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 — "SupplementaryTechnicalArchiveFullTreatise" (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 R⁴ 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 κBE = h∨/ (4g²), Holley–Stroock perturbation, Mosco convergence, and OS reconstruction yield a Wightman QFT withmYM ≥ 4 aOS λ* > 0. Void-boundary ergodic invariance (VEI) forcesλᵥoid = π and mYM = π·aOS; Petz saturation (free-fermion sharpMLSI at k = h∨) locks g² = h∨/π, aOS = 1/ (2π), mYM = 1/2. The same VEI yields a tilt-invariant emergent Newton constantGₑff = 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 Gₑff > 0and the entanglement-package identity.