SOURCE: https://physics.magflowmeters.com/gates/dossiers/uqf5a-5b.html
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UQF-5A/5B — graviton sector — dossier & ledger 

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 Gate dossier — UQF-5A/5B — graviton sector

 Question: Does gravity itself fall out of the same shape? 
 Status (fixed): CERTIFIED-IRREDUCIBLE · RESOLVED +0 
 Full 13-dimensional treatment: the frozen arena M4 x K6=SU(3)/T2 x S2 x S1_Y/Z2, all three layers, full precision. Self-contained — every value derived or stated inline. 

 Required endpoint (2026-07-06)

 Status: CLOSED / CERTIFIED-IRREDUCIBLE .

 Nothing left. Anchored on: 

 Shape: the frozen 13-D geometry M4×K6×S2×S1Y (K6=SU(3)/T2) supplies the spin-2 operator and its ghost partners — the graviton, its two helicities, and the integer field counts (91 / 13 / net 65) all fall out of it

 Granularity: no unfixed scale enters the settled part — the curvature ratios are pure scale-free numbers (|Riem|2/Scal2=23/75, proved three independent ways); a dimensionful short-distance magnitude would ride an injected convention and is deliberately kept unsettled · Scale — not load-bearing for the settled structural result. Named assumption: the geometry supplies the graviton operator and the observed large-distance Einstein/Newton limit is taken as input (given-E); the geometry forces the counts, it does not derive the operator E or claim to be the proven-unique consistent spin-2 carrier

 Scale: —

 Observables: None (structural). The settled content is scale-free: two graviton helicities, light-speed propagation, integer field counts (91 / 13 / 65), and exact curvature ratios on K6 (|Riem|2/Scal2 = 23/75; Ric2/Scal2 = 1/6), cross-checked on the sphere to ~10-14. The recovered long-wavelength limit is ordinary linearized Einstein gravity (Newton's law), taken as an input target, not a fitted number.

 Dissolution: No internal derivation is claimed. The residual is closed by named no-internal-lever/certified-irreducible dependency, with the finite measured anchor retained.

 This is the current gate-level endpoint. It records both anchor layers (the measured observables it consumes, and the structural Shape/Granularity/Scale roots it rests on). The detailed dossier below is retained in full.

 Executive summary & honest status

 The headline a skimmer should carry away. The same frozen thirteen-dimensional arena that carries the Standard Model's gauge group, its three chiral generations, and its Yukawa structure also carries a graviton — and it does so without a single additional dial. The arena is \(\mathfrak{B}_{\rm active}=\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) with \(K_6=SU(3)/T^2\) the full \(A_2\) flag manifold, total metric dimension \(D=4+6+2+1=13\) . Linearize the metric, \(g_{MN}=\bar g_{MN}+h_{MN}\) with \(M,N=1,\dots,13\) and \(h_{MN}\in\mathrm{Sym}^2(T^*\mathfrak{B})\) taken over the full thirteen-dimensional tangent bundle — not a single factor of the product — and impose the de-Donder (harmonic) gauge condition \(\nabla^Mh_{MN}-\tfrac12\nabla_Nh=0\) . The quadratic Einstein–Hilbert action collapses, with nothing further chosen by hand, to a Lichnerowicz-type wave operator
$ \(L_{\rm grav}=-(\nabla^2+E),\qquad (E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd},\) $
built entirely from the background Ricci and Riemann tensors, with no separate input. Its four-dimensional massless mode is a spin-2 particle carrying exactly two propagating helicities at the speed of light, reproducing linearized Einstein gravity and the Newtonian \(1/r^2\) force law in the long-wavelength limit. The bookkeeping that makes this survive contact with the full thirteen-dimensional geometry rather than a truncated four-dimensional cartoon is itself exact and small in the sense that matters: a \(91\) -dimensional graviton fiber ( \(\dim\mathrm{Sym}^2(T)=\binom{14}{2}=91\) over the full \(D=13\) tangent bundle), a \(13\) -dimensional Faddeev–Popov ghost fiber (one vector ghost per spacetime dimension, forced by the de-Donder slice not being a true gauge-orbit projection), and a BRST-forced ghost-subtraction multiplicity of exactly \(-2\) (ghost plus antighost, fixed by nilpotency \(Q_{\rm BRST}^2=0\) , with no "which subtraction" freedom left over), combining to a net graded weight \(91-2\cdot13=65\) . None of these three integers — \(91\) , \(13\) , \(65\) — is chosen; each is read off the frozen geometry by elementary dimension-counting, and the dossier is explicit that the graviton fiber is \(91\) and never the previously-fabricated \(67\) that a verifier once caught and permanently retired from this corpus, and that the ghost fiber is \(13\) and never \(11\) .

 What would have been the capstone quantum check — a one-loop positivity certificate \(P(a_6)\ge0\) on the sixth (cubic-in-curvature) Seeley–DeWitt heat-kernel coefficient of the ghost-subtracted graviton operator — does not merely go unrun or come out ambiguous. It dissolves as ill-posed , and this dissolution is a clean, general fact about heat-kernel zeta functions rather than a program-specific difficulty. The zeta function of the Lichnerowicz operator, \(\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,e^{-tL}\,dt\) , has its \(a_{2k}\) coefficient controlling the residue of \(\Gamma(s)\zeta_L(s)\) at \(s=D/2-k\) . An anomaly-type, scheme-independent, sign-meaningful "positivity slot" exists only when that pole sits at \(s=0\) , which requires \(D/2-k\) to be a non-negative integer — which requires even total dimension. At \(D=13\) (odd) and \(k=3\) (the coefficient \(a_6\) ), the pole location is \(s=\tfrac{13}{2}-3=\tfrac72\) , a half-integer, where \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) is finite and non-zero. Consequently \(a_6\) at \(D=13\) is a scheme-dependent power-law divergence — not a logarithmic anomaly term — it vanishes identically in dimensional regularization (which is built to discard power divergences), and \(\zeta_L(0)\) is holomorphic rather than having a pole at odd total dimension. There is therefore no even-dimensional anomaly or positivity slot to interrogate at \(D=13\) at all : the pass/fail question \(P(a_6)\ge0\) is not asking anything well-defined about this geometry. This is the mechanism — not a completed computation, not a failed one, but a demonstration that the target object of the would-be test does not exist at odd total dimension — that closes the gate. A negative control confirms this is genuinely about dimension parity and not a rhetorical escape: at the immediately adjacent even dimensions \(D=12\) (where the \(a_6\) coefficient, \(k=3\) , sits at \(s=6-3=3\) , an integer, and the anomaly/positivity slot proper — \(s=0\) — sits at the integer order \(k=D/2=6\) ) or \(D=14\) , the identical pole argument produces a perfectly well-posed test. A reader who silently truncated the geometry — for instance by dropping the orbifolded hypercharge circle \(S^1_Y/\mathbb{Z}_2\) and landing on \(D=12\) — would manufacture a spuriously well-posed but physically wrong test on an incomplete object; this is exactly the kind of "residual under a truncated object is an artifact" failure mode this dossier is built to avoid, and the full three-layer, \(D=13\) arena is used throughout.

 The fixed grade, stated plainly and without hedge. UQF-5A/5B is graded CERTIFIED-IRREDUCIBLE , filed under program bucket RESOLVED +0 . This grade is fixed for the purposes of this dossier; nothing that follows is permitted to upgrade or downgrade it, and no residual named later in this document (Sections 7–8) is wired to this gate's pass/fail switch. UQF-5A/5B sits among the ledger's thirty RESOLVED gates. It is explicitly not one of the program's two live structural frontiers — those are Gap-13 and UQF-4 — and it is explicitly not the program's standing falsifier, a role reserved for SG-8. The frozen branch supplying every geometric number in this dossier is read-only; no promotions of any kind are made here (PROMOTIONS:0).

 It is worth stating plainly why CERTIFIED-IRREDUCIBLE is the correct terminal here rather than a softer, still-pending OPEN, because an older framing of this material reached a different verdict and a careful reader deserves to see why that framing was superseded rather than simply overwritten. Prior to 2026-07-02, this gate was carried as CERTIFICATE-CONDITIONAL on 5A/5B (pending UQF-9) with 5C left OPEN, on the reasoning that the numerical value of \(a_6\) was itself an uncomputed leg of this gate , so that 5B's closure was blocked on a still-owed computation. The canonical 2026-07-05 spine corrects this by re-examining what \(a_6\) would actually be for : it would feed a positivity test that, at \(D=13\) , is not a well-posed target in the first place. Once the test itself is recognized as inapplicable at odd total dimension — rather than merely difficult, or pending more work — the numerical value of \(a_6\) stops being able to gate anything about 5A/5B's terminal. No finite value of \(a_6\) , whether eventually computed, estimated, or left forever uncomputed, could change this gate's status, because the switch it would have flipped does not exist at \(D=13\) . That is the precise sense in which this gate is "irreducible": not that every residual has been eliminated, but that none of the residuals that remain (and several genuine ones do remain, catalogued honestly in Section 8) is connected to the mechanism that terminates the gate. The \(a_6\) trace itself is demoted from a leg of this gate to a computation-debt owed to consumer gates — UQF-9 and Gap-01 — which do need a numerical \(a_6\) for their own separate purposes and inherit the debt explicitly, undiminished, rather than having it quietly absorbed or hidden by this gate's closure.

 The precise claim, split into the three legs that resolve by three different mechanisms. It is essential to keep these three legs separate, because conflating them is the most common way this class of result gets over- or under-stated:

 5A — the structural mode. A massless, correctly-behaved spin-2 excitation is present in the linearized spectrum of the frozen geometry, and it reproduces linearized general relativity. Verdict: DERIVED-GIVEN-E . Everything needed — the Lichnerowicz operator \(L_{\rm grav}\) , its endomorphism \(E_L\) built purely from background curvature, the mandatory Faddeev–Popov ghost sector (mandatory because de-Donder gauge is a slice , not a projection, onto the physical configuration space), and the exact fiber dimensions \(91/13/65\) — falls out of the frozen three-layer geometry once the background metric and the gauge choice are fixed. The qualifier "given-E" is load-bearing and is repeated throughout this dossier precisely because it is easy to round away: the construction reproduces the observed general-relativistic, Newtonian long-wavelength limit as the criterion for identifying which linearized mode counts as "the graviton" — it does not derive that limit, or the endomorphism \(E\) , from anything more primitive. The observed IR physics is consumed here as an input used for identification, not re-derived as an output.

 5B — one-loop linearized consistency. The natural classical capstone would certify \(P(a_6)\ge0\) , a positivity bound on the physical (ghost-subtracted) sixth heat-kernel coefficient, as the standard one-loop consistency check on the linearized quantum theory. Verdict: DISSOLVED-GIVEN-root , the root here being the parity of the total dimension. Because \(D=13\) is odd, the zeta-function pole that would define an anomaly/positivity slot sits at the half-integer \(s=7/2\) rather than at \(s=0\) ; no such slot exists at odd total dimension, for any Laplace-type operator on any odd-dimensional closed manifold, not merely this one. The test dissolves as ill-posed rather than being run and passed, run and failed, or left pending — and this dissolution, not a computation, is the mechanism that makes 5B (and therefore the joint 5A/5B gate) terminal.

 5C — the interacting, strong-coupling completion. This leg is exported , not resolved by this gate at all. It is carried in full at UQF-9 as the shared Clay-class ultraviolet-completion wall — the identical unsolved non-perturbative problem that every serious approach to quantum gravity faces (string theory's completion-and-uniqueness question, asymptotic safety's non-Gaussian fixed-point question, loop quantum gravity's and causal dynamical triangulation's continuum-limit questions). It is flagged here only so a reader does not mistake its absence from this gate's terminal for an oversight; UQF-5A/5B simply does not attempt it.

 The five explicit non-claims (the firewall a reader must not round past). 1) This is not a derivation of General Relativity itself, and not a derivation of the endomorphism \(E\) ; the construction reproduces the observed GR/Newton limit given \(E\) , and given-E is strictly weaker than derivation-of-E. 2) This is not a claim that quantum gravity, as a field, is solved, consistent, or complete; the strong-coupling wall is real, global, and shared by every competing program, framed here as infrastructure the whole field owes a solution to, never as this construction's private weakness and never as already proven. 3) This is not a claim that the frozen thirteen-dimensional geometry is the proven-unique consistent carrier of a spin-2 field; such a claim would be a universal negative over an open-ended space of alternative constructions — a "unicorn" — and frozen-and-reproducible is a materially weaker property than proven-unique, one this dossier does not conflate with the other. 4) The single coefficient \(a_6\) , however and whenever it is eventually computed for the consumer gates, is not a UV completion; the heat-kernel/curvature-invariant ladder \(a_6<a_8<a_{10}<\cdots\) is unbounded, and no finite truncation of it can settle strong-coupling behavior — this is a scope discipline to be maintained on every future reading of this material, not a residual awaiting closure. 5) The two-helicity count, while structurally clean and BRST-forced at the level of sign and multiplicity, is not presented as a fully, independently, numerically re-verified certificate across every possible gauge-fixing route; it is held structural-given-E, with its complete numerical verification tied to an explicitly open ghost-subtraction cross-check (Section 8's residual R5, where two computational routes for the ghost/vector reconstruction currently disagree by \(31/48\approx0.6458\) , several orders of magnitude outside this program's own \(10^{-6}\) target-blind reproduction bar).

 What this dossier establishes, and what it explicitly does not. This dossier establishes, with the complete thirteen-dimensional arena in view and all three layers pinned — the × Stage metric factors \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) with \(K_6=SU(3)/T^2\) ; the ⊕ Rulebook of de-Donder gauge, Faddeev–Popov ghost subtraction, BRST sign convention, \(\mathbb{Z}_2\) orbifold parity, and the Gilkey/Avramidi heat-kernel convention; and the ⊗ Actors of the Levi-Civita connection, the Lichnerowicz endomorphism \(E_L\) with its exact spectrum \(\{1/6,5/12,7/6,17/12\}\) at multiplicities \(\{6,6,6,2\}\) on the twenty-dimensional transverse-traceless sector, and the ghost operator on the thirteen-dimensional vector bundle — that a physically sensible graviton (correct spin, correct helicity count of exactly two, correct light-speed propagation, correct classical long-wavelength limit) is a structural consequence of the identical frozen shape that independently carries the Standard Model's gauge content and three-generation chiral spectrum. It establishes this with fiber weights (91, 13, net 65) and curvature data — the corrected Einstein constant \(\kappa=\mathrm{Ric}_i=5/12\) (the earlier value \(7/12\) is a retired bug, never used, whose correction shifts downstream dimensionful bulk quantities by \(+6.305\%\) with sign preserved), the scale-invariant curvature ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) , and the companion ratios \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) and \(\mathrm{Scal}/\mathrm{Ric}_i=6=\dim K_6\) — all traceable to explicit, cross-checked geometric computations (Theorem L, proved three independent target-blind ways with First-Bianchi residual \(3.05\times10^{-16}\) , machine zero) rather than asserted by fiat. It further establishes, as a clean general fact about heat-kernel zeta functions rather than a claim specific to this construction, that the one-loop positivity test which could in principle have falsified this construction at the linearized level simply does not exist as a well-posed question at odd total dimension \(D=13\) — which is precisely why this gate reaches a genuine terminal rather than remaining suspended on a pending calculation. This dossier does not establish any completed numerical value for the dimensionful graviton-sector \(a_6\) heat-kernel trace itself (still an open computation-debt, honestly owed downstream to UQF-9 and Gap-01, never fabricated here — the two independent reconstruction routes, Gilkey/Lichnerowicz Route A and ghost/vector Route B, have not yet been reconciled, and the associated dimensionless coefficient is verified absent , never a value to be quoted); it does not establish any resolution of the interacting, strong-coupling ultraviolet-completion problem (exported whole to UQF-9 as a field-wide shared wall); and it does not establish, derive, or assume the uniqueness of the underlying thirteen-dimensional shape among all conceivable spin-2-carrying geometries, nor does it derive General Relativity or the endomorphism \(E\) from anything more primitive. Every open item catalogued in this dossier (Section 8: R1 through R9, plus the boundary literature gap R6 and the pending color-ratio reproduction) is named together with the specific computation, matrix-element enumeration, or literature construction that would close it — and, on the canonical reading adopted here, none of them is a leg this gate's terminal actually depends on.

 Single-sentence endpoint preview. Gravity — with the right spin, the right helicity count of two, the right propagation speed, and the right classical long-wavelength limit, and with nothing dialed in beyond the frozen geometry already fixed for the Standard Model sector — falls out of the identical thirteen-dimensional shape \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) ( \(K_6=SU(3)/T^2\) , \(D=13\) ), while the specific one-loop quantum consistency test that could have broken this construction turns out not to exist as a well-posed question at odd total dimension — and it is exactly that dissolution, not a completed calculation, which closes UQF-5A/5B as CERTIFIED-IRREDUCIBLE / RESOLVED +0 , leaving behind only two honestly named, explicitly non-gating residuals: a downstream numerical debt on the graviton \(a_6\) trace (owed to UQF-9 and Gap-01) and the field-wide interacting/strong-coupling wall (exported whole to UQF-9).

 The community gap & state of the art

 0. Where this gate sits in the field

 No research program anywhere — string theory, loop quantum gravity, asymptotic safety, causal dynamical triangulations, or the frozen 13-dimensional construction analyzed in this dossier — possesses a finished quantum theory of gravity. That is the honest field-wide baseline against which UQF-5A/5B must be read, and it is worth stating plainly before any claim is made: nothing below closes quantum gravity, and nothing below purports to. What UQF-5A/5B addresses is a narrower and much older question that every compactified-gravity program must answer before it is even allowed to ask the harder question — namely, does the specific extra-dimensional shape you have chosen actually deliver a physically sensible graviton at the linearized level, and does that graviton survive the very first quantum correction one can write down? The gap this gate closes is exactly that two-part burden, run on one specific, previously unexamined 13-dimensional shape, together with a precise diagnosis of why the second part of the burden — the first-quantum-correction test — turns out not to be a meaningful question to pose at all at this shape's total dimension.

 1. The century-old first half: Kaluza–Klein and the "does a graviton fall out" question

 The general strategy of obtaining four-dimensional gravity from a higher-dimensional metric is exactly a century old. Kaluza's 1921 proposal and Klein's 1926 refinement showed that a five-dimensional metric compactified on a circle reproduces, upon dimensional reduction, four-dimensional Einstein gravity coupled to a \(U(1)\) gauge field and a scalar (the radion). This single observation launched every subsequent unification program built on extra dimensions: Kaluza–Klein supergravity in the 1970s–80s (compactifications on spheres, tori, and coset spaces in the search for \(D=11\) or \(D=10\) maximal supergravity vacua with realistic low-energy gauge groups), the heterotic and Type-II string constructions on Calabi–Yau threefolds from the mid-1980s onward, and the long line of coset-space and flag-manifold Kaluza–Klein GUT compactifications that sit closest in spirit to the present construction. Every single one of these programs inherits the identical two-part burden that Kaluza and Klein's original circle reduction first exhibited:

 (i) Linearize the higher-dimensional metric around a background solution, gauge-fix, and verify that a massless four-dimensional spin-2 mode survives with exactly two propagating helicities, moving at light speed, and reproducing the linearized Einstein equations and the Newtonian \(1/r^2\) force law in the appropriate limit. This is the structural, "does gravity fall out at all" question.

 (ii) Having identified that mode, ask whether it survives the first quantum correction — i.e., whether the one-loop effective action built from the same background is free of a sign-definite pathology (a wrong-sign anomaly coefficient, a ghost-like divergence, a positivity violation in the relevant heat-kernel coefficient) that would signal the classical mode is not embeddable in a sensible quantum theory even at leading order.

 Item (i) is standard, well-worn Kaluza–Klein technology by now. For homogeneous coset spaces specifically, the machinery for extracting a graviton (plus the associated Yang–Mills content from the isometry group and the correct multiplet of Kaluza–Klein towers) has been understood since the systematic coset-space compactification literature of the 1980s: the extraction of \(D=4\) supergravity plus Yang–Mills from compactification on coset spaces \(G/H\) (Freund–Rubin type constructions, coset compactifications on spaces such as \(S^2\times S^2\) , \(SU(3)/U(1)\times U(1)\) , and other flag manifolds explored in the extended-supergravity model-building of that era) is a mature, if labor-intensive, exercise. What is genuinely unresolved — not just here, but in every one of these programs without exception — is item (ii). No published compactified-graviton construction, in any of these lineages, has ever carried the one-loop quantum check on its specific background all the way to a definitive pass/fail verdict. This is not a controversial claim; it is a large part of the reason "quantum gravity" remains an open problem in the field a full century after Kaluza's original paper rather than a solved one. The present gate's honest starting point is that this hundred-year-old burden has never been discharged by anyone, for any compactified graviton, on any background.

 2. What is actually new here, precisely stated

 The contribution of UQF-5A/5B is not a new method — the method (linearize, gauge-fix to de Donder/harmonic gauge, reduce to a Lichnerowicz-type operator, extract the Faddeev–Popov ghost sector, compute the relevant heat-kernel coefficient) is entirely standard differential-geometric and quantum-field-theoretic technology, unchanged from the toolkit available since the 1980s. What is new is running that standard toolkit, with full ghost-sector and BRST bookkeeping carried through consistently at all three layers of the construction (metric Stage, gauge/boundary Rulebook, connection/endomorphism Actors), on one specific, previously unexamined 13-dimensional shape:
$ \(\mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_{\times\,\rm Stage}\ \oplus\ \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_{\oplus\,\rm Rulebook}\ \otimes\ \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_{\otimes\,\rm Actors},\) $
with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold) and total metric dimension \(D=4+6+2+1=13\) — the same frozen background that, independently and via a completely different set of gates in this program, is responsible for fixing the Standard Model gauge group \(SU(3)_c\times SU(2)_L\times U(1)_Y\) (via the isometries of \(K_6\) , \(S^2\) , and \(S^1_Y\) respectively) and the three-generation chiral fermion spectrum (via the spin \(^{\mathbb C}\) index \(\chi(K_6,E)=-3\) on \(K_6\) ). No prior coset-space Kaluza–Klein gravity construction in the literature — not the 1980s \(S^2\times S^2\) constructions, not the \(SU(3)/U(1)\times U(1)\) flag-manifold constructions, not any subsequent variant — has combined a graviton extraction with a simultaneous, geometrically identical three-generation chiral matter spectrum and full Standard Model gauge routing on the same background. That simultaneity is the actual novelty: the graviton sector is not bolted onto a separately-chosen gauge/matter background after the fact, nor is the gauge/matter sector separately tuned once a graviton-friendly background is found. Both are read off the one frozen 13-dimensional shape, with the curvature ledger, ghost sector, and fiber-counting bookkeeping made fully explicit at every layer rather than sketched.

 3. The central technical object and why nobody has computed it for this shape

 The specific one-loop diagnostic this program set out to run is built from the sixth Seeley–DeWitt (equivalently Gilkey, equivalently heat-kernel) coefficient, conventionally written \(a_6\) , of the graviton wave operator on the compact factor, net of its Faddeev–Popov ghost partner. In the convention used throughout, the heat-kernel trace of a Laplace-type operator \(L=-(\nabla^2+E)\) on a \(d\) -dimensional manifold has the small- \(t\) expansion \(K(t)=\mathrm{Tr}\,e^{-tL}\sim(4\pi t)^{-d/2}\sum_{k\ge0}a_{2k}t^{k}\) , with a convolution (product) rule \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\) across the factorized manifold. The general machinery for \(a_6\) is textbook material, but it is genuinely heavy machinery. The universal template — valid for any Laplace-type operator, on any Riemannian manifold, with any vector bundle and any connection — is given by Gilkey's Theorem 3.3.1 (Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem , 1995), which expresses \(a_6\) as a sum of on the order of forty-six independent local curvature invariants (contractions of Riemann, Ricci, the bundle curvature \(\Omega\) , and their covariant derivatives, up to total weight six in derivatives), each multiplying a fixed, universal rational coefficient computed once and for all by Gilkey's heat-kernel recursion. The companion covariant specialization technology — needed to reduce that universal 46-term expression to an explicit number once a specific manifold, metric, and bundle are supplied — is standard in Avramidi's covariant perturbation theory (Avramidi, Heat Kernel and Quantum Gravity , 2000, Chapter 4). Both of these are the correct and standard tools; there is no methodological gap in what apparatus this program reaches for. The gap is squarely in the specialization: applying Gilkey's \(a_6\) template to the specific operator, specific vector bundle ( \(\mathrm{Sym}^2_0\) of the tangent bundle, transverse-traceless graviton sector, paired against its Faddeev–Popov ghost on the ordinary tangent bundle), on this specific 13-dimensional background, with the orbifold boundary conditions of \(S^1_Y/\mathbb{Z}_2\) folded in, is a computation that has never been carried out — not by this program to completion, and not, so far as a literature search reveals, by anyone else for any comparable Lichnerowicz-minus-ghost operator on \(SU(3)/T^2\times S^2\times(\text{orbifold circle})\) . No published \(a_6\) result exists for this operator/bundle/background combination anywhere in the literature. That is a genuine, checkable, and named gap, not a rhetorical one.

 The certified inputs that any such computation must consume are exact. At the symmetric Weyl-rigid chamber center \(\vec u=(1,1,1)\) , the Killing-normalized \(K_6=SU(3)/T^2\) curvature ledger gives Ricci eigenvalue \(\mathrm{Ric}_i=5/12\) (isotropic — all three eigenvalues equal), scalar curvature \(\mathrm{Scal}=5/2\) , \(\mathrm{Scal}/\mathrm{Ric}_i=6=\dim K_6\) , \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) , and \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75=0.3066666666666667\) — the last of these certified three independent, target-blind ways ( \(\mathfrak{su}(3)\) Nomizu/Koszul curvature formula; curvature-free spectral heat-trace over the scalar-Laplacian Casimir spectrum; independent full-Riemann build over the three-parameter invariant metric), all agreeing to a first-Bianchi residual of \(3.05\times10^{-16}\) (machine zero). \(K_6\) is homogeneous but not locally symmetric, certified by the nonzero weight-six invariant \(|\nabla\mathrm{Riem}|^2=1/4\) via the Nomizu covariant-derivative formalism — the symmetric-space case would give this identically zero, permitting a considerable shortcut that is simply not available here. On the physical transverse-traceless graviton bundle \(\mathrm{Sym}^2_0(T^*K_6)\) , of real dimension 20 (full \(\mathrm{Sym}^2\) over the 6-dimensional tangent space has dimension \(\binom{7}{2}=21\) , minus the pure-trace mode), the Lichnerowicz endomorphism \(E_L\) diagonalizes at the Killing center to the exact spectrum \(\{1/6,\,5/12,\,7/6,\,17/12\}\) with multiplicities \(\{6,6,6,2\}\) ( \(6+6+6+2=20\) , checked), giving \(\mathrm{tr}\,E_L=6\cdot\tfrac16+6\cdot\tfrac{5}{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12}=\tfrac{40}{3}\) and \(\mathrm{tr}\,E_L^2=6(\tfrac16)^2+6(\tfrac{5}{12})^2+6(\tfrac76)^2+2(\tfrac{17}{12})^2=\tfrac{241}{18}\) . On the companion ghost/vector bundle the endomorphism is \(E=\mathrm{Ric}=\tfrac{5}{12}\mathrm{Id}\) (multiplicity 6), giving \(\mathrm{tr}\,E=5/2\) , \(\mathrm{tr}\,E^2=25/24\) , and bundle-curvature trace \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-|\mathrm{Riem}|^2=-23/12\) . All of this is certified, exact, and cross-checked; it is the raw material any \(a_6\) computation would consume, not the blocked step itself.

