# Gap-02 — The Yang-Mills Mass Gap

## What this gate must establish

This gate must show that pure SU(3) Yang-Mills theory in four dimensions has a **positive mass gap** — a strictly nonzero lowest excitation above the vacuum — surviving the continuum limit. This is one of the Clay Mathematics Institute's seven Millennium Prize Problems. The gap is what makes the strong force short-ranged and the lightest glueball massive rather than massless; it is *observed* (short-range nuclear force, massive hadrons, the measured running of the strong coupling), so this is an observation-locked obligation, not an artifact. The standing claim here is that the theory does **not** close it.

## The geometry's role (load-bearing, not validation)

The frozen 13D geometry fixes where the gap's *origin* comes from — SU(3) emerges from the shape (K₆ = SU(3)/T²) — and it pins the boundary data (the scale Λ_YM, the renormalization conditions) and collapses the search space. But an internal "Lemma 1" descent audit found that every 13D-specific structure it could supply (the coset color source, the KK tower, the spin-c index, the internal-circle fold) is **UV-origin and admissibility discipline only** and supplies *no infrared proof lever*: by Wilsonian universality the IR confining dynamics is indifferent to the UV completion. The honest reading: the geometry settles the gap's *origin* but provides **no shortcut to its proof**. The live mathematical object is therefore ordinary 4D SU(3) Yang-Mills — the actual Clay object, nothing 13D-specific.

## Current honest status

**Badge: Precisely-OPEN.** This is the Clay wall, and it is unsolved here. No status was ever upgraded. No number was derived; no wall was broken.

*Consistency note — two honest framings of one fact.* The homepage scoreboard badges this gate **Reduced-to-axiom (granularity)**: the program's granularity posit (the Finite Operational Cell Law — a smallest physical step, so the a→0 continuum limit is never physically taken) **dissolves the continuum (Clay) half by choice**, leaving only a *finite*, checkable mass-gap residual plus the granularity axiom itself. This brief states the identical situation from the Clay side — the **continuum** problem is Precisely-OPEN, and the cost-floor reframe **changes the question rather than solving it**. Either way: **dissolved ≠ solved; axiom-conditional; NOT a Clay solution.**

What *is* established (rigorous, citable): the implication **"uniform clustering ⇒ the gap"** holds via an all-operator lemma, and the remaining open object has been **localized to a single inequality** — the uniform-gap bridge
> Δ(a,L) / Λ_YM ≥ c > 0, uniformly as a→0, L→∞ (same constant, same measure for which existence, reflection positivity, and nontriviality also hold).

No such uniform lower bound is known; this lone inequality, holding through the d=4 marginal band where the coupling is order one, *is* the entire wall. Two further gates remain separately open even if the bridge held: continuum reflection-positivity survival (OS-reconstruction) and BRST/Gribov positivity (SL-3). One narrow geometric route — a possible mixed 't Hooft anomaly between the spin-c bundle and the Z₃ one-form center (the "R4" row) — is held **OPEN_CANDIDATE**, not a lever: prior evidence tilts against it, and a separate result shows a 't Hooft anomaly does not by itself force a gap (it can be satisfied by a gapless theory). The non-perturbative-QCD ledger row (UQF-11) stays **AUDIT**: the framework supplies UV data only and is **used, not proven** here.

A finite-lattice Monte-Carlo evaluation of an *activation-cost* sufficient condition (a "master inequality") was run honestly on a fixed 4⁴, β=6.0 ensemble; it returned **INCONCLUSIVE** (the decidability margin came out robustly negative across binnings). This is explicitly **not** a refutation of the gap (the condition is sufficient, not necessary) and **not** a Clay solution — it is one horn of one sufficient route failing on one ensemble.

## What would close it

A proof of that one uniform inequality (or, as the Clay statement itself anticipates, most likely a genuinely new idea in constructive quantum field theory). Every observation-preserving geometric modification examined here *relocates* into a strictly harder subproblem; the difficulty — d=4 marginal coercivity at order-one coupling — is invariant under every legal move, and the geometry supplies no constructive lever. A "cost-floor" reframe can put the continuum out of scope *by choice*, leaving a finite, checkable-but-could-fail inequality, but that does not solve the Clay problem; it changes the question. The honest near-term path is incremental (block-spin RG fallback, alternative reflection-positivity/Osterwalder-Seiler routes), with no expectation of closure absent a new idea.

## Sources

- **Gaps & Walls Register** (programme-wide honest status map), the Gap-02 / continuum-Yang-Mills row — the canonical disposition (`Precisely-OPEN`).
- **Gap-02 current-status reconciliation** and the **uniform-gap bridge** proof-lane map — the centerpiece document isolating the exact missing inequality, and the Lemma-1 "no-shortcut" descent audit.
- **Gap-02 master-inequality result** — the finite-lattice Monte-Carlo evaluation returning INCONCLUSIVE (with its plaquette-reproduction hard gate).
- **Clay Mathematics Institute**, "Yang-Mills and Mass Gap" (Millennium Prize Problems) — the external problem statement this gate inherits.

Ceiling on every claim: **serious candidate / partial unification — NOT validated.** Given the observed structure as input is not a derivation of it. No status was ever upgraded.
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