Gap-01 — a6 Seeley-DeWitt wall (one-loop finiteness): the gate anchor ledger

The honest one-line: Gap-01 has a real, checkable scale-free win — the dimensionless heart of the sixth heat-kernel coefficient $\mathrm{tr}[a_6]$ on the frozen 13D graviton+ghost operator is genuinely derived and dual-validated by four exact-rational sphere cross-checks — and on the ratified board (2026-07-08) the gate stands DERIVED-GIVEN-anchor · RESOLVED +0, with the residual family shown openly: the dimensionful magnitude is dissolved-as-ill-posed at odd $D=13$, and the two routes for the $K_6$ graded value currently disagree grossly (carried as a named, bounded computation-debt, not hidden).

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-01 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 and universal table as the canonical SG-4 ledger.

This gate touches none of the famous walls. It is a finite-matching / one-loop-finiteness question, not a route across the Yang–Mills mass gap or the cosmological constant.

1. Gate status header

• Gate-level Gap-01 roll-up (ratified board 2026-07-08): DERIVED-GIVEN-anchor · RESOLVED +0 — the scale-free derivation banked; residual family shown below. Historical roll-up under the superseded least-closed-residual rule: OPEN — partial-derivation banked, kept as history.
• The closed local leg: the scale-free sector. The dimensionless invariants of the sixth Seeley–DeWitt coefficient are forced, and the four sphere cross-checks vanish their residual to machine precision: $$\big|\,a_6^{\rm spectral}(S^n) - a_6^{\rm Gilkey}(S^n)\,\big| \;=\; 0 \ \text{to}\ \sim 4\times10^{-14}, \qquad n=2,4,6 .$$
• Status, split so it cannot be misread:
  • Scale-free sector: DERIVED-GIVEN-E — the dimensionless ratios and signs are forced by Gilkey invariance theory plus standard QFT, given the frozen operator, and are dual-validated target-blind. The scale-free leg is not itself axiom-open.
  • Gate-level Gap-01: DERIVED-GIVEN-anchor · RESOLVED +0 (historical label under the superseded rule: OPEN).
  • Dimensionful magnitude: DISSOLVED-as-ill-posed at odd $D=13$ — reported, where at all, only as a labeled consistency coefficient.
  • The graded value (binding): OPEN / route-INCONSISTENT ($R7 =$ FAIL_VALUE_MISMATCH).

Given $E_{\rm frozen}$, the scale-free invariants are fixed by exact-rational arithmetic, with no per-object retuning. What stays open is everything beyond the scale-free skeleton — the dimensionful magnitude, the two-route reconciliation of the graded value, the positivity functional, and field-level sufficiency.

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

2. Frozen inputs (what Gap-01 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is READ-ONLY. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics. The frozen engine is unmodified; every correction below is recorded as a proposed change against the read-only branch, not applied to it.
• Frozen 13D geometry $M_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2$, with $K_6 = SU(3)/T^2$ the color carrier. It enters Gap-01 as given / charged / inherited. Gap-01 does not derive the geometry or the spectrum. Every "passes" below is a statement about this $E_{\rm frozen}$, not a derivation of it.

3. The operator anchor (given-E)

The Laplace-type object is the de-Donder (harmonic gauge, $\alpha=1$, Lichnerowicz) graviton on $\mathrm{Sym}^2(T)$, with the Faddeev–Popov vector ghost subtracted and the ultralocal Nakanishi–Kugo third ghost contributing nothing at $a_6$: $$ a_6^{\rm phys} \;=\; a_6[\text{grav}] \;-\; 2\,a_6[\text{ghost}] \;+\; 0 , $$ $$ \dim_{\rm fibre}\mathrm{Sym}^2(T) = 91,\qquad \dim_{\rm fibre} T = 13 \quad (\text{in }13D). $$ Status: GIVEN-E / upstream-inherited — read off the frozen branch, not Gap-01-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for Gap-01:

Causal order is not a primary load-bearing anchor for the Gap-01 arithmetic.

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the declared heat-kernel scheme object · open-residual discipline.

