Honest status (binding, must match the live popup chip): CERTIFICATE-CONDITIONAL for the linearized legs 5A/5B (conditional on UQF-9), OPEN for the interacting / strong-coupling leg 5C. STATUS-UPGRADES:0. Frozen 13-D branch
dcc66f1b2685/ manifest metaa5b1e6f9d951— READ-ONLY. Nothing here is applied, nothing is deployed.Built by synthesizing and expanding the gate brief (
articles/GATE_BRIEF_GRAVITON.md), the per-gate dossier (rendered/TOE/PER_GATE_DOSSIERS/GRAVITON_COMPLETION_HANDOFF/01_DOSSIER.md,02_CURRENT_STATE.md), the Gap-01 decomposition + subsystems (rendered/TOE/gap_01_uv_quantum_gravity/), the a₆ reconciliation verdict (GAP01_A6_RECONCILIATION_2026-06-24.md), and the current ledgers (GATE_REGRADE_BESTCASE_AND_HOLES_2026-06-29.md,SPECIALIST_HOLE_QUEUE_2026-06-29.md,PLUG_RESULTS_FIRSTPASS_2026-06-29.md). Every number below is cited to one of these. Where a quantity is uncomputed it is marked OPEN — never fabricated.
Headline. The same frozen 13-dimensional geometry that carries the Standard Model also carries a real graviton. In the weak-field (linearized) limit, the metric fluctuation on the frozen background, gauge-fixed in de-Donder (harmonic) gauge, yields the standard Lichnerowicz spin-2 operator whose four-dimensional massless mode is the graviton — exactly two physical helicities moving at light speed, reproducing Newton and linearized Einstein gravity in the long-wavelength limit, with the heavy Kaluza-Klein tower riding above. That is a genuine, checkable structural certificate (leg 5A), held openly given-E (the observed gravitational limit is the input, not a from-nothing output) and conditional on the still-open UV story (UQF-9).
The honest grade. The binding label is CERTIFICATE-CONDITIONAL (5A/5B, conditional on UQF-9) · OPEN (5C). This is set by the rubric's least-closed-residual rule (06_GATE_GRADING_RUBRIC.md): one open piece on a leg means that leg is not closed. The two genuinely terminal positive legs are (a) the geometry-supplied operator as structure, given-E (5A), and (b) — one leg below, in the a₆ machinery 5B feeds on — a single exact dimensionless geometric input that is now pinned by independent routes. Everything that would certify the quantum theory — the computed a₆ heat-kernel coefficient, its sign and magnitude, the positivity test, the UV completion — is OPEN / not-atomic (02_CURRENT_STATE.md §0).
What this dossier establishes, and what it does not. It establishes, at working-physicist depth: the construction of the de-Donder graviton + ghost operator on the frozen background and why its 4-D massless spin-2 mode is the graviton (5A); the Faddeev-Popov ghost subtraction whose sign and multiplicity are forced by BRST nilpotency (well-posed in principle, but not yet verified against a computed a₆); the exact, route-independent geometric input |Riem|²/Scal² = 23/75 = 0.306667 on K₆ = SU(3)/T², proved three ways and not target-fitted; the precise residual register (R1-R9); and a concrete specialist work plan. It does not establish: that a₆ is computed (the one banked bulk magnitude, −2.818×10⁹⁴ GeV⁶, was refuted/retracted at decision grade); that the full graviton-minus-ghost coefficient (even its sign, the corpus's C ≈ −6.39) is reproduced (it is not — verified absent); that the positivity functional P is even selected (three competing readings remain); or that the theory is UV-complete (5C is a global open problem shared with every approach to quantum gravity). given-E ≠ derivation of E; AXIOM-CLOSED ≠ atomic; a captured log ≠ an independent reproduction; a₆ is the missing object for UV completion, never is it.
Quantum gravity has one structurally famous obstruction: general relativity, quantized perturbatively around a background metric, is non-renormalizable by power counting — the gravitational coupling carries negative mass dimension, so each loop order demands new counterterms with ever-higher curvature. The decisive early result (’t Hooft–Veltman 1974 for pure gravity; Goroff–Sagnotti 1986 for two loops) is that the divergences are not absorbable into the Einstein-Hilbert action: at two loops a genuine, non-vanishing cubic-curvature counterterm appears. The coefficient of that cubic-curvature structure is exactly the kind of object the heat-kernel / Seeley-DeWitt expansion organizes: the sixth coefficient a₆ (the t³ term in the short-time expansion of Tr e^{−tL}) is the one-loop datum that decides whether the cubic-curvature sector of the linearized theory is finite and sign-consistent on a given background.
So the gate splits honestly into three legs (GATE_BRIEF_GRAVITON.md):
tr[a₆(L_grav)] at d = 13 on the de-Donder graviton + ghost operator, projected onto the cubic-curvature invariant basis, run through a positivity test.No one anywhere has a finished quantum theory of gravity. This is not a defect of this programme; it is the field's standing condition, and the dossier states it as such. The leading research approaches each cross some part of the wall but leave the rest open:
01_DOSSIER.md §3.1 R8) — it is suggestive, not a proof.The honest "best bound" common to all of them: each can certify a piece (a finite amplitude at a given order, a fixed point at a given truncation, a discrete kinematics) but none certifies the full strong-coupling theory. This gate's specific, finite, finishing object on the weak-field leg is sharper than the field average: it is one definite number — the d = 13 graviton-minus-ghost a₆ vector — run through a positivity predicate. That number is not yet honestly computed, and the positivity predicate itself is not yet pinned down. Naming the finishing object that precisely is the gate's contribution; computing it is the open work.