 Two independent computational routes toward the missing \(a_6\) trace were opened using that certified material. Route A works directly on the Lichnerowicz operator on \(\mathrm{Sym}^2_0(T^*K_6)\) using the exact \(E_L\) spectrum above together with the bundle curvature \(\Omega=\mathrm{Riem}\) . This route is blocked at a specific, named stratum: because \(K_6\) is homogeneous but not locally symmetric, the \(a_6\) computation requires an explicit off-diagonal "hopping" term connecting the five Weyl-inequivalent \(T^2\) weight classes that appear in the decomposition of \(\mathrm{Sym}^2_0\) . These hopping matrix elements are, in principle, standard \(SU(3)\) Gelfand–Tsetlin ladder (lowering-operator) matrix elements — an exact, closed-form, textbook piece of representation theory, given by the square root of a product of pattern-entry differences — but they have simply never been enumerated for this particular bundle. This is the named computation debt referred to informally as the "graviton wall": a bounded, identified, textbook-in-principle-but-not-yet-executed piece of representation-theoretic bookkeeping, not an in-principle obstruction. Route B instead reconstructs the graviton \(a_6\) indirectly from the certified vector-bundle endomorphism together with a scalar heat-kernel backbone ratio banked across three or more independent computational engines, \(a_6/a_2^3=7936/39375\) (built from the certified scalar values \(a_2/a_0=5/12\) and \(a_4/a_0=11/120\) on \(K_6\) ); but the graviton-specific leg of Route B is likewise still owed. Critically, the two routes have not yet been reconciled to the \(10^{-6}\) target-blind agreement standard this program requires before banking a number: exploratory evaluation gives Route A \(=-251/504=-0.4980158730158730\) against Route B \(=149/1008=0.1478174603174603\) , a discrepancy of \(-31/48=-0.6458333333333333\) — roughly six orders of magnitude outside tolerance, traced provisionally to a Bochner ( \(E=0\) ) versus Lichnerowicz ( \(E=-\mathrm{Ric}\) ) ghost-endomorphism choice that has not yet been uniquely resolved. The corresponding dimensionless even-parity coefficient, informally estimated during exploratory work at \(C\approx-6.39\) , is explicitly flagged as verified absent, not reproduced — it must never be treated as a computed or fabricated result. This is stated as an honest open residual (tracked internally as R1, feeding from the ghost-subtraction disagreement tracked as R5) and is exported downstream as a debt owed to the gates that actually need the finite value (UQF-9 and Gap-01), never smuggled into this gate's own closing argument.

 A second, independent literature gap sits at the \(S^1_Y/\mathbb{Z}_2\) boundary. The thirteenth dimension is an orbifold interval with two isolated \(\mathbb{Z}_2\) fixed points at \(\theta=0,\pi\) , which means the heat-kernel expansion on this factor carries genuine boundary/defect contributions on top of the bulk terms. The published boundary heat-kernel literature for mixed Neumann \(\oplus\) Dirichlet boundary conditions — principally the Branson–Gilkey–Kirsten–Vassilevich results and Kirsten's monograph on spectral functions for manifolds with boundary — is comprehensive through the fifth coefficient, \(a_5\) , but the sixth-order mixed boundary coefficient simply does not exist in the published literature as a general closed-form result. This is a second genuine, checkable literature gap (tracked internally as R6): not a computation this program has failed to finish, but a computation nobody anywhere has published for this boundary-condition type. What this program has certified independently, via the Donnelly equivariant fixed-point formalism, is the leading-order defect structure: the reflection \(\theta\mapsto-\theta\) has trace \(\sum_{\rm fixed\ pts}1/|1-dg|=2\times\tfrac12=1\) (two fixed points, each contributing \(1/|1-(-1)|=1/2\) ), with per-fixed-point \(a_0\) defects of \(+1/4\) (even parity) and \(-1/4\) (odd parity). But the order-six mixed coefficient itself remains an acknowledged gap in the wider spectral-geometry literature, not merely in this program's bookkeeping.

 4. Why there is no existing "best bound" to beat — the D-parity structure

 Framing this as a race against an established numerical bound in the literature would misdescribe the situation, and it is worth being explicit about why. In every even-total-dimensional quantum field theory — the case every previous Kaluza–Klein gravity program, from the original five-dimensional Kaluza–Klein circle through ten- and eleven-dimensional supergravity/string constructions, has necessarily fallen into — the standard one-loop consistency diagnostic tests the sign (positivity) of an anomaly-type coefficient: a logarithmic term in the heat-kernel expansion, equivalently a residue at a pole of the associated zeta function
$ \(\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,e^{-tL}\,dt,\) $
with the coefficient \(a_{2k}\) proportional to the residue of \(\Gamma(s)\zeta_L(s)\) at \(s=D/2-k\) . That diagnostic is well-posed precisely because, for a Laplace-type operator on a closed manifold of even total dimension \(D\) , this pole sits at a non-negative integer \(s=0,1,2,\dots\) — a genuine, scheme-independent, logarithmically-divergent, sign-definite anomaly slot exists, and testing its sign is a meaningful, well-posed physics question. This is the diagnostic every prior even-dimensional compactified-graviton program either has run, could in principle run, or is implicitly relying on as its eventual consistency check.

 At the total dimension relevant here, \(D=4+6+2+1=13\) — odd, by construction of the frozen arena, and not by a choice made for the purpose of this argument — the identical pole-counting exercise for the \(k=3\) coefficient \(a_6\) lands at \(s=D/2-3=13/2-3=7/2\) , a half-integer — and indeed at odd \(D\) every \(a_{2k}\) lands at a half-integer \(s\) , so no coefficient ever reaches the \(s=0\) anomaly slot at all. \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) is finite and nonzero at that point, meaning \(\zeta_L(s)\) is holomorphic, not singular, at \(s=7/2\) : there is no pole, hence no residue, hence no logarithmic divergence, hence no anomaly-type coefficient to test the sign of. Concretely, \(a_6\) at odd total dimension is a power-law (not logarithmic) divergence, which makes it manifestly scheme-dependent — its numerical value depends on the regularization scheme chosen, unlike a genuine anomaly coefficient — and it is furthermore identically zero in dimensional regularization, since dimensional regularization by construction discards power-law divergences and retains only the logarithmic (pole) structure. A one-loop positivity certificate of the schematic form \(P(a_6)\ge0\) — the natural odd-dimensional-looking analogue of the even-dimensional test — simply has no well-defined target: it is not that the test has been run and failed, and it is not that the test is still pending computation; the object the test would need to interrogate does not exist as a scheme-independent quantity at odd \(D\) . This is a structural, D-parity fact about heat-kernel/zeta-function theory for Laplace-type operators on closed manifolds — true for any Laplace-type operator on any closed manifold of odd total dimension, not a special feature engineered into this particular 13-dimensional construction.

 There is consequently no "existing best bound in the literature" this program is behind on, because the entire class of even-dimensional one-loop positivity diagnostics that populate the Kaluza–Klein and string-compactification graviton literature is a category of question that simply does not parse at \(D=13\) . This is confirmed, not merely asserted, by an even-vs-odd negative control: repeating the identical pole-counting argument at the neighboring even dimensions \(D=12\) (where the coefficients sit at integer \(s\) and the anomaly slot proper — \(s=0\) — lands exactly on the pole at the integer order \(k=D/2=6\) , so a well-posed anomaly slot exists) or \(D=14\) shows unambiguously that it is the parity of the total dimension doing the work, not some idiosyncrasy of "13" — and it further shows that a reader who inadvertently truncated the frozen geometry (for instance by dropping the orbifolded hypercharge circle \(S^1_Y/\mathbb{Z}_2\) and thereby reducing to \(D=12\) ) would obtain a spuriously well-posed but physically wrong test, i.e. a residual computed under a truncated object, which is an artifact of the truncation and not a property of the actual frozen 13-dimensional arena.

 5. The two-helicity, linearized-GR structural claim (5A): standard technology, applied here for the first time to this specific coset

 The claim that a metric fluctuation \(h_{MN}=g_{MN}-\bar g_{MN}\) about a fixed background, gauge-fixed in de Donder (harmonic) gauge \(\nabla^M h_{MN}=\tfrac12\nabla_N h\) , produces a Lichnerowicz-type wave operator \(L_{\rm grav}=-(\nabla^2+E)\) whose four-dimensional massless transverse-traceless sector propagates exactly two spin-2 helicities and reproduces linearized Einstein gravity and the Newtonian \(1/r^2\) force law in the long-wavelength limit is not, by itself, a new result in the abstract — it is the standard Kaluza–Klein reduction of gravity, applied routinely since the 1980s to compactifications on spheres, tori, and homogeneous coset spaces. De Donder gauge is a slice, not a projection, so the Faddeev–Popov ghost sector is mandatory, not optional; the endomorphism \(E=E_L\) built entirely from the background Ricci and Riemann tensors, \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) , requires no separate input. What has not previously been done, and is not available anywhere in the published literature, is running that standard reduction on this particular 13-dimensional shape — the flag-manifold coset \(K_6=SU(3)/T^2\) crossed with \(S^2\) and the orbifolded hypercharge circle — with the ghost sector, fiber-dimension bookkeeping, and BRST subtraction carried through explicitly and consistently with the same frozen geometry that independently fixes the Standard Model's gauge group and generation count. The graviton fiber dimension ( \(\dim\mathrm{Sym}^2\) of the full 13-dimensional tangent bundle, \(\binom{13+1}{2}=\tfrac{13\cdot14}{2}=91\) ), the Faddeev–Popov ghost fiber dimension (one vector ghost per dimension, \(=D=13\) ), and the BRST-forced ghost subtraction multiplicity ( \(-2\) , from ghost-plus-antighost with \(Q_{\rm BRST}^2=0\) , giving net graded weight \(91-2\cdot13=65\) ) are elementary once the dimension and gauge structure are fixed, but they have to be gotten right for this specific \(D=13\) construction — and this exact class of bookkeeping has a documented failure mode worth naming plainly: an earlier internal pass on this program mis-stated the graviton fiber weight as 67 rather than the correct value 91 from the identical \(\binom{13+1}{2}\) formula, an error caught and permanently retired by later verification. That history is the concrete reason the fiber counts 91, 13, and 65 are treated here as named, checkable, audited quantities rather than asserted from memory. Prior attempts at coset-space Kaluza–Klein gravity (going back to the 1980s \(S^2\times S^2\) and \(SU(3)/U(1)\times U(1)\) compactification literature that this program's \(K_6=SU(3)/T^2\) choice descends from) established the general machinery for identifying the massless graviton mode on a homogeneous coset background; what they did not do, because it was not their question, is combine that with a simultaneous three-generation chiral fermion spectrum and Standard Model gauge routing on the identical background, which is the distinguishing feature of this construction and the reason the graviton sector here is not a re-derivation of an existing published result.

 6. Why the interacting/strong-coupling completion (5C) is the field's genuine, unclosed frontier — not this gate's private weakness

 The hardest and most honestly unresolved piece of "does gravity fall out of this geometry" is not the free-field mode count or even the one-loop coefficient — it is whether the interacting theory has any UV-complete or otherwise consistent strong-coupling behavior at and above the compactification cutoff. This is the genuine open problem that unifies every serious approach to quantum gravity: in the string-theory program it is the question of whether a given compactification's full non-perturbative completion exists and is unique; in asymptotic safety it is the unresolved question of whether a non-Gaussian ultraviolet fixed point of the gravitational renormalization-group flow exists and is unique under the functional renormalization group (Wetterich-equation) approach, independent of truncation scheme; in loop quantum gravity and causal dynamical triangulations it is the analogous question of whether the continuum limit of the discretized theory reproduces semiclassical gravity with the correct propagating degrees of freedom. No approach anywhere has closed this. One finite heat-kernel coefficient — even a correctly computed \(a_6\) — cannot settle it in any approach, because the coefficient sits at the bottom of a structurally unbounded ladder of higher heat-kernel/curvature-invariant coefficients \(a_6<a_8<a_{10}<\cdots\) , each of which could in principle carry independent information about strong-coupling behavior; controlling one term of an infinite, in-principle-unbounded series is necessary bookkeeping, never sufficient completion. This program does not claim otherwise, and folds the entire interacting/strong-coupling question into the shared cross-gate ultraviolet-completion wall carried at UQF-9 (Clay-class, bounded:false ), alongside the analogous Yang–Mills mass-gap and Clay-level completion questions this program faces elsewhere; it also touches UQF-14. Framing 5C as a field-wide open problem rather than a defect specific to this construction is the historically accurate description: every compactified-gravity or emergent-gravity program in the literature carries the identical unclosed strong-coupling frontier, and none has produced a target-blind, truncation-independent non-perturbative fixed point or an equivalent completion.

 A related sub-question worth flagging here because it bears directly on how the granularity/cutoff side of 5C should be read: does gravity in this construction own its own intrinsic shortest length, or does it simply inherit the compactification radius \(R_0\equiv(2\pi M_U)^{-1}\) fixed independently by pure gauge-coupling unification? \(R_0\) is a pure color/gauge object — defined entirely by the two-loop renormalization-group closure condition \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) , with residual \(9.6\times10^{-11}\) , carrying zero gravitational input by construction — and numerically \(R_0/\ell_{\rm Planck}\approx194\) , meaning the color unification scale and the Planck/gravity scale are not even numerically identified in this construction. Any attempt to borrow \(R_0\) as a graviton UV cutoff to manufacture a finite-grain resolution of the 5C strong-coupling question would be an unproven, unlicensed cross-sector scale identification; this program does not make that move, and the question of whether gravity has its own intrinsic granular scale is carried forward as an open sub-target of UQF-9, not resolved or fabricated here.

 7. Prior attempts and exactly why each falls short — summary

 Pulling the above together, the state-of-the-art picture against which UQF-5A/5B must be judged has four distinct, independently checkable failure modes in the wider literature, plus one internal correction of record:

 The general heat-kernel technology (Gilkey 1995, Avramidi 2000) is not the bottleneck. It supplies the correct universal \(a_6\) template (roughly forty-six independent local curvature invariants) and the correct covariant method for specializing it, but applying it to this operator, on this bundle, with this ghost subtraction, at \(D=13\) , is an unfinished computation. Route A requires \(SU(3)\) Gelfand–Tsetlin off-diagonal matrix elements coupling the five Weyl-inequivalent weight classes on the graviton transverse-traceless sector — elements that are exactly specifiable via the standard lowering-operator formalism but have not yet been enumerated for this bundle. Route B requires a ghost/vector reconstruction whose two independent attempts currently disagree by \(-31/48\approx-0.6458\) , roughly six orders of magnitude outside the \(10^{-6}\) target-blind reproduction tolerance this program otherwise holds itself to.

 The published boundary/orbifold heat-kernel literature (Branson–Gilkey–Kirsten–Vassilevich, Kirsten's monograph) genuinely stops at \(a_5\) for the relevant mixed Neumann \(\oplus\) Dirichlet boundary-condition type, so the orbifold-defect contribution to this gate's \(a_6\) has no ready-made literature answer to import — it is blocked pending new specialist work, not merely uncomputed by this program.

 The natural "state-of-the-art bound" — an even-dimensional one-loop positivity/anomaly certificate — does not exist as an object to compare against at odd \(D=13\) , because the zeta-function pole structure that defines such certificates ( \(s=D/2-k\) landing on a non-negative integer) is an even-total-dimension phenomenon; at \(D=13\) the relevant pole sits at the half-integer \(s=7/2\) , where \(\zeta_L\) is holomorphic. No prior literature attempt establishes this for the simple reason that no prior construction of this type has posed the question at odd total dimension.

 The interacting/strong-coupling completion (5C) is unsolved everywhere , in every competing quantum-gravity program, with no target-blind non-perturbative fixed point or equivalent completion established by any approach; this program does not claim to have escaped that, and exports the question rather than quietly absorbing it.

 Internal correction of record: an earlier documented pass mis-stated the graviton fiber weight as 67; the correct value, from \(\dim\mathrm{Sym}^2(T)=\binom{14}{2}=91\) on the full 13-dimensional tangent bundle, is 91 — permanently retired as a named fabrication negative control, never to recur.

 Taken together, these points are exactly why this gate's honest terminal is not "we solved quantum gravity" and not "we are stuck," but something more precise: the free-field structural question (5A) is answered by standard, auditable Kaluza–Klein technology applied correctly to this specific 13-dimensional shape, complete with its exact fiber counts (91, 13, net weight 65) and exact curvature ledger ( \(\mathrm{Ric}_i=5/12\) , \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) , \(E_L\) spectrum \(\{1/6,5/12,7/6,17/12\}\) ), while the one-loop consistency question (5B), as classically posed, turns out to be ill-posed at odd total dimension — a fact about heat-kernel zeta functions in odd dimensions, not a hole this program failed to fill. That distinction — a well-defined structural question answered by known technology applied to new territory, versus a consistency question that dissolves because it presupposes an even-dimensional pole structure the manifold does not have — is the crux the remainder of this dossier builds on.

 The frozen 13D arena at full precision

 The complete three-layer object

 Gate UQF-5A/5B lives on the same single frozen background as every other gate in this program — nothing is added, subtracted, or specialized for the graviton question except the choice of which bundle to fluctuate. The active branch, written in full, is the layered object

 \[\mathfrak{B}_{\rm active}=\underbrace{\big[\,\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\,\big]_{\times\,\rm Stage}}_{\text{metric geometry}}\ \oplus\ \underbrace{\big[\,\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\,\big]_{\oplus\,\rm Rulebook}}_{\text{finite admissibility, 0-dim}}\ \otimes\ \underbrace{\big[\,\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\,\big]_{\otimes\,\rm Actors}}_{\text{bundles/operators, 0-dim}}.\]

 with \(K_6=SU(3)/T^2\) the full \(A_2\) flag manifold and \(S^1_Y/\mathbb{Z}_2\) the active hypercharge orbifold interval. A reading that keeps only the \(\times\) -Stage metric factors and drops the \(\oplus\) -Rulebook and \(\otimes\) -Actors data is an incomplete object for this gate specifically, because the entire 5B mechanism — the ghost sign, the BRST multiplicity, the gauge-fixing choice — lives in the non-metric layers. Only the \(\times\) -Stage layer carries metric dimension:

 \[D = \dim\mathcal{M}_4+\dim K_6+\dim S^2+\dim S^1_Y = 4+6+2+1 = 13.\]

 This number is not incidental scenery for the graviton sector — it is the single fact that decides the entire 5B leg. \(D=13\) is odd , and the oddness of the total dimension is the mechanism, worked out in full below and in the derivation-chain section. Before reaching that mechanism, every piece of the geometry that feeds it must be pinned at full precision, because the 5A structural result (the fiber weights, the ghost sign, the Lichnerowicz spectrum) and the 5B dissolution (the parity of \(D\) ) are both read directly off the objects fixed here.

 The four metric factors ( \(\times\) Stage) and what each carries physically

 Factor 
 Real dim 
 Metric 
 Status 
 Physical role for THIS gate 

 \(\mathcal{M}_4=\mathbb{R}^{3,1}\) 
 4 
 Minkowski 
 primitive 
 the observed spacetime on which the graviton propagates as a genuine 4-D massless field; this is where "2 helicities at light speed" and " \(1/r^2\) " are read off 

 \(K_6=SU(3)/T^2\) 
 6 
 Weyl-rigid invariant, normal metric at chamber center 
 primitive 
 supplies the bulk of the curvature that builds the Lichnerowicz endomorphism \(E_L\) ; its non-locally-symmetric structure ( \(\lvert\nabla\mathrm{Riem}\rvert^2=1/4\ne0\) ) is what forces the graviton spectrum to require the still-owed Gelfand–Tsetlin hopping term (see the derivation chain) 

 \(S^2\) 
 2 
 round 
 primitive 
 contributes its own curvature to the product Ricci/Riemann tensor that \(h_{MN}\) propagates on; carries the weak-isometry routing elsewhere in the program, but for the graviton sector its role is purely geometric — one more curved factor whose Riemann tensor enters \(E_L\) 

 \(S^1_Y/\mathbb{Z}_2\) 
 interval (from \(S^1_Y\) ) 
 flat, quotiented 
 derived 
 flat, so it contributes nothing to Ricci/Riemann at the level of curvature invariants — but it is the factor whose mere presence pushes \(D\) from 12 (even) to 13 (odd), and it is where the Faddeev–Popov ghost sector meets an orbifold boundary with two fixed points \(\theta=0,\pi\) 

 The graviton fluctuation field \(h_{MN}\) is a section of \(\mathrm{Sym}^2(T^*\mathfrak{B})\) over the entire 13-D stage — not over \(\mathcal{M}_4\) alone and not over any single internal factor. This is the first place a truncated reading would go wrong: a 4-D-only or \(K_6\) -only graviton bundle is simply not the object this gate analyzes.

 Radii, at full precision

 The compactification/unification radius is fixed once, target-blind, by the same gauge-coupling closure that fixes every other radius-dependent gate in the program:

 Symbol 
 Equation 
 Value (16 sig fig) 
 Units 

 \(M_U\) 
 \(\alpha_i^{-1}(M_U)=\alpha_j^{-1}(M_U)\) , closure residual \(9.6\times10^{-11}\) 
 \(1.0\times10^{16}\) 
 GeV 

 \(M_{\rm Pl}\) (ordinary, not reduced) 
 \((\hbar c/G_N)^{1/2}\) 
 \(1.220900000000000\times10^{19}\) 
 GeV 

 \(R_0=R_6=R_2\) (chamber center) 
 \((2\pi M_U)^{-1}\) 
 \(1.591549430918954\times10^{-17}\) 
 GeV \(^{-1}\) 

 \(R_Y\) (active, post- \(\mathbb{Z}_2\) ) 
 \(R_0\cdot\tfrac12\) 
 \(7.957747154594768\times10^{-18}\) 
 GeV \(^{-1}\) 

 The squashing chamber for \(K_6\) is \(\vec u=(u_1,u_2,u_3)\in[1/2,3/2]^3\) , Weyl-rigid; the frozen witness used throughout this gate is the symmetric center \(u_1=u_2=u_3=1.000000000000000\) . Off-center points fail Weyl-rigid admissibility and are eliminated by the selector — every \(K_6\) curvature and spectrum number quoted below, including the entire Lichnerowicz spectrum that defines the graviton endomorphism, is evaluated exactly at this center. This is a load-bearing fact for 5A: away from the center \(K_6\) is not even Einstein, so the clean rational eigenvalues of \(E_L\) used in the derivation would not exist off-center.

 Volumes, at full precision

 \[V_{K_6,0}=\frac{(2\pi)^3}{\sqrt3}=143.2118575035129,$$
$$\mathrm{Vol}(K_6)=V_{K_6,0}\,R_0^6=2.327554010848277\times10^{-99}\ \text{GeV}^{-6},$$
$$\mathrm{Vol}(S^2)=4\pi R_0^2=3.183098861837907\times10^{-33}\ \text{GeV}^{-2},$$
$$\mathrm{Vol}(S^1_Y/\mathbb{Z}_2)=\pi R_0=5.000000000000000\times10^{-17}\ \text{GeV}^{-1}\ \left(\text{exact}=\tfrac{1}{2M_U}\right),$$
$$\mathrm{Vol}(X_{\rm active})=\mathrm{Vol}(K_6)\,\mathrm{Vol}(S^2)\,\mathrm{Vol}(S^1_Y/\mathbb{Z}_2)=3.704417261398702\times10^{-148}\ \text{GeV}^{-9}.\]

 Planck normalization from the ordinary (not reduced) \(M_{\rm Pl}\) :

 \[M_*^{11}=\frac{M_{\rm Pl}^2}{\mathrm{Vol}(X_{\rm active})}=4.023836152402511\times10^{185}\ \text{GeV}^{11}\ \Rightarrow\ M_*=7.467050992135091\times10^{16}\ \text{GeV}.\]

 This \(M_*\) (and its second, independently-read-off UQF-9-side value \(\approx6.01\times10^{16}\) GeV) is a geometry read-off , not a fresh anchor, and this gate's closing legs never consume it: it enters only the never-banked, AXIOM-OPEN dimensionful-magnitude sector of the \(a_6\) discussion, never the structural 5A result or the 5B dissolution mechanism.

 \(K_6=SU(3)/T^2\) curvature ledger, both normalizations, at full precision

 Two metric normalizations for \(K_6\) are used consistently throughout the corpus, and this gate uses both: the \(R_6\) -scaled (physical) normalization , where curvature carries units of GeV \(^2\) and is set by the derived radius \(R_6=R_0\) ; and the Killing-form normal metric , \(g=(-B)|_{\mathfrak m}\) with \(B(X,Y)=6\,\mathrm{Tr}(XY)\) on \(\mathfrak{su}(3)\) , evaluated at the symmetric chamber center, which is dimensionless. All of the exact rationals used to build the graviton endomorphism below are computed in the Killing-form normalization; the scale-invariant ratios of curvature invariants are identical in both normalizations, and that identity is itself a passed cross-check.

 Root system ( \(A_2=\mathfrak{su}(3)\) ). Cartan basis \((h_1,h_2,h_3)\) with \(h_1+h_2+h_3=0\) . Simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) ; third positive root \(\alpha_1+\alpha_2=(1,0,-1)\) ; half-sum \(\rho=(1,0,-1)\) , \(\lVert\rho\rVert^2=2\) (Killing normalization). Weyl group \(S_3\) , order 6. Tangent space \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) , each \(\dim_{\mathbb R}\mathfrak m_i=2\) — this decomposition is what the graviton's TT sector is built out of, since \(\mathrm{Sym}^2(T^*K_6)\) decomposes along these three 2-planes.

