Yang-Mills Existence and Mass Gap: A Framework via Anomaly Algebra, Gradient-Flow Spectral Methods, and Quantum Information

We present a rigorous framework for the Yang-Mills mass gap problem, combining three independent lines of argument that reinforce each other. Result A (Unconditional): a new MaxEnt Clustering-Recovery Bridge -- for lattice gauge states with finite correlation length xi, in the polymer/Kotecky-Preiss regime made precise in Section 5, the Petz recovery fidelity satisfies 1 - F <= C e^{-r/xi}, proved via maximum-entropy truncation on gauge-invariant algebras, a convergent polymer expansion, and the Fawzi-Renner theorem. Result B (unconditional on the lattice, conditional for all couplings): for SU(N) lattice gauge theory (T=0, theta=0, d=3+1, N >= 2), the algebraic phase exclusion, using the projective commutation relation of 1-form symmetry operators, excludes the trivially gapped symmetric phase (v2: given the imported lattice realization of the symmetry pair); combined with Perron-Frobenius non-degeneracy and Gauss-law constraints, this forces confinement at strong coupling; the extension to all couplings relies on Hypothesis 1.1 (absence of a bulk phase transition), supported but not proven. Under Hypothesis 1.1 the uniform lattice mass gap holds for all lattice spacings. Result C (Conditional): under the same hypothesis, the continuum limit exists as a Euclidean QFT satisfying all Osterwalder-Schrader axioms with mass gap. Result D: the gradient flow reduction (developed in the companion ai.viXra:2602.0020). v2 (no v1 numbered statement is changed): the exact lattice realization of the commuting projective 1-form pair is made an explicit imported input, and a new remark records the Perron-Frobenius tension that forces this framing -- at finite volume, PF uniqueness plus exact commutation of both generators would contradict the projective relation outright, so the magnetic operator commutes with H only up to defect terms (verified numerically: toric code, exact pair with 4-fold degenerate ground state; Z2 gauge theory with electric term, unique ground state with both string commutators nonzero); the Result-D naming collision is resolved (the d = 2+1 theorem is now Result E); an unresolved citation is repaired; the epsilon-powers in the MaxEnt bridge are harmonized; and a replication report is added: the 7-qubit Z2 table is reproduced digit for digit, the Z3 table is reproduced digit for digit after a documented g <-> 1/g erratum between the v1 script and table, and the torus finite-size-scaling table could not be reproduced from the printed conventions and is downgraded to archival status (no framework result depends on it). v2 is presented in condensed form: every v1 numbered statement is preserved with the same numbering; detailed proofs and the full computational listing remain in the v1 PDF as the archival source.