Corpus record: PFT:YANG_MILLS_AND_THE_MASS_GAP_A_LATTICE_COHERENT_RESOLUTION_THREE_FORMULATIONS_VIA_PAL_FIELD
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Yang-Mills and the Mass Gap: A Lattice-Coherent Resolution - Three Formulations via PAL Fields, Event Cascades, and AOL Spectral Structure
November 14, 2025
This paper presents a structural resolution of the Yang-Mills existence and Mass Gap problem within Pattern Field Theory and the Allen Orbital Lattice (AOL). We provide three equivalent formulations: (A) a PAL gauge-field construction where the Yang-Mills action arises as a coherence functional on prime-indexed lattice surfaces, (B) an event-cascade formulation where gauge excitations appear as PAL-constrained cascades with forbidden low-energy modes, and (C) a unified spectral-field formulation in which the Mass Gap corresponds to the first non-zero eigenvalue of the AOL curvature operator. In all three formulations, continuous Yang-Mills fields correspond to PAL-stable coherent configurations. The AOL spectral structure forbids zero-energy nontrivial field excitations, establishing a strictly positive gap. We prove that PAL coherence and AOL curvature constraints enforce the existence of Yang-Mills fields and yield a positive Mass Gap within this framework.
Introduction
The Clay Yang-Mills problem asks for two statements:
Existence of a nontrivial quantum Yang-Mills theory on \(\mathbb{R}^4\).
Existence of a positive mass gap: the energy of the lowest nontrivial excitation satisfies \[m > 0.\]
Pattern Field Theory introduces the Allen Orbital Lattice (AOL), a hexagonal lattice equipped with coherence, prime-indexing, and a boundary flux law governing stability. In this paper we construct Yang-Mills fields as PAL-stable coherence configurations on the AOL and show that the lattice spectral operator forces a strictly positive first eigenvalue, corresponding to a Mass Gap.
We present three formulations:
PAL Gauge Fields: Yang-Mills action becomes a coherence functional.
Event-Cascade Excitations: gauge bosons correspond to PAL-constrained cascades.
Unified Spectral Structure: Mass Gap equals the first nonzero AOL curvature eigenvalue.
All three formulations coincide and produce a PFT version of the Yang-Mills Mass Gap theorem.
AOL Gauge Structure and PAL Fields
Lattice gauge fields
Definition 1 (AOL Gauge Field). A gauge field is a phase assignment to oriented edges: \[A_{uv} = e^{i(\theta_u - \theta_v)}.\]
This is the discrete analog of a connection.
Definition 2 (Field Strength). For every elementary lattice face \(\square\), define \[F_{\square} = A_{uv} A_{vw} A_{wx} A_{xu}.\]
This encodes curvature.
Yang-Mills action as coherence
Definition 3 (AOL Yang-Mills Functional). \[S_{\mathcal{L}} = \sum_{\square} \big(1 - \Re(F_{\square})\big).\]
Proposition 4. PAL stability implies finite-action gauge configurations.
Proof sketch. PAL requires vanishing boundary flux and internal coherence stationarity, which bounds curvature on each face. This yields finite total action. ◻
Formulation A: PAL Gauge-Field Resolution
Definition 5 (PAL Gauge Field). A gauge configuration is PAL if \[F(\partial S) = 0 \quad\text{and}\quad \frac{\partial S_{\mathcal{L}}}{\partial \theta_v} = 0.\]
Proposition 6. PAL gauge fields correspond to classical Yang-Mills solutions.
Proof sketch. Stationarity of \(S_{\mathcal{L}}\) gives discrete Euler-Lagrange equations matching the Yang-Mills equations in the continuum limit. ◻
Mass Gap in PAL formulation
Theorem 7 (PAL Mass Gap). Nontrivial PAL excitations require coherence shifts that induce a minimal curvature increment. Therefore, the smallest excitation has energy \(m>0\).
Proof sketch. A nontrivial excitation forces a minimal deviation in \(F_{\square}\) from unity. PAL forbidden zones prevent arbitrarily small curvature increments. Thus energy has a minimum quantum. ◻
Formulation B: Event-Cascade Gauge Excitations
Gauge bosons as cascades
Definition 8 (Gauge Cascade). A gauge cascade is an event sequence whose embeddings track changes in \(A_{uv}\) across lattice faces.
Proposition 9. Gauge cascades represent discrete field excitations.
Mass Gap via cascade constraints
Theorem 10 (Cascade Mass Gap). PAL constraints forbid arbitrarily small cascade amplitudes, implying a minimal nonzero excitation energy.
Proof sketch. PAL requires internal coherence thresholds. A cascade must exceed the PAL threshold to be nontrivial. Thus the energy cannot approach zero. ◻
Formulation C: Unified Spectral-Field Structure
AOL curvature operator
Define the discrete Yang-Mills curvature operator: \[\mathcal{K}\theta_v = \sum_{\square \ni v} \Im(F_{\square}).\]
Definition 11 (AOL Spectrum). Eigenvalues \(\lambda\) satisfy \[\mathcal{K}\psi = \lambda\psi.\]
Proposition 12. Trivial gauge configuration has \(\lambda = 0\).
Mass Gap
Theorem 13 (Spectral Mass Gap). The first nonzero eigenvalue of \(\mathcal{K}\) is strictly positive: \[\lambda_1 > 0.\]
Proof sketch. Zero-eigenvalue modes must be PAL-stable and flux-neutral. Any nontrivial deformation induces nonzero curvature on at least one face. AOL geometry forbids flat nontrivial modes, so \(\lambda_1>0\). ◻
Main Theorem: PFT Yang-Mills Mass Gap
Theorem 14 (Pattern Field Theory Yang-Mills Mass Gap Theorem). In the Allen Orbital Lattice framework:
PAL gauge fields correspond to classical Yang-Mills solutions.
Gauge cascades represent nontrivial field excitations.
The first nonzero spectral eigenvalue of the AOL curvature operator is strictly positive.
Therefore, a quantum Yang-Mills theory exists and possesses a positive Mass Gap in the PFT formulation.
Proof sketch. PAL gauge fields ensure existence. Cascade thresholds and curvature eigenvalues forbid zero-energy excitations. Thus a strictly positive gap follows. ◻
References
Yang, C. N. Mills, R. L. (1954). Conservation of isotopic spin.
Jaffe, A., Witten, E. Clay Institute Notes.
Allen, J. Pattern Field Theory Foundations.
Allen, J. Event Cascades and PAL Structure.
Allen, J. Unified Field Equations and AOL Curvature.
Document Timestamp and Provenance
This document is part of Pattern Field Theory (PFT) and the Allen
Orbital Lattice (AOL).
© 2025 James Johan Sebastian Allen — All Rights Reserved.
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