This programme has attacked the a₆ object directly, and the honest record of what broke is itself the most useful prior-art map for a specialist:
bulk tr[a₆] = −2.817995812×10⁹⁴ GeV⁶, labeled COMPLETE_CROSSCHECKED (GAP01_A6_FINAL.md, via computational_runs_2026-06-23/a6_computation/a6_compute.py). The 2026-06-24 reconciliation initially ruled PROMOTE-WITH-CAVEAT — state it as a labeled consistency coefficient only (GAP01_A6_RECONCILIATION_2026-06-24.md §1).GATE_REGRADE_…2026-06-29.md line 379; B1-RECLASSIFICATION 2026-06-26). The handoff's bright-line is explicit: do not claim a₆ is computed / cross-checked.08_TWO_ROUTE_AGREEMENT.md, 13_FINAL_ROLLUP.md, 2026-06-26) records R7 = FAIL_VALUE_MISMATCH: Route A (physical ghost) = −251/504 vs Route B = 149/1008, with |A − B| = 31/48 = 0.646 — about six orders of magnitude outside the pre-fixed 10⁻⁶ tolerance — and Route A additionally fails its own canonical anchor (−43/504 vs the canonical −16/315) (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 24).So the prior attempts fall short in a specific, instructive way: the scale-free skeleton of the calculation is sound and reproducible, but the dimensionful magnitude and the route agreement are not yet trustworthy. That is exactly why the gate sits at CERTIFICATE-CONDITIONAL and not higher.
This is where the 30-50 pages live. The construction has a fixed anatomy (00_decomposition.md §1): OPERATOR → UNIVERSAL FORMULA → SPECIFIC TRACE → INTERPRETATION. We walk each stage on the frozen geometry.
Branch dcc66f1b2685 / manifest meta a5b1e6f9d951 (READ-ONLY). The background is
M₄ × K₆ × S² × S¹_Y/ℤ₂ , K₆ = SU(3)/T² , M,N = 1 … 13 .
The four non-compact dimensions M₄ carry the observed Lorentzian spacetime; K₆ = SU(3)/T² is the six-dimensional flag manifold that carries the colour sector; S² and the ℤ₂-orbifolded circle S¹_Y/ℤ₂ carry the remaining gauge/hypercharge structure. The total internal+external dimension is d = 13 (03_FROZEN_GEOMETRY_CONTEXT / 01_DOSSIER.md §5.3). The geometry's role here is precise and load-bearing but limited: it supplies the operator whose spectral properties decide consistency. It does not, by itself, certify that the resulting quantum theory is finite or unitary.
Linearize the metric, g_{MN} = ḡ_{MN} + h_{MN}, about the frozen background ḡ. The fluctuation h_{MN} is a symmetric rank-2 tensor field. Before constraints it has
13 · 14 / 2 = 91 components
(S1.1_dedonder_graviton_operator.md: "the fiber is the symmetric-traceless ⊕ trace part of rank-2 tensors at d=13, dimension 13·14/2 = 91 components before constraints"). This is the corpus's graviton fiber dimension — 91 — and the dossier uses it. (A first-pass attempt that quoted a graviton σ-supertrace weight of "67" and a ghost weight of "11" was caught as fabricated; those numbers appear nowhere in the scripts or the corpus. See §6 trap-list.)
Impose the de-Donder (harmonic) gauge condition ∇^M h̄_{MN} = 0 on the trace-reversed fluctuation h̄_{MN} = h_{MN} − ½ ḡ_{MN} h. This gauge choice reduces the linearized Einstein operator to Lichnerowicz / Laplace-type (Bochner) form:
L_grav = −(∇² + E) , E = the curvature endomorphism .
Schematically (S1.1),
L_grav h_{MN} = −∇² h_{MN} − 2 R_{M N}{}^{P Q} h_{PQ} + (Ricci · h , scalar-curvature · h terms) ,
i.e. the curvature endomorphism E_{MN}{}^{PQ} is built from the Riemann, Ricci and scalar curvature of the frozen background. The form is standard textbook structure (grade: diagnostic, [A]/[S] — writable as structure). The named missing object is the explicit, dimension-checked E_{MN}{}^{PQ} specialized to d = 13 on M₄×K₆×S²×S¹_Y, verified to reproduce the known d = 4 limit (S1.1 closure condition, grade certificate-symbolic-setup). The 4-D massless spin-2 mode of L_grav is the graviton; the heavier Kaluza-Klein tower (the modes carrying internal momentum on K₆×S²×S¹_Y) rides above it.
The physical-helicity count. Of the 91 fiber components, gauge invariance (de-Donder) and the residual on-shell constraints remove the unphysical polarizations, leaving — in the 4-D massless sector — exactly two physical spin-2 helicities, propagating at light speed. This is the public certificate-conditional claim (01_DOSSIER.md §0.1; banked item 2). But the helicity count is only well-posed once the ghost subtraction is in place (next stage): a graviton-only fiber trace is gauge-dependent. So the two-helicity statement is structural, given-E — not yet a verified certificate, because the subtraction that fixes it has not been checked against a computed a₆ (this is residual R5; see §3.3, §6).
De-Donder gauge-fixing of gravity introduces an anticommuting vector Faddeev-Popov (FP) ghost; depending on the gauge-fixing weight a bosonic third ("Nielsen-Kallosh") ghost can also appear (S1.2_ghost_sector.md). Each ghost operator is itself Laplace-type (a vector Laplacian + curvature) and so carries its own a₆.
Why the ghost is not optional. A graviton-only a₆ is gauge-dependent and physically meaningless; the positivity criterion (Stage 4) is only well-posed on the ghost-corrected combination. The physically meaningful coefficient is
tr[a₆]^phys = tr[a₆(graviton)] − 2 · tr[a₆(FP ghost)] (+ third-ghost term if present) .
The factor and sign — −2 for the anticommuting FP vector ghost — and the bulk fiber bookkeeping are fixed by BRST nilpotency: the subtraction sign and multiplicity are forced, not chosen (02_CURRENT_STATE.md, banked: "the ghost-corrected coefficient is the physical one, with sign and multiplicity forced by BRST nilpotency"). In bulk fiber-dimension terms the bookkeeping reads (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 48, verify_grading.py):
graviton fiber dim 91 , ghost fiber dim 13 , bulk graded weight = 91 − 2·13 = 65 .
