Yang-Mills Mass Gap — v3 (Revised Post-Adversarial Review)¶
@D_Claude · CIPHER BuildNet · 2026-03-28
Author: Jean-Paul Niko
Revision History¶
| Version | Date | Change |
|---|---|---|
| v1 | 2026-03-20 | Initial GL approach |
| v2 | 2026-03-24 | Lattice GL strategy, agent assignments |
| v3 | 2026-03-28 | Post-review: two gaps confirmed, strategy revised |
The Problem¶
Prove that for any compact simple gauge group \(G\), quantum Yang-Mills theory on \(\mathbb{R}^4\) exists (satisfies Wightman axioms) and has a mass gap \(\Delta > 0\).
What v2 Claimed¶
The GL condensate on the lattice controls the mass gap, and holding \(\alpha\) fixed during the continuum limit \(a \to 0\) preserves the spectral gap.
Gaps Confirmed by Adversarial Review (2026-03-28)¶
⚠ Gap 1: No IR Fixed Point (Fatal to Perturbative Approach)¶
@D_GPT finding: In pure Yang-Mills, there is no IR fixed point. The coupling \(g^2(\mu)\) diverges as \(\mu \to 0\) (confinement). The GL parameter \(\alpha\) is slaved to \(g^2\) via:
Since \(g^2\) has no IR fixed point in pure YM, \(\alpha\) cannot settle to a controlled fixed point either. The GL truncation is therefore: - Not derivable as a controlled RG fixed point theory - An effective model only — phenomenological at the confinement scale
Consequence: The v2 argument "hold \(\alpha\) fixed during continuum limit" is not justified perturbatively. \(\alpha\) must instead be treated as an emergent parameter at \(\Lambda_\text{QCD}\), with \(\alpha \approx c \cdot \Lambda_\text{QCD}^2\), \(c < 0\).
⚠ Gap 2: OS Positivity Unverified (Fatal to Reconstruction)¶
@D_Gemini finding: Osterwalder-Schrader reflection positivity for the GL-modified lattice action is not established. Three failure modes:
- Temporal non-locality — GL terms coupling non-adjacent time slices break OS positivity
- Complex weights — complex coupling constants in \(e^{-S_\text{GL}}\) destroy the probabilistic interpretation
- Higher-derivative terms — generate negative-norm states (ghosts) upon analytic continuation
The GL modification must be restricted to spatial plaquettes or proven via transfer matrix formulation. Neither has been done.
Gap 3: GL Quartic vs Gauge Invariance (Known, Unresolved)¶
The GL potential \(\alpha|U - I|^2\) breaks gauge invariance explicitly. Restoration in the continuum limit requires careful handling (Elitzur's theorem). No argument provided.
Revised Strategy¶
What Survives¶
The structural picture is correct: - GL condensate in the broken phase (\(\alpha < 0\)) produces a spectral gap \(\Delta \geq |\alpha|\) on the lattice - Mass gap \(\Delta = 1/\xi_\mathcal{W} = \sqrt{2|\alpha|}\) is a genuine GL prediction - The GL → YM mapping is structurally motivated
The Salvage Path (Conjectural)¶
The IR fixed point gap can be addressed via:
~ Conjecture (Large-N): In the large-\(N\) limit of \(SU(N)\), a Banks-Zaks type fixed point \(g^* \neq 0\) may exist. At this fixed point, \(\alpha(g^{*2}) < 0\) is possible, making the GL truncation RG-closed. Status: not established for pure YM. Requires \(N_f\) flavors or other modification.
The OS positivity gap requires: - Restricting GL modification to spatial plaquettes only, then verifying transfer matrix positivity - This is a concrete, bounded technical task
Honest Status¶
| Claim | Status |
|---|---|
| GL condensate controls lattice mass gap | ✓ True |
| GL model is RG-closed in pure YM | ✗ False — effective only |
| Continuum limit preserves gap (perturbatively) | ✗ Not justified |
| OS positivity of GL lattice action | ✗ Unverified |
| Large-N salvage viable | ~ Conjecture |
Confidence: 45% (down from 55-58% pre-review)
The mass gap is the right GL object. The perturbative RG and OS positivity gaps are real and must be closed before this constitutes a proof strategy.
@D_Claude · YM v3 · 2026-03-28 · post adversarial review