Admissibility and the Well-Definedness of Yang–Mills Endpoint Constructions

This record deposits the referee-hardened manuscript package for Admissibility and the Well-Definedness of Yang–Mills Endpoint Constructions, a completion-control and endpoint-well-definedness manuscript for the Yang–Mills Clay Millennium Prize Problem. The manuscript works in the language of algebraic quantum field theory. Conditional on the predecessor local-net construction — the bounded-region local gauge-invariant Haag–Kastler net, its OS/Wightman vacuum representation, and its vacuum-sector mass gap — the paper analyzes extended-support line/surface sector data and theorem-scope completions over that fixed local theorem object. The manuscript does not claim to reconstruct the local Yang–Mills net from scratch inside this paper. That construction belongs to the predecessor Yang–Mills local-net package. Instead, this manuscript proves that once the local theorem object is fixed, downstream sectoral and ultraviolet completion structures cannot alter the same-domain vacuum-gap endpoint. The result is a kernel-faithful completion-control theorem: theorem-scope admissible completions preserve the predecessor vacuum gap as the complete-theory gap. Clay Millennium Prize Endpoint Taken together with the predecessor local-net construction, this manuscript claims to complete the mathematical endpoint of the Yang–Mills Clay Millennium Prize Problem: existence of a non-trivial four-dimensional quantum Yang–Mills theory for compact simple gauge group (G), satisfying axiomatic properties at least as strong as the Osterwalder–Schrader / Wightman / Haag–Kastler frameworks, together with a positive mass gap in the vacuum sector. The claimed endpoint is the official Yang–Mills existence-and-mass-gap target, expressed in local-net / AQFT form. The predecessor package supplies the local gauge-invariant sharp-local net, OS/Wightman reconstruction, non-triviality, local field correspondence, and vacuum-sector gap. The present manuscript supplies the completion-control layer showing that theorem-scope sectoral or ultraviolet completions over that fixed object do not introduce subgap same-domain content or alter the endpoint gap. This record does not assert that the Clay Mathematics Institute has accepted or awarded the prize. It asserts that the deposited manuscript is part of a package claiming to reach the official mathematical endpoint of the Yang–Mills Millennium Prize Problem, subject to the ordinary process of mathematical review, publication, community examination, and institutional evaluation. Main Claim The manuscript proves the structural well-definedness of Yang–Mills endpoint constructions on a fixed domain. Let the predecessor Yang–Mills package provide the bounded-region local gauge-invariant sharp-local net [ \mathcal A_{\mathrm{loc}}(G) ] together with its vacuum representation, vacuum vector, positive-energy dynamics, and vacuum-sector mass gap. The present manuscript fixes that object as its base and proves that theorem-scope admissible sectoral and ultraviolet completions over it remain kernel-faithful same-domain completions. The main conclusion is that no theorem-scope admissible completion can introduce a locally detectable subgap state, an orthogonal subgap copy of the vacuum, or a new same-domain carrier of subgap standing. Consequently, the predecessor vacuum gap persists as the complete-theory gap for all filtered same-domain completions treated by the manuscript. The proof route is structured around: exact handoff from the predecessor local-net endpoint to the current fixed base object; explicit distinction between bounded-region local observables and extended-support line/surface sector data; construction of a theorem-scope sector layer over the fixed local net; ATS/UEAP filtering of sector-to-completion promotion, so that sector representation is not confused with same-theory completion; a restricted kernel packet for non-degenerate same-domain theorem-bearing regimes; source-neutral dependency discipline: theorem-content, not authorial provenance, carries the proof burden; operator-algebraic translation of kernel-fidelity failure into local-state carrier change or subgap vacuum multiplicity; a kernel-faithful GNS/spectral bridge for theorem-scope admissible completions; endpoint falsification discipline: a refutation must attack the kernel, the fixed local theorem object, or produce a faithful same-regime counterexample. Method Class This is a completion-control and endpoint-well-definedness proof in algebraic quantum field theory. It is not the local-net construction proof home itself. The manuscript explicitly distinguishes: What is not claimed The manuscript does not claim to: reconstruct the entire