Full Treatise — Supplementary Technical Archive to "Yang–Mills Mass Gap, Emergent Newton Constant, and Black-Hole Information from Void Circuit Closure (0≡∞)"

Supplementary Technical Archive (Full Treatise) — companion to theMain Manuscript submitted to Communications in Mathematical Physics(DOI: 10.5281/zenodo.20854143). This 831-page treatise provides the complete proof architectureunderlying the Main Manuscript: (a) full derivations with all intermediate steps and domain verifications; (b) alternative proof routes (lattice, sector, entropy, and population approaches); (c) extended technical exposition (operator-algebraic infrastructure, Mosco convergence details, BRST cohomology, DHR sector analysis, Bakry–Émery curvature machinery, Tomita–Takesaki modular theory); (d) historical attribution and priority labelling. This treatise provides the full technical depth behindeach step. It is provided strictly as expanded background andreference for the lemmas cited in the Main Manuscript; it is nota second manuscript. Technical spine: Wilson lattice → Osterwalder–Schrader reconstruction → Tomita–Takesaki modular theory → void circuit closure (0 ≡ ∞). Central claim. A single strictly positive Bakry–Émery curvatureconstant κ_BE = h∨/(4g²) on Wilson lattice Ricci geometry, validfor every compact simple gauge group G, simultaneously: 1. creates the Yang–Mills mass gap m_YM² ≥ 16·a_OS²·λ*² > 0, where λ* = κ_BE·exp(−2B); 2. sets the Hawking temperature T_H = κ/(2π) = a_OS (emergent from KMS + Tomita–Takesaki, not assumed); 3. bounds the information-recovery time t_Page^info ∼ √(S_BH)·log S_BH at Petz saturation; 4. controls exponential clustering of spacelike correlations; 5. makes 0 ≡ ∞ a theorem via Tomita J and void-boundary ergodic invariance (VEI); 6. fixes the emergent Newton constant G_eff = π²/16 (tilt-invariant, parameter-free, independent of the tilt B and of the collar; universal across all compact simple global forms G at the coupling lock g² = h∨/π forced by the product law), via VEI which forces λ_void = π universally for all B ≥ 0 with κ_BE > 0; 7. forces the product law k(R)·k(1/R) = (h∨)² closing the modular calibration circuit. Downstream theorems (all derived inside the same framework,not added as independent constructions): • Einstein equations emerge from the entanglement-package identity; • gravitationally induced entanglement (GIE) follows from G_eff > 0 and the same entanglement-package identity (a testable consequence); • black-hole information theorems (the framework — void-circuit closure together with scoped MLSI, ledger conservation, Petz QNEC, and Mosco convergence — yields all twelve): Page curve, information-loss resolution, AMPS/monogamy, firewalls, cloning, complementarity, censorship/no-mining, trans-Planckian, stable remnant exclusion, Bekenstein bound, no-hiding, and third law / unattainability of extremality. Related records: • Main Manuscript — DOI 10.5281/zenodo.20854143 • Full Treatise (this record) — DOI 10.5281/zenodo.20854162 • Submitted to Communications in Mathematical Physics. Author: Fulvio Bennato (Independent Researcher, Napoli, Italy)Email: [email protected]: https://orcid.org/0009-0006-4343-3967