SU(2) lattice Yang–Mills: vacuum, energy, spectral bounds and continuum investigations

This collection studies the vacuum, excitations and observable correlations of interacting SU(2) lattice Yang–Mills theory. It starts from the original link variables, Haar measure, vertex gauge constraints, lattice spacing and physical coupling. The manuscripts develop explicit coordinate and state maps, exact local calculations, estimates uniform in the size of the spatial box, and an examination of possible continuum paths. Where to start. The new Heat correlations and the full physical complement reader introduces the mathematical objects and results, then includes the five complete heat and volume proof manuscripts. The preceding fourth-order vacuum and sixth-order energy reader and source-equation and spectral-return reader supply the earlier calculations. Five foundation papers cover quantum coarse-graining, interacting tensor bands, non-Abelian vertices, volume-independent vacuum estimates and spatial-continuum investigations. The reading guide and living sources connect these manuscripts. Vacuum and energy. The logarithm of the positive vacuum converts the original eigenvalue equation into a coupled source equation. The fourth-order continuation gives two exact presentations of its fourth coefficient, related by explicit tree-coordinate maps: trace words and rational quaternion polynomials. The coefficient comparison covers all 78 connected classes and 8,621 anchored face multisets. Both calculations obtain the sixth-order ground energy, including boundary terms and the elementary-cube contribution −83/1944. The papers retain the scalar energy terms, signed responses to local couplings, and the distinct domains of their analytic remainder estimates. Higher-source and response continuations are preserved with their own stated scope and verification records. Heat correlations and growing boxes. For H = κ[K + ξ∑ₚ(2 − Wₚ)], with κ = 2g²/a and ξ = 1/(4g⁴), the heat calculation supplies 84 exact time functions with rational Laplace transforms across 17 geometric cases. It retains the nonidentity Gram matrix of the first excitation band. The subsequent proof controls the entire absolute row sum of the heat remainder, uniformly over all original open boxes L ≥ 2: the fourth-order error is at most 512 exp(−3τ/2) θ⁶/(1 − θ²), where τ = κt, θ = |ξ|/(3/256) < 1. Local coefficient agreement and this bound construct a spatial-volume limit at fixed lattice spacing. The full complementary physical space. The inverse-energy plaquette states need not form an invariant subspace. Their exact energy minimization over the entire complementary space gives Efull = E − κ²B*D⁻¹B, with the maps, operator domain and inverse established in the proof. For every L ≥ 2, a > 0 and g² ≥ 16, the resulting quadratic forms satisfy (999/1000)E ≺ Efull ≼ E. Writing G(2) for the original state Gram, the restored Gram satisfies G(2) ≼ Grestored ≺ (1001/1000)G(2). Thus relaxation into the full complement changes both quantities by less than one part in a thousand. The edition also gives a genuine positive-correlation interval, inverse-energy moments, and finite-time comparisons for original state and energy quotient metrics. Originating mathematics. Levent Alpöge’s Jacobian counterexample announcement, crediting Akhil for the question and Fable for the construction, supplies the polynomial used in the earlier material-tensor branch: a constant nonzero Jacobian with global noninjectivity. Tao’s exposition is credited separately. The distinct S6 construction circulated by Alpöge and produced with Claude supplies the regular-fibre periods used in the cusp and magnetic-background maps. Engel’s S6 exposition is a separate source. The attribution clarification identifies the exact uses and distinguishes these originating inputs from the workbench’s later constructions. The new heat estimates use their explicitly stated source and semigroup arguments, not an identification of arithmetic or geometric objects with physical energy levels. Scope and reproducibility. These are fixed-spacing lattice results; this edition does not construct a nontrivial four-dimensional continuum field or establish a finite positive continuum mass. Readable PDFs accompany editable sources, exact-arithmetic programs, coefficient tables and dated execution records. Fresh checks of the new packages include 2,425 volume checks, 879 heat checks and 52 route checks, run in both ordinary and optimized Python, with the associated false-formula controls. Those finite executions are distinguished from the supplied analytical proofs and historical producer runs; they are not represented as complete formal or independent external certification. Earlier editions and their original bytes are retained. PolyClank. This is an open, AI-assisted mathematical workbench, including work with ChatGPT 5.6 Sol and GPT-6 Astra. To contribute, fork the GitHub repository, add an argument, calculation or correction with enough explanation to understand and reproduce it, and open a pull request. The contribution guide explains the procedure. The related S6 and Navier–Stokes collections retain their separate archival records.