Existence and Mass Gap for Yang–Mills Theory

Field

Mathematical physics

Problem overview

The Yang–Mills Existence and Mass Gap problem asks whether non-abelian gauge theories in four dimensions admit both:

  • global mathematical existence in a rigorous quantum-field framework
  • a strictly positive mass gap separating the vacuum from the lowest nontrivial physical excitation

Yang–Mills theory lies at the core of modern particle physics, yet the mathematical origin of confinement-scale mass generation remains unresolved. A central challenge is to understand how nonlinear gauge interactions can generate stable positive spectral scales without inserting them by hand.

Why the problem is difficult

The problem sits at the intersection of nonlinear partial differential equations, spectral theory, gauge theory, and quantum field theory.

The main difficulties include:

  • strong nonlinear self-interaction of gauge fields
  • gauge redundancy, which complicates the identification of physical variables
  • controlling behavior across many spatial and energy scales
  • passing rigorously from finite approximations to continuum theories
  • constructing a reflection-positive Euclidean theory and recovering a Hamiltonian theory
  • proving that a positive mass scale emerges dynamically rather than by assumption

Related physical theories strongly suggest that such a gap should exist. One does not observe freely propagating massless strong-force gauge carriers at low energies, and lattice gauge calculations indicate a positive nonzero excitation scale. The open problem is to establish this rigorously in four-dimensional Yang–Mills mathematics.

Current documents on this page

This page currently links two related documents.

The first is a short structural paper:

Local Coherence Hessians: A Structural Classification of Spectral Gaps

This 15-page paper studies local coherence Hessians and identifies when uniform spectral gaps can or cannot persist in large-volume limits. It is not a proof of the Yang–Mills mass gap. Its role is to isolate a structural mechanism: locality alone generally produces gapless Laplacian-type spectra, while multi-channel locking can create uniform curvature on relative modes. A full gap still requires curvature in the final surviving soft sector.

The second is a long candidate proof manuscript:

A Source-to-Endpoint Construction for Compact-Simple Yang–Mills Existence and Mass Gap

This 303-page manuscript attempts a full source-to-endpoint construction for compact-simple Yang–Mills existence and positive mass gap. It builds a finite-type Yang–Mills source, proves a normalized source-level terminal gap, constructs a reflection-positive Euclidean source functional, recovers the ordinary compact-simple Yang–Mills endpoint algebra, applies Osterwalder–Schrader reconstruction, and transfers the source gap to physical Hamiltonian spectral exclusion.

The two papers have different roles. The Hessians paper gives a compact structural model for gap formation. The source-to-endpoint manuscript attempts to lift that type of structural mechanism into a full Yang–Mills proof architecture.

Coherence-geometric approach

In the Coherence Geometry framework, gauge-type systems are modeled through source-side admissibility, interacting coherence channels, local transport, obstruction closure, quotienting, and terminal readout.

The approach does not begin by treating endpoint-visible field configurations as automatically physical. Instead, it asks which structures are generated by the source, survive closure and quotienting, and remain visible after terminal readout and endpoint projection.

The guiding distinction is:

endpoint visibility is not source admissibility.

For Yang–Mills, this means that gauge-fixed soft modes, Hessian-visible directions, continuum shadows, or endpoint-completion artifacts are not counted as physical spectral data merely because they can be displayed in a projected description. They must survive the source construction and the reconstruction pipeline.

Key organizing ideas

  • local coupling alone need not generate a persistent positive gap
  • relative-channel locking can create stable gapped internal modes
  • remaining synchronized soft modes determine whether the full system remains gapless or becomes massive
  • source closure decides which endpoint-visible patterns are actually admissible
  • ordinary endpoint recovery is required so that the construction does not prove a result only for a smaller source-generated theory
  • Hamiltonian mass-gap transfer requires calibrated comparison between source terminal energy and reconstructed spectral energy

From the Hessian paper to the source-to-endpoint manuscript

The Local Coherence Hessians paper studies a simpler projected setting. It shows a sharp structural dichotomy:

nearest-neighbor gradient locality produces Laplacian-type spectra whose lowest nonzero eigenvalue typically scales like \(L^{-2}\), becoming gapless in infinite volume

  • multi-channel locking can gap relative internal sectors uniformly
  • a full uniform gap requires positive quadratic curvature in the final remaining soft sector

This identifies a possible structural origin of mass scales: not locality alone, but locking plus curvature in the surviving low-energy sector.

The longer Yang–Mills manuscript uses this lesson as motivation, but does not simply identify the Hessian gap with the Yang–Mills mass gap. Instead, it attempts a full source-to-endpoint argument.

The manuscript’s proof architecture is:

$$
\Sigma_{G,4}
\to
S_{\mathrm{YM}}(G)
\to
\Delta_{\mathrm{CG}}(G)>0
\to
\omega_{\mathrm{CG}}^E
\to
\omega_{\mathrm{YM}}^E
\to
(\mathcal H_G,\Omega_G,\mathcal A_G,H_G)
\to
\Delta_{\mathrm{YM}}(G)>0.
$$

In words:

  • construct the compact-simple Yang–Mills source
  • define primitive terminal readout and strict source closure
  • prove a normalized source-level no-accumulation theorem
  • construct a reflection-positive Euclidean source functional
  • project to the public Euclidean endpoint
  • prove ordinary endpoint recovery
  • apply Osterwalder–Schrader reconstruction
  • transfer the source gap to the physical Hamiltonian spectrum

Author’s Intuition

Many systems remain dynamically active even after reaching stability. They may continue to fluctuate or rebalance internally at low cost while still resisting larger structural changes unless additional energy is supplied.

A useful analogy appears in protein folding: a folded protein does not become motionless, but continues to move within a stable basin. Small adjustments remain possible, while larger rearrangements require overcoming stronger barriers. Another familiar analogy is the photoelectric effect, where internal activity may persist while a definite energy input is required to enter a qualitatively different state.

The Yang–Mills mass-gap question asks whether the gauge field behaves similarly: local motion may remain possible, yet any genuinely new physical excitation must cross a positive minimum energy threshold.

The short Hessian paper studies this intuition in a local spectral model. The longer source-to-endpoint manuscript attempts to turn it into a full Yang–Mills construction.

Status

Candidate proof manuscript available for technical review

The 15-page Hessians paper is a structural precursor. The 303-page source-to-endpoint manuscript is a candidate proof and should be read as a preprint submitted for independent inspection, criticism, correction, and verification.

Technical Review

These documents are shared for technical evaluation. Corrections, counterexamples, simplifications, literature pointers, and independent verification are welcome.

The author does not assume that a long manuscript is correct merely because it is internally detailed. The purpose of release is to make the argument inspectable.

Available documents

The Zenodo-hosted PDFs are the authoritative technical records. This page is descriptive only.

Relationship to the Clay problem

This work engages the Yang–Mills problem in the spirit for which it was established: identifying whether deeper structural principles clarify the existence and mass gap questions that resist resolution under traditional formulations.

The financial prize associated with the Clay Millennium Problems is not a motivating factor. The motivating factor is determining whether coherence-regulated structure provides a natural resolution of one of the central open problems in mathematical physics.

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