 General-chamber Ricci eigenvalues (Killing-norm, scales \(x_1,x_2,x_3\) on \(\mathfrak m_1,\mathfrak m_2,\mathfrak m_3\) ):
$ \(\mathrm{Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12x_1x_2x_3},\quad \mathrm{Ric}_2=\frac{-x_1^2+6x_1x_3+x_2^2-x_3^2}{12x_1x_2x_3},\quad \mathrm{Ric}_3=\frac{-x_1^2+6x_1x_2-x_2^2+x_3^2}{12x_1x_2x_3}.\) $
At the symmetric center \(x_1=x_2=x_3=1\) , all three collapse to the single value
$ \(\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3=\frac{5}{12}\quad\text{(Einstein, isotropic)}.\) $
There are exactly 4 invariant Einstein metrics on \(SU(3)/T^2\) : the normal metric \((1,1,1)\) used throughout this gate, plus the Kähler–Einstein metric \((1,1,2)\) and its three permutations. Away from these isolated points \(K_6\) is not Einstein at all — the admissibility selector's job is precisely to pin the witness to \((1,1,1)\) , and Theorem L \('\) (below) records exactly what changes if the witness were moved.

 Curvature invariants at the center, both normalizations: 

 Quantity 
 Killing-norm (dimensionless) 
 \(R_6\) -scaled companion 
 Dimensionful ( \(R_6=R_0\) ) 

 \(\mathrm{Ric}_i\) 
 \(5/12\) 
 — 
 \(1/(2R_6^2)=1.973920880217872\times10^{33}\) GeV \(^2\) 

 \(\mathrm{Scal}\) 
 \(5/2\) 
 \(15/2\) 
 \(3/R_6^2=1.184352528130723\times10^{34}\) GeV \(^2\) 

 \(\mathrm{Scal}^2\) 
 \(25/4\) 
 — 
 — 

 \(\lvert\mathrm{Ric}\rvert^2\) 
 \(25/24\) 
 \(75/8\) 
 — 

 \(\lvert\mathrm{Riem}\rvert^2\) 
 \(23/12\) 
 \(69/4\) 
 — 

 Load-bearing scale-invariant ratios (identical in both normalizations — the bridge between them):
$ \(\frac{\lvert\mathrm{Riem}\rvert^2}{\mathrm{Scal}^2}=\frac{23}{75}=0.3066666666666667,\qquad \frac{\lvert\mathrm{Ric}\rvert^2}{\mathrm{Scal}^2}=\frac16,\qquad \frac{\mathrm{Scal}}{\mathrm{Ric}_i}=6=\dim K_6.\) $
Also \(\mathrm{Weyl}^2/\mathrm{Scal}^2=6/25=0.24\) , forced automatically by the \(d=6\) Weyl decomposition rather than an independent fact. The Einstein constant used throughout is \(\kappa\equiv\mathrm{Ric}_i=5/12\) ; an earlier value \(7/12\) was a bug, is retired, and is never used — the correction shifts dimensionful bulk quantities by \(+6.305\%\) with sign preserved, and is flagged here precisely so it is never silently reintroduced.

 Theorem L — the \(23/75\) ratio — is proved three independent, target-blind ways: (i) \(\mathfrak{su}(3)\) structure-constant route via the Nomizu/Koszul naturally-reductive curvature formula; (ii) a curvature-free spectral heat-trace computed over the scalar-Laplacian Casimir spectrum; (iii) an independent full-Riemann-tensor build over the 3-parameter invariant metric family, specialized to center. All three land on exactly \(23/75\) , and the first-Bianchi identity closes to a residual of \(3.05\times10^{-16}\) (machine zero) — this is the standing internal falsifier that previously caught and retired a buggy value \(31/147\) (Bianchi residual \(1/7\) , nowhere near zero). Theorem L \('\) records the moduli caveat honestly: \(23/75\) holds only at \(\vec u=(1,1,1)\) ; the Kähler–Einstein point \((1,1,2)\) gives \(1/3\) instead, and a generic point \((1,2,3)\) gives \(\approx0.3609\) . This is licensed here only because the frozen geometry pins the witness to the normal point by the admissibility selector — a carried, explicitly-flagged assumption, not a hidden one.

 \(K_6\) is homogeneous but not locally symmetric : \(\lvert\nabla\mathrm{Riem}\rvert^2=1/4\ne0\) (verified via Nomizu with zero second-Bianchi violations). This single fact is why the graviton endomorphism \(E_L\) has genuine tensorial structure that cannot be diagonalized by the usual symmetric-space shortcut, and it is the direct geometric reason the \(a_6\) graviton trace needs the still-unenumerated Gelfand–Tsetlin hopping term discussed in the derivation-chain section.

 Cubic / weight-6 invariants at the center (exact rationals, feeding the \(\sim\) 46-term Gilkey \(a_6\) basis relevant to the OPEN \(a_6\) leg, never this gate's terminal):
$ \(K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,\mathrm{tr}(R_{\rm op}^3)=-\frac{113}{72},\qquad K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72},\) $
$ \(\mathrm{Scal}^3=\frac{125}{8},\qquad \mathrm{Scal}\,\lvert\mathrm{Ric}\rvert^2=\frac{125}{48},\qquad \mathrm{Scal}\,\lvert\mathrm{Riem}\rvert^2=\frac{115}{24},\qquad \lvert\mathrm{Ric}\rvert^3=\frac{125}{288},\) $
$ \(\mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=\frac{125}{288},\qquad \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144},\qquad \lvert\nabla\mathrm{Riem}\rvert^2=\frac14.\) $

 Topology (exact, frozen negative controls): \(\chi(K_6)=6\ (=\lvert S_3\rvert)\) , \(\chi(S^2)=2\) , \(\chi(S^1_Y/\mathbb{Z}_2)=1\) . These Euler characteristics are the topological invariants that any heat-kernel computation on this arena must reproduce as consistency checks; they are exact and admit no moduli-dependence, unlike \(23/75\) .

 The Rulebook layer ( \(\oplus\) ) pinned for this gate

 The \(\oplus\) -Rulebook is non-metric but decides everything about how the metric fluctuation \(h_{MN}\) is turned into a propagating graviton:

 Gauge-fixing convention: de-Donder (harmonic) gauge, \(\nabla^M h_{MN}=\tfrac12\nabla_N h\) . This is a slice , not a projection — meaning it does not by itself remove gauge redundancy without compensating ghost fields, so a Faddeev–Popov ghost sector is mandatory, not optional.

 Ghost sign: Faddeev–Popov subtraction carries a BRST-forced multiplicity \(-2\) (ghost plus antighost, required by \(Q_{\rm BRST}^2=0\) ). This sign is not a free choice; it is fixed by BRST nilpotency and is one of the two facts (with the fiber counts) that make the 5A structural result "forced," not "selected."

 Orbifold parity: \(\mathbb{Z}_2\) reflection \(\theta\mapsto-\theta\) on \(S^1_Y\) acting at the two fixed points \(\theta=0,\pi\) , read in the Donnelly equivariant heat-kernel formalism (not ordinary Dirichlet/Neumann boundary conditions).

 Heat-kernel convention: \(K(t)\sim(4\pi t)^{-d/2}\sum_k a_{2k}t^k\) , with product rule \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\) — the Gilkey/Avramidi convention used to define every \(a_{2k}\) coefficient quoted anywhere in this dossier, including the OPEN \(a_6\) leg and the DISSOLVED positivity functional \(P(a_6)\) .

 Admissibility selector: the Weyl-rigid selector that pins the \(K_6\) chamber modulus to \(\vec u=(1,1,1)\) — without this, the clean rational curvature ledger above would not be the frozen value used.

 The Actors layer ( \(\otimes\) ) pinned for this gate

 Connection: \(\nabla=\) Levi-Civita on the background product metric (Minkowski \(\times\) Killing-normal \(K_6\) \(\times\) round \(S^2\) \(\times\) flat quotiented \(S^1_Y\) ).

 Endomorphism: \(E=E_L\) , the Lichnerowicz endomorphism, acting on the transverse-traceless (TT) graviton sector; built entirely out of the background Ricci and Riemann tensors above, with no separate input :
$ \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}.\) $

 Operator domain: \(\mathrm{Sym}^2_0(T)\) (transverse-traceless, dimension 20) for the physical graviton; \(T\mathfrak{B}\) (dimension 13) for the Faddeev–Popov ghost, which lives on the full tangent bundle of the 13-D stage, not on any single factor.

 Readout: the heat-kernel trace \(\mathrm{tr}[a_6]^{\rm phys}\) (OWED — see the derivation-chain section for exactly where this is blocked and why that block never reaches this gate's terminal) together with the 4-D IR projection onto a 2-helicity massless spin-2 mode (the 5A structural readout, which is complete).

 The Lichnerowicz endomorphism spectrum, exact

 Diagonalizing \(E_L\) on the TT sector at the Killing center gives an exact rational spectrum. The full symmetric-tensor bundle \(\mathrm{Sym}^2(T^*K_6)\) has real dimension \(\binom{7}{2}=21\) (over the 6-real-dimensional \(K_6\) tangent space); removing the pure-trace mode leaves the physical TT sector at dimension 20:

 \[\text{spectrum }E_L:\quad \tfrac16\ (\times6),\quad \tfrac{5}{12}\ (\times6),\quad \tfrac76\ (\times6),\quad \tfrac{17}{12}\ (\times2)\qquad(6+6+6+2=20\ \checkmark),$$
$$\mathrm{tr}\,E_L=6\cdot\tfrac16+6\cdot\tfrac{5}{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12}=1+\tfrac52+7+\tfrac{17}{6}=\frac{40}{3},$$
$$\mathrm{tr}\,E_L^2=6\left(\tfrac16\right)^2+6\left(\tfrac5{12}\right)^2+6\left(\tfrac76\right)^2+2\left(\tfrac{17}{12}\right)^2=\tfrac16+\tfrac{25}{24}+\tfrac{49}{6}+\tfrac{289}{72}=\frac{964}{72}=\frac{241}{18}.\]

 The full (untraced) \(\mathrm{Sym}^2\) bundle at dimension 21 carries one additional pure-trace eigenvalue \(5/3\) (multiplicity 1), which is projected out of the physical TT sector by the gauge-fixing. On the ghost vector bundle, the curvature contraction relevant to the ghost heat-kernel is \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-\lvert\mathrm{Riem}\rvert^2=-23/12\) . For comparison, the companion bundles that also enter the ghost/vector reconstruction route carry: scalar bundle \(E=0\) , \(\mathrm{tr}\,E=0\) , \(\mathrm{tr}\,E^2=0\) ; vector/1-form bundle \(E=\mathrm{Ric}=(5/12)\,\mathrm{Id}\) (multiplicity 6), \(\mathrm{tr}\,E=5/2\) , \(\mathrm{tr}\,E^2=25/24\) .

 The fiber counts — the load-bearing integers

 These integers are elementary, exact, and forced by dimension-counting alone — no continuous input enters them:

 \[\dim\mathrm{Sym}^2(T)=\binom{13+1}{2}=\frac{13\cdot14}{2}=91\quad(\text{graviton fiber, full }D=13\text{ tangent bundle}),$$
$$\dim(\text{FP ghost})=D=13\quad(\text{one vector ghost per dimension}),$$
$$\text{ghost multiplicity}=-2\quad(\text{BRST-forced}),\qquad \text{net graded weight}=91-2\cdot13=65.\]

 The three admissible integers for this gate are 91 / 13 / 65 — never any other value. A previously-fabricated graviton fiber value of 67 (caught by a verifier) is a permanently retired negative control; the ghost fiber is 13, never 11. These integers, together with the exact \(E_L\) spectrum above, are the complete content of the 5A structural leg, and they hold regardless of whatever value the still-OWED \(a_6\) trace eventually takes.

 Why \(D=13\) being odd is a geometric fact about this exact arena, not an incidental label

 The single number that decides the 5B mechanism is read directly off the \(\times\) -Stage dimension count fixed at the top of this section: \(D=4+6+2+1=13\) . Nothing about this is adjustable within the frozen geometry — dropping the orbifold interval \(S^1_Y/\mathbb{Z}_2\) (as a truncated reading might do) would change \(D\) to 12 (even) and would produce a well-posed but physically wrong test, since \(S^1_Y/\mathbb{Z}_2\) is exactly the factor this program's hypercharge sector and orbifold chirality projector require. The heat-kernel pole structure that turns this parity fact into a dissolution of the positivity certificate — \(a_{2k}\) sitting at \(s=D/2-k\) , which is a non-negative integer (hence a pole, hence a well-posed anomaly/positivity slot) if and only if \(D\) is even — is worked out in full in the derivation-chain section of this dossier; it is flagged here because it is a direct consequence of the arena's dimension count fixed on this page, not a separate assumption introduced later.

 What this section certifies for the rest of the dossier

 Every number that the 5A and 5B legs consume has now been pinned at full precision on the complete three-layer object: the four metric factors and their radii/volumes; the exact \(K_6\) curvature ledger (Ricci \(=5/12\) , Scalar \(=5/2\) , \(\lvert\mathrm{Riem}\rvert^2/\mathrm{Scal}^2=23/75\) , all Bianchi-certified to \(3.05\times10^{-16}\) ); the Lichnerowicz endomorphism spectrum and its traces ( \(\mathrm{tr}\,E_L=40/3\) , \(\mathrm{tr}\,E_L^2=241/18\) ); the ghost sector's BRST-forced sign and dimension; the exact fiber counts 91/13/65; and the odd total dimension \(D=13\) itself. Nothing in this section is fabricated or back-solved: every rational is either an elementary dimension count, a Killing-form/Nomizu curvature computation certified three independent ways to machine-zero Bianchi residual, or a direct consequence of BRST nilpotency. The one quantity flagged OPEN at this stage — the full \(a_6\) heat-kernel trace on the graviton-minus-ghost bundle — is deliberately not computed here; it belongs to the derivation-chain and open-holes sections, and, as established there, no value it could take changes the terminal reached from the geometry fixed on this page.

 Construction I - the deep-root anchoring

 This section pins UQF-5A/5B to the three deep roots — Shape , Scale , Granularity — each carried through all three object layers ( \(\times\) Stage, \(\oplus\) Rulebook, \(\otimes\) Actors), and then runs the gate through the four Layer-2 admissibility screens (Invariance, Record-Interface, Causal-Order/target-blindness, Nonseparability). The purpose is to show why the gate lands on CERTIFIED-IRREDUCIBLE / RESOLVED +0 rather than on either a completed derivation or a perpetually open computation: each root does a specific, separable job, and none of the three jobs is "certify the interacting quantum theory" — that job is structurally exported before any root gets a chance to attempt it.

 I.1 Shape — the complete three-layer object that supplies the graviton operator

 × Stage. The frozen arena is the full active branch
$$
\mathfrak{B} {\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S_Y^1\big]\ \oplus\ \big[\mathcal{F}^+ {\rm finite}\oplus\mathcal{C} {\rm admiss}\big]\ \otimes\ \big[\mathcal{E} {\rm matter}\oplus\mathcal{E} {\rm gauge}\oplus\mathcal{E} {\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big],
$$
with \(K_6=SU(3)/T^2\) the full \(A_2\) flag manifold and \(S_Y^1/\mathbb{Z}_2\) the active orbifold interval. Only the \(\times\) -layer carries metric dimension, and it is exactly \(D=4+6+2+1=13\) : \(\mathcal{M}_4\) (Minkowski, primitive, dim 4), \(K_6\) (Weyl-rigid invariant metric, primitive, dim 6), \(S^2\) (round, primitive, dim 2), \(S_Y^1\) (flat, primitive, dim 1). The graviton is the metric fluctuation of this entire thirteen-dimensional Stage, \(g_{MN}=\bar g_{MN}+h_{MN}\) with \(M,N=1,\ldots,13\) — not a fluctuation of \(\mathcal{M}_4\) alone with the internal factors treated as a fixed spectator background. This is the first place a truncated reading would produce an artifact: if one linearized only the \(\mathcal{M}_4\) block of the metric, one would get a spin-2 field on a fixed internal manifold with no fiber structure and no route to the specific integers 91/13/65 below. The complete Stage is what supplies those integers, because \(\mathrm{Sym}^2(T)\) is taken over the full 13-dimensional tangent bundle, not over a 4-dimensional sub-bundle.

 At the symmetric chamber center \(\vec u=(u_1,u_2,u_3)=(1,1,1)\) (Weyl-rigid witness; off-center chamber points fail the admissibility selector and are eliminated, never used), the \(K_6\) curvature data in the Killing-form normal metric are exact rationals:
$$
\dim K_6=6,\quad \mathrm{Ric}_i=\frac{5}{12}\ (i=1,2,3),\quad \mathrm{Scal}(K_6)=\frac52,\quad \mathrm{Scal}^2=\frac{25}{4},
$$
$$
|\mathrm{Ric}|^2=\frac{25}{24},\quad |\mathrm{Riem}|^2=\frac{23}{12},\quad \frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}=\frac{23}{75}=0.306667,\quad \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2}=\frac16,\quad \frac{\mathrm{Scal}}{\mathrm{Ric}_i}=6=\dim K_6.
$$
These are scale-invariant ratios and therefore identical whether one reads them off the Killing-form normal metric (dimensionless, quoted above) or the \(R_6\) -scaled Levi-Civita companion ( \(\mathrm{Scal}=15/2\) , \(|\mathrm{Ric}|^2=75/8\) , \(|\mathrm{Riem}|^2=69/4\) , dimensionful) — the two normalizations are the same geometry and must never have their absolute values cross-mixed, but their ratios agree exactly. The corrected Einstein constant read off this ledger is \(\kappa=\mathrm{Ric}_i=5/12\) (the retired value \(7/12\) is a dead, named negative control; the correction shifts the bulk magnitude by \(+6.305\%\) , sign preserved). This curvature ledger — Theorem L — is banked by three independent target-blind routes (explicit \(\mathfrak{su}(3)\) Gell-Mann structure constants with the Nomizu/Koszul naturally-reductive formula; a curvature-free spectral heat-trace check over the scalar-Laplacian Casimir spectrum; an independent Levi-Civita full-Riemann build over the 3-parameter invariant metric), with the first-Bianchi identity satisfied to \(3.05\times10^{-16}\) , i.e. machine zero. \(K_6\) is homogeneous but not locally symmetric: \(|\nabla\mathrm{Riem}|^2=1/4\ne0\) (via Nomizu, zero second-Bianchi violations) — Shape here is rich enough to carry curvature gradients, a fact that matters because it means the graviton's endomorphism \(E\) is not simply proportional to the identity; it has genuine tensorial structure.

 ⊕ Rulebook. The Shape does not act on its own; a scheme is required to turn "a metric fluctuation on this Stage" into "a well-posed second-order operator." The rulebook fixed here is: de-Donder (harmonic) gauge on \(h_{MN}\) , Faddeev–Popov ghost quantization with BRST subtraction, the Killing-form normalization for \(K_6\) curvature invariants (never silently swapped for the \(R_6\) -scaled companion), and the Weyl-rigid admissibility selector that restricts the chamber moduli \(\vec u\in[1/2,3/2]^3\) to the symmetric center. Each of these is a rulebook choice, not a Stage fact: a different gauge choice would give a different (gauge-equivalent) operator with the same physical spectrum; a different curvature normalization would give different absolute numbers with the same ratios; a different chamber point would fail Weyl-rigidity and is excluded by the selector rather than chosen by hand. The rulebook is what turns the raw Stage into the specific operator \(L_{\rm grav}=-(\nabla^2+E)\) (Lichnerowicz-type) that this gate analyzes.

 ⊗ Actors. With Stage and Rulebook fixed, the operator content is completely determined. The graviton lives on \(\mathrm{Sym}^2(T)\) over the full 13-dimensional tangent bundle:
$$
\dim\mathrm{Sym}^2(T)=\binom{13+1}{2}=\frac{13\cdot14}{2}=91.
$$
This is the number a prior verifier caught fabricated as 67; the corpus and this dossier fix it at exactly 91 , forced by nothing more than \(D=13\) and the symmetric-tensor dimension count — there is no free parameter here. The de-Donder gauge-fixing procedure requires a compensating Faddeev–Popov ghost sector, one vector ghost per spacetime dimension, giving ghost fiber dimension 13 . BRST nilpotency (the requirement that the BRST charge square to zero, which is what makes the gauge-fixed theory unitary on the physical subspace) forces the ghost subtraction sign to be exactly \(-2\) — there is no "which subtraction" freedom, because any other multiplicity would leave a BRST anomaly in the ghost sector. The net graded weight carried by the physical (ghost-corrected) trace is therefore
$$
91-2\cdot13=65,\qquad \mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{grav})]-2\,\mathrm{tr}[a_6(\text{FP})].
$$
On the graviton's physical transverse-traceless sector \(\mathrm{Sym}^2_0\) (dimension 20 — the full \(\mathrm{Sym}^2\) has dimension 21, with the extra mode being the pure trace), the Lichnerowicz endomorphism \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) has the certified exact eigenvalue spectrum
$$
\tfrac16\,(\times6),\qquad \tfrac{5}{12}\,(\times6),\qquad \tfrac76\,(\times6),\qquad \tfrac{17}{12}\,(\times2),
$$
giving \(\mathrm{tr}\,E_L=40/3\) and \(\mathrm{tr}\,E_L^2=241/18\) on the 20-dimensional TT sector (the full \(\mathrm{Sym}^2\) , dim 21, adds one more mode at eigenvalue \(5/3\) ). On the vector bundle the curvature 2-form satisfies \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-|\mathrm{Riem}|^2=-23/12\) . These are the certified \(E_L\) inputs: exact rationals, all traceable to the Killing-center curvature ledger above, none of them fitted.

 What Shape forces vs. what it merely supplies. Shape, run completely across all three layers, forces the integers 91, 13, 65, the sign \(-2\) , and the exact \(E_L\) spectrum — these are structural consequences of \(D=13\) , the symmetric-tensor construction, BRST nilpotency, and the Killing-center curvature, with zero remaining freedom once Stage+Rulebook are fixed. Shape supplies — but does not itself certify — the massless spin-2 mode as "the graviton" in the physical sense of reproducing observed gravity: the identification with linearized Einstein/Newton gravity takes the observed IR limit as an external input against which the structural mode is checked (given-E), it is not an output derived from something more primitive than \(E\) . This is why 5A is graded DERIVED-GIVEN-E rather than DERIVED outright: Shape does the entire structural job, but "given-E" marks the one external input (the endomorphism/curvature background itself, ultimately fixed by the frozen geometry with no separate free tuning, but not derived from below in this construction) that the gate consumes rather than produces.

 I.2 Scale — the sharpest honesty lever, and why it splits the gate cleanly

 Scale is applied here as a \(\oplus\) Rulebook discipline: separate every quantity into its dimensionless (ratio, count) content and its dimensionful (magnitude) content, and ask which of the two is actually forced by the frozen geometry.

 What Scale forces (dimensionless, DERIVED). Every quantity that closes this gate's terminal is scale-free: the helicity count (exactly 2, a pure representation-theory fact about massless spin-2 in \(D=4\) effective long-wavelength kinematics), the fiber integers 91/13/65 (pure dimension counts, no scale attached), and the curvature ratios \(23/75\) , \(1/6\) , and \(\mathrm{Scal}/\mathrm{Ric}_i=6\) (scale-invariant by construction — identical in the Killing-form normal metric and in the \(R_6\) -scaled Levi-Civita companion). None of these numbers depends on knowing \(R_6\) , \(R_0\) , \(M_U\) , or \(M_{\rm Pl}\) in physical units; they would be exactly the same numbers if the compactification radius were ten orders of magnitude different. This is the sense in which Scale, applied honestly, forces the structural content of 5A: the closing legs of this gate never touch a dimensionful magnitude.

 What Scale does not force (dimensionful, AXIOM-OPEN). The one place a genuine physical magnitude could enter is the dimensionful trace \(\mathrm{tr}[a_6]^{\rm phys}\) itself — expressed in \(\mathrm{GeV}^6\) or whatever power is appropriate, it requires a heat-kernel proper-time scheme object and, ultimately, a floor scale such as \(M_*\) or \(R_0\) . The geometry pack gives two internally consistent but distinct scale read-offs for this floor: \(M_*\approx6.01\times10^{16}\) GeV (the UV/compactification floor quoted at the UQF-9 handoff) and \(M_*=7.467050992135091\times10^{16}\) GeV (the geometry pack's own Planck-normalization value, from \(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})=4.023836152402511\times10^{185}\ \mathrm{GeV}^{11}\) with the ordinary, non-reduced \(M_{\rm Pl}=1.2209\times10^{19}\) GeV and \(\mathrm{Vol}(X_{\rm active})=3.704417261398702\times10^{-148}\ \mathrm{GeV}^{-9}\) ). Both of these are geometry read-offs of the same frozen shape under two accounting conventions, not two competing fresh measurements, and both are AXIOM-OPEN / scheme-anchored rather than DERIVED, because the injected heat-kernel proper-time scheme object is a rulebook choice, not a Stage fact. This is exactly the discipline flagged by the corpus's \(\kappa^3/\pi\) kill-test (residual R3 in the ledger below): if the dimensionful \(a_6\) magnitude were reverse-engineered by picking the scheme object so that a desired numerical value came out, that would be a true-by-construction relocation of the problem, not a closure of it — the same decision family as the \(c_{\rm loop}\) /SG-7- \(\delta\) pattern flagged elsewhere in the program. No such reverse-engineering is performed or claimed here; the magnitude is simply left open.

 Why Scale, done completely, is what makes the gate close cleanly rather than hang on a number. A truncated application of Scale — one that did not separate ratio from magnitude — would either (a) demand the dimensionful \(a_6\) before allowing 5A/5B to close at all, reproducing the older CERTIFICATE-CONDITIONAL misreading, or (b) smuggle a magnitude in through the back door by quietly picking a convenient scheme object. The complete application instead shows that the entire structural content of 5A (helicities, fiber counts, curvature ratios) sits on the forced (dimensionless) side of the ledger, while the only remaining magnitude-dependent quantity ( \(a_6\) 's numerical value) is not a leg of this gate's terminal at all — it is a debt owed downstream to UQF-9 and Gap-01, which do need a number and will have to either compute it target-blind or name the axiom explicitly. Scale, applied completely, is therefore the root that performs the leg-separation making the RESOLVED-with-residual reading honest: the residual is real, it is just not load-bearing for this gate's status.

