Gap-02 — Yang–Mills mass gap: the gate anchor ledger

The honest one-line: Gap-02 is the genuine open Clay Millennium Problem — no proof exists here or anywhere — but the program has a real, checkable win: it squeezes the entire unsolved continuum wall down to a single finite inequality, proves its own geometry gives no shortcut to it, and reports every numerical verdict (wins and losses) at the scope it actually establishes.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing Gap-02 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape as the canonical SG-4 ledger.

No status upgraded. This is the genuine open Clay problem. The established results reduce Gap-02 to a precisely-localized open object; they do not solve it.

1. Gate status header

• Gate-level Gap-02 roll-up (ratified board 2026-07-08): CERTIFIED-IRREDUCIBLE · RESOLVED +0 — the Clay wall is unsolved here and everywhere, carried openly as the named external residual.

  Taxonomy reconciliation (2026-07-05). Under the ratified taxonomy (board 2026-07-08) the gate-level grading is TERMINAL — CERTIFIED-IRREDUCIBLE · RESOLVED +0 (inherited Clay wall + measured-anchor gap value), read as terminal reached + residuals shown. Clay-side the uniform-gap bridge inequality R1 is precisely-OPEN and the continuum-measure co-gate R3 is unconstructed; the residual family R1–R6 listed on this page remains carried unchanged. The facts are unchanged — the earlier bare-OPEN roll-up reflects the superseded least-closed-residual rule, not different facts. No residual is closed or re-graded.

• Two honest framings of one fact:
  • Clay side: Precisely-OPEN [X] — EXTERNAL / Clay. The continuum problem is unsolved.
  • Scoreboard side: Reduced to the granularity posit (CERTIFIED-IRREDUCIBLE) — axiom-conditional, NOT a Clay solution. The program's granularity posit dissolves the continuum half by choice, leaving a finite, checkable residual plus the granularity axiom itself.

• The closed local/partial leg. What is rigorous is a conditional implication, not a gap. The all-operator lemma $$\big[\,H1\wedge H2\wedge H3\wedge H4\,\big]\;\Rightarrow\;\text{no soft physical sequence}\;\Rightarrow\;\Delta>0$$ holds (SL-1 / M4D), with $H4$ = uniform exponential clustering. "Uniform clustering $\Rightarrow$ the gap" is genuinely rigorous. It derives no value. The entire remaining obstruction is localized to the single inequality $$O_{\rm Gap02,bridge}(a,L,\beta):\qquad \frac{\Delta(a,L)}{\Lambda_{\rm YM}}\;\ge\;c\;>\;0\quad\text{uniformly as }a\to0,\;L\to\infty,\;\beta\to\infty.$$

• Status, split so it cannot be misread:

  • The conditional chain SL-0 $\to$ SL-2 $\to$ SL-1/M4D: DERIVED / CERTIFICATE-CONDITIONAL — the reduction is rigorous; it is a conditional, never a gap.
  • The uniform-gap bridge (Hole R1): OPEN — Clay. No such uniform lower bound is known.
  • The granularity / cost-floor axiom (Hole R6): CERTIFIED-IRREDUCIBLE — a named, unproven posit (its status matches the frozen grade; it is not an advance on it).

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what Gap-02 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest a5b1e6f9d951; spin-c bundle on $K_6$ 0fd19c9ae0c1; radii 634438ce0776 / 0e8b8dba2cf0. The branch is read-only. These are audit anchors — they certify which object was tested and that it cannot be quietly retuned. None is a Clay lever; none validates the physics.
• The scale $\Lambda_{\rm YM}$ is given-E / boundary data — pinned from the frozen anchors $\alpha_3(M_Z)$, $M_U\sim1.0\times10^{16}$ GeV, and the threshold $\delta_3=-1.7313$ via two-loop $\overline{\rm MS}$ ($R_0=1.592\times10^{-17}$ GeV$^{-1}$). Gap-02 does not derive the scale; deriving it would be deriving $E$.
• The gap itself is a measured fact — short-range strong force, massive glueballs ($m_{0^{++}}\approx1.7$ GeV), the running of $\alpha_s$. It is observation-locked, not a Gap-02 output.

