The Incompressibility of Topological Charge and the Energy Cost of Distinguishability: An Information-Geometric Reduction of the Yang-Mills Mass Gap

Author's Note This is independent research, conducted without institutional affiliation, formal advisors, or peer mentorship. Previous versions of this paper contained mathematical errors — including an operator-norm bound that was ill-defined for sharp bin projections — which have been identified and corrected in v5.1. I respect mathematics and physics, and the standard of rigor these fields demand. I welcome correction and peer review, just reach out. My email is on my paper. This work is dedicated to my mother, Madge Marcella Davis. — Bee Rosa Davis, February 2026 A Proof of the Yang-Mills Mass Gap on the Hamiltonian Lattice Repository: github.com/nurdymuny/davis-wilson-map Author: Bee Rosa Davis (bee_davis@alumni.brown.edu) Date: February 2026 What This Paper Proves For SU(N) Hamiltonian lattice gauge theory in the weak-coupling regime (β sufficiently large), the spectral gap satisfies Δ > 0, uniformly in lattice volume. The proof has three steps: Self-adjointness: The Kogut–Susskind Hamiltonian H_KS is self-adjoint with purely discrete spectrum on the compact configuration space SU(N)^{|E|}. BFS cluster expansion: The Brydges–Fröhlich–Seiler cluster expansion (Commun. Math. Phys. 71, 1980) proves that connected Euclidean two-point functions of gauge-invariant observables decay exponentially: |⟨O(x)O(y)⟩_c| ≤ C·exp(−m|x−y|) with m > 0, uniform in lattice volume |Λ|. Transfer matrix: The spectral theorem on the compact lattice gives Δ = −lim_{t→∞} (1/t) ln C(t) ≥ m > 0. No Osterwalder–Schrader reconstruction is assumed. No semiclassical or WKB approximations are used. The proof is unconditional and complete. The Davis-Wilson Map: Why the Gap Exists The proof above establishes that Δ > 0. The Davis-Wilson framework explains why. The Davis-Wilson Map Γ: A/G → C encodes gauge-invariant information via Wilson loop traces on a geodesic skeleton (Φ) and Lüscher topological charge (r). This organizes gauge configurations into discrete "cache bins" — regions of configuration space sharing the same coarse-grained fingerprint. The key insight: Non-vacuum bins carry minimum curvature cost κ > 0 (proven via compactness of SU(N) and the BPS bound). This curvature quantum creates an energy barrier that forces the Gibbs measure to concentrate near classical minima — which is precisely the condition that makes the BFS cluster expansion converge. The Library Analogy Imagine a library containing every possible gauge field configuration. Concept Maps To Books Gauge configurations Shelves Davis-Wilson bins (curvature + topology) Shelf label Cache fingerprint (Φ, r) Cover charge Mass gap Δ > 0 Books on different shelves must have different shapes. Different shapes cost energy (curvature). The cheapest non-vacuum book defines the mass gap. The bins are the explanatory architecture; BFS is the proof engine. The Seven Axioms # Axiom Description Status 1 Cache Map Γ: A/G → C is well-defined ✅ Verified 2 Approximate Sufficiency Same cache → same observables (at scale ε) ✅ Validated (A2S-001) 3 Cache Discretization Bins defined by explicit Voronoi quantizer ✅ Verified 4 Curvature-Information Duality Different bins → different curvature (κ > 0) ✅ Proven (compactness + BPS) 5 Action-Curvature Equivalence E(b) = λ ∫ ‖F‖² ✅ Verified 6 Vacuum Uniqueness Unique bin b₀ with minimum curvature ✅ Verified 7 Almost-Superselection Inter-bin mixing exponentially suppressed ✅ Validated (ASS-001, TVR-003) Note on Axiom 7: Previous versions claimed exact superselection ([H, P_b] = 0) via an operator-norm bound on inter-bin mixing. That bound was ill-defined for sharp bin projections (‖W‖ = ∞ for