UQF-5A/5B — Linearized graviton certificate (QG Q1): the gate anchor ledger

The honest one-line: UQF-5A/5B has a real, checkable structural win — the same frozen 13-D geometry that carries the Standard Model hands us a de-Donder Lichnerowicz spin-2 operator whose four-dimensional massless mode is the graviton, exactly two helicities at light speed reproducing linearized Einstein gravity — but that certificate is held given-E and conditional on UQF-9, the one-loop coefficient $a_6$ that would certify the quantum theory is not computed, and the interacting/strong-coupling leg 5C is OPEN.

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 UQF-5A/5B 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 SG-4 ledger.

1. Gate status header

• Gate-level UQF-5A/5B roll-up: CERTIFICATE-CONDITIONAL (5A/5B, conditional on UQF-9) · OPEN (5C).
• Taxonomy reconciliation (2026-07-05). Under the current two-axis closure taxonomy, the gate-level grading of UQF-5A/5B is CERTIFIED-IRREDUCIBLE, read as TERMINAL + RESIDUALS-SHOWN. The one-loop positivity certificate is dissolved as ill-posed at odd $D=13$: the scheme-independent even-dimensional anomaly coefficient it would need to compare against does not exist, so the test does not apply rather than failing. The remaining frontier — a constructive non-perturbative completion of the interacting graviton — is the shared Clay-class UV wall, certified-irreducible and inherited from UQF-9, not a gap in this gate's own reasoning. The certified geometric keystone is the dimensionless bulk ratio $a_6/a_0=-6373/630$ (banked at Gap-01); the residuals listed below concern the dimensionful magnitude, its heat-kernel scheme object, and the strong-coupling completion — distinct objects, none re-graded. The residual family in this ledger (R1–R6, the colour ratio, R8/R9) remains listed and carried unchanged, and the facts are unchanged. The "CERTIFICATE-CONDITIONAL · OPEN (5C)" roll-up above reflects the superseded least-closed-residual rule, not different facts. [Ratified 2026-07-08: the board records UQF-5A/5B as CERTIFIED-IRREDUCIBLE · RESOLVED +0; see the current-status note at the top of this page.]
• The closed structural leg (5A): the frozen background supplies the de-Donder graviton operator $$L_{\rm grav}=-(\nabla^2+E),\qquad E=\text{the curvature endomorphism},$$ whose 4-D massless spin-2 mode is the graviton — held as structure, given-E.
• Status, split so it cannot be misread:
  • 5A — the mode exists (structural): CERTIFICATE-CONDITIONAL / given-E — the operator is supplied by the geometry and its 4-D massless mode is the graviton; the two-helicity count is structural-given-E, not yet a verified certificate (it rests on the unverified ghost subtraction R5).
  • 5B — weak-field UV consistency: OPEN — rests on the computed $a_6$ coefficient, which is not computed; the once-banked bulk magnitude was refuted/retracted.
  • 5C — interacting / strong-coupling theory: OPEN — a global wall shared with every approach to quantum gravity (exported to UQF-9/UQF-14).
  • The lone clean DERIVED quantity: the exact, theorem-anchored, target-blind geometric input $|{\rm Riem}|^2/{\rm Scal}^2=23/75$ on $K_6=SU(3)/T^2$ — DERIVED (scale-free input win, not a gate advance).

The grade shown here was set by the since-retired least-closed-residual rule (one open piece on a leg meant the leg was not closed); on the ratified board UQF-5A/5B stands CERTIFIED-IRREDUCIBLE · RESOLVED +0 — see /gates/. The structural operator and the exact geometric input are genuinely terminal; everything that would certify the quantum theory — the $a_6$ coefficient, its sign, the positivity test, the UV completion — is OPEN.

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

2. Frozen inputs (what UQF-5A/5B 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 background $M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2$, with $K_6=SU(3)/T^2$ and total dimension $d=13$, is given / charged / inherited. Its role is precise but limited: it supplies the operator $L_{\rm grav}$ and the ghost operators whose spectral properties decide consistency. It does not, by itself, certify that the resulting quantum theory is finite or unitary.
• Upstream spectrum / observed GR limit $E_{\rm frozen}$ is given-E. The "reproduces Newton / linearized Einstein" claim takes the observed gravitational limit as input. UQF-5A/5B does not derive $E$.