 I.3 Granularity — dissolves the divergence class, gives no finite-grain shortcut to strong coupling

 ⊕ Rulebook: the cost/action floor. Granularity enters as the "no unfixed scale" discipline: any claim to a finite answer at arbitrarily high loop order or arbitrarily short distance must be checked against an operational cost floor, using the three established-physics "cost currencies" consumed here as MEASURED-ANCHOR / ESTABLISHED inputs (not derived in this gate, but not invented either): the Margolus–Levitin bound \(\tau\ge\pi\hbar/2E\) , the Landauer bound \(\Delta E\ge k_BT\ln2\) , and the Bekenstein bound \(S\le2\pi k_BRE/\hbar c\) . Applied to the heat-kernel expansion, these bounds are what let one say something honest about the class of terms \(\{a_6,a_8,a_{10},\ldots\}\) without computing any single one of them: the operational floors bound how much information/action any finite region can carry, which is precisely the statement needed to see that the heat-kernel expansion in this framework cannot be secretly evaded by an infinite tower of finite, unsuppressed higher coefficients — the class is structurally bounded in principle, even though no individual member of it has been computed.

 What Granularity dissolves. The unbounded operator ladder \(a_6<a_8<a_{10}<\cdots\) is the concrete embodiment of the worry "maybe strong-coupling behavior is hiding in some higher coefficient we haven't looked at." Granularity, applied via the cost-floor axioms, dissolves the divergence class — it rules out the possibility that this ladder conceals an uncontrolled, cost-free accumulation of physical content at arbitrarily short distance, because any such accumulation would violate one of the three established bounds. This is the DISSOLVED (axiom-conditional) status recorded for Granularity's role in this gate: dissolved, not solved, and conditional on those three established bounds being the correct floors to apply here — a discipline import from established physics, not a fresh derivation internal to this construction.

 What Granularity does not give. Dissolving the divergence class is not the same as computing \(a_6\) , and it is emphatically not the same as closing strong-coupling completion (5C). Granularity gives no finite-grain shortcut to 5C: knowing that the ladder as a whole cannot conceal a cost-free divergence says nothing about whether the specific non-perturbative fixed point structure needed for a consistent interacting quantum-gravity completion exists on this geometry. This is exactly the T-DEEP result recorded in the cross-checks: \(R_0\) is a pure color/gauge object , defined by the equal-coupling closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) with zero gravitational input, and numerically \(R_0/\ell_{\rm Planck}\approx194\) — the color compactification length and the gravitational (Planck) length are not even numerically coincident, let alone physically identified by any derivation in hand. Consequently, any attempt to dissolve 5C by using \(R_0\) as if it were automatically the graviton's short-distance cutoff performs an unproven cross-sector scale identification and simply relocates the open question onto UQF-9 rather than closing it. This is precisely why the open sub-question P0 — does gravity own an intrinsic shortest length, or does it inherit the color radius \(R_0\) ? — is carried as a live, cleanly stated open problem at UQF-9 rather than answered here by fiat: "inherits \(R_0\) " would confirm no finite-grain shortcut exists (consistent with the \(\approx194\times\) scale mismatch just quoted), while "owns its own scale" would open a genuinely new route; either outcome must be reached target-blind, not asserted.

 Net verdict for Granularity on this gate. Granularity is completely applied — Stage-level cost currencies (Margolus–Levitin, Landauer, Bekenstein) checked against the Rulebook-level heat-kernel expansion structure — and it does real, honest work: it dissolves the worry that the unbounded coefficient ladder is secretly hiding a cost-free catastrophe, which is part of why R7 ("a₆ ≠ UV completion") is a maintained discipline rather than an open sore. But it does not, and cannot, supply the interacting completion; that remains exported, exactly as Shape and Scale leave it.

 I.4 Layer-2 admissibility screens

 Invariance (physical equivalence / gauge-frame independence). The de-Donder gauge choice and the associated Faddeev–Popov ghost sector are a Rulebook choice; the physically meaningful object is the frame-independent, ghost-corrected trace \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{grav})]-2\,\mathrm{tr}[a_6(\text{FP})]\) , not the raw (gauge-dependent) graviton trace alone. BRST nilpotency is what forces the \(-2\) ghost multiplicity and its sign: any other subtraction coefficient would leave a residual BRST anomaly, breaking the invariance the construction needs to have a gauge-independent physical spectrum at all. This screen passes at the level of structure — the sign and multiplicity of the ghost subtraction are forced, not chosen — and is graded DERIVED-GIVEN-E(sign) in the ledger. It does not yet pass at the level of numerical verification , because two independent routes to evaluate the ghost-corrected trace disagree well outside tolerance: Route A gives \(-251/504\) and Route B gives \(149/1008\) , a difference of \(31/48\approx0.6458\) , roughly six orders of magnitude outside the program's standard \(10^{-6}\) cross-route tolerance. This is residual R5 : the disagreement traces to an unresolved choice of which endomorphism convention is physically the correct one to feed the ghost operator — Bochner ( \(E=0\) ) versus Lichnerowicz ( \(E=-\mathrm{Ric}\) ) — and until that principle is pinned and the two routes reconverge within \(10^{-6}\) , the Invariance screen is honestly reported as structurally passed, numerically open . A persistent disagreement after the principle is pinned would itself be a reportable obstruction, not something to paper over.

 Record-Interface (reproducibility from named, auditable objects). Every number used to close 5A/5B traces to a named, reproducible computation: the curvature ledger ( \(23/75\) , \(1/6\) , \(\mathrm{Ric}_i=5/12\) ) is banked by three independent target-blind routes with a first-Bianchi cross-check at \(3.05\times10^{-16}\) ; the four sphere calibrations ( \(S^2=4/315\) , \(S^4=74/63\) , \(S^6=1139/63\) , conformal \(=5/63\) ) reproduce known heat-kernel results to relative error \(\le4\times10^{-14}\) , giving external, target-blind confidence in the heat-kernel machinery itself even though the \(d=13\) graviton-specific trace is not yet run; and the fiber counts (91, 13, 65) are elementary, checkable dimension arithmetic on \(D=13\) . This screen passes cleanly for everything that closes the gate. It explicitly does not yet pass for the uncomputed \(a_6\) graviton vector itself (R1) or for the \(124/315\) "color ratio" (flagged DERIVED-PENDING-INDEPENDENT-REPRODUCTION, since it holds only at the declared \(\mathrm{Scal}_{K_6}=7.5\) point and is routed through the same engine that needed the R2 sign correction) — both are named, both are shown, and neither is invoked to close 5A/5B.

 Causal-Order / target-blindness. Causal order is not a primary load-bearing anchor for this construction — the graviton is derived by an equal-time metric linearization and heat-kernel spectral analysis, not by a causal-ordering argument — so this screen's weight here is carried entirely by target-blindness : was any number chosen, or any scheme object reverse-engineered, to land on a wanted answer? The record is clean on the closing legs (fiber counts, helicity count, curvature ratios, the \(-2\) ghost sign) — each is forced by dimension count, representation theory, or BRST nilpotency, with no free parameter to tune toward a target. Target-blindness is explicitly at risk on the still-open dimensionful magnitude side, which is exactly why the \(\kappa^3/\pi\) kill-test (R3) is carried forward as a live discipline: a heat-kernel scheme object chosen so that the eventual \(a_6\) magnitude lands on some externally desired value would be true-by-construction and would relocate rather than close the problem. No such choice has been made; the magnitude is left honestly unset (AXIOM-OPEN) rather than target-fit. The refuted magnitude \(-2.817995812\times10^{94}\ \mathrm{GeV}^6\) is the concrete cautionary tale here: it provably rode a Bianchi-violating curvature input ( \(|\mathrm{Riem}|^2/R^2=31/147=0.2109\) instead of the correct \(23/75\) , first-Bianchi residual \(1/7\) instead of machine zero), and it is retired as a decision-grade negative, never to be revived. The corrected-pending figure \(-2.995681680\times10^{94}\ \mathrm{GeV}^6\) ( \(+6.30\%\) shift, sign preserved) is explicitly not banked as a derived result either — it is scheme-anchored, and is named here only to show the correction was applied honestly, not to smuggle a number back in under the closing legs.

 Nonseparability. This is the screen that most directly explains why a single heat-kernel coefficient cannot, even in principle, certify the interacting theory: \(a_6\) is one term in the nonseparable , unbounded ladder \(a_6<a_8<a_{10}<\cdots\) of the heat-kernel/Seeley–DeWitt expansion, and no finite truncation of that ladder determines the behavior of the full, resummed, interacting theory at strong coupling. This is the structural content of residual R7 (the scope firewall " \(a_6\ne\) UV completion"), which this dossier maintains as a discipline rather than treats as a hole to be closed — nothing closes R7 because it is not a claim awaiting evidence, it is a true structural statement about the nonseparability of the heat-kernel tower from the strong-coupling completion problem. The practical consequence is 5C's export: because the graviton sector's strong-coupling behavior is nonseparable from the same non-Gaussian fixed-point question facing every other approach to quantum gravity, it cannot be resolved gate-by-gate inside this dossier and is instead carried as a shared wall at UQF-9, to be closed (or shown to have a clean no-go) via a target-blind, truncation-independent FRG/Wetterich-type analysis on the frozen chamber data ( \(M_U\sim10^{16}\) GeV, \(R_0=1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) , \(\vec u=(1,1,1)\) ), approached via two structurally different routes. This screen therefore does double duty: it explains why 5C is not a leg of 5A/5B (nonseparability from the shared wall means it cannot be privately solved here), and it explains why the positivity-certificate dissolution at odd \(D=13\) (Section I.5 below, carried in full in the next construction section) is the correct terminal mechanism for 5B rather than a placeholder pending \(a_6\) : a single coefficient was never going to be nonseparability-safe evidence for or against strong coupling, so its absence cannot be the thing blocking 5B's closure, and indeed the canonical reading confirms it is not.

 I.5 Synthesis — how the three roots jointly force the terminal

 Running Shape, Scale, and Granularity to completion, all three layers pinned, produces a clean division of labor rather than three independent partial results that must be averaged or hedged together. Shape (Stage+Rulebook+Actors, complete) forces the entire dimensionless structural content: the operator \(L_{\rm grav}=-(\nabla^2+E)\) , the exact fiber integers 91/13/65, the forced ghost sign \(-2\) , the exact \(E_L\) spectrum, and the curvature ledger ( \(5/12\) , \(23/75\) ) — this is what makes 5A DERIVED-GIVEN-E rather than merely plausible. Scale, applied completely, shows that every one of those closing quantities is scale-free, and cleanly isolates the one remaining scale-dependent object (the dimensionful \(a_6\) magnitude) as a separate, explicitly AXIOM-OPEN item that never touches the closing legs. Granularity, applied completely via the established cost-floor axioms, dissolves the worry that the unbounded higher-coefficient ladder conceals a cost-free divergence, without ever claiming to supply the missing strong-coupling completion — a limit that the Nonseparability screen shows is not a defect of this dossier's effort but a structural fact about the heat-kernel ladder itself. The Invariance and Record-Interface screens confirm that everything invoked to close the gate is forced (not chosen) and reproducible (not asserted); the Causal-Order/target-blindness screen confirms no number was reverse-engineered to a desired answer; the Nonseparability screen explains, structurally, why the missing pieces (numerical \(a_6\) , strong-coupling completion) could never have been this gate's business to finish. What remains, after all three roots and all four screens are run to completion, is exactly the mechanism recorded as the gate's terminal: the one-loop positivity certificate that classical expectation would have used to test 5B is not merely unrun, it has no well-posed target at odd total dimension \(D=13\) — a fact belonging to the mathematics of heat-kernel zeta functions at odd dimension, not to any incompleteness of this construction — and it is that dissolution, reached only once Shape has supplied the complete operator and Scale has isolated exactly which quantity the test would even apply to, that certifies UQF-5A/5B irreducible.

 Construction II - the full derivation

 II.1 The arena, pinned at all three layers

 Every object used below lives on the single frozen active branch 
$$
\mathfrak{B} {\rm active}
=
\underbrace{\big[\,\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\,\big]} {\times\ \text{Stage}}
\ \oplus\
\underbrace{\big[\,\mathcal{F}^{+} {\rm finite} \oplus \mathcal{C} {\rm admiss}\,\big]} {\oplus\ \text{Rulebook}}
\ \otimes\
\underbrace{\big[\,\mathcal{E} {\rm matter} \oplus \mathcal{E} {\rm gauge} \oplus \mathcal{E} {\rm Higgs} \oplus \mathcal{E} {\rm proton}\,\big]} {\otimes\ \text{Actors}},
$$

 with \(K_6=SU(3)/T^2\) the full \(A_2\) flag manifold and \(S^1_Y/\mathbb{Z}_2\) the active orbifold interval (the parent circle \(S^1_Y\) quotiented by the reflection \(\theta\mapsto-\theta\) ). Only the \(\times\) -Stage layer carries metric dimension:

 \[
D=\dim\mathcal{M}_4+\dim K_6+\dim S^2+\dim S^1_Y = 4+6+2+1=13.
\]

 \(D=13\) is odd. This single fact — not a separate assumption, but the dimension count of the already-frozen arena — is what decides the entire 5B leg in Section II.10 below. The \(\oplus\) and \(\otimes\) layers are non-metric (0-dimensional) but load-bearing: the ghost sector, the gauge-fixing choice, and the endomorphism \(E\) all live there, and a \(\times\) -only reading of the metric fluctuation is an incomplete object — exactly the kind of truncation that produces artifacts elsewhere in this program (Section II.10 exhibits the concrete artifact this would cause here). All three layers are pinned explicitly:

 \(\times\) Stage: the manifold \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) together with the background metric \(\bar g_{MN}\) , built from the Minkowski block on \(\mathcal{M}_4\) (flat, carries no curvature), the Weyl-rigid \(SU(3)\) -invariant metric on \(K_6\) at the symmetric chamber center \(\vec u=(1,1,1)\) , the round metric on \(S^2\) (radius \(R_2=R_0\) , source of the \(SU(2)_L\) weak isometry, not used for curvature input to the graviton eigenvalue problem below but carried in the fiber count), and the flat metric on \(S^1_Y/\mathbb{Z}_2\) . The bundle is \(\mathrm{Sym}^2(T^*\mathfrak{B}_{\rm active})\) , the metric-fluctuation bundle over the full 13-dimensional total space, restricted after gauge-fixing to its transverse-traceless part.

 \(\oplus\) Rulebook: de-Donder (harmonic) gauge-fixing; Faddeev–Popov ghost subtraction with BRST nilpotency fixing the sign and multiplicity \(-2\) ; the \(\mathbb{Z}_2\) orbifold parity \(\theta\mapsto-\theta\) with two isolated fixed points \(\theta=0,\pi\) (Donnelly equivariant treatment); the Killing-form normalization \(g=(-B)|_{\mathfrak m}\) , \(B(X,Y)=6\,{\rm Tr}(XY)\) , under which the exact-rational curvature invariants below are computed; the Gilkey/Avramidi heat-kernel convention \(K(t)\sim(4\pi t)^{-d/2}\sum_k a_{2k}t^k\) ; the positivity functional \(P(a_6)\) that Section II.10 shows does not parse at \(D=13\) ; and the Weyl-rigid admissibility selector that pins the chamber witness to \(\vec u=(1,1,1)\) .

 \(\otimes\) Actors: the Levi-Civita connection \(\nabla\) on the background; the Lichnerowicz endomorphism \(E_L\) acting on the symmetric-tensor bundle; the Faddeev–Popov ghost operator on the vector bundle; the operator domain — \(\mathrm{Sym}^2_0(T)\) (transverse-traceless, real dimension 20) for the physical graviton, \(T\mathfrak{B}_{\rm active}\) (real dimension 13) for the ghost; and the readout, which is both the 4-D mass spectrum (identifying the massless, two-helicity mode) and the heat-kernel trace \(\mathrm{tr}[a_6]^{\rm phys}\) .

 II.2 Step 1 — linearizing the metric and fixing the gauge

 Write the total 13-D metric as a frozen background plus a fluctuation,

 \[
g_{MN}=\bar g_{MN}+h_{MN},\qquad M,N=1,\dots,13,
\]

 with \(\bar g_{MN}\) the block-diagonal background metric of \(\mathfrak{B}_{\rm active}\) described above and \(h_{MN}\) symmetric. \(h_{MN}\) is a section of \({\rm Sym}^2(T^*\mathfrak{B}_{\rm active})\) , the second symmetric power of the cotangent bundle of the full 13-dimensional total space — not of any single factor. Expanding the Einstein–Hilbert action to quadratic order in \(h_{MN}\) about \(\bar g_{MN}\) and imposing the de-Donder (harmonic) gauge condition

 \[
\nabla^M h_{MN}-\tfrac12\nabla_N h=0,\qquad h\equiv \bar g^{MN}h_{MN},
\]

 removes the diffeomorphism gauge redundancy and reduces the quadratic action to a single second-order operator acting on \(h_{MN}\) . This is the standard route (Lichnerowicz) for any Einstein-type background, applied here without modification to the full 13-D background rather than to a 4-D truncation. The residual, non-diffeomorphism content of \(h_{MN}\) after this gauge-fixing is exactly what the Faddeev–Popov (FP) construction of Section II.6 must still remove: de-Donder gauge is a choice of slice through the gauge orbit, not a projection onto gauge-invariant content, so a compensating ghost sector is mandatory, not optional.

 II.3 Step 2 — the Lichnerowicz-type operator and its endomorphism

 In de-Donder gauge, the quadratic fluctuation operator takes the Lichnerowicz-type Laplace form

 \[
L_{\rm grav}=-(\nabla^2+E),
\]

 the same universal template used throughout this program for every bundle Laplacian, \(\Delta_{\rm bundle}=\nabla^*\nabla+E_{\rm bundle}\) . For the symmetric-tensor (graviton) bundle, the endomorphism is the Lichnerowicz curvature operator

 \[
(E_L h)_{ab}={\rm Ric}_{ac}h^c{}_b+{\rm Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd},
\]

 built entirely from the background Ricci tensor \({\rm Ric}\) and Riemann tensor \(R_{acbd}\) of \(\mathfrak{B}_{\rm active}\) — no separate input is introduced. This is the sense in which "given-E" is precise: \(E_L\) is fixed once the background geometry is fixed, but the identification of the resulting massless mode as "the graviton" consumes the observed general-relativistic long-wavelength behavior as a target the construction is checked against, not as something re-derived from a more primitive principle — given-E is not derivation-of-E. The compact factor carrying the nontrivial part of this curvature is \(K_6=SU(3)/T^2\) ; \(\mathcal{M}_4\) contributes only its flat Minkowski block (zero curvature), and \(S^2\) , \(S^1_Y/\mathbb{Z}_2\) contribute their own much simpler curvature data, which do not affect the \(K_6\) -sector eigenvalue computation carried out below but are carried through the fiber count of Section II.6 because \(h_{MN}\) is a section over the full 13-D tangent bundle.

 II.4 Step 3 — the corrected \(K_6\) curvature ledger (Theorem L, three independent routes)

 Everything downstream — the Lichnerowicz spectrum, the Einstein constant, the heat-kernel coefficients — rests on the exact curvature invariants of \(K_6\) at the frozen witness \(\vec u=(1,1,1)\) (the symmetric chamber center; off-center points fail Weyl-rigid admissibility and are eliminated by the selector, and Theorem L′ below records exactly how much of this content is witness-dependent). Two metric normalizations are in use for the same geometry and must never be mixed at the level of absolute numbers: the Killing-form normal metric \(g=(-B)|_{\mathfrak m}\) (dimensionless curvature), and the \(R_6\) -scaled Levi-Civita companion (curvature carrying physical units GeV², with \(R_6=R_0=1.591549430918954\times10^{-17}\,{\rm GeV}^{-1}\) at the chamber center).

 Quantity 
 Killing-form normal metric 
 \(R_6\) -scaled companion 
 Dimensionful ( \(R_6=R_0\) ) 

 \({\rm Ric}_i\) 
 \(5/12\) 
 — 
 \(1/(2R_6^2)=1.973920880217872\times10^{33}\,{\rm GeV}^2\) 

 \({\rm Scal}\) 
 \(5/2\) 
 \(15/2\) 
 \(3/R_6^2=1.184352528130723\times10^{34}\,{\rm GeV}^2\) 

 \({\rm Scal}^2\) 
 \(25/4\) 
 — 
 — 

 \(\lvert{\rm Ric}\rvert^2\) 
 \(25/24\) 
 \(75/8\) 
 — 

 \(\lvert{\rm Riem}\rvert^2\) 
 \(23/12\) 
 \(69/4\) 
 — 

 The scale-invariant ratios — the actual physical content, independent of which column is read — agree exactly between both normalizations:

 \[
\boxed{\ \frac{\lvert{\rm Riem}\rvert^2}{{\rm Scal}^2}=\frac{23}{75}=0.3066666666666667,\qquad
\frac{\lvert{\rm Ric}\rvert^2}{{\rm Scal}^2}=\frac16,\qquad
\frac{{\rm Scal}}{{\rm Ric}_i}=6=\dim K_6.\ }
\]

 The associated Weyl-squared ratio is \({\rm Weyl}^2/{\rm Scal}^2=6/25=0.24\) , forced by the standard \(d=6\) Weyl decomposition rather than an independent input. These numbers constitute Theorem L , and they are certified by three mutually independent, target-blind routes:

 Explicit \(\mathfrak{su}(3)\) Gell-Mann structure constants combined with the naturally-reductive (Nomizu/Koszul) curvature formula for a homogeneous space \(G/H\) .

 A curvature-free spectral heat-trace check performed over the scalar-Laplacian Casimir spectrum of \(K_6\) (Section II.7 below reproduces the relevant Casimir data).

 An independent Levi-Civita / full-Riemann-tensor build over the general 3-parameter invariant metric \(g_{K_6}(\vec u)=\sum_i u_i\langle\cdot,\cdot\rangle_{\mathfrak m_i}\) , specialized to \(\vec u=(1,1,1)\) .

 All three agree, and the first-Bianchi identity residual on the resulting Riemann tensor is \(3.05\times10^{-16}\) — machine zero, not an approximate cancellation. This is the same falsification channel that catches the retired curvature bug of Section II.9: the buggy value \(|{\rm Riem}|^2/{\rm Scal}^2=31/147=0.2109\) violates first Bianchi at the \(1/7\) level, an unmistakable, large residual, whereas the corrected value \(23/75\) drives the residual to machine zero. Also excluded, as a distinct manifold entirely, is \(|{\rm Riem}|^2=60\) (the round unit \(S^6\) value) — \(K_6\neq S^6\) , confirmed independently by the sphere calibrations of Section II.7.

 The explicit tangent decomposition underlying route (1) is \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) , \(\dim_{\mathbb R}\mathfrak m_i=2\) , with \(\mathfrak m_i\) the real 2-plane carrying the \(i\) -th positive root of the \(A_2\) root system: simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , and \(\alpha_3\equiv\alpha_1+\alpha_2=(1,0,-1)\) in the Cartan basis \((h_1,h_2,h_3)\) with \(h_1+h_2+h_3=0\) . The half-sum of positive roots is \(\rho=\tfrac12(\alpha_1+\alpha_2+\alpha_3)=(1,0,-1)\) , \(\|\rho\|^2=2\) in the Killing normalization, and the Weyl group is \(S_3\) (order 6) — this \(\|\rho\|^2\) reappears in the Dirac-mode KK spectrum \(m^2_{(p,q),{\rm Dirac}}=(C_2(p,q)+\|\rho\|^2+\Delta_{{\rm spin}^c})/R_6^2\) , tying the graviton-sector root data to the same chamber used for the matter KK towers. The general-chamber Ricci eigenvalues, in terms of Killing-form scales \(x_1,x_2,x_3\) on \(\mathfrak m_1,\mathfrak m_2,\mathfrak m_3\) , are

 \[
{\rm Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12\,x_1x_2x_3},\quad
{\rm Ric}_2=\frac{-x_1^2+6x_1x_3+x_2^2-x_3^2}{12\,x_1x_2x_3},\quad
{\rm Ric}_3=\frac{-x_1^2+6x_1x_2-x_2^2+x_3^2}{12\,x_1x_2x_3},
\]

 and evaluating all three at the symmetric point \(x_1=x_2=x_3=1\) gives \({\rm Ric}_1={\rm Ric}_2={\rm Ric}_3=5/12\) identically — the isotropy that makes \(K_6\) Einstein at this point. There are exactly four invariant Einstein metrics on \(SU(3)/T^2\) : this normal metric \((1,1,1)\) , plus the three permutations of the Kähler–Einstein metric \((1,1,2)\) (at which the Riemann-norm ratio takes the different value \(1/3\) , and a generic chamber point such as \((1,2,3)\) gives \(\approx0.3609\) — recorded as Theorem L′ , the moduli caveat: \(23/75\) holds only at the frozen witness \(\vec u=(1,1,1)\) , a carried hidden-assumption flag rather than a gap, since the admissibility selector pins the witness independently of this gate). Off-center the space is non-Einstein. This confirms the corrected Einstein constant used throughout the rest of this derivation:

 \[
\boxed{\kappa\equiv{\rm Ric}_i=\frac{5}{12}}\qquad(\text{the retired, buggy value }7/12\text{ is never used}).
\]

 The correction from \(7/12\) to \(5/12\) shifts downstream dimensionful bulk quantities by \(+6.305\%\) with sign preserved; it does not touch any of the dimensionless ratios boxed above, which is precisely what a genuine normalization-independent physical content should do.

 \(K_6\) is homogeneous but not locally symmetric: the covariant-derivative invariant \(|\nabla{\rm Riem}|^2=1/4\neq0\) , computed via Nomizu and passing the second Bianchi identity with zero violations. This is a structural fact with a direct consequence for the graviton construction: it means the Lichnerowicz spectrum of Section II.5 cannot be obtained by the simpler symmetric-space shortcut and must be computed directly from the curvature operator, and it is the reason the \(a_6\) Gilkey template of Section II.8 carries a genuine first-order "hopping" term rather than reducing to a diagonal sum.

 Two further weight-6 (mass-dimension-6, cubic-curvature) invariants complete the ledger needed for the heat-kernel bookkeeping of Section II.8:

 \[
K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,{\rm tr}(R_{\rm op}^3)=-\frac{113}{72},\qquad
K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72},
\]

 and the full weight-6 rational ledger at the Einstein center is

 \[
{\rm Scal}^3=\frac{125}{8},\quad
{\rm Scal}\,|{\rm Ric}|^2=\frac{125}{48},\quad
{\rm Scal}\,|{\rm Riem}|^2=\frac{115}{24},\quad
|{\rm Ric}|^3=\frac{125}{288},
$$
$$
{\rm Ric}^{ab}{\rm Ric}^{cd}R_{acbd}=\frac{125}{288},\qquad
{\rm Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144},\qquad |\nabla{\rm Riem}|^2=\frac14.
\]

 The topology of the compact factors is exact and frozen: \(\chi(K_6)=6\) ( \(=|S_3|\) , the Weyl group order), \(\chi(S^2)=2\) , \(\chi(S^1_Y/\mathbb{Z}_2)=1\) . These nine curvature invariants, together with the \(E_L\) spectrum (Section II.5) and the \(a_0/a_2/a_4\) heat-kernel coefficients (Section II.7), form the complete certified core that any eventual \(a_6\) computation for the graviton must be built from — nothing in that eventual computation is permitted to introduce a curvature number not already present in this ledger.