The honest open part (R5). The subtraction is well-posed — its structure and sign are forced — but it has not been verified against an actually computed graviton a₆. Worse, the program's own two-route cross-check on the ghost a₆ disagreed: the physical FP ghost (E_ghost = −Ric, the Lichnerowicz/Hodge vector Laplacian per E_OMEGA_OPERATOR.md S2.2 and a6_compute.py line 387) gives a₆/a₀ = −251/504, while a Bochner reading (E = 0) gives 149/1008, and |−251/504 − 149/1008| = 31/48 = 0.646 exactly (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 24). The two values are the corpus's two disagreeing routes. A first-pass attempt to declare the ghost "reconciled" was caught: the apparent agreement at 149/1008 was obtained only by silently redefining the physical ghost operator from E = −Ric to E = 0 — a target-fitting move toward agreement (same source, line 24). The honest standing: the ghost subtraction is structurally forced but numerically unverified, and the operator identity (Bochner E = 0 vs Lichnerowicz E = −Ric) is itself a live ambiguity.
The Seeley-DeWitt coefficient a₆ of a Laplace-type operator L = −(∇² + E) is a universal local invariant: a fixed linear combination of curvature monomials of mass dimension 6, with universal numerical coefficients (the Gilkey theorem). The two pieces:
S2.1_cubic_curvature_basis.md):R³, R·R_{MN}R^{MN}, R·R_{MNPQ}R^{MNPQ}, R_{MN}R^{NP}R_P{}^M, the two independent triple-Riemann contractions R_{MNPQ}R^{MP}{}_R{}^Q R…, etc.;R□R, R_{MN}□R^{MN}, (∇R)², (∇Riem)²;E³, E□E, E·Ω², (∇Ω)², Ω³, R·E².
At generic dimension there are ~17 pure-gravity invariants plus the bundle terms; the corpus works with the standard ~46-term reduced Gilkey basis at d = 13 (01_DOSSIER.md §1, R1). The named missing object is the explicit identity-reduced independent basis (after Bianchi and integration-by-parts reduction), with the reduction identities recorded (S2.1).01_DOSSIER.md §5.2). This is the symbolic template to be specialized to the frozen geometry's (E, ω, Ω) data.tr[a₆] (subsystems S3.1-S3.3; residual R1, the wall)This is the actual missing number. Three sub-steps:
tr[a₆] (the wall).The result is a coefficient vector over the ~46-term basis, not a single scalar; "the magnitude" means the dimensionful contraction of that vector with the background curvatures. The corpus grades S3.3 "perturbative but heavy" — a definite, finite (if enormous) symbolic computation, not an undecided existence problem (00_decomposition.md §0 scope-fact 2). It is [S] (set-up-able / machine-lane), never [A]/AI-closeable.
The load-bearing geometric input (the banked exact win). The single curvature ratio that controls the K₆ pure-curvature sector is
|Riem|² / Scal² on K₆ = SU(3)/T² = 23/75 = 0.306667… (exact rational) .
This is PROVED three independent ways, all target-blind (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 86; GATE_REGRADE_…2026-06-29.md line 1007). The three witnesses:
Ric = 5/12, Scal = 5/2, |Riem|² = 23/12, hence |Riem|²/Scal² = (23/12)/(5/2)² = 23/75, first-Bianchi residual < 1×10⁻¹⁵.k6_fd_nablaR.py) independently prints Ric = 5/12, |Riem|² = 1.91667 = 23/12, first-Bianchi residual 0.a6_07_k6_indep.py) and the spectrum-demanded exact heat-trace a₄ route (a6_08_reconcile.py) both land on 0.306667 — "the two independent routes agree" (11_a6_BLIND.md; GATE_REGRADE_…2026-06-29.md lines 1054, 1094).The corpus also records a Theorem L label (in the 2026-06-25 consolidated GRAVITON.md) summarizing the multi-route proof of 23/75. Honesty note carried from the verifier: the verifiable artifacts establish 23/75 by two exact independent routes plus a direct connection build (three witnesses total), and no standalone document literally named "Theorem L" exists outside that consolidated summary (GATE_REGRADE_…2026-06-29.md lines 1094-1096). The robust, defensible statement is therefore: 23/75 is pinned exactly, by multiple independent target-blind routes, criterion = the first Bianchi identity (a theorem), so it is not target-fitted. That is a real banked input win — not a gate advance.
The confirmed engine error (residual R2). The live a₆ engine did not use 23/75. Its K₆ Riemann builder (a6_compute.py :: k6_riem_gilkey :: Rop) writes two naturally-reductive quarter-bracket terms with a sign that violates the first Bianchi identity (max residual 1/7 = 0.142857 in the engine's declared Ric = 1/2 normalization), giving |Riem|²/Scal² = 31/147 (≈ 0.2109) — not a valid Riemann tensor (PLUG_RESULTS_FIRSTPASS_2026-06-29.md lines 66, 72, 86). Flipping the two signs to the Besse-7.38 / KN-II naturally-reductive form restores first-Bianchi to machine zero (3×10⁻¹⁶) and yields 23/75. The deficit is
1 − (31/147)/(23/75) ≈ 1 − 0.2109/0.306667 = 31.23% ("the confirmed ~31.2% Riemann-norm deficit") .
This is a real, quantified, localized internal error — confirmed by an independent rebuild that matches the engine to maxdiff 0.0 (so the bug is real, not a strawman), and by a third witness that builds Ricci from the sectional ground truth (raw eigenvalue 5/12, matching the corrected tensor, not the engine's 7/12). The engine's Ricci/Scalar were right, which is why a Ricci-only self-check "passed" and masked the Riemann-norm bug (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 72).
