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 Fifth-reference source, physical spectrum and heat estimates reader introduces the completed source equation and its physical consequences, followed by both full new proofs. The Heat correlations and the full physical complement reader includes the five preceding heat and volume manuscripts. The fourth-order vacuum and sixth-order energy reader and the 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 all 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. The completed fifth-reference source. For H = κ[K + ξ∑ₚ(2 − Wₚ)], with κ = 2g²/a and ξ = 1/(4g⁴), the fifth-order reference is followed by its entire residual, a bounded inverse of the actual linearization and a convergent nonlinear correction. The endpoint is included through an explicit vanishing Catalan-series tail. The source domain reaches g² ≥ 1/(2√α₅), where 0.018424953576117616681 < α₅ < 0.018424953576117616682; the threshold is approximately 3.6835519839857273. The source returns to the actual positive vacuum, scalar ground energy and full physical spectral gap. In particular, g² ≥ 3.7 gives Δ > 1.6207κ, uniformly over all original boxes L ≥ 2 and a > 0. The finite coefficient calculation retains all 662 fifth-source representatives and 124,864 anchored multisets, with 5,726 rational primal/dual certificates and an orientation-sensitive trace coefficient bound. Heat correlations and growing boxes. The heat calculation supplies 84 exact time functions with rational Laplace transforms across 17 geometric cases. Its degree-zero, degree-two and degree-four matrices are unchanged. The fifth-source estimate enlarges the complex heat circle to |ξ| < 1/55. In dimensionless time τ = κt, the complete absolute-row remainder is bounded by (67896/169) exp(−13τ/8) (55|ξ|)⁶/[1 − (55|ξ|)²], for every τ ≥ 0 and every original box. 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. Minimizing their energy over the entire complementary space gives Efull = E − κ²B*D⁻¹B, with the operator domain and inverse established in the proof. Write G(2) for the original state Gram. For every L ≥ 2, a > 0 and g² ≥ 13, the quadratic forms satisfy (1999/2000)E ≺ Efull ≼ E and G(2) ≼ Gfull ≺ (2001/2000)G(2). Thus both complementary corrections are below one part in two thousand. For g² ≥ 16 they are below one part in twenty-five thousand. These estimates also carry to the constructed fixed-spacing spatial limit. The proofs retain the original state and energy metrics, signed heat-moment weights, mixed terms and support-quotient maps. 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. Sources and reproducibility. The exponential-vacuum and character/Casimir framework is credited to D. Schütte, Zheng Weihong and C. J. Hamer, The Coupled Cluster Method in Hamiltonian Lattice Field Theory. Eymard supplies the Fourier-algebra antecedent; Lumer and Phillips supply the semigroup theorem used after explicit domain and range checks. Readable PDFs accompany editable sources, exact-arithmetic programs, coefficient tables and dated execution records. Fresh ordinary and optimized fifth-reference replay passed 51 named checks and 11 false-formula controls, including 316 integer tensor calculations and all 124,864 anchored returns. The supplied 30-run history, including full polynomial/certificate audits, remains separately attributed and was not repeated in full during publication. Earlier fresh volume, heat and route checks are retained with their own receipts. These finite checks are not represented as formal or independent external certification of the analytic proofs. Mathematical scope. These are fixed-spacing lattice results. The manuscripts examine explicit continuum paths and state where those paths leave the proved strong-coupling domains; this edition does not construct a nontrivial four-dimensional continuum field or establish a finite positive continuum mass. 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.