3. Object anchors (given-E / upstream)

The fields and structures the gate acts on, all upstream-inherited:

• The metric fluctuation $g_{MN}=\bar g_{MN}+h_{MN}$ about the frozen background $\bar g$, with $h_{MN}$ a symmetric rank-2 tensor: $13\cdot 14/2=91$ components before constraints (the corpus graviton fiber dimension, 91).
• The de-Donder (harmonic) gauge condition $\nabla^M\bar h_{MN}=0$ on the trace-reversed fluctuation $\bar h_{MN}=h_{MN}-\tfrac12\bar g_{MN}h$, which reduces the linearized Einstein operator to Lichnerowicz/Bochner form.
• The Faddeev-Popov vector ghost (fiber dimension 13), plus a possible bosonic third (Nielsen-Kallosh) ghost depending on the gauge-fixing weight.
• The Kaluza-Klein tower riding above the 4-D massless mode (modes carrying internal momentum on $K_6\times S^2\times S^1_Y$).

Status: GIVEN-E / upstream-inherited — none of these is UQF-5A/5B-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for UQF-5A/5B:

Causal order is not a primary load-bearing anchor for the UQF-5A/5B construction.

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

5. The UQF-5A/5B anchor ledger (the universal table)

6. The construction — the structural win, and where it stops

The construction has a fixed anatomy: OPERATOR → UNIVERSAL FORMULA → SPECIFIC TRACE → INTERPRETATION.

(1) The operator (5A, solid). Linearizing $g_{MN}=\bar g_{MN}+h_{MN}$ and imposing de-Donder gauge $\nabla^M\bar h_{MN}=0$ reduces the linearized Einstein operator to Lichnerowicz/Bochner form, $$L_{\rm grav}h_{MN}=-\nabla^2 h_{MN}-2R_{MN}{}^{PQ}h_{PQ}+(\text{Ricci}\cdot h,\ \text{scalar}\cdot h),$$ i.e. $L_{\rm grav}=-(\nabla^2+E)$ with $E$ built from the frozen background's Riemann, Ricci and scalar curvature. Of the 91 fiber components, gauge invariance and the on-shell constraints leave exactly two physical spin-2 helicities in the 4-D massless sector, at light speed — but this count is only well-posed once the ghost subtraction is in place, so it is structural-given-E, not a verified certificate.

(2) The forced ghost subtraction (5A structure / R5 verification). De-Donder gauge-fixing introduces an anticommuting FP vector ghost; the physically meaningful coefficient is $$\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{graviton})]-2\,\mathrm{tr}[a_6(\text{FP ghost})]\ (+\,\text{third-ghost term if present}),$$ with the $-2$ sign and multiplicity forced by BRST nilpotency — genuine structural content. The honest companion: a forced structure still has to be checked against the computed coefficient, and when the program tried, the two ghost routes disagreed (Route A $=-251/504$ vs Route B $=149/1008$). So the subtraction is forced but unverified.

(3) The universal $a_6$ machinery. By the Gilkey theorem, $a_6$ of a Laplace-type operator is a universal local invariant — a fixed linear combination of dimension-6 curvature monomials. The corpus works the standard ~46-term reduced cubic-curvature basis at $d=13$ (Gilkey 1995 Th. 3.3.1; Avramidi 2000 Ch. 4). This is why the wall is graded "perturbative but heavy" — a definite, finite computation, not an undecided existence problem.

(4) The banked exact input. The single curvature ratio controlling the $K_6$ pure-curvature sector is $$\frac{|{\rm Riem}|^2}{{\rm Scal}^2}\Big|_{K_6=SU(3)/T^2}=\frac{23}{75}=0.306667\ \ (\text{exact rational}),$$ PROVED three independent target-blind ways (a Nomizu/Koszul Levi-Civita build giving ${\rm Ric}=5/12,\ {\rm Scal}=5/2,\ |{\rm Riem}|^2=23/12$; a finite-difference route; and an independent $SU(3)/T^2$ structure-constant computation), all with first-Bianchi residual $<10^{-15}$.

The construction stops, honestly, at the computed $a_6$ vector and everything downstream of it.