The corrected magnitude — and why it is still not a closure. Monkeypatching the engine to the Bianchi-correct K₆ curvature (on-disk untouched) shifts the algebraic graviton(91)−2·ghost(13) a₆ from −2.8306×10⁹⁴ to −3.017783×10⁹⁴ GeV⁶ (+6.61%), and the COMPLETE bulk from −2.817995812×10⁹⁴ to −2.995681680×10⁹⁴ GeV⁶ (+6.30%, sign preserved negative) (PLUG_RESULTS_FIRSTPASS_2026-06-29.md lines 66, 72). Crucially: no value of C ≈ −6.39 was recovered — the corpus's full graviton-minus-ghost dimensionless coefficient C is verified absent from the corrected run; the corrected magnitude stays negative but was not tuned to make any positivity test pass (PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 72). And the dimensionful magnitude remains scheme/scale-anchored (R3, below): even Bianchi-corrected, it rides an injected scale, so the number is never gap-closing.
The decision-grade falsifier. The criterion converts "positive-definiteness for physical Hilbert-space closure" into one named functional P(tr[a₆]) ≥ 0 whose sign is the test (S4.1_positivity_functional.md). No reading is yet selected — there are three candidates:
σ⁶-type term does not produce a wrong-sign (ghost/tachyon) higher-derivative kinetic term breaking unitarity;Because no functional is selected, P(a₆) ≥ 0 is not yet a well-defined predicate (GATE_REGRADE_…2026-06-29.md line 1001, R4). The decision wiring (S4.2) is set up in principle — a violation refutes the gate at decision grade — but it cannot be run until both (i) P is selected and (ii) a₆ is computed target-blind. This is why the gate's headline bet is, honestly, a falsifier-in-principle, not yet a runnable falsifier (§7).
The TOTAL a₆ is bulk + defect. The defect lives at the S¹_Y/ℤ₂ fixed points — a totally-geodesic reflection locus M₄ × K₆ × S² × {θ = 0, π} with mixed Neumann ⊕ Dirichlet (ℤ₂-reflection) projectors. The required object is the order-6 boundary heat-kernel coefficient there. In the literature the boundary tower is known only through a₅ (BGKV; Kirsten) (GAP01_A6_RECONCILIATION_2026-06-24.md §1.3; PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 92, obstruction O1). So the defect total is genuinely BLOCKED — a real literature gap, not a fabrication-able piece. The corpus refuses to emit any TOTAL (bulk + defect) and refuses to fabricate the defect. (A later note records a candidate reframing of the defect's identity as a Donnelly equivariant (1/2)c₃^γ, recorded in the Gap-01 sibling-gate handoff as a route, not a settled result; this dossier keeps R6 BLOCKED until that route is independently delivered.)
h_{MN} on M4×K6×S²×S¹_Y --de-Donder gauge-fix--> L_grav = −(∇²+E) (Lichnerowicz) [5A structure, given-E]
+ FP-ghost sector (−2× subtraction, sign forced by BRST nilpotency) [R5: forced but UNVERIFIED]
--Gilkey Th.3.3.1 / Avramidi Ch.4 a₆ template (d=13, ~46-term cubic basis)-->
--d=13 holonomy-projected fiber trace, K6 input |Riem|²/Scal²=23/75--> tr[a₆] [R1: NOT computed; bulk REFUTED]
[R2: engine had 31.2% deficit]
[R3: magnitude scheme-anchored]
--positivity functional P(a₆) ≥ 0 ?--> weak-field UV consistency (5B) [R4: P UNSELECTED, test UNRUN]
--(bulk + Z2 orbifold defect = TOTAL)--> [R6: defect BLOCKED in literature]
==strong-coupling / UV completion (5C / UQF-9)== [R8/R9: global wall]
The construction is solid as far as the structural operator and the exact geometric input go; it stops, honestly, at the computed a₆ vector and everything downstream of it.
These are the moves that produced the believable, reproducible progress — shared at working-physicist depth so a reader can both check them and build on them.
(I) The geometry's role is to supply the operator, not to certify the theory. The single clarifying move is to stop asking "does the geometry prove quantum gravity is consistent?" and ask instead "what operator does the geometry hand us, and what are its properties?" The frozen background fixes L_grav and the ghost operators uniquely (given-E); consistency is then a property of that operator (its a₆), separable from the open question of whether one coefficient certifies the whole theory. This is what makes 5A genuinely terminal as structure while keeping 5B/5C honestly open.
(II) Heat-kernel coefficients are universal — so the calculation has a fixed skeleton. Because a₆ is a universal local invariant (Gilkey), the hard part is bookkeeping, not invention: fix the basis (S2.1), fix the universal functional (S2.2), feed the background's (E, ω, Ω) (S1.3), trace and project (S3). This is why the corpus can grade the wall "perturbative but heavy" — [S], set-up-able — rather than Clay-hard [X]. The discipline that follows: fully specify the computation, never claim to run it unless it is genuinely run on the machine lane, and never fabricate an entry.
(III) Scale-free first; magnitude last. The sharpest honesty lever in the whole gate. The calculation cleanly splits into a dimensionless / scale-free part (curvature ratios, colour factors — which terminate on geometry) and a dimensionful magnitude (which rides an injected scale). The scale-free heart is where the real, checkable wins live: 23/75, and the four exact-rational sphere cross-checks (§5). The magnitude is where the mystery is relocated, not eliminated. Keeping the two strictly separate is what lets the dossier claim 23/75 confidently while refusing to claim the GeV⁶ magnitude.
(IV) Validate the machinery on a soluble laboratory before trusting it on K₆. The curvature-derivative engine was checked on the Berger 3-sphere, where the closed form (∇Riem)² = 256·a²(a²−1)² (null at the round point a = 1) is known exactly. Three independent codes agree (berger_symbolic.py SymPy-exact; an independent numpy rebuild indep_berger.py; a Bianchi tester bianchi_test.py), and the earlier "retraction" of this result was pinned as a (1,3)-tensor differentiation verifier bug, not a wrong closed form (GAP01_A6_RECONCILIATION_2026-06-24.md §1, §4). Lesson: a soluble cross-check both builds trust in the derivative sector and localizes where the engine is wrong (the Riemann-norm sector, not the derivative sector).