5. The Gap-01 anchor ledger (the universal table)

6. The arithmetic — the scale-free win, in full

For a Laplace-type operator $D$ on a $d$-manifold the heat trace organizes as $$ \mathrm{Tr}\,e^{-tD} \;\sim\; (4\pi t)^{-d/2}\sum_{k\ge0} t^{k}\!\int\!\sqrt{g}\,\,\mathrm{tr}_V\!\big[a_{2k}(x)\big],\qquad t\to0^+, $$ and $a_6$ is the $k=3$, mass-dimension-6, cubic-curvature term. The earned win is that the dimensionless content of $a_6$ on the frozen operator is forced, and dual-validated two structurally independent ways:

• Route A (Gilkey contraction): plug the curvature into the eq.(4.29) functional and contract.
• Route B (spectral peel): sum the actual $S^n$ Laplacian eigenvalue heat trace and extract the $t^3$ coefficient by a 60-digit Vandermonde peel — no reference to the Gilkey formula.

The two routes agree on four exact rationals: $$ a_6(S^2)=\tfrac{4}{315},\quad a_6(S^4)=\tfrac{74}{63},\quad a_6(S^6)=\tfrac{1139}{63},\quad a_6^{\rm conf}(S^6)=\tfrac{5}{63}, $$ to relative error $0$ to $\sim4\times10^{-14}$. The rationals are recovered from the spectral side with no reference value fed in; they are computed, not injected.

The $K_6$ color factor $124/315$ is dual-validated but metric-selected — it holds at the normalization $\mathrm{Scal}_{K_6}=7.5$. It must be carried with that caveat, not sold as "the one ratio that drops out of nothing." The genuinely normalization-robust statements are the ratios such as $a_4/a_2^2 = 66/125$, exact independent of $R_6$ because scalars are Levi-Civita–immune.

Diagnostic — the correction is specific, not trivial. The $\sim31\%$ curvature error was caught by a theorem, not a target. The engine's $K_6$ builder wrote a first-Bianchi-violating sign: $$ \underbrace{|\mathrm{Riem}|^2/\mathrm{Scal}^2 = 31/147}_{\text{1st-Bianchi residual }=1/7\ \text{(VIOLATES)}} \quad\longrightarrow\quad \underbrace{|\mathrm{Riem}|^2/\mathrm{Scal}^2 = 23/75}_{\text{1st-Bianchi residual }=3.05\times10^{-16}}. $$ One sign choice cannot simultaneously fake an exact rational and Bianchi-exactness; that double coincidence is what makes the fix trustworthy with no target. The Einstein constant moves $\kappa = 7/12 \to 5/12$ and the bulk shifts $+6.305\%$, sign preserved (negative). Reproduced four target-blind ways (Nomizu rebuild; covariant 2nd-Bianchi; the corpus's own finite-difference reference; sectional-curvature sums).

The obstruction split. The scale-free leg closes: $$ O_{\rm Gap01,\,scalefree}(E_{\rm frozen}) = 0 \quad(\text{ratios, signs, sphere checks}), $$ but the full gate obstruction additionally carries the dimensionful and graded-value legs, which we do not assert vanish: $$ O_{\rm Gap01}(E) = \big(O_{\rm scalefree},\ O_{\rm magnitude},\ O_{\rm graded\text{-}value},\ O_{\rm positivity}\big),\qquad O_{\rm Gap01}(E_{\rm frozen}) \ne 0 . $$

7. The $\mathbb{Z}_2$ "defect," split into honest objects

The single phrase "the $\mathbb{Z}_2$ boundary defect" hid the wrong object. Split honestly:

1. The boundary-coefficient framing — a "missing order-6 mixed Neumann/Dirichlet coefficient" (the published mixed-boundary tower stops at $a_5$). Status: DISSOLVED / wrong-object artifact. A global isometric reflection on a closed manifold is not a manifold-with-boundary BVP. The decisive diagnostic: the twisted trace $\mathrm{Tr}_\sigma(e^{-tD})$ on $S^1_R/\mathbb{Z}_2 = 1$ exactly, $t$-independent — an integer-power $t^0$ Donnelly/Lefschetz series with no $1/\sqrt{t}$ boundary tower.
2. The Donnelly equivariant reduction $\ \mathrm{tr}[a_6]^{\mathbb{Z}_2} = \tfrac12\,c_3^\gamma$, with $\det(I-d\sigma|_N)=2$, totally-geodesic fixed locus, angle deficit $0$. Status: STRUCTURE-BUILT — verified by six target-blind tests to machine precision (including the spinning-field T5 through all Seeley–DeWitt orders, diff $\sim1\times10^{-13}$; curved scalar defect $a_6 = 2/315 = \tfrac12\cdot\tfrac{4}{315}$ to rel. $1.3\times10^{-14}$).
3. The grading constants $\ \mathrm{tr}\,\gamma_{\rm grav}=67,\ \mathrm{tr}\,\gamma_{\rm ghost}=11$, Block-A weight $67-2\cdot11=45$. Status: DERIVED-GIVEN-E — pure linear algebra forced by $D=13$ plus the reflection; all $[\gamma,E],[\gamma,\Omega]$ vanish on the actual engine matrices.
4. The numeric $c_3^\gamma$. Status: OPEN — it inherits Hole 1 (the LC correction); the structure is built, the number is not emitted.

This is REDUCE-not-RELOCATE: a graded $a_6$ of a 12d base via the already-built, sphere-validated engine is strictly easier than deriving a new research-grade boundary functional — but "no total emitted" remains dominant, for a runnable computation-debt reason, not a literature wall.

8. The value leg — and the two-route inconsistency

The dimensionful magnitude has two compounding problems.

(R1) Odd-$D$ dissolution. There is no canonical finite dimensionful $a_6$ at odd $n$: $a_6$ sits at the half-integer $\zeta$-pole $s=7/2$ (a scheme-dependent power divergence, zero in dim-reg), with no log term and no conformal-anomaly slot ($a_{d/2}=a_{6.5}$ is non-integer). The banked figure is a local density $\times\,(4\pi)^{-13/2} = 7.1637\times10^{-8}$ — neither the canonical $A_6$ (units GeV$^{-7}$) nor an object that legitimately pairs with that prefactor. So the magnitude is dissolved-as-ill-posed, admissible only as a labeled consistency coefficient.

(R7, binding) Route inconsistency. $K_6 = SU(3)/T^2$ is naturally reductive but non-symmetric, so the Levi-Civita connection differs from the canonical (Peter–Weyl/Casimir) one by $\Lambda(X)Y = \tfrac12[X,Y]_m$. The canonical spectral method is exact at $a_0,a_2$ but misses higher coefficients on a non-symmetric coset. The gap is located exactly: $$ \text{vector } a_4 \text{ on } K_6:\quad \underbrace{\tfrac{23}{10}}_{\rm canonical} \to \underbrace{\tfrac{281}{120}}_{\rm Levi\text{-}Civita},\qquad \tfrac{281}{120}-\tfrac{23}{10} = \tfrac{1}{24}\ \text{EXACTLY}, $$ with scalars LC-immune (which is why $a_4/a_2^2 = 66/125$ is exact). The two graded routes therefore disagree, in both sectors: $$ \text{physical ghost (Lichnerowicz }E=-\mathrm{Ric}):\quad \Big|{-\tfrac{251}{504}} - \tfrac{149}{1008}\Big| = \tfrac{31}{48} \approx 0.646, $$ roughly six orders outside the pre-fixed $10^{-6}$ tolerance; and the graviton (Route A $K_6$-bundle anchor $-43/504$ vs $-16/315$; two graviton $a_4$ anchors $893/210$ vs $1643/210$). The honest statement is route-INCONSISTENT ($R7 =$ FAIL_VALUE_MISMATCH), which is stronger than "uncomputed."