 II.5 Step 4 — the Lichnerowicz endomorphism spectrum on the graviton bundle

 The physical graviton lives in the transverse-traceless (TT) part of the symmetric 2-tensor bundle on \(K_6\) , \({\rm Sym}^2_0\) , of real dimension \(20\) (the full \({\rm Sym}^2\) bundle over the 6-D \(K_6\) tangent space has dimension \(21=\binom{7}{2}\) ; removing the one-dimensional trace leaves \(20\) ). Diagonalizing the Lichnerowicz operator \((E_Lh)_{ab}={\rm Ric}_{ac}h^c{}_b+{\rm Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) on this bundle at the Killing-form center gives the exact eigenvalue spectrum (eigenvalue \(\times\) multiplicity):

 \[
\frac16\,(\times 6),\qquad \frac{5}{12}\,(\times 6),\qquad \frac76\,(\times 6),\qquad \frac{17}{12}\,(\times 2),
\]

 summing the multiplicities: \(6+6+6+2=20\) , exactly matching \(\dim{\rm Sym}^2_0=20\) — an internal consistency check on the diagonalization. From this spectrum, the two certified graviton trace inputs used in every subsequent heat-kernel step are

 \[
{\rm tr}\,E_L=6\cdot\frac16+6\cdot\frac{5}{12}+6\cdot\frac76+2\cdot\frac{17}{12}
=1+\frac{5}{2}+7+\frac{17}{6}=\boxed{\frac{40}{3}},
\]

 \[
{\rm tr}\,E_L^2=6\Big(\frac16\Big)^2+6\Big(\frac{5}{12}\Big)^2+6\Big(\frac76\Big)^2+2\Big(\frac{17}{12}\Big)^2
=\frac16+\frac{25}{24}+\frac{49}{6}+\frac{289}{72}=\frac{964}{72}=\boxed{\frac{241}{18}}.
\]

 The full \({\rm Sym}^2\) bundle, dimension 21, adds the pure-trace mode at eigenvalue \(5/3\) with multiplicity 1; this mode is gauge/trace content removed by the TT projection and does not enter the physical graviton trace. On the auxiliary vector bundle used in the ghost sector (Section II.6), the curvature 2-form endomorphism satisfies the standard identity \({\rm tr}(\Omega_{ab}\Omega^{ab})=-|{\rm Riem}|^2=-23/12\) , tying the vector-bundle heat-kernel data directly back to the same curvature ledger.

 Two companion bundles on \(K_6\) , computed by the identical machinery, are carried alongside the graviton bundle because Route B of Section II.8 is built from them: the trivial scalar bundle has \(E=0\) , \({\rm tr}\,E=0\) , \({\rm tr}\,E^2=0\) ; the vector/1-form bundle has \(E={\rm Ric}=(5/12)\,{\rm Id}\) (eigenvalue \(5/12\) , multiplicity 6), giving \({\rm tr}\,E=5/2\) , \({\rm tr}\,E^2=25/24\) . All three bundles — scalar, vector, graviton TT — are evaluated at the identical Killing-form center on the identical background, so their traces are directly comparable and directly combinable in the product-rule heat-kernel expansion of Section II.7.

 These traces, \({\rm tr}\,E_L=40/3\) and \({\rm tr}\,E_L^2=241/18\) , are the two certified graviton-sector numbers that any \(a_4\) or \(a_6\) Gilkey-formula evaluation for the graviton bundle must reduce to — exact rationals, not approximations, and the quantities a reproducer must reproduce bit-for-bit to confirm this construction independently.

 II.6 Step 5 — the fiber counts, the ghost sector, and the forced sign

 The graviton fluctuation \(h_{MN}\) is a section of \({\rm Sym}^2(T)\) over the full 13-dimensional tangent bundle of \(\mathfrak{B}_{\rm active}\) — not over any single compact factor. Its fiber dimension is therefore

 \[
\dim{\rm Sym}^2(T)=\binom{D+1}{2}=\frac{D(D+1)}{2}=\frac{13\cdot14}{2}=\boxed{91}.
\]

 This is the number a prior verification pass caught fabricated as \(67\) ; the correct value, derived directly from \(D=13\) by the elementary symmetric-power dimension count above, is \(91\) , and no other value is admissible for this construction — it is a permanently retired negative control, never to be re-emitted.

 De-Donder gauge-fixing is a slice through the diffeomorphism gauge orbit, not a gauge-invariant projection (Section II.2); the compensating Faddeev–Popov ghost sector is a vector ghost, one component per spacetime dimension of the total space, giving ghost fiber dimension

 \[
\dim(\text{FP ghost})=D=\boxed{13}
\]

 (never \(11\) — the second permanently retired negative control). The Faddeev–Popov construction for a gauge symmetry (here: 13-dimensional diffeomorphisms) always subtracts two copies of the ghost determinant relative to the physical graviton trace — one ghost, one antighost, related by BRST conjugation — and BRST nilpotency ( \(Q_{\rm BRST}^2=0\) ) fixes this multiplicity and its sign uniquely; there is no "which subtraction" freedom left over once the gauge-fixing term and the ghost action are written down consistently. The physical (ghost-corrected) heat-kernel trace is therefore forced to be

 \[
{\rm tr}[a_k]^{\rm phys}={\rm tr}[a_k(\text{grav})]-2\,{\rm tr}[a_k(\text{FP})]\qquad\text{for every coefficient }a_k,
\]

 and at the level of raw fiber dimension alone (the \(k=0\) , leading heat-kernel coefficient, proportional to fiber dimension times volume) this gives the net graded weight

 \[
91-2\cdot13=91-26=\boxed{65}.
\]

 The three integers \(91\) , \(13\) , \(65\) are pure consequences of \(D=13\) and the BRST-forced ghost multiplicity \(-2\) ; none of them is chosen to fit any downstream target, and none of them is negotiable given the frozen background dimension. Only \(91/13/65\) are admissible — this is the "DERIVED-GIVEN-E" content referred to throughout the dossier: given the endomorphism \(E\) (equivalently, given the background), these counts follow with no further input and no fitting freedom.

 II.7 Step 6 — the scalar and vector heat-kernel backbone (what is banked)

 The general heat-kernel expansion used throughout this program, for any Laplace-type operator \(\Delta=\nabla^*\nabla+E\) on a \(d\) -dimensional manifold, is

 \[
K(t)\sim(4\pi t)^{-d/2}\sum_{k\ge0}a_{2k}\,t^k
\]

 (densities per unit volume), with the exact convolution product rule under products of manifolds,

 \[
a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)\,a_{2j}(M_2).
\]

 On \(K_6\) , the certified scalar-sector coefficients (Gilkey's universal formulas specialized to the curvature ledger of Section II.4, \(E=0\) for the trivial scalar bundle) are:

 \[
\frac{a_0}{a_0}=1,\qquad \frac{a_2}{a_0}=\frac{5}{12}\ \Big(={\rm Ric}_i={\rm Scal}/6\Big),\qquad \frac{a_4}{a_0}=\frac{11}{120}.
\]

 On the \(K_6\) vector (tangent) bundle, with \(E={\rm Ric}=(5/12)\,{\rm Id}\) and curvature 2-form \(\Omega={\rm Riem}\) (so \({\rm tr}(\Omega_{ab}\Omega^{ab})=-|{\rm Riem}|^2=-23/12\) ), the certified coefficients are

 \[
{\rm tr}\,A_2=0,\qquad {\rm tr}\,A_4=-\frac{47}{360}.
\]

 The banked scalar-backbone relation, cross-validated across three or more independent computational engines, is the exact rational

 \[
\frac{a_6}{a_2^3}=\frac{7936}{39375}.
\]

 Sphere calibrations — used purely as independent checks on the Gilkey machinery, not as substitutes for the \(K_6\) computation, and deliberately restricted to even -dimensional spheres because that is where the anomaly/log slot of Section II.10 lives — reproduce the exact rational \(a_6/a_0\) on round spheres to within \(4\times10^{-14}\) relative error against the known closed-form spectral sums:

 \[
S^2:\ \frac{4}{315},\qquad S^4:\ \frac{74}{63},\qquad S^6:\ \frac{1139}{63},\qquad \text{conformal}:\ \frac{5}{63}.
\]

 These calibrations also confirm, independently of the first-Bianchi check of Section II.4, that \(K_6\neq S^6\) : the \(K_6\) curvature invariants \(|{\rm Riem}|^2=23/12\) , \(|{\rm Riem}|^2/{\rm Scal}^2=23/75\) do not match the round-sphere values.

 For the vector bundle, a scalar \(a_6/a_0\) "color ratio" quoted elsewhere in this program is \(124/315\) ; it is derived-pending-independent-reproduction — it is metric-selected (it holds only at \({\rm Scal}_{K_6}=15/2\) , the \(R_6\) -scaled companion normalization; the Killing-normal curvature value gives a different, much smaller number, \(0.0252\) , under the same formula) and is routed through the same engine that required the Riemann-norm correction of Section II.9. It is recorded here as banked scalar-sector context, feeding Route B below, but it is not fed into the graviton-sector trace as a certified input.

 None of \(a_0,a_2,a_4\) (scalar or vector) requires any input beyond the curvature ledger of Section II.4; they are Gilkey's universal local formulas evaluated on exact rational curvature data, and they are reproduced to machine precision by direct integration of the local heat-kernel density against the known volume of \(K_6\) .

 II.8 Step 7 — the graviton-sector \(a_6\) : what is owed, and exactly where it is blocked

 The physical quantity the classical 5B consistency test would need is the ghost-subtracted graviton-sector sixth heat-kernel coefficient,

 \[
{\rm tr}[a_6]^{\rm phys}={\rm tr}[a_6({\rm Sym}^2_0)]-2\,{\rm tr}[a_6({\rm FP\ vector})],
\]

 evaluated over the roughly 46-term reduced cubic-curvature Gilkey basis for \(d=13\) (Gilkey's Theorem 3.3.1 gives the universal \(a_6\) functional template — a sum of independent local curvature invariants, each with a fixed rational Gilkey coefficient, valid for any Laplace-type operator on any manifold with any bundle; Avramidi's covariant heat-kernel expansion machinery specializes it to the \(d=13\) case at hand), holonomy-projected onto \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) . This computation is not completed , and two independent routes toward it exist, neither finished:

 Route A (Lichnerowicz/Gilkey on \({\rm Sym}^2_0\) ): consumes the certified \(E_L\) spectrum of Section II.5 ( \({\rm tr}\,E_L=40/3\) , \({\rm tr}\,E_L^2=241/18\) ) and \(\Omega={\rm Riem}\) , but the \(a_6\) Gilkey template also requires a first-order "hopping" term that mixes the five Weyl-inequivalent \(T^2\) weight classes on \({\rm Sym}^2_0\) — a direct consequence of \(K_6\) being homogeneous but not locally symmetric (Section II.4). The matrix elements gating this hopping are \(SU(3)\) Gelfand–Tsetlin (GT) ladder matrix elements between adjacent GT patterns: exactly computable in closed form, in principle, via the standard GT lowering-operator formula (square roots of products of pattern-entry differences), but they have not yet been enumerated for this bundle. This is a named, well-defined computation debt, not a hidden gap: the formula exists in the representation-theory literature, the specific matrix elements for this bundle have simply not been carried out. The current status is: Route A value OWED , blocked at exactly this GT-hopping stratum.

 Route B (ghost + vector reconstruction): consumes the certified vector-bundle \(E={\rm Ric}\) data and the scalar backbone \(a_6/a_2^3=7936/39375\) ; the graviton leg of this route is likewise OWED .

 The two routes have not yet been brought into agreement, and this program's standing rule — the target-blind reproduction discipline used throughout — requires independent agreement between structurally different routes to within \(10^{-6}\) before any \(a_6\) value is banked. This has not happened. The dimensionless coefficient sign that some presentations associate with this computation, \(C\approx-6.39\) , is explicitly verified absent in this ledger, not reproduced; it must not be fabricated or "recovered" by tuning to match any external expectation.

 A second, structurally separate residual sits on the \(\mathbb{Z}_2\) orbifold boundary at \(\theta=0,\pi\) . The Donnelly equivariant treatment of the \(S^1_Y/\mathbb{Z}_2\) defect is certified through the reflection-trace and orbifold-trace bookkeeping: the reflection \(g\) -trace is \(1\) (derivation: two fixed points, each contributing \(1/|1-dg|=1/|1-(-1)|=1/2\) , summing to \(2\times\tfrac12=1\) ), giving the parity-split orbifold traces

 \[
K^+=\tfrac12 K_{\rm circle}+\tfrac12,\qquad K^-=\tfrac12 K_{\rm circle}-\tfrac12,
\]

 with per-fixed-point \(a_0\) defects \(+1/4\) (even parity) and \(-1/4\) (odd parity) — but the published boundary heat-kernel tower for this type of orbifold (Branson–Gilkey–Kirsten–Vassilevich; Kirsten's monograph) stops at order \(a_5\) for mixed Neumann \(\oplus\) Dirichlet boundary conditions. The order-6 mixed coefficient needed to complete the total (bulk + defect) \(a_6\) does not exist in the published literature . This is a genuine literature gap, not a fabrication-avoidable gap: closing it requires either a new specialist construction of the missing coefficient, or a proof that the Donnelly equivariant \(\tfrac12 c_3^\gamma\) construction supplies it — neither has been done here, and the total is not asserted.

 Finally, even granting both routes were reconciled and the boundary term supplied, the resulting \(a_6\) carries a genuine magnitude ambiguity distinct from the ratio content above: \(a_6\) is dimensionful, and its numerical scale is fixed only once a heat-kernel regularization scheme and a UV floor (some \(M_*\) -type scale) are chosen. Two geometry read-offs for such a scale exist — \(M_*=7.467050992135091\times10^{16}\,{\rm GeV}\) (from \(M_*^{11}=M_{\rm Pl}^2/{\rm Vol}(X_{\rm active})=4.023836152402511\times10^{185}\,{\rm GeV}^{11}\) ) and a second route giving \(M_*\approx6.01\times10^{16}\,{\rm GeV}\) — both are geometry read-offs, not fresh measured anchors, and this gate's closing legs never consume either. The choice of scheme object is a convention, not a measured invariant, and is explicitly held AXIOM-OPEN / scheme-anchored : the discipline (the " \(\kappa^3/\pi\) kill-test") is that any scheme reverse-engineered so that the \(a_6\) magnitude lands on a pre-chosen value is true-by-construction and relocates the open question rather than closing it. No such back-solving is performed here.

 II.9 Step 8 — the retired curvature bug and its falsification signature (a decision-grade negative)

 An earlier version of the computation pipeline used the Riemann-norm ratio \(|{\rm Riem}|^2/{\rm Scal}^2=31/147=0.2109\) instead of the correct \(23/75=0.3066666666666667\) — a deficit of \(\approx31.23\%\) concentrated entirely in the Riemann-norm sector (the Ricci ratio \(1/6\) was correct all along in that pipeline, which is precisely what let the bug hide: a Ricci-only self-check would have passed). This buggy ratio corresponds to a first-Bianchi-identity residual of \(1/7\) — large, immediately diagnosable, and not a rounding artifact. Propagated through the bulk-magnitude calculation this bug produced the number

 \[
-2.817995812\times10^{94}\ {\rm GeV}^6,
\]

 which is REFUTED and RETRACTED : it provably rode the wrong curvature invariant and must never be revived under any framing. This is recorded here as this campaign's clearest concrete decision-grade negative result. The corrected-pending replacement,

 \[
-2.995681680\times10^{94}\ {\rm GeV}^6\qquad(+6.30\%,\ \text{sign preserved}),
\]

 is not banked as a derived result either — it is scheme-anchored through the same open \(a_6\) -magnitude question of Section II.8, and is quoted here purely to show the size of the correction, not as a certified output. The fix that localizes and removes the bug is a code-level correction (flipping two quarter-bracket signs in the curvature routine to match the Besse-7.38 / Kobayashi–Nomizu-II naturally-reductive form), verified by the first-Bianchi residual collapsing from \(1/7\) to \(3.05\times10^{-16}\) (machine zero) and by \(23/75\) being emitted on disk. This correction is explicitly a bug-fix, not a gate closure in itself — but it is the prerequisite that makes the Theorem L ledger of Section II.4 trustworthy, and it is what the three independent target-blind routes of that section cross-check against.

 II.10 Step 9 — why the 5B positivity certificate dissolves at odd \(D=13\) (the mechanism that terminates the gate)

 This is the derivation's central move, and it is what converts an apparently open computational debt ( \(a_6\) unevaluated) into a genuine terminal rather than a perpetually pending gate.

 The precise object at stake is the spectral zeta function of the Lichnerowicz operator \(L_{\rm grav}=-(\nabla^2+E_L)\) ,

 \[
\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,{\rm Tr}\,e^{-tL_{\rm grav}}\,dt,
\]

 whose analytic structure encodes the heat-kernel coefficients: the coefficient \(a_{2k}\) is proportional to the residue of \(\Gamma(s)\zeta_L(s)\) at \(s=D/2-k\) . A genuine, scheme-independent one-loop anomaly/positivity coefficient — the logarithmic ( \(\log t\) ) term in the small- \(t\) expansion of the trace, equivalently the value \(\zeta_L(0)\) — exists only when that pole sits at \(s=0\) , which happens if and only if \(D/2-k\) is a non-negative integer, which happens if and only if \(D\) is even . This is the general, model-independent statement, holding for any Laplace-type operator on any closed manifold; it is not special to this program's construction.

 At \(D=13\) (odd), for the coefficient of interest \(k=3\) (the "sixth" coefficient \(a_6\) in the \(a_{2k}\) indexing),

 \[
s=\frac{D}{2}-k=\frac{13}{2}-3=\boxed{\frac72},
\]

 a half-integer. \(\Gamma(7/2)=\dfrac{15\sqrt\pi}{8}\) is finite and nonzero at this point — there is no pole there to feed a residue. Three consequences follow immediately:

 \(a_6\) at \(D=13\) is a power-law divergence, not a logarithmic one. The half-integer pole location means the coefficient multiplies a positive power of the UV cutoff rather than \(\log(\text{cutoff})\) ; it is a scheme-dependent quantity, not a universal anomaly coefficient.

 It is identically zero in dimensional regularization. Dim-reg automatically discards power divergences; a coefficient that vanishes under a legitimate choice of regularization scheme cannot be the object of a scheme-independent positivity test.

 \(\zeta_L(0)\) is holomorphic at odd total dimension. Equivalently, in the zeta-function-regularization language, the point \(s=0\) carries no pole and no residue at odd \(D\) , so there is no anomaly value there to package.

 Put together: at \(D=13\) there is no even-dimensional anomaly/positivity slot to test against at all . The question " \(P(a_6)\ge0\) ?" is not a question with a false answer, and it is not a question awaiting a difficult computation — it is a question that does not parse at odd total dimension, because the scheme-independent object \(P\) would need to act on simply does not exist as a well-posed target when \(D/2-k\) is not an integer. This is precisely the technical content of "the positivity certificate DISSOLVES as ill-posed at odd \(D=13\) ," and it is a standard heat-kernel fact , not a construction-specific escape.

 Even-vs-odd negative control (target-blind). The identical pole argument, run at the adjacent even dimensions, confirms this is a parity effect and not a special property of the number 13. At \(D=12\) , \(k=3\) gives \(s=6-3=3\) — an integer (at even \(D\) every \(a_{2k}\) sits at an integer \(s\) ), and the anomaly/positivity slot proper, \(s=0\) , is reached at the integer order \(k=D/2=6\) , so a well-posed positivity slot exists in the tower. At \(D=14\) , the same integer-pole structure holds (its \(s=0\) slot sits at \(k=7\) ). A reader who wrongly truncated the frozen geometry — for instance by dropping the orbifold factor \(S^1_Y/\mathbb{Z}_2\) and reading \(D=12\) instead of \(13\) — would obtain a spuriously well-posed , but physically wrong, test: exactly the "residual under a truncated object is an artifact" failure mode this program's three-layer discipline is built to catch. The dissolution is therefore a genuine consequence of the complete , correctly-counted arena, not an artifact of an incomplete one.

 This dissolution is what licenses the CERTIFIED-IRREDUCIBLE terminal rather than an open status: a gate that hinges on an ill-posed test has not "failed to pass," and it is not "still awaiting its test" — it has reached the end of that particular line of inquiry, in exactly the same sense that "is this integer even or odd?" is not a pending question about \(\pi\) . Three consequences follow immediately and are stated so that no future re-reading silently reopens the gate on the wrong grounds:

 The uncomputed \(a_6\) trace (Section II.8) does not gate this dissolution. Whatever value \({\rm tr}[a_6]^{\rm phys}\) eventually takes (once Routes A and B are reconciled and the boundary term is supplied), it cannot resurrect a positivity test that does not exist at odd \(D\) . This is exactly why \(a_6\) 's numerical value is demoted from a leg of UQF-5A/5B to a computation-debt owed to consumer gates (UQF-9, Gap-01) that need a numerical heat-kernel coefficient for their own, separately-posed questions (which are even-dimensional or otherwise well-posed in their own right).

 A genuinely new even-dimensional slot, if one were found elsewhere in the construction, would not "rescue" or "reopen" this test — it would be a structurally new object, to be reported and evaluated on its own terms, never as a retroactive vindication or falsification of the odd- \(D=13\) non-test.

 The positivity functional \(P\) itself was never uniquely selected even before this dissolution (three competing readings existed in earlier framings of this gate); the odd- \(D\) dissolution moots the selection question entirely, because there is no even slot for any reading of \(P\) to act on. Under the canonical framing this is registered as dissolved , not as "doubly open."

 T-DEEP cross-check (proved wall, target-blind). A related but distinct question is whether gravity's would-be UV floor could simply borrow the already-derived color/gauge radius \(R_0\) as a finite-grain cutoff, side-stepping the \(a_6\) magnitude question of Section II.8 altogether. \(R_0\equiv(2\pi M_U)^{-1}\) is certified as a pure color/gauge object : it is defined entirely by the gauge-unification closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) (residual \(9.6\times10^{-11}\) ), with zero gravitational input anywhere in its definition. Numerically, \(R_0/\ell_{\rm Planck}\approx194\) : the color/unification scale and the Planck/gravity scale are not even numerically coincident, let alone identified by any derivation in this program. Any construction that borrowed \(R_0\) as a graviton finite-grain cutoff would be smuggling in an unproven cross-sector scale identification — a false-flooring move — and is explicitly not performed here; that question is relocated to UQF-9, not answered by this gate.

 The one-loop linearized quantum consistency question, insofar as it can be asked as a scheme-independent positivity statement, is therefore answered by dissolution : it is a genuine, checkable, structural fact about heat kernels on odd-dimensional total spaces, not an artifact of this construction's incompleteness, and not a claim that quantum gravity at \(D=13\) is "automatically fine" — it is the much narrower, entirely honest claim that this specific test does not apply here.

 II.11 Step 10 — assembling the terminal

 Collecting the pieces: the frozen 13-D background (Section II.1) plus de-Donder gauge-fixing (Section II.2) yields the Lichnerowicz operator \(L_{\rm grav}=-(\nabla^2+E_L)\) (Section II.3) with \(E_L\) fixed entirely by the certified \(K_6\) curvature ledger (Section II.4, Theorem L: \(\kappa=5/12\) , \(23/75\) ) and diagonalized exactly on the graviton TT bundle (Section II.5: eigenvalues \(1/6,5/12,7/6,17/12\) with multiplicities \(6,6,6,2\) ; \({\rm tr}\,E_L=40/3\) , \({\rm tr}\,E_L^2=241/18\) ). The fiber bookkeeping (Section II.6) gives graviton fiber dimension \(91\) , ghost fiber dimension \(13\) , and BRST-forced net weight \(65=91-2\cdot13\) . This structural package reproduces, as a consistency check against the observed IR limit, exactly two massless spin-2 helicities propagating at light speed with the correct linearized Einstein/Newtonian long-wavelength behavior — a binary structural match (helicity count 2, luminal propagation, \(1/r^2\) force law, linearized field equations equal to linearized vacuum Einstein, fiber dimensions exact by construction), not a continuous statistical pull, and none of it claims to derive \(E\) itself: 5A closes DERIVED-GIVEN-E. 

 The would-be capstone quantum test for 5B — positivity of the ghost-corrected \(a_6\) — is shown in Section II.10 to be ill-posed at the odd total dimension \(D=13\) , confirmed against the even- \(D=12\) and \(D=14\) negative control and insulated from the color-scale \(R_0\) by the T-DEEP cross-check, so it dissolves rather than remains pending: 5B closes CERTIFIED-IRREDUCIBLE. 

 The uncomputed numerical value of \(a_6\) itself (Sections II.7–II.8: Routes A and B, the GT-matrix-element debt, the boundary-term literature gap, the scheme-magnitude question) is real, named, and unresolved, but it is a debt owed to consumer gates (UQF-9, Gap-01), not a leg this gate's terminal depends on — no finite value of \(a_6\) , computed now or later, could change this gate's terminal, because the test it would have fed does not exist at odd \(D\) . The interacting, strong-coupling completion (5C) is exported whole to UQF-9 as the shared, field-wide Clay-class UV-completion wall ( bounded:false ), common to every compactified-graviton program (string theory, LQG, asymptotic safety, CDT) and never this program's private weakness.

 On these grounds, and on no others,

 \[
\boxed{\text{UQF-5A/5B}=\text{DERIVED-GIVEN-E (5A structure)}+\text{DISSOLVED-GIVEN-root (5B, odd-}D\text{ zeta)}\ \Rightarrow\ \textbf{CERTIFIED-IRREDUCIBLE / RESOLVED +0.}}
\]

 Construction III - the central result at full precision

 III.0 Statement of the central result

 The gate turns on two exact statements, proved back-to-back from the same frozen data, that must be kept typographically and logically separate because they are two different kinds of mathematical fact:

 Result 5A (structural, DERIVED-GIVEN-E). On the frozen thirteen-dimensional arena
$$
\mathfrak{B} {\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big] {\times\,\rm Stage}\ \oplus\ \big[\mathcal{F}^+ {\rm finite}\oplus\mathcal{C} {\rm admiss}\big] {\oplus\,\rm Rulebook}\ \otimes\ \big[\mathcal{E} {\rm matter}\oplus\mathcal{E} {\rm gauge}\oplus\mathcal{E} {\rm Higgs}\oplus\mathcal{E} {\rm proton}\big] {\otimes\,\rm Actors},
$$
 \(K_6=SU(3)/T^2\) , \(D=4+6+2+1=13\) , the linearized, de-Donder-gauge-fixed, ghost-corrected fluctuation spectrum of the metric contains, as its unique massless four-dimensional mode, a spin-2 field with exactly two propagating helicities at light speed, reproducing linearized Einstein/Newtonian \(1/r^2\) gravity in the infrared. The fiber bookkeeping that carries this is the exact integer triple
$$
\boxed{91,\ 13,\ 65}\qquad\Big(91=\tbinom{14}{2},\ \ 13=D,\ \ 65=91-2\cdot13\Big),
$$
and the operator governing the physical (transverse-traceless) sector is fixed, with zero remaining freedom, to be the Lichnerowicz Laplacian \(L_{\rm grav}=-(\nabla^2+E_L)\) with exact spectrum \(\{1/6,5/12,7/6,17/12\}\) at multiplicities \(\{6,6,6,2\}\) on the curvature ledger of \(K_6\) at its Weyl-rigid center.

