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

This deposit is the Main Manuscript (189 pp). The companion Supplementary Technical Archive — Full Treatise (831 pp), DOI10. 5281/zenodo. 20854162 — is available on Zenodo on request. ABOUT THIS DEPOSIT This record presents an independent manuscript proving a positive Yang–Millsmass gap on ℝ⁴ for every compact global form G of every compact simple Liealgebra, together with a unified derivation of emergent gravity andblack-hole information results. The proof chain runs from the Wilson latticeformulation through Osterwalder–Schrader reconstruction and Tomita–Takesakimodular theory to the void-circuit closure theorem (0 ≡ ∞): onereconstructed Hamiltonian HOS, read three ways. The manuscript develops three main results. 1. Yang–Mills mass gap (four dimensions, every compact simple global form) A complete chain from the Wilson lattice action to a Wightman QFT with apositive mass gap. The lower bound is analytic: a Bakry–Émery curvaturebound, stable under Holley–Stroock perturbation, feeds a volume-uniformclustering rate that passes through Osterwalder–Schrader reconstruction andMosco convergence to the spectrum of the reconstructed Hamiltonian. The exactvalue is a bulk–edge pinch on that one spectrum: mYM = 3π · aOS, attained by the Cartan three-form in the physical edge sector (Whitehead'sinvariant degree three), while physical-sector exhaustion and amagnetic-branch certificate, mₘag (G) > 3π · aOS for every global form, exclude every lower competitor. At the coupling lock g² = h^∨ / π, βOS = 2π, itself forced — void-circuit closure forces the self-dual product lawk (R) k (1/R) = (h^∨) ², hence the level k = h^∨, and free-fermion sharp-MLSIPetz saturation closes the bound to equality — the gap takes the explicitvalue mYM = 3/2. The same product law calibrates the running coupling: g² (R) → 0 (asymptotic freedom). 2. Emergent gravity Nothing gravitational is added. KMS periodicity forces κβ = 2π, theBisognano–Wichmann modular Hamiltonian is derived rather than imported, andthe horizon gap is the same spectrum: mₕor = mYM = 3π · aOS, one spectrum in two interpretations. The same ergodic invariance fixes auniversal, dimensionless, tilt-invariant Newton constant Gₑff = π² / 16, independent of the tilt and of the collar, universal across all compactsimple global forms. Einstein's equations Gₐb = 8π Gₑff Tₐb are a theoremof the same construction, derived from the entanglement first law with thestress implementer constructed as the unique coercive solution of aDirichlet-to-Neumann matching problem. Gravitationally induced entanglementfollows from Gₑff > 0 as a testable consequence. 3. Black-hole information The dissipative closure of the same circuit yields four quantitative laws: anexact conservation ledger for radiation relative entropy plus surface entropyalong modular time; the generalized second law with nonnegative entropyproduction; a Petz-sharp, two-sided Page time tPage ∼ √ (SBH) · log SBH; and an internally derived island formula for the radiation entropy. Fromthese laws twelve theorem-level black-hole information results follow asfacets of one structure: the Page curve and information-loss resolution; theexclusion of firewalls, cloning, monogamy violation, and stable remnants;observer complementarity and no-hiding; the Bekenstein bound, the third lawof unattainability, trans-Planckian regularity, and a finite no-free-lunchrecovery cost — together with cosmic censorship, a no-mining strong converse, and gravitationally induced entanglement. METHODOLOGY AND HYPOTHESES The derivation uses only two explicit inputs: (H1) the Wilson lattice action on a finite hypercubic sublattice of ℤ⁴; (H2) a compact simple gauge group G, with normalized Haar measure on each link. Osterwalder–Schrader reflection positivity of the Wilson measure is itself atheorem of the construction (a link-Gram argument valid for every compact G), not an assumption. No continuum action, no background spacetime geometry, andno Einstein equations are assumed as inputs. Structures such as Bakry–Émery curvature, Holley–Stroock perturbation, MLSIergodicity, Mosco convergence, Sugawara/Kac–Moody grading, and Petzsaturation enter as derived theorems or standard tools applied to (H1) – (H2), not as additional postulates. CONCEPTUAL CORE: VOID-CIRCUIT CLOSURE The conceptual engine combines two mechanisms. First, under MLSI ergodicity, the GKSL-dissipated modular transport sendsboth radial limits — r → 0 (singularity) and r → ∞ (spatial infinity) — tothe same void state ωᵥoid. Here ωᵥoid denotes the unique ergodic stationarystate of the GKSL semigroup on the collar algebra, not to be confused withthe reversible vacuum, which is recovered only in the limit λ_* → 0. Second, the Tomita modular conjugation J acts on the geometric dilationobservable X: = log (r/R_*) as the operator identity J X J = −X + 2π, equivalently J Ageom J = e^ (2π) · Ageom^ (−1). This maps r → 0 to r → ∞ as J-conjugate mirror images. Together, these two facts yield the void-circuit closure theorem, written0 ≡ ∞ in the text: the singular and asymptotic endpoints are simultaneouslydriven to the same void state and identified as J-conjugates, compactifyingthe radial parameter as ℝ ∪ ∞ ≅ S¹. The symbol 0 ≡ ∞ is an equivalencelabel for this identification — an ergodic state-equality — not a numericalequality. This closure, together with scoped MLSI, ledger conservation, Petz QNEC, andMosco convergence, is then used to derive the emergent geometric sector, theEinstein equations, and the black-hole information results inside the samealgebraic framework. INTENDED AUDIENCE This work is intended for researchers in algebraic and constructive QFT, mathematical gauge theory, noncommutative analysis, modular theory ofoperator algebras, and the algebraic side of black-hole information. Focusedtechnical feedback on theorem dependencies is welcome. STATUS This record is a preprint. The 831-page Full Treatise holds the expandedderivations behind the manuscript's Full-pointers — full-length proofs, thesupporting lemma infrastructure, and the numerical certificate registers —and is available on request at DOI 10. 5281/zenodo. 20854162. Related records: • Full Treatise (companion archive) — DOI 10. 5281/zenodo. 20854162 • Main Manuscript (this record) — DOI 10. 5281/zenodo. 20854143 Author: Fulvio Bennato (Independent Researcher, Napoli, Italy) Email: fulvio. bennato@gmail. comORCID: https: //orcid. org/0009-0006-4343-3967 LICENSE / COPYRIGHT Copyright © 2026 Fulvio Bennato. All rights reserved. This work may be read, downloaded, cited, and used for non-commercialscholarly review, research, and teaching, provided that proper citation tothe DOI and author is maintained. No commercial use, republication, redistribution, derivative work, text-and-data mining, machine-learning training, model evaluation, datasetinclusion, or automated extraction is permitted without prior writtenpermission from the copyright holder.