int \langle F\wedge *F\rangle +\int \langle A\wedge *J\rangle \] with respect to compactly supported variations of \(A\) gives \[ \delta S \mathrel{=} \int \langle \delta A\wedge (g^{-2}D{*}F-{*}J)\rangle \] up to a boundary term. Stationarity gives \(D{*}F=g^2{*}J\). The source-free equation is the case \(J=0\). ◻

Proof. On the exact-Markov branch, repair preserves exactly the repaired visible datum \(\rho_C\), changes only complementary invisible fiber data, and acts on the quotient-first physical algebra rather than on microscopic representatives. Let \(\Phi_C\) be the Heisenberg repair map and let \(\mathcal N_C\) be the repaired local fixed algebra. Exact repair semantics require \(\mathcal N_C\) to be exactly the \(\rho_C\)-measurable subalgebra. They give \[ \Phi_C(a)=a \quad (a\in\mathcal N_C), \] \[ \Phi_C(\mathcal A_r^{G,\mathrm{sv}})\subseteq\mathcal N_C, \qquad \omega_r\circ\Phi_C=\omega_r, \] and \(\Phi_C\) is \(\mathcal N_C\)-bimodular. Therefore, for \(a\in\mathcal N_C\) and \(x\in\mathcal A_r^{G,\mathrm{sv}}\), \[ \omega_r\!\left(a^*\Phi_C(x)\right)=\omega_r(a^*x). \] Since \(\Phi_C(x)\in\mathcal N_C\), this equation characterizes its class in \(K_r\) as the orthogonal projection of \(x\) onto the \(\rho_C\)-measurable subspace, uniquely modulo \(\pi_r\)-null functions. That orthogonal projection is the \(\pi_r\)-preserving conditional expectation \(E_C\). ◻

Proof. This is the complete-fiber homogeneity receipt. Quotienting removes declared implementation labels, but the proof must also check that no remaining source constraint or relaxation rate distinguishes points in the fiber. Under that check, the conditioned state and primitive relaxation carry the stated full measure-preserving symmetry. ◻

Proof. For a finite uniform fiber, the symmetric-group representation on the mean-zero subspace is irreducible. For a standard atomless fiber, use equal-measure finite partitions. Automorphisms within pieces and permutations of pieces force every bounded commutant operator to be scalar on the mean-zero step functions; compatibility under refinements gives one scalar, and these step functions are dense in \(L^2_0(F,\nu)\). Positivity and the kernel condition make the scalar strictly positive. The formula follows. ◻

Proof. The finite receipt supplies the ground-state transform and its primitive local pieces. Each \(D_C\) is supported on one collar \(C\), preserves exactly the repaired visible datum \(\rho_{C,r}\), and relaxes the complete complementary conditional fiber. Hence \[ \ker D_C=\operatorname{Ran}(E_C). \]

Lemma 7 gives the full hidden-fiber permutation symmetry. Applying Lemma 8 fiberwise gives \(D_{C,y}=c_C(y)(I-E_{C,y})\). The cross-fiber receipt makes \(c_C(y)=c_C\), so \[ D_C=c_C(I-E_C). \] The scalar is positive because \(C\) is active. Branch homogeneity makes it a collar-type scalar, so summing over the active collars gives (1), and exponentiation of the positive self-adjoint generator gives (2). ◻

Proof. The type tuple is a complete source signature. Items (G1)–(G2) therefore identify precisely a finite image with a positive finite minimum, while item (G4) prevents refinement from producing a new or rate-degenerate type. ◻

Proof. Every \(P_{v,r,b}\) is an orthogonal conditional expectation. The Dobrushin comparison applied to item (G3) gives (AT), so for \(f\perp\mathbf1\), \[ \langle f,L_{r,b}^{\mathrm{rep}}f\rangle =\sum_vc_{v,r,b}\|(I-P_{v,r,b})f\|_2^2 \ge\frac{c_*}{A_*}\|f\|_2^2. \] This proves the operator bound, and the spectral theorem gives the transfer bound. The four uniformities follow because items (G1)–(G4) use one type table and one approximate-tensorization modulus on the entire cofinal tower. ◻

Proof. For \(q\preceq r\preceq s\), coarse-shadow functoriality gives \[ (\omega_s\circ\iota_{rs})\circ\iota_{qr} =\omega_s\circ\iota_{qs}, \] proving (i).

For a fixed local cylinder algebra, its state space is weak-* compact by Banach–Alaoglu. The chosen local algebras are separable because compact metrizable \(G\) has a countable Peter–Weyl test algebra and a finite regulator has only finitely many record and sector generators. Their state spaces are therefore weak-* metrizable. Successively extract a subsequence for the first, second, and later cylinders, choosing the \(k\)-th diagonal index beyond the \(k\)-th regulator stage. The diagonal subsequence is cofinal and converges on all cylinders. For a general directed regulator set, use a convergent subnet in the compact product of the local state spaces; unresolved early coordinates may be filled arbitrarily because each fixed coordinate is genuinely resolved on a cofinal tail. Weak-* continuity of precomposition and part (i) yield \[ \overline\omega_r\circ\iota_{qr} =\lim_\alpha\omega_{s_\alpha\downarrow r}\circ\iota_{qr} =\lim_\alpha\omega_{s_\alpha\downarrow q} =\overline\omega_q, \] which proves (ii).