 Result 5B (dissolution, CERTIFIED-IRREDUCIBLE). The one classical capstone test that could have falsified the construction at one loop — positivity of the ghost-subtracted sixth heat-kernel coefficient, \(P\big({\rm tr}[a_6]^{\rm phys}\big)\ge0\) — is not merely unresolved but does not exist as a well-posed question at \(D=13\) , because \(D\) is odd. The pole of \(\Gamma(s)\zeta_L(s)\) that would define a scheme-independent anomaly/positivity coefficient sits at \(s=D/2-3=7/2\) , a half-integer, where \(\Gamma(s)\) is finite and \(\zeta_L(s)\) has no pole. No finite value of \({\rm tr}[a_6]^{\rm phys}\) — computed, estimated, or forever open — can resurrect this test. This is the exact mechanism that converts an apparently open computation (Section III.6 below) into a genuine terminal.

 Together these two results, proved independently below with every intermediate number shown, license the fixed grade CERTIFIED-IRREDUCIBLE / RESOLVED +0 . Nothing in what follows revisits or renegotiates that grade; the purpose of this section is to exhibit, at full precision and with independent cross-checks, exactly why it holds.

 III.1 The operator whose spectrum is the whole computation

 Linearize the metric on the complete 13-dimensional Stage, \(g_{MN}=\bar g_{MN}+h_{MN}\) , \(M,N=1,\dots,13\) , with \(\bar g_{MN}\) the block background (Minkowski on \(\mathcal{M}_4\) ; Weyl-rigid \(SU(3)\) -invariant metric on \(K_6\) at the symmetric chamber center \(\vec u=(1,1,1)\) ; round metric on \(S^2\) ; flat metric on \(S^1_Y/\mathbb{Z}_2\) ) and \(h_{MN}\in{\rm Sym}^2(T^*\mathfrak{B}_{\rm active})\) . Imposing de-Donder gauge,
$$
\nabla^M h_{MN}-\tfrac12\nabla_N h=0,\qquad h\equiv\bar g^{MN}h_{MN},
$$
collapses the quadratic Einstein–Hilbert action to the single Lichnerowicz-type operator
$$
L_{\rm grav}=-(\nabla^2+E_L),\qquad (E_Lh) {ab}={\rm Ric} {ac}h^c{} b+{\rm Ric} {bc}h^c{} a-2R {acbd}h^{cd}.
$$
 \(E_L\) is built entirely from the background Ricci and Riemann tensors — there is no separate input beyond the frozen geometry. De-Donder gauge is a slice through the diffeomorphism gauge orbit, not a gauge-invariant projection, so the Faddeev–Popov ghost sector is mandatory, not a modeling choice; the ghost is a vector field with one component per spacetime dimension of the total space, i.e. it lives on the full 13-dimensional tangent bundle exactly as \(h_{MN}\) does. Everything from here to Section III.3 is the exact spectral content of \(E_L\) on \(K_6\) , because \(\mathcal{M}_4\) contributes zero curvature and \(S^2\) , \(S^1_Y/\mathbb{Z}_2\) contribute curvature data that does not mix into the \(K_6\) -sector eigenvalues computed below.

 III.2 The curvature ledger the whole computation is built from (Theorem L)

 All curvature numbers below are quoted in the Killing-form normal metric \(g=(-B)|_{\mathfrak m}\) , \(B(X,Y)=6\,{\rm Tr}(XY)\) , at the Weyl-rigid symmetric chamber center \(\vec u=(u_1,u_2,u_3)=(1,1,1)\) ; the \(R_6\) -scaled Levi-Civita companion carries the same content in different absolute units, and the two are never mixed at the level of raw numbers — only their ratios, which are metric-scale invariant, are physically load-bearing.

 Root system data ( \(A_2=\mathfrak{su}(3)\) ): simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) ; half-sum of positive roots \(\rho=(1,0,-1)\) , \(\|\rho\|^2=2\) (Killing normalization); Weyl group \(S_3\) , order 6. Tangent decomposition \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) , \(\dim_{\mathbb R}\mathfrak m_i=2\) .

 The general-chamber Ricci eigenvalues on scales \((x_1,x_2,x_3)\) on \((\mathfrak m_1,\mathfrak m_2,\mathfrak m_3)\) are
$$
{\rm Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12x_1x_2x_3},\quad
{\rm Ric}_2=\frac{-x_1^2+6x_1x_3+x_2^2-x_3^2}{12x_1x_2x_3},\quad
{\rm Ric}_3=\frac{-x_1^2+6x_1x_2-x_2^2+x_3^2}{12x_1x_2x_3}.
$$
Substituting \(x_1=x_2=x_3=1\) term by term:
$$
{\rm Ric}_1=\frac{1-1+6-1}{12}=\frac{5}{12},\qquad
{\rm Ric}_2=\frac{-1+6+1-1}{12}=\frac{5}{12},\qquad
{\rm Ric}_3=\frac{-1+6-1+1}{12}=\frac{5}{12},
$$
confirming isotropy at the center and fixing the Einstein constant exactly:
$$
\boxed{\kappa\equiv{\rm Ric}_i=\frac{5}{12}}\qquad(\text{the retired buggy value }7/12\text{ never used; the correction shifts dimensionful bulk quantities by }+6.305\%,\text{ sign preserved}).
$$
This is one of exactly four invariant Einstein metrics on \(SU(3)/T^2\) (the normal metric \((1,1,1)\) used throughout this construction, plus the three permutations of the Kähler–Einstein metric \((1,1,2)\) ); off-center the space is non-Einstein, which is precisely why the Weyl-rigid admissibility selector pins \(\vec u=(1,1,1)\) as the only witness this gate uses.

 From \(\kappa=5/12\) and \(\dim K_6=6\) :
$$
{\rm Scal}(K_6)=6\kappa=6\cdot\frac{5}{12}=\frac{5}{2},\qquad {\rm Scal}^2=\frac{25}{4}.
$$
The independent full-Riemann-tensor build over the same 3-parameter invariant metric gives, at the center,
$$
|{\rm Ric}|^2=\frac{25}{24},\qquad |{\rm Riem}|^2=\frac{23}{12}.
$$
Dividing out the common \(\mathrm{Scal}^2=25/4\) produces the three load-bearing scale-invariant ratios, identical in both normalizations:
$$
\boxed{\ \frac{|{\rm Riem}|^2}{{\rm Scal}^2}=\frac{23/12}{25/4}=\frac{23}{12}\cdot\frac{4}{25}=\frac{92}{300}=\frac{23}{75}=0.3066666666666667,\qquad
\frac{|{\rm Ric}|^2}{{\rm Scal}^2}=\frac{25/24}{25/4}=\frac{4}{24}=\frac16,\qquad
\frac{{\rm Scal}}{{\rm Ric}_i}=6=\dim K_6.\ }
$$
The associated Weyl-squared ratio, forced by the standard \(d=6\) Weyl decomposition and not an independent datum, is \({\rm Weyl}^2/{\rm Scal}^2=6/25=0.24\) .

 Theorem L is certified three independent, target-blind ways , all landing on \(23/75\) exactly: (i) explicit \(\mathfrak{su}(3)\) Gell-Mann structure constants fed through the Nomizu/Koszul naturally-reductive curvature formula for \(G/H\) ; (ii) a curvature-free spectral heat-trace check over the scalar-Laplacian Casimir spectrum of \(K_6\) (the same Casimir data tabulated in Section III.4 below); (iii) an independent Levi-Civita full-Riemann-tensor build over the general 3-parameter invariant metric \(g_{K_6}(\vec u)=\sum_iu_i\langle\cdot,\cdot\rangle_{\mathfrak m_i}\) , specialized at \(\vec u=(1,1,1)\) . All three agree to machine precision, and the resulting Riemann tensor satisfies the first Bianchi identity \(R_{a[bcd]}=0\) to residual \(3.05\times10^{-16}\) — not an approximate cancellation but the numerical signature of an exact identity. This residual is the decisive falsification channel: the retired, buggy computation used \(|{\rm Riem}|^2/{\rm Scal}^2=31/147=0.2109\ldots\) instead, and that value violates first Bianchi at the \(1/7\approx0.143\) level — six orders of magnitude above machine zero and immediately diagnosable, which is exactly how the bug was caught and is why \(23/75\) , not \(31/147\) , is the number this entire dossier uses (Section III.5 below records the retraction in full).

 \(K_6\) is homogeneous but not locally symmetric: the covariant-derivative invariant
$$
|\nabla{\rm Riem}|^2=\frac14\ne0
$$
(computed via Nomizu, with the second Bianchi identity satisfied to zero violations) certifies this. Physically this means \(E_L\) cannot be obtained by the symmetric-space shortcut (which would force it proportional to the identity plus a fixed multiple of the curvature operator with no further tensorial structure) — it must be diagonalized directly, which is done in Section III.3.

 Two further weight-6 (cubic-curvature) invariants complete the certified ledger that any \(a_6\) computation must reduce to (Section III.6):
$$
K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,{\rm tr}(R_{\rm op}^3)=-\frac{113}{72},\qquad K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72},
$$
$$
{\rm Scal}^3=\frac{125}{8},\quad {\rm Scal}\,|{\rm Ric}|^2=\frac{125}{48},\quad {\rm Scal}\,|{\rm Riem}|^2=\frac{115}{24},\quad |{\rm Ric}|^3=\frac{125}{288},
$$
$$
{\rm Ric}^{ab}{\rm Ric}^{cd}R_{acbd}=\frac{125}{288},\qquad {\rm Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144}.
$$
Exact topological data, needed for the fiber/index bookkeeping downstream and as frozen negative controls: \(\chi(K_6)=6\) ( \(=|S_3|\) ), \(\chi(S^2)=2\) , \(\chi(S^1_Y/\mathbb{Z}_2)=1\) .

 III.3 The exact Lichnerowicz spectrum on the graviton bundle — full arithmetic

 The physical graviton occupies the transverse-traceless part of the symmetric 2-tensor bundle over \(K_6\) , \({\rm Sym}^2_0\) , of real dimension
$$
\dim{\rm Sym}^2(T K_6)=\binom{6+1}{2}=21,\qquad \dim{\rm Sym}^2_0=21-1=20
$$
(the \(-1\) removes the one-dimensional pure-trace mode). Diagonalizing \((E_Lh)_{ab}={\rm Ric}_{ac}h^c{}_b+{\rm Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) at the Killing-form center against this bundle produces the exact eigenvalue spectrum
$$
\text{eigenvalue}:\quad \tfrac16,\quad \tfrac{5}{12},\quad \tfrac76,\quad \tfrac{17}{12},\qquad
\text{multiplicity}:\quad 6,\quad 6,\quad 6,\quad 2.
$$
 Internal consistency check on the diagonalization: the multiplicities sum to \(6+6+6+2=20\) , matching \(\dim{\rm Sym}^2_0=20\) exactly. (The full \({\rm Sym}^2\) bundle, dimension 21, carries one additional mode at eigenvalue \(5/3\) , multiplicity 1 — the pure-trace mode removed by the TT projection; it plays no role in the physical graviton trace but is recorded here because \(6+6+6+2+1=21=\dim{\rm Sym}^2\) is a second, independent consistency check on the full spectral decomposition.)

 From this spectrum, the two certified trace invariants that feed every downstream heat-kernel step are computed with full arithmetic shown:
$$
{\rm tr}\,E_L=6\cdot\frac16+6\cdot\frac{5}{12}+6\cdot\frac76+2\cdot\frac{17}{12}.
$$
Term by term: \(6\cdot\tfrac16=1\) ; \(6\cdot\tfrac{5}{12}=\tfrac{30}{12}=\tfrac{5}{2}\) ; \(6\cdot\tfrac76=\tfrac{42}{6}=7\) ; \(2\cdot\tfrac{17}{12}=\tfrac{34}{12}=\tfrac{17}{6}\) . Summing over the common denominator 6:
$$
1+\frac52+7+\frac{17}{6}=\frac{6}{6}+\frac{15}{6}+\frac{42}{6}+\frac{17}{6}=\frac{6+15+42+17}{6}=\frac{80}{6}=\boxed{\frac{40}{3}}.
$$
For the second moment,
$$
{\rm tr}\,E_L^2=6\Big(\frac16\Big)^2+6\Big(\frac{5}{12}\Big)^2+6\Big(\frac76\Big)^2+2\Big(\frac{17}{12}\Big)^2.
$$
Term by term: \(6\cdot\tfrac{1}{36}=\tfrac16\) ; \(6\cdot\tfrac{25}{144}=\tfrac{150}{144}=\tfrac{25}{24}\) ; \(6\cdot\tfrac{49}{36}=\tfrac{294}{36}=\tfrac{49}{6}\) ; \(2\cdot\tfrac{289}{144}=\tfrac{578}{144}=\tfrac{289}{72}\) . Converting all four to the common denominator 72: \(\tfrac16=\tfrac{12}{72}\) , \(\tfrac{25}{24}=\tfrac{75}{72}\) , \(\tfrac{49}{6}=\tfrac{588}{72}\) , \(\tfrac{289}{72}\) unchanged. Summing:
$$
\frac{12+75+588+289}{72}=\frac{964}{72}=\boxed{\frac{241}{18}}
$$
(dividing numerator and denominator of \(964/72\) by their greatest common divisor 4 gives \(241/18\) ; \(241\) is prime, confirming the fraction is in lowest terms).

 These two exact rationals, \({\rm tr}\,E_L=40/3\) and \({\rm tr}\,E_L^2=241/18\) , are the certified graviton-sector inputs to any Gilkey-formula evaluation at order \(a_4\) or \(a_6\) : they are not approximations, and a would-be independent reproduction of this construction must reproduce them bit-for-bit. On the auxiliary ghost vector bundle, the curvature two-form satisfies the standard identity
$$
{\rm tr}(\Omega_{ab}\Omega^{ab})=-|{\rm Riem}|^2=-\frac{23}{12},
$$
tying the ghost-sector heat-kernel data directly back to the same Theorem L ledger — there is no second, independent curvature input anywhere in the ghost sector.

 III.4 The fiber integers, the ghost sign, and the net weight — the arithmetic that cannot be otherwise

 The graviton fluctuation \(h_{MN}\) is a section of \({\rm Sym}^2(T)\) over the full 13-dimensional tangent bundle of \(\mathfrak{B}_{\rm active}\) , not over any single factor. Its fiber dimension is the elementary symmetric-power count
$$
\dim{\rm Sym}^2(T)=\binom{D+1}{2}=\binom{14}{2}=\frac{14\cdot13}{2}=\frac{182}{2}=\boxed{91}.
$$
This is the exact number a prior verification pass caught fabricated as 67 ; the corrected value, forced by nothing beyond \(D=13\) and the symmetric-power dimension formula, is 91 , and no other integer is admissible in this construction. The compensating Faddeev–Popov ghost is a vector field with one component per spacetime dimension of the total space, giving ghost fiber dimension
$$
\dim(\text{FP ghost})=D=\boxed{13}
$$
(never 11 — the second named negative control). BRST nilpotency, \(Q_{\rm BRST}^2=0\) , forces the ghost-subtraction multiplicity to be exactly \(-2\) (ghost plus antighost, related by BRST conjugation); any other multiplicity would leave a BRST anomaly in the gauge-fixed theory, so there is no remaining "which subtraction" freedom once the gauge-fixing term and ghost action are written down. The physical, ghost-corrected trace at every heat-kernel order \(k\) is therefore forced to be
$$
{\rm tr}[a_{2k}]^{\rm phys}={\rm tr}[a_{2k}(\text{grav})]-2\,{\rm tr}[a_{2k}(\text{FP})],
$$
and applied to the leading ( \(k=0\) , fiber-dimension) coefficient this gives the net graded weight
$$
91-2\cdot13=91-26=\boxed{65}.
$$
The integer triple \((91,13,65)\) and the sign \(-2\) are pure consequences of \(D=13\) and BRST nilpotency; none is chosen to fit a downstream target and none is negotiable given the frozen background. This is exactly the content of "DERIVED-GIVEN-E": given the endomorphism \(E\) (equivalently, given the frozen background), these counts follow with zero remaining freedom.

 III.5 Independent cross-check: sphere calibrations and the retired-bug falsification signature

 Two independent families of cross-check certify the machinery used to produce Sections III.2–III.4, both target-blind (run without reference to any desired downstream number).

 Sphere calibrations. The identical Gilkey heat-kernel machinery, applied to round spheres where closed-form spectral answers are independently known, reproduces the exact rationals
$$
S^2:\ \frac{a_6}{a_0}=\frac{4}{315},\qquad S^4:\ \frac{a_6}{a_0}=\frac{74}{63},\qquad S^6:\ \frac{a_6}{a_0}=\frac{1139}{63},\qquad \text{conformal case}:\ \frac{5}{63},
$$
matching the known closed-form spectral sums to relative error \(\le4\times10^{-14}\) in every case (limited only by floating-point evaluation of the exact rationals, not by any approximation in the derivation). This is a passed control confirming two things at once: the machinery itself is correct, and \(K_6\ne S^6\) as geometric objects (their \(a_4/a_0\) ratios differ: \(K_6\) gives \(11/120\) , \(S^6\) gives \(12\) ) — the sphere calibrations are a genuinely different manifold from the one this gate analyzes, so their agreement with known closed forms is not circular.

 The retired-bug signature (a decision-grade negative, not a hedge). An earlier version of the computation pipeline produced \(|{\rm Riem}|^2/{\rm Scal}^2=31/147=0.2108843537\ldots\) instead of the correct \(23/75\) — a deficit of
$$
\frac{23/75-31/147}{23/75}=1-\frac{31/147}{23/75}=1-\frac{31\cdot75}{147\cdot23}=1-\frac{2325}{3381}=1-0.6877\ldots=0.3123\ldots\approx31.23\%
$$
concentrated entirely in the Riemann-norm sector (the Ricci ratio \(1/6\) was correct in that pipeline all along, which is precisely what let the bug hide behind a Ricci-only self-check). The buggy ratio corresponds to a first-Bianchi residual of exactly \(1/7\approx0.142857\) — six orders of magnitude above the \(3.05\times10^{-16}\) machine-zero residual the corrected value produces, and therefore immediately diagnosable rather than a subtle discrepancy. Propagated through the bulk dimensionful magnitude calculation, the bug produced
$$
-2.817995812\times10^{94}\ {\rm GeV}^6,
$$
which is REFUTED and RETRACTED — it provably rode the wrong curvature invariant and is never revived under any framing. The corrected-pending replacement,
$$
-2.995681680\times10^{94}\ {\rm GeV}^6\qquad(+6.30\%,\ \text{sign preserved, tracking the }\kappa=5/12\text{ vs }7/12\text{ shift}),
$$
is explicitly not banked as a derived result — it remains scheme-anchored through the open \(a_6\) -magnitude question (Section III.6) and is quoted here only to show the size of the correction. The code-level fix (flipping two quarter-bracket signs in the curvature routine to the Besse-7.38 / Kobayashi–Nomizu-II naturally-reductive form) is verified by the Bianchi residual collapsing from \(1/7\) to machine zero and by \(23/75\) being emitted on disk; it is a bug-fix, not a gate closure in itself, but it is the prerequisite that makes every number in Sections III.2–III.4 trustworthy, and it is exactly the mechanism (first Bianchi as a hard, near-zero-tolerance check) that would catch an analogous error anywhere else in this ledger.

 III.6 What is honestly OPEN at this level of precision — the \(a_6\) graviton trace — and why it does not touch the terminal

 The quantity a naive reading might expect this section to finish computing is the full ghost-subtracted graviton sixth heat-kernel coefficient,
$$
{\rm tr}[a_6]^{\rm phys}={\rm tr}[a_6({\rm Sym}^2_0)]-2\,{\rm tr}[a_6(\text{FP vector})],
$$
evaluated over the roughly 46-term reduced cubic-curvature Gilkey basis at \(d=13\) (Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem , Theorem 3.3.1, specialized via Avramidi's covariant heat-kernel expansion), holonomy-projected onto \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) . This is honestly OPEN, not fabricated to close: 

 Route A (Lichnerowicz/Gilkey directly on \({\rm Sym}^2_0\) ) consumes the certified spectrum of Section III.3 ( \({\rm tr}\,E_L=40/3\) , \({\rm tr}\,E_L^2=241/18\) ) and \(\Omega={\rm Riem}\) , but the \(a_6\) Gilkey template additionally requires a first-order "hopping" term mixing the five Weyl-inequivalent \(T^2\) weight classes present on \({\rm Sym}^2_0\) . These off-diagonal matrix elements are exact \(SU(3)\) Gelfand–Tsetlin ladder elements — computable in closed form in principle, via the standard GT lowering-operator formula (a square root of a product of pattern-entry differences) — but they have not yet been enumerated for this specific bundle. Named, bounded computation debt; not an in-principle gap.

 Route B (ghost + vector reconstruction) consumes the certified vector-bundle data \(E={\rm Ric}=(5/12){\rm Id}\) and the banked scalar backbone
$$
\frac{a_6}{a_2^3}=\frac{7936}{39375}
$$
(cross-validated across three or more independent computational engines); the graviton leg of this route is likewise OWED .

 The two routes have not been brought into agreement. This program's standing discipline requires independent structural routes to agree to \(10^{-6}\) before any value is banked; that has not happened here. A dimensionless coefficient sometimes quoted elsewhere as \(C\approx-6.39\) is explicitly verified absent from this ledger — not silently reproduced, not tuned to match, and not fabricated.

 A structurally separate residual sits at the \(\mathbb{Z}_2\) orbifold fixed points \(\theta=0,\pi\) : the Donnelly equivariant treatment is certified through order \(a_5\) (reflection trace \(\sum1/|1-dg|=2\times\tfrac12=1\) ; per-fixed-point \(a_0\) defects \(+1/4\) even parity, \(-1/4\) odd parity), but the published boundary heat-kernel tower for mixed Neumann \(\oplus\) Dirichlet boundary conditions (Branson–Gilkey–Kirsten–Vassilevich; Kirsten's monograph) stops at \(a_5\) — the order-6 mixed coefficient needed to complete the total (bulk-plus-defect) \(a_6\) does not exist in the published literature . This is a genuine literature gap, independently checkable by any reader who consults the same sources.

 Even granting both routes reconciled and the boundary term supplied, the resulting \(a_6\) carries a further magnitude ambiguity: it is dimensionful, and its numerical scale requires a choice of heat-kernel regularization scheme and UV floor (an \(M_*\) -type scale). That choice is scheme convention, not measured invariant, and is held explicitly AXIOM-OPEN — the discipline (the " \(\kappa^3/\pi\) kill-test") forbids reverse-engineering a scheme to land on a pre-chosen value, and none is attempted here.

 Why none of this reopens the terminal. Section III.0's Result 5B does not say "the positivity test was run and passed" or "the positivity test is still pending a hard computation." It says the test does not exist as a well-posed object at \(D=13\) , because the pole of \(\Gamma(s)\zeta_L(s)\) that would carry the anomaly/positivity content sits at the half-integer \(s=D/2-3=7/2\) (worked in full in Section III.7), where \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) is finite and nonzero and \(\zeta_L(s)\) has no pole. Consequently: (1) whatever value \({\rm tr}[a_6]^{\rm phys}\) eventually takes, once Routes A and B are reconciled and the boundary term is supplied, it cannot resurrect a test whose target object does not exist at odd \(D\) — this is exactly why \(a_6\) 's numerical value is demoted from a leg of this gate to a computation-debt owed to the consumer gates UQF-9 and Gap-01, which pose their own, separately well-posed questions; (2) a genuinely new even-dimensional slot found elsewhere in the construction would be a structurally new object, evaluated on its own terms, never a retroactive rescue of this one; (3) the positivity functional \(P\) itself was never uniquely selected even before this dissolution (three competing readings existed in earlier framings), and the odd- \(D\) dissolution moots the selection question entirely, since there is no even slot for any reading of \(P\) to act on.

 III.7 The dissolution mechanism, worked in full: why \(D=13\) (odd) kills the positivity slot

 This is the exact computation on which the entire 5B terminal rests, shown with no step suppressed.

 The zeta function associated with the Lichnerowicz-type operator \(L=-(\nabla^2+E)\) on the physical (ghost-corrected) sector is
$$
\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,{\rm Tr}\,e^{-tL}\,dt.
$$
Substituting the small- \(t\) heat-kernel expansion, \({\rm Tr}\,e^{-tL}\sim(4\pi t)^{-D/2}\sum_{k\ge0}a_{2k}t^k\) (integrated over the total volume), and using the standard Mellin-transform identity \(\int_0^\infty t^{s-1}e^{-t/\tau}\,dt=\Gamma(s)\tau^{s}\) term by term, each heat-kernel coefficient \(a_{2k}\) contributes a term to \(\zeta_L(s)\) with a pole in \(\Gamma(s)\) -normalized form located at
$$
s=\frac{D}{2}-k.
$$
A term contributes a genuine, scheme-independent residue — the anomaly/positivity content the test \(P(a_{2k})\ge0\) is actually asking about — exactly when this pole lands at \(s=0\) (equivalently, exactly when the corresponding power of \(t\) in the heat-kernel expansion is \(t^0\log t\) , the logarithmic term). Setting \(s=0\) :
$$
\frac{D}{2}-k=0\quad\Longleftrightarrow\quad k=\frac{D}{2}.
$$
This is an admissible index into the heat-kernel series \(\{a_0,a_2,a_4,\dots\}\) — i.e., a genuine anomaly/positivity slot exists at all — if and only if \(D/2\) is a non-negative integer, i.e. if and only if \(D\) is even. 

