Yang-Mills & Mass Gap — Research Program Abstract

Constructive Non-Perturbative Frameworks & Topological Mass Gap Constraints

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Mathematics Root · Millennium Frontiers · Yang-Mills Research Program

Public Status Boundary. This manuscript presents an authorial exploratory research program exploring constructive non-perturbative approaches to gauge field theories and mass gap emergence under the Pasev Gauge Principle. In accordance with the project constitution, this work represents an unverified theoretical proposal (RESEARCH_PROGRAM / UNVERIFIED_HEURISTIC); it does not claim a completed Millennium Prize solution or external peer-reviewed resolution (ACTIVE_P4_SEALS = 0).

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1. Architectural Concept & Research Objective

The Yang-Mills Existence and Mass Gap problem requires constructing a mathematically rigorous Quantum Yang-Mills theory on ${\mathbb{R}}^4$ with compact simple gauge group $G$ satisfying the Wightman or Osterwalder-Schrader axioms, and proving a strictly positive mass gap $\mathrm{\Delta }>0$:

$$S_{\text{YM}}[A]=\frac{1}{4g^2}{\int }_{{\mathbb{R}}^4}\mathrm{Tr}(F_{\mu \nu }{F}^{\mu \nu })d^{4}x,\mathrm{spec}(H)\subseteq \{0\}\cup [\mathrm{\Delta },\mathrm{\infty }),\mathrm{\Delta }>0$$

Within Digital Fabrica Theory, the mass gap is explored through Lattice Wilson Loop Holonomies & Gauge Orbit Invariance under the Pasev Gauge Principle (PGP), investigating whether non-perturbative confinement emerges from discrete plaquette boundary obstructions on relational lattices before taking the continuum limit $a\to 0$.

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2. Formal Definitions & Relational Gauge Operators

• $A_{\mu }\in {\mathrm{\Omega }}^1({\mathbb{R}}^4,\mathfrak{g})$: The non-abelian gauge potential connection 1-form.
• $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }+ig[A_{\mu },A_{\nu }]$: The gauge-covariant field curvature 2-form.
• $U_P=\mathcal{P}\exp (ig{\oint }_{\partial P}A\cdot dx)\in G$: The Wilson loop holonomy operator around elementary lattice plaquette $P$.
• $S_W(U)=\beta \sum _P(1-\frac{1}{N}\mathrm{Re}\mathrm{Tr}{U}_P)$: The lattice Wilson gauge action.
• ${\mathrm{\Delta }}_{\text{gap}}=inf\{E\in \mathrm{spec}(H)\setminus \{0\}\}>0$: The non-perturbative mass gap separating the unique vacuum $\mathrm{\Omega }$ from one-particle excitations.

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3. Epistemic Classification & Lean 4 Formalization Bounds

• Formal Classification: RESEARCH_PROGRAM / UNVERIFIED_HEURISTIC.
• Lean 4 Proof Status: No machine-checked proof of Yang-Mills existence or mass gap exists in the repository. Formalization is strictly restricted to foundational gauge invariance and groupoid lemmas:
  • thm_pgp_gauge_identity (Fabrica.PGP): Identity gauge transformation functional invariance $F({\text{id}}_{\mathrm{\Phi }}\phi )=F(\phi )$.
  • thm_pgp_gauge_composition (Fabrica.PGP): Composite gauge transformation invariance preservation $F(g_2(g_1\phi ))=F(\phi )$.
  • thm_pgp_gauge_orbit_invariance (Fabrica.PGP): Invariance under iterated gauge orbit transformations.
  • thm_one_cycle_linear_combination (Fabrica.InvariantEngineering): Subspace closure of boundary-free 1-cycles.
• Millennium Prize Demarcation: No claim of prize solution, peer-reviewed acceptance, or Clay Mathematics Institute submission is made (ACTIVE_P4_SEALS = 0).

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4. Canonical Continuations

Direction	Target Resource	Purpose
Frontier Hub	Frontier Mathematics & Proofs Hub →	Survey of exploratory mathematical programs and epistemic boundaries
Gauge Principle	Pasev Gauge Principle →	Relational gauge connections and local scale covariance
Proof Gateway	Lean 4 Formalization Roadmap →	26 machine-verified theorem records and verified lemma trees
Review Gateway	Mathematical Review Gateway →	Interactive verification readiness dashboard and atlas

EXTERNAL REFERENCE