(V) Use a theorem, not a target, as the acceptance criterion. The 23/75 win is credible precisely because its acceptance test is the first Bianchi identity — a theorem the curvature tensor must satisfy — not "does it match a desired a₆?" The engine's 31/147 value fails Bianchi; the corrected 23/75 passes it. No target was consulted. This is the κ³/π discipline in action: when the acceptance criterion is an internal theorem, there is no room to back-solve to an answer.
(VI) The ghost subtraction is forced, which is a strength — but forced ≠ verified. BRST nilpotency fixes the subtraction sign (−2 for the FP vector ghost) and multiplicity with no freedom. That is genuine structural content (5A). The honest companion insight: a forced structure still has to be checked against the actual computed coefficient, and when the program tried, the two ghost routes disagreed by 31/48. So the insight cuts both ways — it tells us the subtraction is not a free parameter, and it tells us we have not yet earned the right to call the helicity count a verified certificate.
Everything banked is reproducible from named scripts. This section lists the certificates, the numerical checks (with honest pulls), and exactly how a reader re-runs them.
| Banked fact | Value | Evidence (file) |
|---|---|---|
| Graviton fiber dimension | 91 = 13·14/2 | S1.1_dedonder_graviton_operator.md |
| Ghost fiber dimension; bulk graded weight | 13; 91 − 2·13 = 65 | PLUG_RESULTS_FIRSTPASS_2026-06-29.md line 48 (verify_grading.py) |
| K₆ curvature ratio (exact, target-blind) | |Riem|²/Scal² = 23/75 = 0.306667 | PLUG_… lines 86, 88; 11_a6_BLIND.md; k6_fd_nablaR.py |
| K₆ Levi-Civita curvatures | Ric = 5/12, Scal = 5/2, |Riem|² = 23/12 | PLUG_… line 86 |
| Confirmed engine error (R2) | 31/147 = 0.2109 (Bianchi-violating); deficit 31.23% | PLUG_… lines 66, 72, 86 |
| Sphere cross-checks (S²/S⁴/S⁶/conformal) | 4/315, 74/63, 1139/63, 5/63 | GAP01_A6_RECONCILIATION_2026-06-24.md §1.1; a6_compute.py :: run_crosschecks() |
| Sphere-check relative error | 0 / 0 / 1.97×10⁻¹⁶ / 4.11×10⁻¹⁴ | GAP01_A6_RECONCILIATION_2026-06-24.md §4(A) |
| Berger derivative closed form | (∇Riem)² = 256·a²(a²−1)², null at a=1 | berger_symbolic.py, indep_berger.py, bianchi_test.py |
| Ghost subtraction sign/multiplicity | −2 (FP), forced by BRST nilpotency | 02_CURRENT_STATE.md; 01_DOSSIER.md R5 |
run_crosschecks() computes the spectral side from the actual Sⁿ eigenvalue heat trace via 60-digit mpmath Vandermonde t³ extraction — independent of the Gilkey eq.(4.29) contraction — and matches the algebraic side to relative error 0 to ~4×10⁻¹⁴ (GAP01_A6_RECONCILIATION_2026-06-24.md §1.1, §4A). This anchors the dominant algebraic sector to exact-rational sphere values. Honest pull: this validates the machinery on spheres; it does not validate the d = 13 K₆×S²×S¹_Y magnitude.UQF5A_5B.md bright-line). A later check sharpened this further: 124/315 is metric-SELECTED — it holds only at Scal_K6 = 7.5, and the corpus's own declared curvature gives 0.0252 instead (GATE_REGRADE_…2026-06-29.md line 1157; GAP01 handoff). So 124/315 is NOT the clean target-blind invariant and must carry that caveat — it is not banked here as a settled derived win. (The clean, target-blind, theorem-anchored invariant is 23/75, not 124/315.)−2.817995812×10⁹⁴ GeV⁶ was COMPLETE_CROSSCHECKED on 2026-06-23 and reproduced exactly by a6_compute.py — but it is refuted at decision grade and retracted as 31/147-contaminated (the Bianchi-violating curvature input) (UQF5A_5B.md banked; GATE_REGRADE_…2026-06-29.md line 379). It must never be presented as computed/closing. The Bianchi-corrected value −2.995681680×10⁹⁴ GeV⁶ (sign-preserved, +6.30%) is the corrected magnitude, but it remains scheme-anchored and is not gap-closing.C ≈ −6.39 is unreproduced. The full graviton-minus-ghost dimensionless coefficient C, whose sign the headline falsifier rides, is verified absent from the corrected run (PLUG_… line 72; GATE_REGRADE_…2026-06-29.md line 1043). Reproducing its sign target-blind is part of R1.In the scratchpad lane (…/scratchpad/recon/ and …/scratchpad/R5_clean/), all target-blind, no fitted a₆ (PLUG_RESULTS_FIRSTPASS_2026-06-29.md lines 28, 68):
recon/k6_curv2.py → prints |Riem|²/Scal² = 23/75, first-Bianchi violations 0.recon/nabla_riem.py → |∇Riem|² = 54, second-Bianchi violations 0.r3_verify.py → rebuilds the K₆ curvature both sign conventions: as-written 31/147 (Bianchi max-res 0.142857), sign-flipped 23/75 (Bianchi max-res 3×10⁻¹⁶), deficit 31.23%.r3_bulk_delta.py → imports frozen a6_compute.py, monkeypatches k6_riem_gilkey to the Bianchi-correct builder, re-runs main(): COMPLETE bulk −2.995681680×10⁹⁴ GeV⁶, Δ +6.3054%, G5 Berger gate True in both runs (on-disk corpus untouched).a6_compute.py :: run_crosschecks() → the four exact-rational sphere matches.Cross-check against the corpus certificate …/PER_GATE_DOSSIERS/SHARED_BLOCKER_BUILDS/B1_heatkernel/certificates/B1_Z2_DEFECT_COMPLETION/04_R3_K6_BIANCHI_FIX.md (values agree to 7 digits). Note the stale orphan: a6_sphere_crosscheck.py is a superseded, failing standalone script (it prints "ALL CROSS-CHECKS: FAIL — DO NOT EMIT") and is not invoked by the engine; the operative check is run_crosschecks() inside a6_compute.py, which passes (GAP01_A6_RECONCILIATION_2026-06-24.md §3.1). Remove or mark it SUPERSEDED before any release.