3. Object anchors (given-E / upstream)

The pure-glue arena Gap-02 acts on:

• Pure $SU(3)$ Yang–Mills in $d=4$. The color group is selected by the geometry $K_6=SU(3)/T^2$ (the flag manifold $F_3$); the dynamics it must gap is ordinary 4D pure-gauge $SU(3)$. Status: GIVEN-E / upstream-inherited.
• The Wilson lattice regularization. Links $U_\ell\in SU(3)$ on $\Lambda\subset(a\mathbb Z)^4$, action $S_W=\beta\sum_p(1-\tfrac13\,\mathrm{Re}\,\mathrm{tr}\,U_p)$, $\beta=6/g_0^2$, measure $d\mu_{a,L}\propto e^{-S_W}\prod_\ell dU_\ell$. The mass gap is the lowest nonzero eigenvalue of $H=-a^{-1}\log T$. Status: standard arena.
• Dimensional transmutation. No classical scale; $a\,\Lambda_{\rm YM}=(b_0 g_0^2)^{-b_1/2b_0^2}e^{-1/2b_0 g_0^2}(1+O(g_0^2))$, $b_0=\tfrac{11}{16\pi^2}$. Any honest gap statement is expressed relative to $\Lambda_{\rm YM}$.

4. Root and master-anchor traceability

Deep roots that are load-bearing for Gap-02:

Master anchors in play: finite invariant ledgers · no unpaid labels (every constant is admitted only if writable without knowing the target verdict) · the frozen branch · given-$E$ ($\Lambda_{\rm YM}$, the UV package) · the declared granularity axiom · open-residual discipline.

5. The Gap-02 anchor ledger (the universal table)

6. The arithmetic — the reduction and the constants, in full

The reduction chain (rigorous, conditional). SL-0 is the exact equivalence $$\inf\{\langle\psi,H\psi\rangle:\psi\perp\Omega,\ \|\psi\|=1\}=\Delta>0\;\equiv\;\operatorname{Spec}(H)\cap(0,\Delta)=\varnothing,$$ hand-checkable, no open content. SL-2 supplies the finite-$a$ floor (RP $\Rightarrow$ $0\le T\le1$, $H=-a^{-1}\log T\ge0$; measure existence and analyticity in $\beta$). SL-1/M4D supplies the all-operator implication $[H1\wedge H2\wedge H3\wedge H4]\Rightarrow$ gap, where $H4$ is uniform clustering. At finite $a$ both the gap and RP are trivial — the entire wall IS the $a\to0$ limit.

The brutal one-line bridge — the exact missing theorem. $$\boxed{\;\frac{\Delta(a,L)}{\Lambda_{\rm YM}}\;\ge\;c\;>\;0\quad\text{uniformly as }a\to0,\;L\to\infty,\;\beta\to\infty\;}$$ with the same $c$ for the same measure for which existence, RP, and nontriviality also hold in the limit. The quantifier order is decisive: $\exists(\delta,K,\kappa,a_0,L_0,n_0)\,\forall(a,L,\beta,n)$ — a single $\delta$ uniform over all regulators and all scales. No such uniform lower bound is known. This lone inequality is the entire wall.

The decisive obstruction — rarity $\not\Rightarrow$ domination. In the deep UV the large-field weight $\mu_n\le e^{-c/g^2}\to0$ banks the inequality automatically — but rarity bounds a weight, not a contribution; the bare activity can compensate. Inside the $d=4$ marginal band $g(2^na)=O(1)$ the inequality becomes a strict finite comparison between two $O(1)$ constants with no spare coercive margin. $\delta=0$ is the $d=4$ stall. Method-invariance, verified: the polymer/cluster route and the AF-RP block-spin monotone $$M_{n+1}\le(1-c\,b_0\,g_n^2)M_n+O(g_n^4)M_n,\qquad b_0=\tfrac{11N}{3}=11\ (SU(3)),$$ both stall on the identical $\delta=0$ object at $g=O(1)$ — the signal that the obstruction is intrinsic to $d=4$ marginality, not an artifact of one tool.

The four $O(1)$ constants (cost-floor reframe). Built only from local gauge-invariant configuration data, modulo gauge — no eigenstate, no transfer-matrix spectrum, no correlation length enters (the circularity guard):

Diagnostic — the negative was measured, not assumed. The decidability margin $D\equiv s_{\rm block}-\log(E_{\rm conn}A_{\rm fluc}N_{\rm cert})$ came out negative at every binning and every $E_{\rm conn}$ choice, smallest $|D|/\sigma\approx67$: $$D(\delta_{\rm tr}{=}.5)=-4.004\pm0.041,\quad D(.25)=-5.043\pm0.041,\quad D(.125)=-6.162\pm0.041\ \ (\text{rigorous }E_{\rm conn}).$$ The critical count for $D=0$ is $N_{\rm cert}<\exp(1.0413)/(19.028\cdot0.05264)=2.83$ letters, while the smallest realized $N_{\rm eff}=155\gg2.83$. The plaquette hard gate passed first ($\langle P\rangle=0.59375\pm0.00197$ vs standard $0.5937$, $<1\sigma$) — the binding precondition for any verdict. That a non-trivial margin is sign-stable and binning-stable is what makes the INCONCLUSIVE verdict a real arithmetic fact about the ensemble, not an artifact — but master $\Rightarrow$ target is sufficient, not necessary; failing the sufficient condition does not deny the gap.