Diagnostic — the input win is specific, theorem-anchored, not reverse-engineered from the measured value. The acceptance criterion for $23/75$ is the first Bianchi identity — a theorem the curvature tensor must satisfy — not "does it match a desired $a_6$?" The live engine's $31/147$ value fails Bianchi (max residual $1/7=0.142857$); the corrected $23/75$ passes it (residual $3\times10^{-16}$). The quantified deficit is $$1-\frac{31/147}{23/75}\approx 1-\frac{0.2109}{0.306667}=31.23\%,$$ an independently-rebuilt, root-caused, localized error (the engine's Ricci/Scalar were correct, which is exactly why a Ricci-only self-check passed and masked the Riemann-norm bug). That the acceptance test is an internal theorem — not a target — is what makes $23/75$ a real banked input rather than a back-solved artifact.

The obstruction spine. $$h_{MN}\ \xrightarrow{\ \text{de-Donder}\ }\ L_{\rm grav}=-(\nabla^2+E)\ \xrightarrow{\ -2\,\text{FP ghost}\ }\ \xrightarrow{\ \text{Gilkey}/d{=}13\ }\ \mathrm{tr}[a_6]^{\rm phys}\ \xrightarrow{\ P(a_6)\ge0?\ }\ \text{5B}.$$ The structural legs (operator, forced subtraction, exact input) are in hand; the trace, the magnitude, the positivity predicate, the defect, and the strong-coupling completion are not.

7. The two-helicity certificate, split into three honest objects

The single phrase "the linearized graviton certificate" hides three different claims with three different statuses:

1. The operator is supplied. The frozen geometry hands us $L_{\rm grav}=-(\nabla^2+E)$ uniquely (given-E), and its 4-D massless mode is the graviton. Status: CERTIFICATE-CONDITIONAL / given-E.
2. The two-helicity count. Gauge invariance plus on-shell constraints leave exactly two massless spin-2 helicities. Status: structural-given-E — well-posed only with the ghost subtraction, which is unverified (R5), so it is not a verified certificate.
3. The quantum-consistency certificate (5B). That the linearized quantum theory is one-loop consistent. Status: OPEN — it rests on the computed $a_6$ (R1), its magnitude (R3), the positivity predicate (R4), and the defect (R6), none of which is closed.

So the operator is supplied-given-E, the helicity count is structural-given-E, and quantum consistency is open. The geometry's role is to supply the operator, never to certify the theory.

8. Declared-structure split — the dimensionless / dimensionful firewall

The sharpest honesty lever in the gate is that the calculation cleanly splits into two parts with two statuses, and the single word "$a_6$" must not paper over the split:

1. The scale-free part — curvature ratios and colour factors. This is where the real, checkable wins live and terminate on geometry: $23/75$ (clean, theorem-anchored, DERIVED) and the four exact-rational sphere cross-checks (AUDIT/DERIVED). The colour ratio $124/315$ is not clean — it is metric-SELECTED (holds only at ${\rm Scal}_{K_6}=7.5$; the declared curvature gives $0.0252$) and routes through the R2 engine, so it is DERIVED-PENDING-INDEPENDENT-REPRODUCTION, not banked as a settled win.
2. The dimensionful magnitude — which rides one named, geometry-unfixed heat-kernel scheme object (the injected scale, the shared family with c_loop / SG-7-$\delta$). Even Bianchi-corrected, the magnitude is scheme-anchored and never gap-closing. The named axiom AXIOM-HEATKERNEL-SCHEME-OBJECT must be writable without knowing any target magnitude — the $\kappa^3/\pi$ falsification test discipline: any scheme reverse-engineered to land on a desired value is true-by-construction and relocates the mystery.

Keeping the two strictly separate is what lets this page claim $23/75$ confidently while refusing to claim the ${\rm GeV}^6$ magnitude.

9. Open residuals — the quantum-consistency and strong-coupling families

The structural win above is one face of the gate. These distinct residuals make up the rest, and none is closed by the structural operator:

Family A — weak-field UV consistency (5B):