This is the most load-bearing section. Each open hole is a work-package a specialist can start from immediately. Format per hole: (a) precise statement; (b) why it's hard + prior-attempt traps; (c) exactly what closes it (with the refuting outcome); (d) machinery + inputs (exact paths); (e) leverage.
(a) Precise statement. Compute tr[a₆]^phys = tr[a₆(L_grav)] − 2·tr[a₆(FP ghost)] at d = 13, on the de-Donder operator over the frozen M₄×K₆×S²×S¹_Y background, projected onto the ~46-term cubic-curvature Gilkey basis with K₆×S²×S¹_Y holonomy — as a coefficient vector, target-blind. Include reproducing the sign of the full dimensionless graviton-minus-ghost coefficient C (the corpus value ≈ −6.39, currently unreproduced).
(b) Why it's hard / traps. It is "perturbative but heavy": the ~46-term basis × the full graviton+ghost fiber × d = 13 holonomy makes it a major computer-algebra undertaking. Prior attempts hit two obstructions and the verifier caught two traps:
- The 2026-06-23 bulk number −2.818×10⁹⁴ GeV⁶ was refuted/retracted (Bianchi-violating curvature input). Trap: do NOT re-bank it as computed.
- Route B cannot yet reach the graviton LC a₆: the section-trace moments T_j_LC(p,q) on K₆ are Kostant quasi-polynomial (multiplicity deficits on triangular-tip strata where fiber weights |w|² ≤ 8 exit the weight hexagon); global-polynomial resummation poisons even a₂, and brute peel is cost-bounded (reachable maxC₂ ≈ 124, needs thousands) (GAP01.md first-pass residual). It is a computable gap, not a continuum wall.
- Fabrication trap #1 (caught): a first pass quoted graviton σ-supertrace weight 67 and ghost weight 11 (→ "67 − 2·11 = 45"). These appear nowhere in the scripts or corpus; the corpus uses graviton dim 91, ghost dim 13, bulk graded weight 65 (PLUG_RESULTS_FIRSTPASS_2026-06-29.md lines 11, 23). (The number 67 does legitimately appear as tr Sym²(A) in the ℤ₂-graded defect weight bookkeeping — do not import it as the graviton σ-supertrace weight.) Use 91 / 13 / 65.
- Fabrication trap #2 (caught): do not "recover" C ≈ −6.39 by tuning; the corrected run found it absent. Reproduce it (or whatever sign genuinely falls out) target-blind, or report it OPEN.
(c) Exactly what closes it. Run the full graviton+ghost fiber trace over the reduced cubic basis at d = 13, on the machine lane, reproducibility-gated, after fixing R2 (the curvature input). Success criterion: a target-blind coefficient vector that (i) reproduces the four sphere cross-checks and (ii) survives an independent second-route agreement within the pre-fixed 10⁻⁶ tolerance. A refuting result is valid and decision-grade: if the computed a₆, fed to the selected P (R4), violates positivity, the gate is refuted on the spot.
(d) Machinery & inputs. Gilkey (1995) Th. 3.3.1 + Avramidi (2000) Ch. 4 (the universal functional); the reduced basis (S2.1); the frozen (E, ω, Ω) data (S1.3); the corpus engine …/computational_runs_2026-06-23/a6_computation/a6_compute.py (use only with R2 patched); the scratch reproducers in …/scratchpad/recon/ and …/scratchpad/routeB/PR21_clean_pins.py. The Gap-04 reproducibility infrastructure is the stated prerequisite. The most promising lever for the stuck graviton route: derive the SU(3) Gelfand-Tsetlin off-diagonal matrix elements that gate the 5-class Levi-Civita Lichnerowicz graviton hopping on K₆ (the named upstream object) rather than brute-peeling.
(e) Leverage. Highest in the gate — everything in 5B rides it, and the same GT/graviton-route fix is load-bearing for the sibling Gap-01 R1, SG-6 R5, and SG-7 R2 (GAP01.md §6).
(a) Precise statement. The live engine's K₆ Riemann builder (a6_compute.py :: k6_riem_gilkey :: Rop) uses a sign on the two quarter-bracket terms that violates the first Bianchi identity, giving |Riem|²/Scal² = 31/147 instead of the theorem-correct 23/75 — a 31.23% deficit that poisons every downstream magnitude.
(b) Why it's hard / traps. It isn't hard — it is a localized, already-root-caused sign error. The trap is the opposite: treating it as already closed. A first pass declared the ~31% error RESOLVED; the same-date canonical GAP01.md flags that framing as PROMOTION_RISK — R3/the magnitude stay OPEN/OWED; the curvature input is corrected, the gate is not (GATE_REGRADE_…2026-06-29.md lines 1094-1096). Also: the engine's Ricci/Scalar were correct, so a Ricci-only self-check masks the Riemann-norm bug — do not trust a Ricci self-check as proof the curvature is right.
(c) Exactly what closes it. Owner code-fix to the localized Riemann-norm sector: flip the two quarter-bracket signs to the Besse-7.38 / KN-II naturally-reductive form; retain the confirmed derivative-sector ratio and the now-pinned 23/75 input; re-validate against the sphere cross-checks and the Berger gate. Success = first-Bianchi residual to machine zero and |Riem|²/Scal² = 23/75 from the on-disk engine. Disposition ceiling: DISCLOSED-CORRECTED / computation-debt — a correction owed, not a gate closure.
(d) Machinery & inputs. a6_compute.py; the verification scripts r3_verify.py, r3_bulk_delta.py; the certificate …/B1_heatkernel/certificates/B1_Z2_DEFECT_COMPLETION/04_R3_K6_BIANCHI_FIX.md; cross-witnesses k6_fd_nablaR.py, the sectional ground-truth build.
