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:

1. Self-adjointness: The Kogut–Susskind Hamiltonian H_KS is self-adjoint with purely discrete spectrum on the compact configuration space SU(N)^{|E|}.
2. 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|.
3. 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.

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

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

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:

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