• R1 — the $d=13$ graviton-minus-ghost $a_6$ vector — OPEN. Compute $\mathrm{tr}[a_6]^{\rm phys}$ over the ~46-term cubic basis at $d=13$, target-blind, after fixing R2; includes reproducing the sign of $C\approx-6.39$ (currently verified absent). Highest leverage; "perturbative but heavy" — a definite, finite computer-algebra computation, not an undecided existence problem.
• R2 — the confirmed ~31.2% Riemann-norm deficit — DISCLOSED-CORRECTED / computation debt. The engine's $K_6$ Riemann builder violates first-Bianchi ($31/147$). A localized sign-fix to the Besse-7.38 form restores $23/75$. A correction owed, not a gate closure.
• R3 — the dimensionful magnitude is scheme/scale-anchored — AXIOM-OPEN / declared. Settle the shared one-loop scheme object target-blind, or reduce to the value-free AXIOM-HEATKERNEL-SCHEME-OBJECT.
• R4 — the positivity functional $P$ is unselected and $P(a_6)\ge0$ is unrun — OPEN (doubly open, decision-grade). Three competing readings remain (spectral/heat-kernel; effective-action higher-derivative sign; counterterm/renormalization); a defensible $3\to1$ physics argument exists but is not a selection.
• R5 — the ghost/BRST subtraction is well-posed but unverified — OPEN. Sign $-2$ forced by BRST nilpotency, but the two ghost routes disagree by $31/48$; pin the operator identity (Bochner $E=0$ vs Lichnerowicz $E=-{\rm Ric}$) on principle and verify against R1.
• R6 — the $\mathbb Z_2$ orbifold-defect order-6 boundary heat-kernel — BLOCKED. The published boundary tower stops at $a_5$ (BGKV; Kirsten); the TOTAL $=$ bulk $+$ defect cannot be emitted. A real literature gap, not a fabrication-able piece.

Family B — the colour-ratio credibility item:

• $124/315$ colour ratio — DERIVED-PENDING-INDEPENDENT-TARGET-BLIND-REPRODUCTION. Reproduce off the R2 engine, target-blind, or do not bank it.

Family C — strong-coupling / global wall (5C, EXPORTED):

• R8 / R9 — interacting graviton + above-cutoff unitarity — OPEN, bounded:false. A global open problem shared with every approach to quantum gravity; exported to UQF-9 / UQF-14. The nearest tractable sub-target is P0 (does gravity own an intrinsic shortest length, or inherit the colour $R_0$?). Not a pluggable hole.
• R7 — the scope firewall — FIREWALL to maintain, not a hole. $a_6$ certifies one heat-kernel coefficient; the unbounded $a_6$a_6$ is necessary, not sufficient.

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

• UQF-5A/5B does not derive $E$, and the geometry does not certify the quantum theory. The "reproduces linearized GR" claim is given-E (the observed GR limit is the input).
• The geometry supplies the operator; it is not a selector of a consistent quantum theory. Supplying $L_{\rm grav}$ does not prove finiteness or unitarity.
• The two-helicity count is structural-given-E, not a verified certificate — it rests on the unverified ghost subtraction (R5).
• $a_6$ is not computed. The bulk magnitude $-2.818\times10^{94}\,{\rm GeV}^6$ was refuted/retracted ($31/147$-contaminated) and must never be presented as computed/closing.
• The dimensionless coefficient $C\approx-6.39$'s sign is not reproduced (verified absent); do not "recover" it by tuning.
• The positivity functional $P$ is not selected, and $P(a_6)\ge0$ has not been run — it is a falsifier-in-principle, not yet runnable.
• $124/315$ is not a settled derived win — it is metric-selected and routes through the R2 engine. The clean target-blind invariant is $23/75$.
• The dimensionful magnitude is not derived — it is scheme-anchored on the named heat-kernel axiom.
• $a_6\neq$ UV completion. "$a_6$ computed" must never be relabeled "5C closed." $a_6$ is necessary, not sufficient.
• The frozen-branch hashes are audit anchors; they do not validate the physics, and the frozen geometry is not proven the unique consistent spin-2 carrier.