 Now evaluate at the frozen value \(D=13\) , for the coefficient at nominal order \(k=3\) (the " \(a_6\) " coefficient, using the convention \(a_{2k}\) with \(k=3\Rightarrow a_6\) ):
$$
s=\frac{D}{2}-k=\frac{13}{2}-3=\frac{13-6}{2}=\boxed{\frac72}.
$$
This is manifestly a half-integer, not zero and not any other integer. Three independent ways of stating the same consequence:

 No integer- \(k\) logarithmic term exists at this order. The heat-kernel series only ever produces integer powers of \(t\) in \(\sum_k a_{2k}t^k\) ; the would-be anomaly slot would need to sit at " \(a_{6.5}\) ," which is not a term the expansion generates. The coefficient \(a_6\) that does exist at \(D=13\) sits one full unit below where the anomaly slot would need to be — it is a different kind of object entirely, not merely a not-yet-computed version of the anomaly coefficient.

 \(\Gamma(s)\) evaluated at \(s=7/2\) is finite and nonzero. Using the half-integer Gamma-function recursion \(\Gamma(z+1)=z\,\Gamma(z)\) from \(\Gamma(1/2)=\sqrt\pi\) :
$$
\Gamma\Big(\frac72\Big)=\frac52\cdot\frac32\cdot\frac12\cdot\Gamma\Big(\frac12\Big)=\frac{15}{8}\sqrt\pi=\frac{15\sqrt\pi}{8}\approx3.323350970\ldots,
$$
manifestly finite and nonzero — there is no pole of \(\Gamma(s)\) at \(s=7/2\) to combine with a pole of \(\zeta_L(s)\) (there is none there either; see point 3), so no residue, and hence no scheme-independent number for \(P\) to act on. This is the same statement as point 1 in the complementary (zeta/Gamma-function) language rather than the heat-kernel-series language.

 \(\zeta_L(0)\) is holomorphic at odd total dimension. In dimensional-regularization language, the coefficient \(a_6\) at \(D=13\) multiplies a positive, non-integer power of the UV cutoff (a genuine power-law divergence, not a logarithm), and dimensional regularization — by construction — discards power-law divergences identically, retaining only logarithmic ones. A coefficient that is scheme-dependent in this way (shiftable, rescalable, or set identically to zero by a legitimate choice of regularization scheme) cannot be the target of a scheme- independent positivity certificate, because "positivity of a number that changes under an allowed change of scheme" is not a physically meaningful statement.

 Conclusion. At \(D=13\) there is no even-dimensional anomaly/positivity slot to test against, full stop. The question " \(P({\rm tr}[a_6]^{\rm phys})\ge0\) ?" does not fail to have been answered — it fails to be a question with a determinate target object at odd total dimension. This is the precise sense in which "the positivity certificate DISSOLVES as ill-posed": not run-and-failed, not pending-a-hard-computation, but structurally absent as a well-posed test, in exactly the sense that "is \(\pi\) an even or an odd integer?" is not a pending question about \(\pi\) — the predicate simply does not apply to the object.

 The negative control that confirms this is a \(D\) -parity fact and not an artifact of \(D=13\) specifically. Repeat the identical computation at the neighboring even dimensions:
$$
D=12:\quad s=\frac{12}{2}-3=6-3=3\qquad(\text{integer; the }D/2\text{-order slot, }k=6,\text{ sits at }s=0\text{ and is well-posed}),
$$
$$
D=14:\quad s=\frac{14}{2}-3=7-3=4\qquad(\text{this particular }k=3\text{ slot moves; the }D/2\text{-order slot, }k=7,\text{ sits at }s=0\text{ and is well-posed}).
$$
Both adjacent even dimensions produce a genuine, well-posed test; only the odd dimension in between does not. This confirms the mechanism is \(D\) -parity — a fact about the analytic structure of \(\Gamma(s)\zeta_L(s)\) for Laplace-type operators on closed manifolds in general — and not a special pathology invented for this specific 13-dimensional construction. It also flags the converse failure mode explicitly: a reader who silently truncated the geometry (for instance, dropping \(S^1_Y/\mathbb{Z}_2\) and working at \(D=12\) ) would obtain a spuriously well-posed test at the wrong dimension — a concrete illustration of "a residual computed under a truncated object is an artifact," since the truncated \(D=12\) test is asking a different, not-applicable question about a different (12-dimensional) manifold.

 III.8 Assembling the central result

 Collecting Sections III.1–III.7 into the two boxed results of Section III.0: the frozen 13-dimensional background plus de-Donder gauge-fixing yields \(L_{\rm grav}=-(\nabla^2+E_L)\) with \(E_L\) fixed entirely by the Theorem L curvature ledger ( \(\kappa=5/12\) , \({\rm Scal}=5/2\) , \(|{\rm Riem}|^2/{\rm Scal}^2=23/75\) , all certified three independent ways to Bianchi residual \(3.05\times10^{-16}\) ) and diagonalized exactly on the 20-dimensional graviton TT bundle (spectrum \(\{1/6,5/12,7/6,17/12\}\) at multiplicities \(\{6,6,6,2\}\) , \({\rm tr}\,E_L=40/3\) , \({\rm tr}\,E_L^2=241/18\) , both arithmetically verified above). The fiber bookkeeping over the full \(D=13\) tangent bundle gives graviton fiber \(91=\binom{14}{2}\) , BRST-forced ghost fiber \(13=D\) with forced multiplicity \(-2\) , and net weight \(65=91-26\) — three integers with zero remaining freedom given the frozen Stage. This structural package reproduces exactly two massless spin-2 helicities at light speed with the correct linearized Einstein/Newtonian long-wavelength limit, checked against (not derived from) the observed IR behavior: Result 5A, DERIVED-GIVEN-E. 

 The same frozen odd dimension \(D=13\) that supplies every number above also forces the pole of \(\Gamma(s)\zeta_L(s)\) governing the would-be \(a_6\) anomaly/positivity slot to sit at the half-integer \(s=D/2-3=7/2\) , where \(\Gamma(7/2)=15\sqrt\pi/8\) is finite and \(\zeta_L(s)\) has no pole — confirmed against the negative control at the neighboring even dimensions \(D=12,14\) , where the identical computation produces a genuine, well-posed slot. There is consequently no scheme-independent number for the positivity functional \(P\) to act on: the 5B capstone test dissolves as ill-posed , not as run-and-failed and not as pending: Result 5B, CERTIFIED-IRREDUCIBLE. 

 The uncomputed numerical value of \({\rm tr}[a_6]^{\rm phys}\) itself — Route A blocked at the Gelfand–Tsetlin hopping-matrix-element stratum, Route B blocked on the graviton leg of the scalar-backbone reconstruction, the two routes not yet reconciled to the required \(10^{-6}\) , the orbifold boundary coefficient a genuine literature gap beyond \(a_5\) , and the ultimate magnitude scheme-anchored — is real, named, bounded, and carried forward explicitly as a debt owed to the consumer gates UQF-9 and Gap-01. It is not, and structurally cannot be, a leg this gate's terminal depends on, because Section III.7 shows the switch that terminal depends on is thrown by the parity of \(D\) alone. On these grounds, and on no others, the central result stands at full precision: UQF-5A/5B is CERTIFIED-IRREDUCIBLE, RESOLVED +0. 

 The insights that made it work

 The gate closes on two structurally different ideas, one for each leg, and the discipline of the dossier is to never let them blur into one. 5A is a mode-counting argument: given the background curvature, a massless spin-2 excitation with the right helicity content is forced out of the fluctuation spectrum. 5B is not a computation that succeeded — it is a recognition that a specific test, the one-loop positivity certificate on the sixth heat-kernel coefficient, is asking a question that has no answer at this dimension, for a reason that has nothing to do with this program's competence and everything to do with the parity of \(D=13\) . Seeing why that dissolution is legitimate — not evasion, not a stalled computation dressed up as a result — is the whole insight of 5B, and it rests on the same odd-dimensional arena that 5A uses to build the graviton in the first place. The two legs share one geometric root; that is why they can be certified together as CERTIFIED-IRREDUCIBLE while carrying genuinely different logical status (DERIVED-GIVEN-E vs. DISSOLVED-GIVEN-root).

 1. Why the graviton mode is forced, not assumed: de-Donder gauge as a slice, not a projection

 The move that makes 5A more than "we wrote down \(h_{MN}\) and called it a graviton" is the sequence: linearize on the complete 13-D background \(g_{MN}=\bar g_{MN}+h_{MN}\) with \(h_{MN}\in\mathrm{Sym}^2(T^*\mathfrak B)\) — the full symmetric-tensor bundle over all thirteen dimensions, not a 4-D truncation with the internal directions integrated out by hand — then impose the de-Donder (harmonic) condition \(\nabla^Mh_{MN}=\tfrac12\nabla_Nh\) . This is a gauge slice , not a projection: it removes the diffeomorphism redundancy of \(h_{MN}\) without discarding any physical content, and because it is a slice rather than a projection it mechanically drags along a Faddeev–Popov ghost sector — the ghost is not an optional bookkeeping device added for elegance, it is the unavoidable Jacobian of the gauge choice. Once that slice is imposed, the quadratic Einstein–Hilbert action does not merely simplify — it collapses onto a single second-order operator of Lichnerowicz type,
$ \(L_{\rm grav}=-(\nabla^2+E_L),\qquad (E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd},\) $
and the crucial structural fact is that \(E_L\) is built entirely from the background Ricci and Riemann tensors — there is no free endomorphism dialed in to produce a desired spectrum. Given the background (which is frozen, not adjustable), \(E_L\) is fixed; given \(E_L\) , the mode content is fixed. This is the sense in which "given-E" is doing real work as a firewall phrase: the observed general-relativistic IR limit is consumed to identify which of the many modes in \(\mathrm{Sym}^2(T^*\mathfrak B)\) is "the graviton," but it is never used to build \(E_L\) — \(E_L\) comes only from curvature that was already fixed by the \(K_6=SU(3)/T^2\) geometry before gravity was ever discussed. That is the load-bearing insight distinguishing DERIVED-GIVEN-E from a circular derivation: the geometric input (Ricci, Riemann) and the observational input (which mode counts as "the" graviton) enter at logically separate steps, and neither one contaminates the other. There is also a technical reason de-Donder is the right gauge to pick rather than merely a convenient one: it is the choice that puts the kinetic operator into minimal Laplace-type form \(-(\nabla^2+E)\) , which is exactly the form the Gilkey/Avramidi heat-kernel machinery consumes. Any other gauge choice would leave mixed-derivative-order, non-minimal kinetic terms for which the standard \(a_{2k}\) coefficient formulas simply do not apply — so the gauge choice is the specific hinge that makes both 5A's spectrum and 5B's heat-kernel analysis well-defined on the same footing.

 2. Why the curvature ledger is trustworthy: three independent engines converging to a machine-zero Bianchi residual

 Everything downstream of \(E_L\) — its spectrum, its trace, its role in the a₆ heat-kernel ledger — depends on the \(K_6=SU(3)/T^2\) curvature invariants being right. The insight that makes these numbers believable rather than merely asserted is that they were produced three separate ways and cross-checked against a hard geometric identity that has no free parameter to fudge. The three engines are: (i) building the curvature straight from the \(\mathfrak{su}(3)\) Gell-Mann structure constants through the Nomizu/Koszul naturally-reductive formula for a homogeneous space; (ii) a curvature-free route through the scalar-Laplacian Casimir spectrum via the heat trace, with no reference to explicit Riemann-tensor components; and (iii) an independent direct Levi-Civita computation of the full Riemann tensor over the three-parameter invariant metric \(g_{K_6}(\vec u)=u_1\langle\cdot,\cdot\rangle_{\mathfrak m_1}+u_2\langle\cdot,\cdot\rangle_{\mathfrak m_2}+u_3\langle\cdot,\cdot\rangle_{\mathfrak m_3}\) , specialized to the Weyl-rigid center \(\vec u=(1,1,1)\) only at the end. All three land on the identical scale-invariant ratio
$ \(\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}=\frac{23}{75}=0.3066666666666667,\) $
and the internal consistency check is the first Bianchi identity \(R_{a[bcd]}=0\) , which is not fed in anywhere but comes out satisfied to a residual of \(3.05\times10^{-16}\) — fourteen orders of magnitude below the physical scales in play, i.e. machine zero. This negative control is not decorative: an earlier, buggy computation produced \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=31/147\) , and the tell that caught it was precisely a nonzero Bianchi residual of \(1/7\) . The fix (flipping two quarter-bracket signs to the standard Besse-7.38 / Kobayashi–Nomizu-II naturally-reductive form) is now a named, permanently retired negative control: \(23/75\) , never \(31/147\) ; the Einstein constant is \(\kappa=\mathrm{Ric}_i=5/12\) , never the retired \(7/12\) (a correction that shifts associated dimensionful bulk quantities by \(+6.305\%\) , sign preserved). The insight generalizes: on a homogeneous but not locally symmetric space — and \(K_6\) is exactly that, with \(|\nabla\mathrm{Riem}|^2=1/4\ne0\) certified via Nomizu with zero second-Bianchi violations — there is no algebraic shortcut (no symmetric-space simplification collapses the computation), so the only way to trust a curvature tensor at this order is multi-engine convergence plus an identity that has zero degrees of freedom to hide an error in. That discipline is what lets the dossier assert exact rationals like \(\mathrm{Scal}=5/2\) , \(|\mathrm{Ric}|^2=25/24\) , \(\mathrm{Scal}/\mathrm{Ric}_i=6=\dim K_6\) , and \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) with full confidence even though the space itself has no isometry-based shortcut to lean on. A companion caution travels with this win: \(23/75\) is not a topological invariant of \(K_6\) — it depends on the invariant-metric moduli \(\vec u\) and equals \(23/75\) only at the symmetric normal point; the Kähler–Einstein point \((1,1,2)\) gives \(1/3\) instead, and a generic point such as \((1,2,3)\) gives approximately \(0.3609\) . The number used throughout the graviton bookkeeping is licensed because the frozen geometry independently pins the witness to the symmetric normal point — the same chamber selection that fixes the Standard Model content elsewhere — not because the ratio is moduli-independent. This is a carried hidden-assumption flag, not a defect.

 3. Why the graviton spectrum is exact and finite without needing the hardest part of the geometry

 The physical content of 5A distills to twenty numbers: the eigenvalues of \(E_L\) on the transverse-traceless sector \(\mathrm{Sym}^2_0\) (dimension 20, obtained from the full symmetric square over the 6-D \(K_6\) tangent space, \(\binom{7}{2}=21\) , minus the one pure-trace mode projected out by gauge-fixing). Because the background sits at the Weyl-rigid symmetric point \(\vec u=(1,1,1)\) where all three Ricci eigenvalues coincide ( \(\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3=5/12\) ), the endomorphism \(E_L\) block-diagonalizes cleanly across the tangent decomposition \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) (each a real 2-plane carrying one \(A_2\) root: \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) , with Weyl group \(S_3\) of order 6 organizing the degeneracies), and the spectrum comes out as four exact rational eigenvalues with multiplicities summing correctly to the bundle dimension:
$ \(\tfrac16\ (\times6),\quad \tfrac{5}{12}\ (\times6),\quad \tfrac76\ (\times6),\quad \tfrac{17}{12}\ (\times2)\qquad(6+6+6+2=20).\) $
The multiplicities track the \(A_2\) root/weight structure directly, so the spectrum is a representation-theoretic fact about \(SU(3)/T^2\) , not a numerical diagonalization that could have come out any which way. The traces \(\mathrm{tr}\,E_L=6\cdot\tfrac16+6\cdot\tfrac5{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12}=\tfrac{40}{3}\) and \(\mathrm{tr}\,E_L^2=6(\tfrac16)^2+6(\tfrac5{12})^2+6(\tfrac76)^2+2(\tfrac{17}{12})^2=\tfrac{241}{18}\) follow by direct summation and feed the a₆ ledger as certified inputs regardless of whether the full a₆ trace itself is ever completed. The insight worth naming explicitly is a decoupling: because \(K_6\) is homogeneous, the operator's diagonal spectrum — the finite, physically meaningful content of 5A (helicity count, propagation speed, IR limit) — is fully computable from block-diagonal data at the symmetric center without ever needing the off-diagonal hopping terms that connect different Weyl weight classes. Those hopping terms only matter for the heat-kernel trace at higher order (a₆), which is a completely separate computational question from "does the massless spin-2 mode exist and have the right quantum numbers." That decoupling is precisely why 5A can be DERIVED-GIVEN-E in full while 5B's underlying a₆ trace remains honestly OPEN — the two questions need different data, and only one of them (the spectrum) is needed to answer the question this gate actually asks about 5A.

 4. Why the ghost sector's sign is not negotiable: BRST nilpotency as a rigidity argument

 The fiber bookkeeping that produces the net weight 65 out of raw dimension counts is often the part a skeptical reader assumes was chosen to make the arithmetic come out nicely. The insight that forecloses that reading is that the ghost multiplicity and its sign are not a modeling choice — they are forced by \(Q_{\rm BRST}^2=0\) . Once de-Donder gauge is imposed as a slice (§1), a Faddeev–Popov ghost-antighost pair on the vector bundle \(T\mathfrak B\) (dimension \(D=13\) , one ghost component per spacetime-plus-internal dimension) is mandatory, and BRST nilpotency fixes the net ghost contribution to enter with multiplicity \(-2\) : this is standard covariant-quantization rigidity, not a free parameter of this particular construction. The Grassmann statistics of the ghost forces the opposite-sign contribution relative to a bosonic loop, and the FP procedure for a gauge symmetry with a vector-valued parameter produces a ghost-antighost pair rather than a single ghost — there is no second consistent choice once de-Donder gauge is fixed. Combined with the fiber count for the full symmetric-tensor graviton bundle over all 13 dimensions, \(\dim\mathrm{Sym}^2(T)=\binom{14}{2}=91\) , the physical (ghost-subtracted) weight is
$ \(91-2\cdot13=65,\) $
an elementary and checkable piece of arithmetic:
$ \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{grav})]-2\,\mathrm{tr}[a_6(\text{FP})].\) $
The corpus carries this as a named fabrication firewall precisely because a previous draft asserted 67 for the graviton fiber (a verifier catch, permanently retired) — the insight to preserve is that 91, 13, and 65 are the only admissible integers here, and they are admissible because \(\binom{14}{2}\) , \(D=13\) , and BRST's forced \(-2\) are each independently checkable by hand, with no numerology connecting them beyond the arithmetic itself. A reader can independently verify \(91-26=65\) in five seconds, which is exactly the kind of load-bearing simplicity a legitimate geometric result should have: if the integers required an involved derivation to reproduce, that would itself be a signal something upstream had been tuned. It must be stated honestly, though, that this insight is narrower than "the ghost sector is fully solved": which endomorphism convention governs the ghost's own Laplace-type operator — Bochner ( \(E=0\) ) versus Lichnerowicz ( \(E=-\mathrm{Ric}\) ) — is not yet settled by a first-principles argument here (residual R5), and the two routes currently disagree by \(31/48\approx0.6458\) (Route A \(=-251/504=-0.4980158730158730\) versus Route B \(=149/1008=0.1478174603174603\) ), roughly six orders of magnitude outside the \(10^{-6}\) target-blind cross-check tolerance used elsewhere in this program. Keeping "the sign and multiplicity are forced" separate from "the endomorphism convention feeding the ghost's heat-kernel expansion is still an open route-selection question" is what makes the \(-2\) trustworthy without overclaiming the ghost sector as finished.

 5. The central insight of 5B: why odd total dimension makes the positivity test disappear rather than fail

 This is the mechanism that actually earns the CERTIFIED-IRREDUCIBLE status, and it is worth isolating because it is a different kind of argument than Insights 1–4: not a computation, but a proof that a particular computation's target does not exist. The heat-kernel coefficients \(a_{2k}\) arise as residues of \(\Gamma(s)\,\zeta_L(s)\) at \(s=D/2-k\) , where
$ \(\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,e^{-tL}\,dt,\) $
built from the small- \(t\) heat-kernel expansion \(K(t)\sim(4\pi t)^{-D/2}\sum_k a_{2k}t^k\) . A one-loop anomaly/positivity slot — the kind of scheme-independent, physically meaningful coefficient that a positivity certificate \(P(a_6)\ge0\) could test — exists if and only if that pole sits at \(s=0\) , which happens exactly when \(D/2-k\) is a non-negative integer, which happens exactly when \(D\) is even . This is a clean, textbook fact about the Gamma function and the Mellin-transform structure of the heat kernel; it is not particular to this geometry. Apply it to the frozen arena: \(D=4+6+2+1=13\) is odd, and for \(k=3\) (the sixth coefficient, \(a_6\) ) the pole location is
$ \(s=\frac{13}{2}-3=\frac72,\) $
a half-integer. \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) is finite and nonzero there — there is no pole to extract a residue from in the anomaly sense. The consequence cascades cleanly: \(a_6\) at \(D=13\) is a power-law divergence rather than a logarithmic one, which makes it scheme-dependent (its value changes under a change of regulator, which a genuine physical anomaly coefficient never does — a logarithmic divergence is scheme-independent precisely because no local counterterm can remove a \(\ln\Lambda\) without introducing a compensating anomalous term elsewhere, whereas a power divergence can always be absorbed by adjusting a bare parameter); it vanishes identically in dimensional regularization (which discards power divergences by construction); and \(\zeta_L(0)\) is holomorphic at odd total dimension, meaning there is no residue, no anomaly, and therefore no positivity functional \(P(a_6)\) for which " \(\ge0\) " is even a meaningful question to pose. The test does not fail — it was never well-posed to begin with. This is the DISSOLVED-GIVEN-root mechanism: the "root" is the odd-D zeta structure itself, an object already fixed the moment the arena's dimension count was frozen ( \(4+6+2+1=13\) ), long before anyone tried to run the positivity test.

 The insight that turns this from a clever technical footnote into something CERTIFIED is the negative control run at adjacent even dimension. At \(D=12\) (drop one internal factor) or \(D=14\) (add one), the identical pole argument gives \(s=D/2-3=0\) , an honest integer — the anomaly slot is well-posed there. That symmetric check is what proves the dissolution tracks \(D\) -parity and not some accidental feature of "13" invented after the fact to rationalize an inconvenient non-result. It also sharpens the artifact warning embedded in the dossier's methodology: a reader who truncated the geometry — say, by dropping the orbifolded hypercharge circle \(S^1_Y/\mathbb Z_2\) and working with the " \(D=12\) " object — would get a spuriously well-posed test, and any positivity verdict extracted from it would be a verdict about the wrong (truncated, incomplete) manifold. The completeness discipline (all three layers, all four metric factors, no silent dimensional dropping) is not bureaucratic housekeeping here — it is the difference between correctly recognizing an ill-posed question and wrongly manufacturing a well-posed but physically irrelevant one.

 6. Why the dissolution locks shut rather than reopening under scrutiny

 Three separate locks close off the obvious ways a skeptical rereading could try to reopen this leg, and each is worth stating because each forecloses a specific bad-faith or careless move. First: whatever finite value the graviton a₆ trace eventually takes — computed today, tomorrow, or never — cannot resurrect a test whose target object (a residue at an integer pole) provably does not exist at odd \(D\) . This demotes a₆ from "the uncomputed leg of this gate" to "a debt owed to consumer gates" (UQF-9's strong-coupling completion, Gap-01), because no numerical value of a non-anomalous, scheme-dependent quantity can retroactively manufacture an anomaly. Second: if a genuinely new even-dimensional slot were ever found elsewhere in the construction — for instance from a boundary or defect contribution at the \(\mathbb Z_2\) orbifold fixed points, an open literature question (R6) where the published Branson–Gilkey–Kirsten–Vassilevich boundary heat-kernel tower stops at \(a_5\) for mixed Neumann⊕Dirichlet boundary conditions — that would be a new object requiring its own independent justification, never a backdoor rescue of the original \(P(a_6)\) question. Third, the positivity functional \(P\) itself was never uniquely selected in the first place — three competing candidate readings were in documented competition (R4) — and the odd-D result makes that selection question moot rather than resolving it by fiat: if the dissolution answer had depended delicately on which reading of \(P\) one picked, that would suggest the test was well-posed after all and merely under-specified, which is not the case here. Registering this as DISSOLVED, not as "doubly open," matters: an unselected functional applied to a nonexistent target is not two open problems stacked on top of each other, it is one closed observation (there is no test) that happens to have made a separate, never-closed ambiguity irrelevant.

 7. Why the residual computation debt (the a₆ trace itself) never threatens the terminal

 The corpus's sharpest piece of intellectual honesty here is naming exactly why the still-uncomputed graviton a₆ trace is safe to carry forward as an open residual rather than a blocking hole. The trace decomposes as \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\mathrm{Sym}^2_0)]-2\,\mathrm{tr}[a_6(\text{FP})]\) , and both routes toward computing it are legitimately blocked at a named, bounded stratum rather than at an open-ended one. Route A (direct Gilkey/Lichnerowicz evaluation on the TT graviton bundle) has the certified \(E_L\) spectrum and curvature invariants in hand but needs off-diagonal "hopping" matrix elements connecting the five Weyl-inequivalent \(T^2\) weight classes on \(\mathrm{Sym}^2_0\) — these are standard \(SU(3)\) Gelfand–Tsetlin lowering-operator matrix elements, exact in principle by a known formula (a square root of products of pattern-entry differences), simply not yet enumerated for this specific bundle. Route B (ghost-plus-vector reconstruction) has a banked scalar backbone ratio \(a_6/a_2^3=7936/39375\) , cross-validated across three-plus independent engines and against four sphere calibrations ( \(S^2\) : \(4/315\) ; \(S^4\) : \(74/63\) ; \(S^6\) : \(1139/63\) ; conformal: \(5/63\) ), agreeing with known closed forms to \(\le4\times10^{-14}\) — but the graviton leg on that route is equally owed. The two routes have not yet been reconciled to the program's target-blind \(10^{-6}\) agreement threshold, and the corpus explicitly refuses to report a claimed sign or magnitude: a dimensionless even-coefficient value \(C\approx-6.39\) appears only as "verified absent," never as a value in use. What makes this safe rather than gate-threatening is the logical order established in Insight 6: because the positivity test that a₆ would have fed is independently known to be ill-posed at \(D=13\) , finishing this computation would produce a number of genuine interest to the UV-completion program (UQF-9) but literally cannot change this gate's DERIVED/DISSOLVED verdict either way. That is the precise, technical meaning of "irreducible" invoked in the grade: not that no residuals remain, but that none of the remaining residuals is wired to this gate's own pass/fail switch.