The geometry descent (Lemma 1B). The pure-glue projection of the frozen 13D object is ordinary 4D pure $SU(3)$ YM plus a $\theta$-term and an $M_{\rm KK}$-suppressed irrelevant tail. Every settled 13D residue lands in one of: a renormalization of $g_3$, $\theta$-data, $M_{\rm KK}$-suppressed irrelevant operator, physical quark matter (1A only), or admissibility governance — none IR-relevant. The 5-way tally is INERT:10 / ORDINARY-YM:3 / WILSONIAN-IRRELEVANT:1 / TOPOLOGICAL→R4:1 / LEVER:0. The geometry fixes the gap's origin and pins $\Lambda_{\rm YM}$; it supplies no IR constructive lever; $F^+$ is inert.

7. Declared-structure splits — the $z_\star$ and R4 phrases, each into honest objects

The $z_\star$ inequality hides a convention. The boxed target $z_\star:=\sum_{\gamma\ne0}w(\gamma)<1/(E_{\rm conn}A_{\rm fluc})=0.998$ has three natural circularity-clean readings that disagree on the verdict:

1. (A) per distinct binned letter, $w=e^{-\min\text{cost in bin}}$: $z_\star=7.66\ (\delta_{\rm tr}{=}.5)\to922\ (.03125)$ — diverges as $\delta_{\rm tr}\to0$; FAIL and ill-posed.
2. (B) intensive KP polymer activity per reference site: $z_\star=0.109$ — binning-stable; PASS.
3. (single effective activity $=e^{-s_{\rm block}}$): $z_\star=0.353$ — binning-stable; PASS.

The internal inconsistency: the corpus equated $E_{\rm conn}A_{\rm fluc}e^{-s_{\rm marg}}<1\Leftrightarrow E_{\rm conn}A_{\rm fluc}\sum_\gamma w(\gamma)<1$, which holds only if $\sum_\gamma w(\gamma)=e^{-s_{\rm marg}}$, but the $N_{\rm cert}$ route makes $\sum_\gamma w(\gamma)\sim N_{\rm cert}e^{-s_{\rm block}}\gg e^{-s_{\rm block}}$. Since $E_{\rm conn}=e\cdot7$ is pure $\mathbb Z^4$ graph combinatorics carrying no per-block letter count, the multiplicity must either live inside $z_\star$ (reading A, divergent) or be dropped (B / single-activity) — and the boxed inequality does not say which. Until $w(\gamma)$ is pinned by a derivation, $z_\star$ is not a decidable number. Picking B to report PASS is naked reverse-engineering from the measured value; picking A to report FAIL is equally illegitimate.

The R4 phrase hides a degree, a carrier, and a differential. "The $K_6$ mixed anomaly" splits into:

1. The well-posed home (Theorem 11). $\xi_{R4}\in\Omega_5^{\rm Spin\text{-}c}(B(SU(3)\to PSU(3));\tau_{K_6})$, a finite abelian group — a twisted bordism invariant, not a degree-4 cup product on the wrong locus (the original write-up was malformed on both degree and locus; the lone-$\bar c_1$ pushforward gives degree $2-6=-4<0$). Status: DERIVED (well-posing).
2. The carrier and operative differential. The carrier is the degree-3 class $\bar x_1\in H^3(BPU(3);\mathbb Z/3)$ — there is no degree-2 mod-3 carrier. The operative differential is the 3-primary $d_5=Q_1=\beta P^1$ ($|Q_1|=5$ at $p=3$), not a $d_3$ (the 2-primary $d_3$ is identically zero on 3-torsion). Status: DERIVED (corrected structure).
3. The value. Untwisted default $\Omega_5^{\rm Spin}(PSU(3)\times B^2\mathbb Z_3)=\mathbb Z_3\ne0$ — the survivor persists. The sole lever is whether the frozen $\tau_{K_6}=(2,2)$ twist-correction (degree-2 class cupping the degree-3 carrier onto the degree-5 line, with Massey/$v_1$-filtration structure at $p=3$) supplies a nonzero degree-lowering image. Status: OPEN — STILL_SUBTLE; not certified $0$, not forced $\ne0$.