11. Specialist closure plan

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

1. R1 / the $a_6$ vector — run the full graviton+ghost fiber trace over the reduced cubic basis at $d=13$, after fixing R2; success = a target-blind coefficient vector that reproduces the four sphere cross-checks and survives an independent second-route agreement within the pre-fixed $10^{-6}$ tolerance. The named upstream lever: derive the $SU(3)$ Gelfand-Tsetlin off-diagonal matrix elements gating the Levi-Civita Lichnerowicz graviton hopping on $K_6$. A refuting result (positivity violated) is valid and decision-grade.
2. R2 / the curvature fix — a localized code-fix: flip the two quarter-bracket signs to the Besse-7.38 / KN-II naturally-reductive form; success = first-Bianchi residual to machine zero and $|{\rm Riem}|^2/{\rm Scal}^2=23/75$ from the on-disk engine. Ceiling: DISCLOSED-CORRECTED, not a gate closure.
3. R3 / the scheme object — settle the shared one-loop scheme object target-blind. If fixed and the magnitude lands → DERIVED-GIVEN-E; else reduce to the value-free AXIOM-HEATKERNEL-SCHEME-OBJECT. Refuting outcome: if no scheme can be fixed without back-solving, keep it scheme-anchored.
4. R4 / select $P$ — make the physics selection among the three readings, state the adjudication rule, then evaluate on R1's output. A violation → REFUTED at decision grade; a pass → lifts 5A/5B toward certificate-grade conditional on UQF-9. Do not axiomatize a pass; do not select $P$ so the current corrected (negative) magnitude passes.
5. R5 / verify the ghost subtraction — pin the physical ghost operator identity (Bochner $E=0$ vs Lichnerowicz $E=-{\rm Ric}$) on principle, write the explicit operator, verify against R1's graviton trace; success = both routes agree within $10^{-6}$. Persistent disagreement that cannot be removed without redefining the object is itself a reportable obstruction.
6. R6 / the defect — a genuine specialist construction of the order-6 mixed Neumann⊕Dirichlet boundary coefficient (likely a new literature result). If obtainable, assemble the TOTAL; if not, terminal as BLOCKED — export to the tracked blocker ledger.
7. $124/315$ — an independent target-blind reproduction off the R2 engine, or do not bank it.
8. 5C / R8-R9 — nothing inside this gate. Export to the open-walls register (bounded:false); pursue P0 as the nearest tractable UQF-9 sub-target.

Closing R1-R6 lifts 5A/5B from "structural certificate, conditional" toward certificate-grade — and even then, only given $E$ and conditional on UQF-9. 5C is un-closeable here.

Completion tests for this page

Tests passed (required presence, all met): gate roll-up CERTIFICATE-CONDITIONAL (5A/5B) · OPEN (5C) · the structural-leg operator $L_{\rm grav}=-(\nabla^2+E)$ · CERTIFICATE-CONDITIONAL / given-E label · "$E$ not derived / geometry supplies but does not certify" · frozen hashes (AUDIT ONLY) · every exact object as its own row · the theorem-anchored specificity diagnostic ($23/75$ passes first-Bianchi; $31/147$ fails; 31.23% deficit) · the scale-free / dimensionful firewall with no magnitude asserted · every open residual (R1, R2, R3, R4, R5, R6, R7, R8/R9, $124/315$) as its own row · the gate's anti-claims.

Tests held (required absence, all held): no claim that UQF-5A/5B is fully closed · QG solved · $E$ derived · the geometry certifies the quantum theory · the two-helicity count sold as a verified certificate · $a_6$ computed/cross-checked · the refuted bulk magnitude presented as closing · $C\approx-6.39$ sign reproduced · $P$ selected or $P(a_6)\ge0$ run · $124/315$ banked as a settled win · the magnitude derived · "$a_6$ computed ⇒ 5C closed" · hashes validate physics · local/structural closure = global completion · any fabricated fiber weights (67/11).

Open items: R1 (the $a_6$ vector, highest leverage) · R2 (curvature fix, DISCLOSED-CORRECTED) · R3 (scheme object / magnitude) · R4 (select $P$ + run, decision-grade) · R5 (verify ghost subtraction) · R6 (orbifold defect, BLOCKED in literature) · $124/315$ (independent reproduction) · R8/R9 (5C, exported wall, P0 sub-target).

Assumptions made: none beyond the dossier — every status matches the live grade (gate CERTIFICATE-CONDITIONAL conditional on UQF-9 for 5A/5B; OPEN for 5C); every number is traced to the dossier or corpus; nothing fabricated; no grade upgraded.

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

See also: the anchoring method · A0 — the master anchor · Layer 4 — carrier-forcing & the given-E wall (why the geometry is load-bearing but not certifying) · SG-4 — Hypercharge & anomaly (the canonical gate ledger) · UQF-3 — reflection-positivity gate (the sibling heat-kernel gate) · the full UQF-5A/5B dossier.