(e) Leverage. High and cheap — a prerequisite to any trustworthy a₆ magnitude (R1, R3). The corrected curvature also feeds SG-6/SG-7.
(a) Precise statement. Select the precise functional P(tr[a₆]) ≥ 0 that encodes "positive-definiteness for physical Hilbert-space closure," plus its decision-grade adjudication rule; then evaluate it on the computed a₆ (R1).
(b) Why it's hard / traps. It is physics selection among three competing readings (spectral/heat-kernel positivity; effective-action higher-derivative sign; counterterm/renormalization positivity) — graded [O] owner-physics, not AI-closeable. Until one is chosen, P(a₆) ≥ 0 is not a well-defined predicate (S4.1; GATE_REGRADE_…2026-06-29.md line 1001). Trap (caught): the gate's EDGE rhetoric presented "run it through the positivity functional" as a ready falsifier; the frozen R4 says it is doubly open — naming a decision-grade falsifier that cannot yet be run overstates falsifiability (GATE_REGRADE_…2026-06-29.md line 1008). A first-pass narrowing argued reading (a) principled and ruled (b) wrong-object (propagating-mode sign is a₀/a₂, not a₆) and (c) scheme-dependent (R3 trap) — a defensible physics argument 3→1, NOT a selection (PLUG_… line 72). State it as argued, not selected.
(c) Exactly what closes it. Make the physics selection (owner), state the adjudication rule precisely, then evaluate on R1's output. A violation → REFUTED at decision grade (a valid, valuable negative). A pass → lifts 5A/5B toward certificate-grade conditional on UQF-9. Do not axiomatize a pass; do not select P so that the current corrected (negative) magnitude passes.
(d) Machinery & inputs. S4.1_positivity_functional.md, S4.2_falsifier_adjudication.md; the candidate menu in gap_01_uv_quantum_gravity/03_attempt.md §D; R1's computed vector.
(e) Leverage. High — it is the gate's decision-grade falsifier. Selecting P is also the move that converts the gate's bet from "falsifier-in-principle" to "runnable."
(a) Precise statement. Only the dimensionless ratios (23/75; the sphere checks) are clean. The dimensionful magnitude rides one named, geometry-unfixed heat-kernel scheme object (the injected scale — the same shared decision family as c_loop / SG-7-δ).
(b) Why it's hard / traps. No measured invariant reaches the magnitude. κ³/π falsification test trap: any scheme object reverse-engineered so the magnitude lands on a desired value is true-by-construction and relocates the mystery rather than closing it. The named axiom AXIOM-HEATKERNEL-SCHEME-OBJECT must be writable without knowing any target magnitude.
(c) Exactly what closes it. 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 AXIOM-HEATKERNEL-SCHEME-OBJECT (= still OPEN, honestly disclosed). A refuting outcome: if no scheme can be fixed without back-solving to a known number, say so and keep it scheme-anchored.
(d) Machinery & inputs. GAPS_AND_WALLS_REGISTER.md §4 (the shared scheme decision); the c_loop / SG-7-δ cross-gate scheme family; 01_DOSSIER.md §4.3.
(e) Leverage. Medium-high — it is the difference between "ratio derived" and "magnitude derived," and it ties three gates together via the shared scheme object.
(a) Precise statement. Write the FP- (and any third-) ghost operator explicitly for the corpus de-Donder weight at d = 13, and confirm the subtraction sign (−2) and multiplicity against R1's computed graviton trace.
(b) Why it's hard / traps. The sign/multiplicity are forced by BRST nilpotency — but the program's two ghost routes disagreed by 31/48 (Route A −251/504 vs Route B 149/1008), and Route A failed its own canonical anchor (−43/504 vs −16/315). Trap (caught): the "agreement" at 149/1008 was obtained only by silently redefining the physical ghost operator from E = −Ric (corpus-binding) to E = 0 (Bochner) — target-fitting toward agreement (PLUG_… line 24). Do not relabel the operator to force a match.
(c) Exactly what closes it. Pin the physical ghost operator identity (Bochner E = 0 vs Lichnerowicz/Hodge E = −Ric) on principle, write the explicit operator, and verify the subtraction against R1's graviton trace. Success = a gauge-independent ghost-corrected coefficient with both routes agreeing within 10⁻⁶. A refuting outcome: persistent route disagreement that cannot be removed without redefining the object is itself a reportable obstruction.
(d) Machinery & inputs. S1.2_ghost_sector.md; E_OMEGA_OPERATOR.md S2.2; CONVENTIONS.md; a6_compute.py line 387; the two-route certificates 08_TWO_ROUTE_AGREEMENT.md, 13_FINAL_ROLLUP.md; scratch RECON_ghost_a6.py, routeB/PR21_clean_pins.py.
(e) Leverage. Medium-high — it is a well-posedness prerequisite for any 5B consistency claim and for the verified (not merely structural) two-helicity count.
(a) Precise statement. Construct the order-6 mixed Neumann ⊕ Dirichlet (ℤ₂-reflection) boundary heat-kernel coefficient at the S¹_Y/ℤ₂ fixed locus, needed for the TOTAL a₆ = bulk + defect.
(b) Why it's hard / traps. The published boundary tower stops at a₅ (BGKV; Kirsten). Trap: refuse to fabricate the defect or any TOTAL. The geometry is correctly identified as a reflection (deficit 0, not a Cheeger cone). A candidate identity-reframing as a Donnelly equivariant (1/2)c₃^γ is recorded as a route in the sibling Gap-01 handoff; treat it as unverified until independently delivered.
(c) Exactly what closes it. A genuine specialist construction of the order-6 mixed-boundary coefficient (likely a new literature result). If obtainable, assemble the TOTAL. If not, terminal as BLOCKED — export the missing-literature object to the tracked blocker ledger.
(d) Machinery & inputs. Boundary heat-kernel literature (BGKV, Kirsten); A6_Z2_DEFECT.md; CERT_Z2_DEFECT_STRUCTURE_2026-06-25.md; …/B1_Z2_DEFECT_COMPLETION/03_OBJECT_IDENTITY.md; scratch verify_z2.py, verify_order6.py.
