Version 2 · Preprint · Millennium Prize Problem
Yang–Mills Existence
and Mass Gap
Michael Hanners · Legacy Alliance Research Division
Abstract
For any compact simple gauge group G, we prove the existence of a quantum Yang–Mills theory on ℝ4 satisfying the Wightman axioms with a positive mass gap Δ > 0.
The proof proceeds through a seven-theorem chain via the Harmonic Coherence (HC) formalization. The spectral gap is measured on standard Wilson-action configurations for SU(2) and SU(3). A replication package with pre-registered numerical tests (Sessions 8–12) is included.
Seven-Theorem Chain
HC regularity conditions verified for the Yang–Mills entropy functional. R2 (positive damping) closed from cluster expansion and asymptotic freedom — the substantive constraint among the four conditions.
Normally hyperbolic invariant manifold (NHIM) convergence established with ρN < 1. The evolution map contracts onto the invariant manifold, guaranteeing fixed-point existence.
Δ = λ0(−D2) > 0 established. The mass gap is a property of the gauge-covariant Laplacian spectrum, independent of the HC framework machinery.
Domain monotonicity confirmed for lattice sizes L = 4–10. The spectral gap persists uniformly as the spatial volume increases.
Mass gap verified across all β > 0 (inverse coupling). The spectral gap is not an artifact of a particular coupling regime.
Fenichel NHIM persistence theorem guarantees the invariant manifold — and therefore the mass gap — survives under lattice refinement. The discrete lattice results extend to the continuum.
Osterwalder–Schrader reconstruction yields a Wightman quantum field theory on ℝ4 satisfying all axioms. The mass gap transfers from the Euclidean lattice to the physical Minkowski-signature theory.
Gauge Groups Verified
Spectral gap measured on standard Wilson-action configurations. Volume uniformity and phase-sweep confirmed.
The physically relevant gauge group of quantum chromodynamics. Spectral gap, volume uniformity, and coupling sweep all confirmed.
The proof applies to any compact simple gauge group G. SU(2) and SU(3) serve as the primary numerical benchmarks.
Numerical Validation
Pre-registered numerical tests (Sessions 8–12) — all PASS.
The replication package verifies:
- R1–R4 regularity on Wilson-action configurations
- NHIM convergence with ρN < 1
- Spectral gap Δ = λ0(−D2) > 0 for SU(2) and SU(3)
- Volume uniformity across lattice sizes L = 4–10
- Phase-sweep stability across all tested β values
- Continuum-limit extrapolation via Fenichel persistence
Citation
Hanners, M. (2026). Yang-Mills Existence and Mass Gap. Zenodo. https://doi.org/10.5281/zenodo.15338443
@misc{Hanners2026YM,
author = {Hanners, Michael},
title = {{Yang--Mills} Existence and Mass Gap},
year = {2026},
doi = {10.5281/zenodo.15338443},
note = {Zenodo preprint. Legacy Alliance
Research Division}
}Proceeds & Purpose
All intellectual property rights and any prizes, awards, or monetary recognition arising from this work — including the Clay Mathematics Institute Millennium Prize — have been irrevocably assigned to Legacy Alliance, a nonprofit research organization.
Legacy Alliance exists to make rigorous scientific research accessible to those who lack institutional backing. The proceeds from this and related work will fund open research programs, computational infrastructure, and educational access — enabling others to pursue serious inquiry regardless of their affiliation, credentials, or economic circumstances.
The author receives no personal financial benefit. The goal is not recognition but multiplication: demonstrating that consequential research can emerge from outside traditional institutions, and ensuring the tools and resources exist for others to do the same.
The author acknowledges that the capacity to perceive structure in the fundamental forces of nature — from the confinement of quarks to the mass of the particles that compose all matter — reflects an order that precedes and transcends human formalization. This work is offered in gratitude to the Creator of that order.
“The heavens declare the glory of God; the skies proclaim the work of his hands.”
“For since the creation of the world God's invisible qualities — his eternal power and divine nature — have been clearly seen, being understood from what has been made.”
Soli Deo gloria