Yang–Mills Mass Gap, Emergent Newton Constant, and Black-Hole Information from Void Circuit Closure (0≡∞) Main manuscript PRE PRINT

Main Manuscript (187 pp) — Submitted to Annals of Physics. Supplementary Technical Archive — "Supplementary_Technical_Archive_Full_Treatise" (714 pp) DOI 10.5281/zenodo.19708570 . Expanded proof architecture and technical background. Available on request. We prove a positive mass gap for pure Yang–Mills on R^4 for everycompact simple gauge group G, via the chain Wilson lattice → Osterwalder–Schrader reconstruction → Tomita–Takesaki modular theory → void circuit closure (0 ≡ ∞). No assumption enters beyond the Wilson action and compact simple G.Bakry–Émery curvature κ_BE = h∨/(4g²), Holley–Stroock perturbation,Mosco convergence, and OS reconstruction yield a Wightman QFT withm_YM ≥ 4 a_OS λ* > 0. Void-boundary ergodic invariance (VEI) forcesλ_void = π and m_YM = π·a_OS; Petz saturation (free-fermion sharpMLSI at k = h∨) locks g² = h∨/π, a_OS = 1/(2π), m_YM = 1/2. The same VEI yields a tilt-invariant emergent Newton constantG_eff = π²/16 (pure number, independent of the tilt B, of the collar,and of the gauge group G); the product law k(R)k(1/R) = (h∨)² closescalibration, yielding g²(R) → 0. Einstein's equations emerge fromthe entanglement-package identity. The 0 ≡ ∞ closure yields twelveblack-hole information theorems (Page curve, information loss,AMPS/monogamy, firewalls, cloning, complementarity, censorship/no-mining, trans-Planckian, stable remnant exclusion, Bekensteinbound, no-hiding, third law/unattainability of extremality);gravitationally induced entanglement (GIE) follows from G_eff > 0and the entanglement-package identity.