(e) Leverage. Medium — a real literature gap; closing it would, with R1, allow a genuine TOTAL.
(a) Precise statement. Reproduce 124/315 with an independent, target-blind computation that does not route through the R2-carrying engine.
(b) Why it's hard / traps. Trap (bright-line): do not present 124/315 as a settled "derived dual-validated win" — the same R2 engine produced it, and it is metric-SELECTED (holds only at Scal_K6 = 7.5; the corpus's declared curvature gives 0.0252) (GATE_REGRADE_…2026-06-29.md line 1157). It is not the clean target-blind invariant — 23/75 is.
(c) Exactly what closes it. An independent target-blind reproduction off the R2 engine. Until then it is not an independently-derived win and is not banked as one.
(d) Machinery & inputs. The Peter-Weyl / su(3) spectral-peel route (Route B); 11_a6_BLIND.md.
(e) Leverage. Low-medium — a credibility item, not a gate-mover; 23/75 already carries the clean-invariant role.
(a) Precise statement. A genuine UV completion / strong-coupling closure of the interacting graviton theory, plus graviton-sector unitarity above the cutoff (UQF-14).
(b) Why it's hard / traps. This is a global open problem shared with every approach to quantum gravity; UQF-9's candidate fixed point exists only at a declared truncation; UQF-14 is explicitly blocked by UQF-9. Trap (BS-10 false-openness): do not list "solve QG strong-coupling" as a pluggable hole — it is a wall, not a work-package. Export it; mark it bounded:false. The nearest tractable lever the corpus has isolated is P0: does gravity own an intrinsic shortest length, or inherit the colour R0? — the cleanest UQF-9 sub-target (UQF5A_5B.md open holes R8/R9).
(c) Exactly what closes it. Nothing inside this gate. Export to the open-walls register; pursue P0 as the nearest tractable sub-target under UQF-9.
(d) Machinery & inputs. UQF-9 dossier; the open-walls register; S6_uv_completion_scope.md.
(e) Leverage. Un-closeable here; closing UQF-9 (R8) is what would lift R9.
Not a hole to close but a firewall to maintain. A finite, positive a₆ certifies one heat-kernel coefficient — the unbounded a₆ < a₈ < a₁₀ < … operator ladder is exactly why one coefficient cannot settle strong-coupling. The gate's signature mis-close is relabeling "a₆ computed" as "5C closed"; refuse it. The bounded claim — a₆ is necessary, not sufficient — is the ceiling, not a hedge (S6_uv_completion_scope.md; 01_DOSSIER.md R7).
What is claimed. (1) The frozen 13-D geometry supplies the operator: the de-Donder Lichnerowicz graviton + FP-ghost sector whose 4-D massless spin-2 mode is the graviton, with the KK tower above — leg 5A, as structure, given-E. (2) Exactly two physical massless spin-2 helicities at light speed, reproducing Newton / linearized Einstein in the IR — the public certificate-conditional claim, given-E and conditional on the ghost subtraction (R5) and UQF-9. (3) The exact, theorem-anchored, target-blind geometric input |Riem|²/Scal² = 23/75 on K₆ — a banked input win, not a gate advance.
What is explicitly NOT claimed. That a₆ is computed (the bulk magnitude was refuted/retracted); that C ≈ −6.39's sign is reproduced (it is not); that the positivity functional P is selected (three readings remain) or that P(a₆) ≥ 0 has been run; that 124/315 is a settled derived win (downgraded, metric-selected); that the theory is UV-complete (5C OPEN); that the magnitude is derived (scheme-anchored). The two-helicity count is structural-given-E, not a verified certificate, because R5 is unverified.
dissolved ≠ solved. Two unicorns are dissolved as shared ceilings on all of physics, never claimed as our wins and never listed as our weaknesses:
- That a finite, positive a₆ (one heat-kernel coefficient) constitutes a full UV completion — a universal-sufficiency claim over the unbounded a₆ < a₈ < a₁₀ < … ladder, unprovable in principle for any theory. The bounded claim "a₆ is necessary, not sufficient" is the ceiling.
- That the frozen 13-D background / de-Donder operator is the absolutely unique geometry that could supply a consistent spin-2 sector under any possible mathematics — a universal negative over an open-ended space of constructions, unprovable for any object in any field.
The anchors paid. The structural operator rides the frozen background (READ-ONLY, not proven unique — selection ≠ derivation). The "reproduces linearized GR" claim is given-E (the observed GR limit is the input). The dimensionful magnitude terminates on no measured invariant — it sits on the named heat-kernel scheme axiom. The one clean DERIVED, scale-free, theorem-anchored quantity is 23/75.
The honest edge (stated as it actually stands). The intended testable bet: select P (R4), then compute the d = 13 graviton-minus-ghost a₆ target-blind (R1, after fixing R2, including reproducing the sign of C), then evaluate P on it. If it comes out negative, the gate is refuted on the spot. Once P is selected and a₆ is computed target-blind, that is a falsifiable, decision-grade wager. Until both are done it is a falsifier-in-principle, not yet runnable — and the dossier says so plainly rather than overstating it.
The ceiling. Serious candidate, NOT validated. The geometry supplies the operator; it does not certify the quantum theory. STATUS-UPGRADES:0; frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY; given-E ≠ derivation of E; AXIOM-CLOSED ≠ atomic; a captured log ≠ an independent reproduction; ANCHORED ≠ DERIVED; a₆ is the missing object for UV completion, never is it. Nothing applied, nothing deployed.
Dossier built 2026-06-29 by synthesizing and expanding the frozen corpus (graviton gate brief + GRAVITON_COMPLETION_HANDOFF dossier + Gap-01 decomposition/subsystems + the a₆ reconciliation + the 2026-06-29 regrade/hole-queue/plug ledgers). Frozen 13-D K₆ branch only. Common material referenced to the corpus source-of-truth, not duplicated. Every number traceable to a cited file; uncomputed quantities marked OPEN, never fabricated.