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 compact-gauge branch reconstructs a compact group \(G\) from the zero-obstruction transportable bosonic sector category and its faithful fiber functor . 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. Therefore there is 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 and vanish from the strict Yang–Mills fixed form. 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. Assumption 1 supplies the compatible support-visible weak-\(*\) / GNS cylinder extraction. The resulting Euclidean transfer semigroup is therefore the transfer semigroup of (YM). ◻

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. The repair semantics give \[ \Phi_C(a)=a \quad (a\in\mathcal N_C), \] \[ \Phi_C(\mathcal A^{\mathrm{sv}}_r)\subseteq\mathcal N_C, \qquad \pi_r\circ\Phi_C=\pi_r, \] and \(\Phi_C\) is \(\mathcal N_C\)-bimodular. Therefore, for \(a\in\mathcal N_C\) and \(x\in\mathcal A^{\mathrm{sv}}_r\), \[ \pi_r\!\left(a^*\Phi_C(x)\right)=\pi_r(a^*x). \] Since \(\Phi_C(x)\in\mathcal N_C\) and \(\pi_r\) is faithful, this equation uniquely characterizes \(\Phi_C(x)\) as the orthogonal projection of \(x\) onto \(\mathcal N_C\) in the GNS inner product. That orthogonal projection is the \(\pi_r\)-preserving conditional expectation \(E_C\). ◻

Proof. The quotient removes implementation labels by construction: two representatives with the same repaired datum \(y\) and different hidden fiber coordinates define the same support-visible observable state. The MaxEnt rule conditioned on \(y\) therefore has no admissible support-visible constraint that can distinguish two points of \(F_C(y)\). The unique constraint-compatible conditioned state is the uniform state on that finite fiber. Any primitive collar relaxation preserving the OPH quotient must commute with the resulting full permutation action. ◻

Proof. The permutation representation on \(L^2(F)\) splits as constants plus the zero-sum subspace. The zero-sum subspace is irreducible for the full symmetric group when \(|F|\ge2\). Schur’s lemma therefore makes \(D_F\) a scalar on that subspace. Positivity and the kernel condition make the scalar strictly positive. The formula follows. ◻

Proof. The OPH MaxEnt axiom gives a local-Gibbs state with a quasi-local finite-range generator on the declared finite-constraint branch. After the support-visible quotient, each primitive local Euclidean piece \(D_C\) is supported on one collar \(C\), preserves exactly the repaired visible datum \(\rho_C\), and relaxes only complementary fiber data. Hence \[ \ker D_C=\operatorname{Ran}(E_C). \]

Lemma 5 gives the full hidden-fiber permutation symmetry. Applying Lemma 6 fiberwise gives a positive scalar \(c_C\) on the orthogonal complement of the repaired datum: \[ D_C=c_C(I-E_C). \] 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. Active collars are exactly the collars on which complementary invisible fiber data are genuinely relaxed. Hence \(D_C\neq0\) on \(\ker(E_C)\), so \(c_C>0\). Finite local combinatorial type gives a finite list of bounded collar patterns, including their visible interface alphabets, hidden fiber cardinalities, local constraint templates, and admissible repair maps. Branch homogeneity makes the conditioned MaxEnt relaxation scalar a function of this finite type data. Refinement stability says that refinement replaces a collar by copies of the same bounded type templates with the same normalized local Euclidean repair semantics, so no new rate-degenerating collar type appears along the cofinal family. Taking the minimum of \(c_C\) over the finite active type list gives \(c_*>0\). ◻

Proof. For each color \(a\), define the parallel color expectation \[ E_{r,a}:=\prod_{C\in\mathcal C_{r,a}}E_C. \] Same-color collars are disjoint, so the factors commute. Exact quotient-local gluing makes \(\{E_{r,a}\}_{a=1}^{q}\) a commuting family of orthogonal projections on the vacuum branch.

Repair completeness says that the only support-visible observables fixed by every local repair are constants: \[ \bigcap_{a=1}^{q}\operatorname{Ran}(E_{r,a})=\mathbb C\,\mathbf 1_r. \tag{7} \] Define \[ \widetilde L_r:=\sum_{a=1}^{q} c_*(I-E_{r,a}). \] For commuting projections, \[ I-\prod E_C\le \sum_C(I-E_C), \] so \(\widetilde L_r\le L_r^{\mathrm{rep}}\). The projections \(E_{r,a}\) are simultaneously diagonalizable. On the joint all-ones eigenspace, (7) says the vector is constant. Every nonconstant joint eigenspace has at least one color eigenvalue zero, so \[ \widetilde L_r\ge c_*(I-P_{0,r}). \] This proves (5), and the spectral gap bound (6) follows. ◻

Proof. The compact-gauge ladder in the OPH compact paper proves fixed-cutoff bosonic sector categories, faithful monoidal refinement functors, compatible finite-dimensional fibers, the directed colimit sector category, and compact gauge reconstruction . The support-visible compact-gauge quotient algebra is generated by overlap sector projectors and compact-gauge visible bosonic observables, modulo the maximal repair-invariant overlap-trivial kernel.

Each regulator has finite-dimensional state space and finite local cylinder algebras, so its state space is weak-\(*\) compact. Refinement compatibility makes the local marginals projective: the restriction of a refined cylinder marginal to any coarser cylinder equals the coarser marginal. Choosing a countable cofinal family of support-visible cylinders and applying a diagonal subnet argument gives a limiting state on their algebraic union. Positivity and normalization pass to the limit on every cylinder, and the quotient by the repair-invariant overlap-trivial kernel makes the limit faithful on the support-visible gauge-invariant algebra.

The GNS construction applied to this limiting state gives \(K\). Transporting the same coherent refinement data through the finite-stage intertwiners \(U_r\) gives the physical limiting Hilbert space \(\mathcal H\). These are exactly the support-visible continuum objects used below. ◻

Proof. Theorem 7 gives exact finite-stage intertwining. Proposition 10 supplies the direct-limit and support-visible GNS extraction hypotheses. Therefore the finite intertwiners pass to the continuum. Equality of generators follows from uniqueness of the self-adjoint generator of a strongly continuous contraction semigroup. ◻

Proof. Euclidean covariance is part of the four-dimensional scaling chart and the local-Gibbs cylinder family. Reflection positivity is part of the ordinary vacuum branch in Assumption 1. Regularity follows from the finite local cylinder construction and the support-visible weak-\(*\) limit in Proposition 10. The vacuum vector is cyclic for the GNS closure of the gauge-invariant local cylinder algebra by construction. 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 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. Proposition 10 transports these local cylinder vectors into 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 9 gives \[ L_r^{\mathrm{rep}}\ge c_*(I-P_{0,r}). \] By Proposition 10 and Theorem 11, this lower bound passes to the support-visible continuum: \[ L^{\mathrm{rep}}\ge c_*(I-P_0). \] 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). ◻