The earned partial — kept strictly separate. A Gilkey-free $su(3)$ route earned the Bochner ($E=0$) ghost $a_6/a_0 = 149/1008$ ($= -\tfrac{16}{315}+\tfrac{143}{720}$), dissolving a named $su(3)$ root-shell blocker for that object only. It does not reconcile the physical ghost: the physical Faddeev–Popov ghost is the Lichnerowicz operator with $E=-\mathrm{Ric}$, a different operator from the Bochner $E=0$ object. An apparent "two routes agree exactly" only arises by silently swapping the $E=-\mathrm{Ric}$ ghost for the $E=0$ operator and pairing the reduction's own anchor with the spectral peel — the same route twice, not two independent routes.

The SCHEME-ANCHORED cascade. The Gap-01 magnitude, the SG-6 modulus loop coefficient, and the SG-7 threshold vector ride the same heat-kernel scheme object, and deeper the same SU(3) Gelfand–Tsetlin / Levi-Civita-hopping sector. Reconcile the routes once — fix the GT off-diagonal matrix elements — and the fix propagates to all three. That is the gate's single highest-leverage move.

9. Open residuals — the value and adjudication family

Each is a separate row; none is closed by the scale-free leg.

• Graviton Levi-Civita correction / two-route reconciliation (R4/R5/R6/R7) — OPEN, BINDING. The missing object is the SU(3) Gelfand–Tsetlin off-diagonal matrix elements gating the 5-class LC Lichnerowicz graviton hopping; the routes disagree by $31/48$ in the physical ghost and analogously in the graviton.
• Dimensionful magnitude scheme object (R1) — DISSOLVED-as-ill-posed. Terminates, if pursued, on the named value-free AXIOM-HEATKERNEL-SCHEME-OBJECT, not on a GeV$^6$ number.
• Curvature-input correction (R3) — DISCLOSED-CORRECTED / computation-debt. A one-line sign flip ($31/147\to23/75$), owed not closed; correcting an input is necessary, not sufficient.
• $\mathbb{Z}_2$-defect numeric value (R2 value-leg) — OPEN. Structure built; the number inherits Hole 1; "no total emitted" stays dominant.
• Positivity functional $\Pi(a_6)$ (MO-10/MO-11) — OPEN / UNMADE. No valid realization on this off-shell odd-$D$ branch; no sign asserted.
• Sufficiency (MO-12, field-level) — OPEN (shared QG ceiling). $a_6$ is one term in the unbounded $a_8/a_{10}/\dots$ tower; necessary-not-sufficient, dissolved as a shared ceiling, not a private gap.
• Spectrum given, not derived (given-E) — DISCLOSED / scoped to the selection gates.

These are separate rows under one top-level family: value, magnitude, and adjudication.

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

• Gap-01 does not derive $E$. The operator and bundles are read off the frozen branch (given-E ≠ derivation-of-E).
• The scale-free sector is a forced skeleton, not a selector. $O_{\rm Gap01,scalefree}(E_{\rm frozen})=0$ does not imply $\ker O_{\rm Gap01}=\{E_{\rm SM}\}$, and it does not select a spectrum.
• There is no finite dimensionful $\mathrm{tr}[a_6]$ value — it is ill-posed at odd $D=13$; any GeV$^6$ figure is a labeled consistency coefficient, never gap-closing.
• The two value routes DISAGREE ($R7 =$ FAIL_VALUE_MISMATCH); the graded total is not emitted. The Bochner-ghost $149/1008$ is not the physical ghost and does not reconcile the routes.
• The color factor $124/315$ is metric-selected (at $\mathrm{Scal}_{K_6}=7.5$); it is not normalization-robust.
• The R3 correction is owed, not gate-closing; correcting an input is necessary, not sufficient.
• No positivity sign is asserted. $\Pi$ has no valid realization here (off-shell gauge dependence; odd-$D$ predicate; object/predicate mismatch; no admissible input).
• A finite, positive $a_6$ does not constitute a UV completion (MO-12: necessary-not-sufficient, shared with all of quantum gravity).
• The frozen-branch hashes are audit anchors; they do not validate the physics. A captured log is not an independent reproduction.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package — every one writable without knowing the target value.

