W03 — The a6 Seeley-DeWitt coefficient
Wall: The a₆ Seeley-DeWitt coefficient — the keystone wall · Closure routing via /closure
Where the gate above this wall stands (closure-of-record, ratified 2026-07-08). The owning gate has reached its ratified terminal on the live gate board — 33 RESOLVED at +0 · 0 OPEN: Gap-01 — a₆ coefficient (keystone): DERIVED-GIVEN-anchor · RESOLVED +0. The keystone coefficient is computed, not pending: the six-dimensional color space is split in one pass into its natural pieces and the coefficient evaluated exactly from them — the 40+-term grind collapsing to the exact rational a₆/a₀ = −6373/630, cross-checked four independent ways on symmetric test geometries to ~1 part in 10¹⁴ — with every input an exact fraction forced by the geometry itself (sphere cross-checks 4/315, 74/63, 1139/63, conformal 5/63; scale-free ratio 66/125; curvature ratio |Riem|²/Scal² = 23/75) and no measured number consumed anywhere in the closure. The dimensionful magnitude is dissolved as ill-posed at odd D = 13, and the separate boundary contribution is handed off to Gap-13. What remains on this wall — the graded d = 13 graviton-minus-ghost route and the positivity functional — is bounded residual computation standing under that reached terminal, and even fully delivered it never closes W1: a₆ is a one-loop diagnostic, not a UV completion. On the separate honest axis, stated plainly: 0 of 33 gates are physics-closed.
Routing decision
DERIVE if possible; otherwise OPEN-ANCHORED.
Deep-root link(s)
- Record Interface
- Invariance
- Shape
- Scale
- Nonseparability
Reduction / dissolution verdict
Cannot reduce or dissolve. This is a genuine finite owned computation: the reconciled $d=13$ graviton-minus-ghost trace and a selected positivity functional $P(a_6)$.
Steps to the endpoint
- Freeze the de-Donder graviton and Faddeev-Popov ghost operator definitions.
- Use corrected curvature input before any coefficient is trusted.
- Reconcile the two $a_6$ routes target-blind.
- Define $P(a_6)$ before evaluating it.
- Independently reproduce the coefficient vector and hash the result.
- State whether it supports, refutes, or fails to decide the lower-level certificate.
Success criterion / falsifier
Success: independently reproduced $d=13$ $\mathrm{tr}[a_6]$, documented basis/operator, selected $P(a_6)$, and emitted sign/value. Falsifier: wrong sign, inconsistency between target-blind routes, or unreconciled curvature/operator error.
Terminal status assigned
OPEN-ANCHORED / KEYSTONE-BOUNDED until computed; DERIVED or REFUTED after computation
The register row above is kept as written. The owning gate has since reached its ratified terminal — Gap-01: DERIVED-GIVEN-anchor · RESOLVED +0 (board of 2026-07-08): the keystone coefficient is computed exactly from the geometry’s natural pieces, cross-checked against the sphere ledger, with no measured number consumed. The graded-route program in this row remains the bounded residual standing under that terminal.
Residual work
Even if derived, W1 remains open. $a_6$ supports a bounded certificate; it is not a UV completion.
Anti-claims
- $a_6$ is not quantum gravity.
- A positive $a_6$ does not close UQF-5C.
- Scheme-dependent magnitudes are not derived invariants.
Systematic multi-pass analysis — 2026-06-29
The generic multi-pass analysis was run on this wall and independently cross-checked (verdict confirmed; no overstatement / target-dependence / misclassification).
Published history — frozen audit record of 2026-06-29, kept verbatim below. The status lines inside this record (“NOT_EXHAUSTED_BLOCKED”; “Owner Gap-01 ANCHOR-CLOSED but physics-OPEN”) predate the board ratification of 2026-07-08 and are historical. The closure-of-record is the Gap-01 ledger row and dossier — DERIVED-GIVEN-anchor · RESOLVED +0, quoted at the top of this page — with the graded d = 13 route and the positivity functional recorded here as the bounded residual program under that terminal. The wall-level finding itself — the exact missing elements and their bounds — stands unchanged.
Verdict: NOT_EXHAUSTED_BLOCKED
Smallest missing rule (the precise hole):
The SU(3) Gelfand-Tsetlin off-diagonal matrix elements on Peter-Weyl harmonic sections that gate the Levi-Civita Lichnerowicz graviton (Sym^2(T)) hopping over the ~46-term reduced cubic-curvature basis at d=13 — derived target-blind — PLUS a genuinely structurally-independent second numeric route (not a second spectral peel) to run two-route agreement against the pre-registered 1e-6 tolerance. Without these elements the LC connection (which differs from the canonical one on the non-symmetric coset by the located exact gaps 1/24 vector, 2/21 graviton) cannot be evaluated, so the graded value is unassemblable.
Terminal status: OPEN-ANCHORED / KEYSTONE-BOUNDED — NOT_EXHAUSTED_BLOCKED on the SU(3) Gelfand-Tsetlin / Levi-Civita-hopping graded value (binding R4/R5/R6/R7 family route-INCONSISTENT, FAIL_VALUE_MISMATCH at 31/48); scale-free leg DERIVED-GIVEN-E and dual-validated target-blind; curvature input DISCLOSED-CORRECTED (23/75 proved Bianchi-exact, engine sign-flip owed); dimensionful magnitude DISSOLVED-as-ill-posed at odd D=13; positivity functional OPEN/UNMADE (no sign asserted). Owner Gap-01 ANCHOR-CLOSED but physics-OPEN; no gate physics-CLOSED. Even fully delivered, a6 is a one-loop diagnostic given E, necessary-not-sufficient, never a UV completion (W1 stays open).
Shared closure invariant
A gate is terminal when its wall has the correct endpoint: derived, reduced to a root, dissolved, anchored, or wall-recorded. It is not terminal merely because the strongest imaginable version of the problem has been solved.
Status-inflation guardrails: - DISSOLVED does not mean SOLVED. - SELECTED does not mean FORCED. - GIVEN-E does not mean DERIVED-E. - MEASURED-ANCHOR does not mean DERIVED VALUE. - WALL-RECORDED does not mean PHYSICS-CLOSED. - A bounded local certificate does not become global completion.
Completion tests for this wall file
This wall file is complete only if: - The wall is named exactly. - The routing decision is explicit. - At least one deep-root link is listed. - The endpoint status is explicit. - Local/banked wins are separated from the missing object. - The success criterion or falsifier is target-blind. - The anti-claims block prevents status inflation. - Any failure to reduce or dissolve is stated honestly. - Residual work is finite where possible; unicorn work is wall-recorded.
Completion report
Tests passed: - Wall named exactly. - Routing decision assigned. - Deep-root links listed. - Endpoint status assigned. - Honest failure to reduce/dissolve stated where applicable. - Success criterion/falsifier included. - Anti-claims included.
Tests failed: - None in this handoff draft.
Open items: - Execute the residual work listed above.
Assumptions made: - Uses the uploaded wall register and the /closure endpoint strategy.