Unconditional Construction of Four-Dimensional Euclidean Yang–Mills Theory for Compact Simple Lie Groups with a Positive Mass Gap

Version 3 of this manuscript presents a constructive framework for four-dimensional Euclidean Yang–Mills theory with compact simple Lie group G. Starting from the Wilson lattice gauge measure, reflection positivity, and a bounded countable Osterwalder–Schrader admissible test algebra, the paper develops an explicit polymer/Kotecký–Preiss closure mechanism and proves a continuum Yang–Mills construction with positive mass gap for all bare inverse couplings β0 ≥ βYM, where βYM < ∞ is a deterministic threshold assembled from finite closure constants. The proof is organized through an Independence Ledger: each load-bearing step in the final theorem is either proved inside the manuscript, imported as a standard external theorem with its hypotheses verified, or recorded as a named proof obligation. In the final version, all proof obligations are closed. The bounded-algebra kernel chain is polymer/KP closure ⇒ uniform exponential clustering ⇒ bounded-algebra OS reconstruction ⇒ Hamiltonian spectral gap. The weak-coupling mechanism is not an ordinary strong-coupling expansion. The continuum step is based on scale-normalized effective-action transport. After each shell integration, the post-shell effective action is compared fibrewise over admissible coarse backgrounds. The remaining connected cumulant correction is controlled by a contraction estimate qcum = Ccum τ*^(1)(β), where Ccum is volume- and scale-uniform and τ*^(1)(β) → 0 by the closed polymer/KP bounds. For β0 ≥ βYM this gives a summable scale-defect recurrence, yielding continuum trajectory closure and uniqueness. The manuscript proves exact mixed-colour GOOD/BAD polymer representation, local BAD-factor control, uniform KP summability, observable insertions, physical-unit mass-gap persistence, effective-action transport, uniqueness of the continuum limit, Euclidean invariance, nontriviality through a bounded OS/RP-GNS witness, and the Schwartz/Wightman reconstruction upgrade. Standard external inputs, including Wilson reflection positivity, Osterwalder–Schrader reconstruction, Kotecký–Preiss cluster expansion, finite-range decomposition, and Brascamp–Lieb type estimates, are isolated and their hypotheses are verified in the manuscript. The supplementary file ym_certificate_replay_v1.zip is included as non-load-bearing consistency and audit material. It is a Python standard-library replay package that checks artifact integrity, theorem-label presence, ledger closure, threshold-chain consistency, effective-action/cumulant chain ordering, and external-input declarations. It is not a proof engine, not a simulation proof, and not a numerical proof of Yang–Mills. The mathematical claims rest on the analytic theorems and verified external inputs in the manuscript. License: CC BY-NC-ND 4.0 International. Redistribution is allowed with attribution; commercial use and modifications are prohibited. All commercial rights are reserved to the author.