Supplementary Technical Archive (Full Treatise) — companion to theMain Manuscript submitted to Nuclear Physics B (DOI: 10. 5281/zenodo. 20149357). This 796-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 κBE = h∨/ (4g²) on Wilson lattice Ricci geometry, validfor every compact simple gauge group G, simultaneously: 1. creates the Yang–Mills mass gap mYM² ≥ 16·aOS²·λ*² > 0, where λ* = κBE·exp (−2B) ; 2. sets the Hawking temperature TH = κ/ (2π) = aOS (emergent from KMS + Tomita–Takesaki, not assumed) ; 3. bounds the information-recovery time tPageⁱnfo ∼ √ (SBH) ·log SBH 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 Gₑff = 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 λᵥoid = π universally for all B ≥ 0 with κBE > 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 Gₑff > 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. 20149357 • Full Treatise (this record) — DOI 10. 5281/zenodo. 20149621 • Submitted to Nuclear Physics B (via Elsevier Article Transfer Service, following prior consideration at Annals of Physics). Author: Fulvio Bennato (Independent Researcher, Napoli, Italy) Email: fulvio. bennato@gmail. comORCID: https: //orcid. org/0009-0006-4343-3967
Fulvio Bennato (Wed,) studied this question.