1. Graviton LC correction (binding, highest-leverage). Derive the SU(3) Gelfand–Tsetlin off-diagonal matrix elements on Peter–Weyl harmonic sections target-blind; assemble the LC Lichnerowicz graviton graded $c_3^\gamma$; build a genuinely structurally-independent second numeric route (not a second spectral peel); re-run two-route agreement against the $10^{-6}$ tolerance. Success: routes agree with no number fed in. Refuting result (also terminal): if they cannot reconcile without back-solving, reduce the magnitude leg to AXIOM-HEATKERNEL-SCHEME-OBJECT. Load-bearing for Gap-01, SG-6, and SG-7 at once.
2. Magnitude scheme object (R1). Fix the shared one-loop scheme object target-blind; if it lands → DERIVED-GIVEN-E; if not → the named value-free axiom. The odd-$D$ dissolution makes the named axiom the realistic terminal.
3. Curvature correction (R3). Apply the sign flip in Rop(), re-run downstream, record the disclosed-corrected note. Terminal as DISCLOSED-CORRECTED.
4. $\mathbb{Z}_2$-defect value (R2). After Hole 1, add a $\gamma$-graded variant, evaluate on the 12d base, multiply by $\tfrac12$; emit with a structurally-independent second route. Refuting result: if routes still disagree, the defect value is terminal-BLOCKED and exported.
5. Positivity functional. Either obtain an on-shell background without mutating the frozen geometry (likely impossible) plus the total value plus a sign↔positivity theorem in odd-$D$ off-shell (nonexistent), or accept the certified non-realizability and keep $\Pi$ OPEN/UNMADE. A provable "cannot decide" is the realistic terminal.
6. Sufficiency (MO-12). No short route — a quantum-gravity-wide problem. Keep necessary-not-sufficient binding; do not frame it as a private weakness.

Closing the value-and-adjudication family would upgrade Gap-01 from "scale-free leg derived, gate open" toward a finite one-loop diagnostic — and even then, only given $E$, and never as a claimed UV completion.

Completion tests for this page

Required presence (all met): gate roll-up DERIVED-GIVEN-anchor · RESOLVED +0 (historical OPEN label kept as history) · the scale-free closed leg $O_{\rm Gap01,scalefree}(E_{\rm frozen})=0$ · scale-free DERIVED-GIVEN-E · $E$ not derived · frozen hashes (AUDIT ONLY) · operator $a_6[\text{grav}]-2a_6[\text{ghost}]$ · four sphere rationals $4/315,\ 74/63,\ 1139/63,\ 5/63$ · color factor $124/315$ (metric-selected caveat) · diagnostic $31/147\to23/75$ Bianchi · $\mathbb{Z}_2$ Donnelly reduction $\tfrac12 c_3^\gamma$ · grading constants $67,11,45$ · R1 dissolution · R4 gap $1/24$ · R7 mismatch $31/48$ · positivity UNMADE · MO-12 sufficiency · every open residual as its own row · the gate's anti-claims.

Required absence (all held): no claim Gap-01 is closed · $E$ derived · scale-free sector selects a spectrum · $\ker O_{\rm Gap01}=\{E_{\rm SM}\}$ · a finite dimensionful value emitted · two routes agree / R7 met · $124/315$ normalization-robust · R3 gate-closing · a positivity sign asserted · $a_6$ "is" a UV completion · hashes validate physics · scale-free closure = total/global completion.

Tests passed: all required-presence items present; all required-absence items held. Tests failed: none. Open items: R1 (magnitude scheme object), R2 value-leg, R3 (correction owed), R4/R5/R6/R7 (graviton LC / two-route, binding), positivity $\Pi$, MO-12 sufficiency, given-E scope. Assumptions made: none beyond the dossier; every number traces to the Gap-01 dossier and its cited certificates.

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

See also: the anchoring method · the master anchor · Layer 2 — no unpaid exact labels (why $124/315$ must carry its metric-selected caveat) · Layer 4 — carrier-forcing & the given-E wall · the SG-6 anchor ledger and SG-7 anchor ledger (the shared heat-kernel scheme object) · the full Gap-01 dossier.