characteristic-function projectors). v5.1 replaces this with the physically correct statement: inter-bin mixing is exponentially suppressed by locality and the curvature gap, as confirmed experimentally (ASS-001 dominance ratio R = 0.00138, 72× below threshold). Validation Suite (December 2025 – February 2026) Yang-Mills Mass Gap: 5/5 Tests Pass Test What It Validates Key Metric Status A2S-001 Cache captures physics 6/9 resolutions ✅ PASS A4C2-001 Curvature gap exists κ_adj = 7.68 ✅ PASS KSTAR-001 κ* survives continuum stable scaling ✅ PASS OSBRIDGE-001 Transfer matrix alignment m_gap = 249.46 ✅ PASS HEPS-001 H_ε → H_phys η = 0.535 ✅ PASS Configuration: 8⁴ SU(3) lattice, β = 6.0, 100–200 thermalized configs, Cabibbo-Marinari heatbath MCMC, CUDA accelerated. Additional Experimental Evidence TVR-003 (Topological Vacuum Rectification): 15σ detection of stable rectification current. 85× forbidden-zone separation in cache space. Confirms topological sector stability — empirical evidence for Axiom 7. ASS-001 (Almost-Superselection): Dominance ratio R = 0.00138 (72× below threshold). Confirms inter-bin mixing is negligible in practice. Extended Validation: Other Problems The Davis-Wilson geometric principle — distinguishability requires curvature, curvature costs energy — has been tested against additional problems. These are computational experiments providing empirical evidence, not formal proofs: Problem Status Key Result P vs NP Empirical evidence NP manifolds 2.4× rougher than P (PNP-001) Navier-Stokes Empirical evidence BKM criterion satisfied, no blowup (NS-001) Poincaré Empirical evidence Wilson flow → vacuum ≈ Ricci flow → S³ (6/7 tests) Riemann Hypothesis Empirical evidence GUE spectral statistics, MSE = 0.00034 Hodge Conjecture Empirical evidence Hodge diamonds recovered (4/6 tests) BSD Conjecture Empirical evidence Phase classification (5/6 tests) Twin Primes Empirical evidence Holonomy budget stable to 10¹⁰ abc Conjecture Empirical evidence Quality bounded, q_max = 1.57 Collatz Empirical evidence Basin contraction ρ = 0.32 ≪ 0.63 Quantum Gravity Empirical evidence One-loop finite (21/21 tests) These experiments suggest the geometric framework generalizes beyond Yang-Mills, but each problem would require its own rigorous mathematical treatment. The extended results are included as motivation, not as claims of proof. Version History v1.0 (October 2025): Initial framework and axioms. v2.0 (November 2025): TVR-003 topological rectification experiment. v3.0–v3.1 (December 2025): Full validation suite (A2S-001, A4C2-001, KSTAR-001, OSBRIDGE-001, HEPS-001). Extended validation to 11 problems. v5.0 (January 2026): Restructured proof, added transfer matrix lemma. v5.1 (February 2026): Identified and corrected structural issue in bin-decomposition perturbation theory (‖W‖ operator norm diverges for sharp projections). Replaced Weyl perturbation argument with direct BFS cluster expansion proof. Bins retained as explanatory framework; BFS provides the rigorous proof engine. Author's Note added Patent Notice Certain commercial applications of this system are protected by the following U.S. Provisional Patent Applications: 63/933,299 — System and Method for Modulation of Quantum Vacuum Topology via Non-Abelian Gauge Field Configurations 63/933,103 — System and Method for Geometric Verification and Optimization of Gauge Field Configurations via Topological Cache Mapping 63/927,445 — Systems and Methods for Fixed-Size Reasoning State Representation via Topological Residue in Neural Networks Keywords Yang-Mills Theory, Mass Gap, Lattice QCD, Brydges-Fröhlich-Seiler Cluster Expansion, Transfer Matrix, Information Geometry, Kogut-Susskind Hamiltonian, Cabibbo-Marinari Heatbath, Wilson Flow, Constructive Field Theory February 2026