For a representative \([a]_r\) in the algebraic inductive limit, define \[ \omega_\infty([a]_r):=\overline\omega_r(a). \] Compatibility makes this well defined. Positivity and normalization are checked in one finite algebra containing the element. For every \(t\succeq r\), \[ |\overline\omega_r(a)| =|\overline\omega_t(\iota_{rt}(a))| \le\|\iota_{rt}(a)\|. \] Taking the infimum along the tail gives the inductive-limit norm bound even for noninjective bonding maps, so the state extends uniquely to the \(C^*\)-completion. This proves (iii).

On a commutative local cylinder algebra, the Riesz representation theorem identifies the null ideal with the continuous functions that vanish on the measure support. The support quotient is therefore \(C(\operatorname{supp}\mu_O)\), where \(\mu_O\) is the representing local Radon measure, and its induced state is faithful. If \(O\subseteq O'\), state compatibility gives \[ a\in\mathcal N_\omega(O) \quad\Longleftrightarrow\quad \iota_{OO'}(a)\in\mathcal N_\omega(O'). \] Thus the inclusions descend injectively and the support quotients form an isotone faithful local net. For \(r\preceq t\), define \[ V_{rt}[a]_r:=[\iota_{rt}(a)]_t. \] It is well defined and isometric because \[ \|V_{rt}[a]_r\|^2 =\overline\omega_t(\iota_{rt}(a^*a)) =\overline\omega_r(a^*a). \] These maps compose, preserve \(\mathbf 1\), and intertwine the cylinder representations. Their Hilbert direct limit is the GNS completion of the cylinder union, proving (iv).

Finally, suppressing the \(d=4\) superscript, summing (FS) over \(x\) gives \(Z_s=\alpha_{sr}Z_r\). Division by the partition functions then gives \((p_{sr}^{(4)})_\#\mu_s^{(4)}=\mu_r^{(4)}\), equivalently \(\omega_s^{(4)}\circ\iota_{rs}^{(4)}=\omega_r^{(4)}\). The commuting time-zero square gives the corresponding \(d=0\) identity. ◻

Proof. Generalized Mosco convergence is equivalent to generalized strong-resolvent and transfer-semigroup convergence on the declared Hilbert identifications. In the exact alternative, the coherent finite contraction semigroups define strongly continuous direct-limit semigroups on the dense cylinder images, and uniform contractivity extends them to the completed Hilbert spaces. Their bounded operators are symmetric and positive on the dense finite-stage union, hence self-adjoint and positive after extension. Convergence of \(U_r\) and \(U_r^{-1}\) on dense form cores passes the finite-stage intertwining relation to the limit, so uniqueness of self-adjoint semigroup generators gives \(UHU^{-1}=L^{\mathrm{rep}}\). For a repair-form Mosco recovery sequence \(\psi_r\to\psi\), \[ q_r^{\mathrm{rep}}(\psi_r) \ge \delta_*\bigl(\|\psi_r\|^2-\|P_{0,r}\psi_r\|^2\bigr). \] Taking the limit and using strong convergence of the vacuum projections gives the continuum form inequality. In the exact direct-limit alternative, the finite bound is equivalently \[ \|e^{-tL_r^{\mathrm{rep}}}(I-P_{0,r})\|\le e^{-\delta_*t}; \] coherence passes this estimate to the direct-limit semigroup, and the spectral theorem gives the same continuum form inequality. ◻

Proof. Euclidean covariance, locality, reflection positivity, regularity, clustering, transfer convergence, and nontriviality are precisely the certificate items in Assumption 20. Proposition 18 supplies the four-dimensional Euclidean cylinder state \(K^E\) and its time-zero GNS restriction \(K\); the certificate supplies the missing regularity and transfer controls. The finite/limiting time-zero identity in (Y5) identifies the OS time-zero inner product and translation semigroup with the transfer-form limit \((\mathcal H,H)\). The standard Osterwalder–Schrader reconstruction theorem therefore produces the Hilbert space, vacuum, local algebra, and positive Hamiltonian, and identifies the physical time-translation semigroup with the certified Euclidean transfer semigroup . ◻

Proof. The compact-gauge witness and physical-UV landing theorem in the OPH compact paper supplies a realized nontrivial compact-gauge branch . At finite cutoff this gives a support-visible gauge-invariant local observable, such as a nonconstant Wilson/plaquette cylinder observable, whose vacuum variance is positive. Its GNS vector is orthogonal to the vacuum after subtracting its expectation value. The finite-range local-Gibbs generator assigns finite energy to finite-cylinder excitations. The continuum nontriviality step is not a consequence of weak-\(*\) state convergence alone; it is the variance-floor and finite-energy item (Y6) in Assumption 20. With that certificate, the local cylinder vectors survive in the support-visible continuum, so the continuum local algebra is strictly larger than \(\mathbb CI\) and contains non-vacuum finite-energy local excitations. ◻

Proof. At every finite stage, Proposition 13 gives \[ L_r^{\mathrm{rep}}\ge \delta_*(I-P_{0,r}). \] By Theorem 21, on the extracted cylinder/GNS space, this lower bound passes to the support-visible continuum: \[ L^{\mathrm{rep}}\ge \delta_*(I-P_0^K). \] Using (9), \(UHU^{-1}=L^{\mathrm{rep}}\). Conjugating the lower bound by \(U^{-1}\) gives (10), and the spectral statement (11) follows. Since unitary equivalence preserves the nonzero spectrum and identifies the vacuum with the constant sector, the infimum of the nonzero spectrum is the same on both sides, giving (12). ◻