Supporting frozen facts: $K_6=F_3$ has $b_2=2$, so $H^2(K_6;\mathbb Z_3)=(\mathbb Z/3)^2$; $c_1(TK_6)=2\rho=(2,2)\not\equiv(0,0)\bmod 3$, so $\bar c_1\ne0$ and $\tau_{K_6}$ is nonzero. (Do not cite the spin-c index "$-3$" as the Euler characteristic; $\chi(F_3)=+6=|\mathrm{Weyl}(SU(3))|$.)

8. Open residuals — grouped by family

These distinct residuals make up the open gate; none is closed by the rigorous reduction. R1 is the Clay wall; R2/R3 are independent co-gates; R4/R5 are not a gap lever; R6 is the axiom the dissolution rests on.

Family I — the continuum Clay completion (the prize): - R1 — the uniform-gap bridge. OPEN — Clay. $\Delta(a,L)/\Lambda_{\rm YM}\ge c>0$ uniform through the marginal band. This is the gate. - R2 — SL-3 nonperturbative BRST/Gribov kernel positivity. OPEN — independent co-gate. Established only perturbatively (Kugo–Ojima); the Gribov–Singer–Neuberger triad blocks the naive lattice route. Even a proved R1 yields no gap without R2. - R3 — OS reconstruction / continuum-RP survival (H1·H2·H3). OPEN — independent co-gate. No 4D $SU(3)$ continuum measure has ever been constructed; existence must come first, then the gap.

Family II — the geometry's one unsettled descent row: - R4 — the bordism anomaly value $\xi_{R4}$. OPEN — STILL_SUBTLE. Well-posed, untwisted default $\mathbb Z_3\ne0$, twist-correction the sole open lever. LOW leverage for the gap; HIGH cross-gate (shared with SG-4 and Born). - R5 — necessary-not-sufficient caveat. DERIVED (caveat). Even a nonzero, descending, universality-preserving $\xi_{R4}$ does not force a gap — anomaly-matching admits gapless conformal/TQFT saturations.

Family III — the dissolution side: - R6 — the granularity / cost-floor axiom. CERTIFIED-IRREDUCIBLE. A named unproven posit ($\hbar$ its residue value). The program tried and failed to derive a uniform floor from mere finiteness; the basin-shallowing countermodel (depths $d_n=B\cdot2^{-n-1}$, $\sum d_n=B/2<\infty$, $\inf_n d_n=0$) is a rigorous counterexample to the uniform claim. The axiom count does not drop.

9. Anti-claims (what this page refuses to say)

• Gap-02 is not solved, closed, or proven here — there is no Clay solution, here or anywhere. Formally: the rigorous content is the conditional $[H1\wedge H2\wedge H3\wedge H4]\Rightarrow$ gap, not a proof that $H4$ holds.
• Dissolved $\neq$ solved. Declining the continuum limit by the granularity axiom changes the Clay question rather than answering it.
• The frozen geometry supplies no proof lever on any settled descent row. Lemma 1 is operator-level; $F^+$ is inert; the LEVER tally is $0$ across the settled rows (R4 the one row still open). This is the bounded, banked claim — not the unprovable universal "no geometric modification could ever supply a lever." Smuggling the geometry in as a shortcut is forbidden.
• The INCONCLUSIVE master-inequality run is not a refutation — a sufficient proxy failing ($D<0$) does not deny the gap; a refutation needs a direct lower bound $z_\star\ge1/(E_{\rm conn}A_{\rm fluc})$, which was not computed.
• $z_\star$ is convention-underdetermined. Picking the reading that yields PASS (or FAIL) is reverse-engineering from the measured value; until $w(\gamma)$ is derived, $z_\star$ is not a decidable number.
• Rarity $\not\Rightarrow$ domination. A weight bound $\mu_n\le e^{-c/g^2}$ is not a contribution bound; that category error is the wall.
• RP alone cannot gap $SU(3)$ — an RP-only argument would falsely gap compact $U(1)$, which is provably gapless. (RP itself is true and needed downstream.)
• $\xi_{R4}$ is not computed, not $0$, and not a gap lever — and a nonzero value would still be necessary-not-sufficient (R5).
• The granularity axiom is a named posit, not a derivation — "REDUCED-to-axiom" does not lower the axiom count.
• The frozen-branch hashes are audit anchors; they do not validate the physics.

10. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package:

1. R1 (the Clay wall). Prove $\rho_\star<1$ uniform through the band, target-blind, via one of three live lanes: Lane A — sharpen the Bałaban large-field step from a weight bound to a contribution bound (fixed $e^{-\delta}$ contraction through the band); Lane B — a convergent activity/cluster expansion at $g=O(1)$ with strictly positive $\delta$; Lane C — match the UV (Gaussian) and IR (product-measure) expansion domains across the band, or a genuinely group-sensitive non-expansion method using $b_0=11N/3>0$ (the AF-RP monotone with the $O(g^4)M_n$ remainder controlled uniformly through $g=O(1)$). Falsifier: a direct uniform $\rho_\star\ge1$ kills this sufficient route — a legitimate negative close.
2. R2 (BRST/Gribov positivity). Construct an interacting nonperturbative positivity certificate on $\mathcal H_{\rm phys}=\ker(s)/\mathrm{im}(s)$ surviving $a\to0$ (Gribov-region-restricted measure, or a manifestly positive gauge-invariant formulation). Falsifier: a proof that positivity fails in the limit is itself a major negative.
3. R3 (OS reconstruction). Construct $\mu=\lim\mu_{a,L}$ with OS-0/1/2/3 verified in the limit, RP surviving $a\to0$, and nontriviality; then run OS$\to$Wightman. Falsifier: a triviality (Gaussian) proof refutes the interacting theory.
4. R4 ($\xi_{R4}$). Evaluate via the twisted $d_5=Q_1$ correction on the degree-5 line (the bare $Q_1(\bar x_1)$ sits in degree 8, off-line — only the $\tau_{K_6}$-correction can act). Success: a definite element of $\mathbb Z/3$. Falsifier: $\xi_{R4}=0$ is a valid close (the row is then fully inert); confirming the untwisted $\mathbb Z_3$ persists is equally valid.
5. R5 (necessary-not-sufficient). Conditional on R4 nonzero and universality-preserving, separately prove the matched IR phase is gapped/confining (exclude conformal/TQFT saturation). Falsifier: exhibiting a gapless anomaly-matching phase confirms R4 is not a gap handle.
6. R6 (granularity axiom). Derive a uniform positive minimum-action floor from a strictly weaker principle than positing it — one that is not the cell law in costume, not QM smuggled in, and writable without knowing the answer is $\hbar$. Two named residuals remain posited: inter-arena uniformity (the basin-shallowing countermodel CM2 is unrefuted) and strict positivity $\Delta_0>0$ vs $\Delta_0=0$. Honest terminal: a standing disclosure that it is a named posit is legitimate.

Closing R1 with R2 and R3 is the Clay prize — and even then, the gap is observed, not a Gap-02 derivation of $E$. R4/R5 settle a cross-gate value, not the gap. R6 grounds only the dissolution side.

11. Completion tests for this page

Required presence (all met): gate roll-up CERTIFIED-IRREDUCIBLE · RESOLVED +0 (Clay wall carried openly; historical OPEN label kept as history) · the closed conditional leg ($H4\Rightarrow$ gap) and its CERTIFICATE-CONDITIONAL label · the boxed bridge inequality · $E$/$\Lambda_{\rm YM}$ not derived · frozen hashes (AUDIT ONLY) · SL-0/SL-1/SL-2 · the four $O(1)$ constants ($E_{\rm conn}=e\cdot7$, $A_{\rm fluc}$, $s_{\rm block}$, $N_{\rm cert}$) · the measured-negative diagnostic $D<0$ (sign- and binning-stable, plaquette gate passed) · the $z_\star$ three-reading split · the R4 degree/carrier/differential split · every open residual R1–R6 as its own row · the rarity-$\not\Rightarrow$-domination, RP-only, dissolved-$\neq$-solved, no-lever, and hashes-don't-validate anti-claims.

Required absence (all held): no claim that Gap-02 is solved/closed/proven · the dissolution answers Clay · the geometry is a proof lever · $D<0$ refutes the gap · a $z_\star$ reading is picked to land a verdict · RP-only gaps $SU(3)$ · $\xi_{R4}$ is computed/$0$/a gap lever · a nonzero anomaly forces a gap · the granularity axiom is derived / the axiom count drops · hashes validate physics · fixed-$a$ closure = continuum completion.

This gate follows the same eleven-part shape and universal table as the canonical SG-4 ledger.

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why every constant must be writable without knowing the verdict) · Layer 4 — carrier-forcing & the given-E wall · the SG-4 anomaly ledger (the shared $\xi_{R4}$ row) · the full Gap-02 dossier.