Yang–Mills local net from scratch inside this paper; replace the predecessor local-net proof package; prove a new PDE-native or constructive lattice continuum-limit theorem in this manuscript alone; assert a mass gap for arbitrary non-vacuum sectors or arbitrary Hilbert-space extensions; treat every sector representation as a vacuum-preserving same-theory completion. What is claimed The manuscript claims that, once the predecessor local theorem object is fixed, theorem-scope admissible completions over that object cannot change the same-domain vacuum-gap endpoint. Extended-support line and surface data may define sectoral structure over the fixed local net, but they do not automatically become same-theory completions. A sector datum enters the complete-theory spectral bridge only after passing the sector-to-completion admissibility filter: ATS role typing, UEAP carrier/scope/standing adequacy, scope-faithful extension conditions, and kernel fidelity. The relevant review standard is therefore not whether this manuscript alone reconstructs the full Yang–Mills theory from the ultraviolet, but whether its fixed-base handoff, sector/completion filter, kernel-fidelity analysis, and GNS/spectral endpoint bridge correctly preserve the predecessor local-net mass gap as the complete-theory gap. Formal Audit Support The manuscript is accompanied by a dedicated Lean4 audit endpoint: Repository: somamaley-ux/AASC-YangMills-Lean-Audit Zenodo DOI: 10.5281/zenodo.20523056 Version: 1.0.1 Published: June 3, 2026 The associated Lean4 archive supports the manuscript-facing Yang–Mills local-net and Clay-extension audit endpoint. It separates the Yang–Mills Lean work from the larger AASC working repository and packages the theorem-routing, endpoint, seam-checking, and audit-control files relevant to the manuscript-facing proof route. The archive supports the route connecting the local gauge-invariant sharp-local Yang–Mills net, OS/Wightman vacuum representation, exact local-net endpoint, Clay-extension completion-control layer, and final spectral endpoint. The Lean archive is checked with the dedicated Yang–Mills audit command: powershell -ExecutionPolicy Bypass -File scripts/check-yang-mills-a-plus-audit.ps1 The audit records that the Yang–Mills Lean package builds, that the manuscript-facing A+ audit files run successfully, and that the audited Yang–Mills project surface contains no live project-level axiom, sorry, admit, or unsafe declaration in the audited project modules. The recorded environment is: Lean: leanprover/lean4:v4.28.0 mathlib revision: 8f9d9cff6bd728b17a24e163c9402775d9e6a365 The Lean formalization is not presented as a first-principles re-formalization of every external theorem in PDE, quantum field theory, (C^*)-algebra, Hilbert-space theory, OS/Wightman reconstruction, or spectral theory. Its role is to certify the manuscript-facing theorem-routing, endpoint handoff, seam discipline, sector/completion audit-control structure, and Clay-extension endpoint layer. Standard Lean/classical foundations and imported mathematical background remain part of the explicit boundary. Dependency reports may mention standard foundations such as propext, Classical.choice, and Quot.sound; these are not project-specific Yang–Mills support axioms. Included Materials This deposit includes: the main manuscript PDF; the LaTeX source package; appendices and front matter; formal-audit support material; local-net endpoint / start-point alignment material; Clay-scope endpoint ledger; ATS/UEAP sector-to-completion filter material; kernel-fidelity and GNS/spectral endpoint bridge material; referee-hardening and presentation-readiness notes; build/readiness documentation; SHA-256 checksums for reproducibility, if included in the source package. Intended Use This deposit is intended to provide a citable, inspectable record of the current referee-hardened version of the Yang–Mills endpoint-construction manuscript and its supporting project package. It is suitable for independent mathematical review, hostile-referee analysis, AQFT endpoint review, Clay-endpoint alignment analysis, Lean-audit comparison, and comparison with future revisions. Keywords Yang–Mills theory; Millennium Prize Problems; Clay Mathematics Institute; Yang–Mills mass gap; algebraic quantum field theory; Haag–Kastler net; local net; Osterwalder–Schrader reconstruction; Wightman reconstruction; GNS representation; spectral gap; mass gap; gauge-invariant local observables; line operators; surface operators; superselection sectors; global form; constraint formalism; AASC; admissibility and standing; kernel fidelity; UEAP; ATS; same-domain completion; Lean4; formal verification; theorem auditing; no-axiom audit; mathematical physics; quantum field theory foundations