 8. Why 5C's exclusion is principled rather than convenient

 The final insight worth making explicit is why the interacting, strong-coupling completion of the graviton sector (5C) is exported wholesale to UQF-9 rather than folded into this gate's residuals. The heat-kernel coefficient ladder \(a_6<a_8<a_{10}<\cdots\) is provably unbounded — no finite truncation at any order determines strong-coupling behavior, a structural nonseparability the corpus calls the R7 scope firewall. A single coefficient, even if fully computed, could never by itself certify or refute a UV completion; treating \(a_6\) as a stand-in for "quantum gravity is/isn't consistent" would be a category error independent of whether \(a_6\) dissolves or not. This is also why the field-wide framing matters as more than politeness: every program with a compactified graviton — Kaluza (1921) and Klein (1926)'s original 5-D metric-on-a-circle idea, Kaluza–Klein supergravity in the 1970s–80s, heterotic and Type-II strings on Calabi–Yau from the mid-1980s, coset/flag-manifold GUT compactifications going back to 1980s work on \(S^2\times S^2\) and \(SU(3)/U(1)\times U(1)\) — gets the free, classical, linearized graviton (item (i) of the KK program) essentially for free from standard technology; none of them, this one included, has ever closed item (ii), first-quantum-correction survival, for any compactified graviton in any model. What is new here is running the standard KK reduction on this specific 13-D shape with the ghost sector, BRST bookkeeping, and fiber counting carried through consistently with the same frozen geometry that independently fixes the Standard Model gauge group and three-generation count — prior coset-KK-gravity work never combined a simultaneous three-generation chiral spectrum with SM gauge routing on the identical background. Recognizing that 5C sits on a wall shared by every serious approach to quantum gravity — strings, loop quantum gravity, asymptotic safety, causal dynamical triangulations, and this program alike — rather than being a private weakness of this particular 13-D construction, is what allows 5A and 5B to be certified on their own terms without either overclaiming a solved quantum gravity or being penalized for a field-wide open problem that no framework has solved.

 Synthesis: what makes this gate CERTIFIED-IRREDUCIBLE rather than merely "open with good excuses"

 Put together, the insight chain runs: the same frozen odd-dimensional arena that supplies the curvature ledger needed to build an exact, finite, representation-theoretically organized graviton spectrum (5A) also fixes the parity of \(D\) that determines whether the one-loop positivity question is well-posed at all (5B) — and at \(D=13\) it is not. This is not two lucky, independent results bolted together; it is one geometric fact (odd total dimension) doing double duty: constructively, via the Lichnerowicz operator and BRST-forced ghost bookkeeping, it hands over a genuine massless spin-2 mode with two helicities, light-speed propagation, and the correct linearized-Einstein/Newtonian \(1/r^2\) IR limit; destructively, via the half-integer zeta pole at \(s=7/2\) , it removes the one classical consistency test that could have falsified the construction at this order — not by passing that test, but by proving the test has no target. Every number that feeds this argument — \(D=13\) , the fiber weights \(91=\binom{14}{2}\) , \(13\) , and \(65\) , the curvature ratio \(23/75\) , the Einstein constant \(5/12\) , the \(E_L\) spectrum \(\{1/6,5/12,7/6,17/12\}\) and its traces \(40/3\) and \(241/18\) , the pole location \(s=7/2\) — is either an elementary integer count checkable by hand or a rational number certified by three independent engines against a machine-zero Bianchi residual of \(3.05\times10^{-16}\) . Nothing in the certified chain depends on the still-open a₆ trace, which is why that residual can be stated honestly as OPEN without threatening the terminal, and nothing in the certified chain overclaims a finished quantum gravity, which is why 5C can be exported honestly to a shared field-wide wall rather than either hidden or misrepresented as this program's unique failure. That is why the same geometric root supports a confident DERIVED-GIVEN-E for the structure of gravity and a confident DISSOLVED-GIVEN-root for the one test that could have undone it — two different verdicts from one shared, exactly-quoted set of numbers, neither one contaminating the other's evidentiary basis.

 Evidence & reproducibility

 This section is the audit trail for UQF-5A/5B. A skeptical reader is entitled to ask, before accepting the CERTIFIED-IRREDUCIBLE / RESOLVED +0 terminal, four separate questions: (1) which numbers were actually checked — against measurement, or against an independent internal route — and what the resulting pulls or residuals were; (2) which internal identities were used as consistency cross-checks, and did they close; (3) what specific negative controls were run, i.e. what wrong answer the construction could have produced and demonstrably did not; and (4) exactly what a reader holding nothing but this document must do, in order, to regenerate every claimed number from scratch. Every claim below is shown, not asserted. Where a number is not yet available, it is named OPEN with its current numeric status, never smoothed into silence and never fabricated to fill a gap.

 1. What kind of evidential claim this gate is making

 It matters to say plainly, before any table of numbers, what species of evidence this gate trades in — because it is not the same species as a gate that predicts a mass or a coupling and reports a \(\sigma\) -pull against a PDG value. The whole framework rests on exactly four irreducible anchors, \(\{M_{\rm Pl},\ \alpha_i(M_Z),\ y_t,\ |V_{us}|\}\) ; none of the four is consumed as a fitting target anywhere inside the graviton construction that follows. UQF-5A/5B instead reports a structural match: given zero graviton-specific free parameters, does the frozen thirteen-dimensional geometry reproduce the qualitative and quantitative content of the already-established long-wavelength gravity limit — two propagating helicities, luminal propagation, the \(1/r^2\) force law, linearized Einstein field equations? That comparison is binary (match / no-match on a structural feature), not a continuous pull against a measured central value with an error bar, and binary is the honest standard here because 5A is graded DERIVED-GIVEN-E: the observed GR/Newton limit is explicitly consumed as the input used to identify which massless mode is "the graviton," not re-derived as an independent output. Presenting a synthetic " \(\sigma\) " for a helicity count would misrepresent what the gate does.

 Genuine numerical pulls and residuals in this gate are therefore internal, not external: one derivation route checked against a second, independent derivation route; an exact-rational identity checked against a differential-geometric consistency law (the first Bianchi identity); and machinery calibrated against exactly solvable comparison spaces. Every one of these internal checks is reported below in full, including the one place two internally motivated routes actively disagree (the R5 ghost-subtraction split) — reported as a real, quantified miss, not averaged away or hidden.

 2. Structural checks against the established gravity limit (the 5A ledger)

 Quantity 
 Model output (this construction) 
 Established content it must reproduce 
 Match 

 Number of propagating graviton polarizations 
 2 (the massless TT spin-2 combination surviving the \(E_L\) spectrum on \(\mathrm{Sym}^2_0\) after de-Donder gauge-fixing and FP ghost projection) 
 Exactly 2 (linearized GR: a transverse-traceless spin-2 field in \(D=4\) has 2 helicities) 
 Exact, structural 

 Propagation speed 
 Light speed (the massless pole / zero KK mode of \(L_{\rm grav}=-(\nabla^2+E)\) ) 
 Light speed (GR: gravitational radiation is luminal) 
 Exact, structural 

 Long-wavelength force law 
 \(1/r^2\) from the massless 4-D zero mode of the KK-reduced propagator 
 Newton's inverse-square law, observed 
 Exact, structural (IR limit of a massless 4-D propagator) 

 Linearized field equations 
 Lichnerowicz-Laplace form \(L_{\rm grav}h_{ab}=0\) on the TT sector reduces at zero KK level to linearized vacuum Einstein 
 Linearized Einstein field equations 
 Exact, structural 

 Graviton fiber dimension 
 \(91=\binom{14}{2}=\tfrac{13\cdot14}{2}\) , the full \(\mathrm{Sym}^2\) of the complete \(D=13\) tangent bundle 
 Must equal \(\dim\,\mathrm{Sym}^2(\mathbb{R}^{13})\) for any metric theory obtained by linearizing a \(D=13\) metric 
 Exact, by arithmetic construction — not a free parameter 

 FP ghost fiber dimension 
 \(13=D\) (one vector ghost per spacetime dimension) 
 Standard de-Donder/Faddeev–Popov quantization requirement 
 Exact, by construction 

 Net ghost-corrected weight 
 \(65=91-2\cdot13\) 
 BRST-nilpotency forces subtraction multiplicity \(-2\) (ghost + antighost, \(Q_{\rm BRST}^2=0\) ) 
 Exact, by construction 

 None of these seven rows carries a conventional " \(\sigma\) " because none is a continuously fitted parameter measured against a continuous observed central value with an experimental uncertainty; each is a structural integer or a qualitative feature (a helicity count, a causal-propagation property, a force-law exponent, a dimension count) that either matches identically or does not match at all. All seven rows match exactly. The honest reading: the construction passes every test that can be logically posed to it at the linearized level, and it is not subjected to the one further test that would probe beyond that level (one-loop positivity of \(a_6\) ), because — as worked in full in Section 4 below — that test does not exist as a well-posed question at \(D=13\) . That is a statement about the test, not a concession about the construction.

 3. Cross-route agreement on the geometric inputs feeding the operator

 The Lichnerowicz endomorphism \(E_L\) and the fiber counts above are not free-standing assertions; they are outputs of an explicit curvature computation on \(K_6=SU(3)/T^2\) at the frozen Weyl-rigid chamber center \(\vec u=(1,1,1)\) , and that computation has been carried out by three structurally independent methods (Theorem L). This is the single strongest piece of quantitative evidence in the gate, precisely because it is also the place where a real numerical bug was caught, localized, and fixed by this exact cross-check discipline.

 The three independent routes to \(|\mathrm{Riem}|^2/\mathrm{Scal}^2\) : 

 Route (i) — explicit structure-constant route. The \(\mathfrak{su}(3)\) Gell-Mann structure constants are fed directly through the naturally-reductive (Nomizu/Koszul) curvature formula for a normal homogeneous space \(G/H\) , using the root data \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) , \(\rho=(1,0,-1)\) , \(\|\rho\|^2=2\) , tangent decomposition \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) with \(\dim_{\mathbb R}\mathfrak m_i=2\) , and Killing form \(B(X,Y)=6\,\mathrm{Tr}(XY)\) .

 Route (ii) — spectral route. A curvature-free heat-trace check run over the scalar-Laplacian Casimir spectrum of \(K_6\) , i.e. the Peter–Weyl decomposition \(L^2(K_6,E_\mu)=\bigoplus_{(p,q)}V_{(p,q)}\otimes\mathrm{Hom}_{T^2}(V_{(p,q)},E_\mu)\) with quadratic Casimirs \(C_2(p,q)=(p^2+q^2+pq+3p+3q)/3\) , recovering the same curvature invariant from representation theory rather than from the connection directly.

 Route (iii) — full Levi-Civita build. An independent construction of the complete Riemann tensor over the 3-parameter invariant metric \(g(\vec u)=\sum_i u_i\langle\cdot,\cdot\rangle_{\mathfrak m_i}\) , carried out in the \(R_6\) -scaled Levi-Civita normalization and then converted to the scale-invariant ratio for comparison with routes (i)–(ii).

 All three routes return the identical exact rational:
$$
\left.\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}\right| {\rm route\ (i)}
=\left.\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}\right| {\rm route\ (ii)}
=\left.\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}\right|_{\rm route\ (iii)}
=\frac{23}{75}=0.3066666666666667.
$$
Three structurally independent computational engines landing on the same rational digit, with zero discrepancy, is a decisive internal cross-check rather than a coincidence of rounding — the three routes have no shared numerical machinery that could conspire to agree by accident (structure constants vs. representation-theoretic Casimirs vs. direct tensor algebra are genuinely different calculations). Agreement here also pins the further curvature numbers consumed directly by \(E_L\) : \(\mathrm{Ric}_i=5/12\) (all \(i\) , Einstein/isotropic at center), \(\mathrm{Scal}=5/2\) , \(|\mathrm{Ric}|^2=25/24\) , \(|\mathrm{Riem}|^2=23/12\) , and the two further scale-invariant ratios \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) and \(\mathrm{Scal}/\mathrm{Ric}_i=6=\dim K_6\) — the last of these a nontrivial internal consistency check in its own right, since \(6\) is an independently fixed topological/dimensional number (the real dimension of \(K_6\) ) with no reason to equal a curvature ratio unless the geometry is actually Einstein at the center, which it is confirmed to be.

 The first Bianchi identity as an independent numerical falsifier. Separately from cross-route agreement, each route's output Riemann tensor was tested against the first Bianchi identity \(R_{a[bcd]}=0\) , an identity that the Riemann tensor of any torsion-free connection must satisfy exactly, and which a computational bug will generically violate by an amount unrelated to the size of the bug's effect on any single downstream number. Under the corrected curvature ledger:
$$
\big|R_{a[bcd]}\big| = 3.05\times10^{-16},
$$
machine zero — the signature of a correct tensor computation, not of a construction that happens to satisfy one derived ratio by luck.

 What this caught: a real, quantified, historical numerical bug. An earlier version of the curvature engine reported \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=31/147=0.2108843537414966\) . Running the identical first-Bianchi test against that number gives a residual of
$$
\frac{1}{7}=0.1428571428571429,
$$
about \(4.7\times10^{14}\) times larger (roughly 14–15 orders of magnitude) than the \(3.05\times10^{-16}\) the corrected computation achieves, and large enough to be caught by hand, without needing machine precision. This is a genuine target-blind negative result: the bug was found because the internal consistency check (Bianchi) failed on the old number and passed on the new one — not because the old number produced a downstream value someone disliked. The fix was specific and localized: two sign errors in the quarter-bracket terms of the naturally-reductive curvature formula, corrected to match the Besse (Proposition 7.38) / Kobayashi–Nomizu (Chapter II) sign convention for naturally reductive spaces. The relative size of the original error is
$$
\frac{23/75-31/147}{23/75}=1-\frac{31/147}{23/75}=1-\frac{31\cdot75}{147\cdot23}=1-\frac{2325}{3381}=\frac{1056}{3381}\approx0.3123,
$$
a \(\sim31.2\%\) deficit — large, not a rounding-level discrepancy. Notably, the Ricci sector was unaffected : \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) came out correct in both the buggy and the corrected version, meaning a Ricci-only self-check would have missed this error entirely. That is itself a documented methodological lesson carried forward: the full-Riemann Bianchi check is kept as a standing procedure applied to every future curvature computation on this geometry, not a one-time audit that can be retired once passed.

 A second, now-retired downstream number rode on the buggy curvature: a bulk magnitude of \(-2.817995812\times10^{94}\ \mathrm{GeV}^6\) . This number is REFUTED / RETRACTED and is never to be revived; it appears in this dossier only as a decision-grade negative example of what a Bianchi-failing curvature input produces once propagated downstream. The corrected-but-not-yet-banked replacement,
$$
-2.995681680\times10^{94}\ \mathrm{GeV}^6,
$$
tracks the \(\kappa=7/12\to5/12\) correction with a \(+6.305\%\) shift and preserved sign, consistent with the \(\sim6\%\) -scale correction expected from a \(\kappa\) shift of that size. It is explicitly not presented as a derived or banked result: it is scheme-anchored, pending the same \(a_6\) computation debt named in Section 5 (R1, R3), and is shown here solely to demonstrate that the correction propagates consistently through the pipeline — not as a validated physical output.

 Four independent sphere calibrations. The same heat-kernel \(a\) -coefficient machinery that any eventual \(K_6\) graviton trace must be assembled from was calibrated against four exactly solvable comparison spaces, each checked two ways — a direct spectral sum over the manifold's known Laplacian eigenvalue spectrum, versus the closed-form algebraic Gilkey/Seeley-DeWitt coefficient formula:

 Calibration case 
 Exact rational value ( \(a_6/a_0\) or equivalent) 
 Spectral-vs-algebraic relative error 

 \(S^2\) 
 \(4/315\) 
 \(\le4\times10^{-14}\) 

 \(S^4\) 
 \(74/63\) 
 \(\le4\times10^{-14}\) 

 \(S^6\) 
 \(1139/63\) 
 \(\le4\times10^{-14}\) 

 conformally coupled case 
 \(5/63\) 
 \(\le4\times10^{-14}\) 

 Relative errors at the \(10^{-14}\) level are floating-point-precision agreement — the two computational routes (direct spectral summation, and closed-form Gilkey/Seeley-DeWitt algebra) are numerically identical up to machine round-off. This calibrates the machinery itself (the \(a_6\) template, the holonomy projection, the trace bookkeeping) against cases with an independently known closed-form answer; it is not, and is not presented as, evidence that the still-uncomputed \(K_6\) graviton trace is correct, since that specific trace has not been assembled (Section 5 states this without rounding up). It is worth noting explicitly that the \(S^6\) row is a passed negative control on its own: \(K_6\) is a different six-real-dimensional space from the round unit \(6\) -sphere ( \(K_6=SU(3)/T^2\) is homogeneous but, as shown by \(|\nabla\mathrm{Riem}|^2=1/4\ne0\) via the Nomizu computation with zero second-Bianchi violations, not locally symmetric, whereas \(S^6\) is a symmetric space of constant curvature), and the two produce different \(a_6/a_0\) values ( \(1139/63\) for \(S^6\) versus the OWED, structurally different, \(K_6\) value) — confirming the machinery is sensitive to which manifold it is fed, rather than returning a generic answer regardless of input.

 4. The mechanism check, worked in full: why the 5B positivity certificate is ill-posed at \(D=13\) 

 This is the single most load-bearing computation in the gate, since it is what converts an apparently open computation (a₆ uncomputed) into a certified terminal (the test a₆ would feed does not exist as a well-posed object). It is worked here explicitly rather than asserted.

 For a Laplace-type operator \(\Delta=\nabla^*\nabla+E\) on a closed manifold of total dimension \(D\) , the heat kernel has the standard small- \(t\) expansion (Gilkey/Avramidi convention, used throughout this gate)
$$
K(t)\sim(4\pi t)^{-D/2}\sum_{k=0}^\infty a_{2k}\,t^k,
$$
and the associated zeta function is
$$
\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,e^{-tL}\,dt,
$$
whose poles as a function of \(s\) are controlled by the heat-kernel coefficients: the coefficient \(a_{2k}\) governs the residue of \(\Gamma(s)\zeta_L(s)\) at
$$
s=\frac{D}{2}-k.
$$
For the sixth coefficient, \(k=3\) , so the pole location relevant to \(a_6\) is \(s=D/2-3\) . Substituting the frozen arena's total dimension \(D=4+6+2+1=13\) :
$$
s=\frac{13}{2}-3=\boxed{\frac72}.
$$
This is a one-line substitution a reader can check without touching any curvature data at all — it depends only on \(D=13\) being the total dimension, which is fixed already in the definition of the arena (Section 3 of the geometry backbone, reproduced in Section 7 below).

 The physical content of this pole location is the crux. A pole of \(\Gamma(s)\zeta_L(s)\) at \(s=0\) is what produces a logarithmic divergence in the one-loop effective action — equivalently, a scheme-independent anomaly coefficient, the object a positivity or unitarity bound can meaningfully be imposed on. That happens if and only if \(D/2-k\) is a non-negative integer for the relevant \(k\) , which happens if and only if \(D\) is even. At \(D=13\) (odd), \(D/2-k=7/2\) is a half-integer for \(k=3\) (indeed for every integer \(k\) , since \(13/2\) is itself already a half-integer and subtracting an integer preserves that). \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) is finite and nonzero there — there is no pole of \(\Gamma(s)\) to combine with a pole of \(\zeta_L(s)\) at that point. Three consequences follow directly:

 \(a_6\) at \(D=13\) contributes to the effective action as a power-law divergence , not a logarithmic one, and is therefore scheme-dependent (its numerical value depends on the regularization scheme chosen, unlike a true anomaly coefficient);

 it is identically zero in dimensional regularization , since dim-reg by construction discards power-law divergences and keeps only logarithmic ones;

 \(\zeta_L(0)\) itself is holomorphic — a standard fact for the zeta function of a Laplace-type operator on a closed manifold of odd total dimension: the odd-dimensional Seeley–DeWitt/heat-kernel coefficients that would produce a residue at \(s=0\) are simply not present in the relevant slot, so \(\zeta_L(s)\) is entire (pole-free) at \(s=0\) .

 The 5B positivity certificate is a statement of the form "the coefficient of the logarithmic divergence — equivalently, the residue at the anomaly pole — must satisfy \(P(a_{2k})\ge0\) ." That statement presupposes the existence of a logarithmic-divergence coefficient, i.e. presupposes an even total dimension in which \(a_D\) sits at \(s=0\) and genuinely controls a scheme-independent anomaly. At \(D=13\) , the object occupying the analogous slot for \(k=3\) ( \(a_6\) , sitting at \(s=7/2\) ) is by construction a finite, scheme-dependent number with no anomaly interpretation. Asking " \(P(a_6)\ge0\) ?" at \(D=13\) is therefore not a question with an unknown or a wrong answer — it is a question that does not parse, because the mathematical object \(P\) is defined to act on (a scheme-independent anomaly residue) simply is not present at odd \(D\) . This is the exact numerical and analytic content behind "the positivity certificate DISSOLVES as ill-posed at odd \(D=13\) ," and a reader can verify the whole chain with the single pole-location computation above — no curvature data, no fiber counts, nothing beyond \(D=13\) itself.

 Cross-check on the mechanism: does it generalize correctly, or is it special pleading for 13? The claim "odd total dimension \(\Rightarrow\) no even-slot anomaly at that order" is the standard statement about heat-kernel zeta functions on odd-dimensional closed manifolds — the identical structural fact behind, for instance, the absence of perturbative gravitational or gauge anomalies in odd bulk spacetime dimension generally. It is not tuned to \(D=13\) . The four sphere calibrations of Section 3 are all even -dimensional ( \(S^2,S^4,S^6\) ) precisely because those are the cases where \(a_{2k}\) at \(k=D/2\) is the scheme-independent anomaly coefficient worth calibrating in the first place; \(D=13\) was never expected to produce an analogous log-anomaly slot at \(k=3\) , and the computation above confirms it does not, for a structural reason rather than by accident.

 The even/odd swap test — the sharpest available negative control on this mechanism. If a reader repeats the identical pole-location computation at an adjacent even total dimension, the outcome flips:
$$
D=12:\ s=\frac{12}{2}-3=6-3=3\quad(\text{integer; the }s=0\text{ anomaly slot itself sits at the integer order }k=D/2=6 \Rightarrow \text{well-posed}),
$$
$$
D=14:\ s=\frac{14}{2}-3=7-3=4\quad(\text{integer; the }s=0\text{ slot sits at }k=D/2=7 \Rightarrow \text{still well-posed}).
$$
This confirms explicitly that the dissolution is driven by the parity of \(D\) , not by any property peculiar to the number 13. It also flags a specific, named failure mode for anyone reconstructing this gate: a reader who truncated the frozen geometry — for instance by dropping the \(S^1_Y/\mathbb{Z}_2\) orbifold factor and working on the resulting \(D=12\) arena — would obtain a spuriously well-posed positivity test, because \(D=12\) genuinely does have an \(s=0\) anomaly slot in its heat-kernel tower (at the integer order \(k=D/2=6\) ), while the odd \(D=13\) arena has none at any order. That test would be well-posed but would apply to the wrong manifold; it is exactly the "a residual seen under a truncated object is an artifact" warning made concrete for this gate. The correct, frozen arena has \(D=13\) , is odd, and the certificate dissolves; nothing about that changes if a different, truncated object happens to make the analogous question well-posed.

 5. What remains honestly open: the residual numbers, stated without rounding up

 Several numerical objects are explicitly not computed, or not yet reconciled between routes. Each is stated here with its actual current numeric status, not gestured at as generic "future work," and none of them is wired to this gate's terminal — that is what "irreducible" means operationally (Section 6 below makes the load-bearing distinction explicit).

 R1 — the physical graviton \(a_6\) trace. The object
$$
\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\mathrm{Sym}^2_0)]-2\,\mathrm{tr}[a_6(\text{FP vector})],
$$
to be assembled from the certified \(E_L\) spectrum ( \(\mathrm{tr}\,E_L=40/3\) , \(\mathrm{tr}\,E_L^2=241/18\) ) and the curvature invariants of Section 3, over the roughly 46-term reduced cubic-curvature Gilkey basis at \(d=13\) holonomy-projected onto \(K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) , has not been computed . Route A (Lichnerowicz/Gilkey on \(\mathrm{Sym}^2_0\) ) is blocked specifically at the first-order "hopping" term mixing the 5 Weyl-inequivalent \(T^2\) weight classes on \(\mathrm{Sym}^2_0\) — exact in principle via the standard \(SU(3)\) Gelfand–Tsetlin lowering-operator formula, but not yet enumerated for this bundle. Route B (ghost + vector reconstruction) consumes the certified vector endomorphism \(E=\mathrm{Ric}\) and the certified scalar backbone ratio \(a_6/a_2^3=7936/39375\) (banked across 3+ engines), but its graviton leg is likewise OWED. A naive extrapolated sign, \(C\approx-6.39\) , is explicitly verified absent in the actual \(D=13\) construction — stated here so that no reader mistakes the absence of a computed value for a placeholder value, and so this number is never later "recovered" by fitting. Marked OPEN.

 R2 — resolved, flagged for auditability. The \(\sim31.2\%\) Riemann-norm deficit of Section 3 ( \(31/147\) vs. \(23/75\) ) has been diagnosed (two sign errors in the quarter-bracket terms of the naturally-reductive curvature formula) and corrected (Besse-7.38 / Kobayashi–Nomizu-II sign convention), with the fix independently verified by the first-Bianchi residual dropping from \(1/7\approx0.1429\) to \(3.05\times10^{-16}\) . Status: DISCLOSED-CORRECTED — carried here as the named prerequisite that R1's curvature inputs depend on, not as an open item in its own right.

 R3 — the dimensionful magnitude. Even once R1's dimensionless trace is eventually computed, converting it to a physical energy density requires injecting a heat-kernel scheme object — effectively a choice of UV floor or renormalization scheme, tied to \(M_*\) . Two geometry read-offs for \(M_*\) exist, \(6.01\times10^{16}\) GeV and \(7.467050992135091\times10^{16}\) GeV (from \(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})=4.023836152402511\times10^{185}\ \mathrm{GeV}^{11}\) ); both are geometry read-offs, not fresh anchors, and neither is consumed as measured input by this gate's closing legs. This magnitude sector is AXIOM-OPEN / scheme-anchored , never presented as derived. The applicable kill-test (R3, per the source ledger): if a scheme object were chosen so the resulting magnitude matched a pre-selected target value, that would be a relocation of the assumption dressed as a closure, and is explicitly disallowed — the \(\kappa^3/\pi\) back-solving check exists precisely to catch this.

 R5 — an active, quantified two-route disagreement. Two candidate conventions for the ghost operator's endomorphism — Bochner ( \(E=0\) ) versus Lichnerowicz ( \(E=-\mathrm{Ric}\) ) — give
$$
\text{Route A}=-\frac{251}{504}=-0.4980158730158730,\qquad \text{Route B}=\frac{149}{1008}=0.1478174603174603,
$$
$$
\text{Route A}-\text{Route B}=-\frac{251}{504}-\frac{149}{1008}=-\frac{502}{1008}-\frac{149}{1008}=-\frac{651}{1008}=-\frac{31}{48}=-0.6458333333333333.
$$
A genuine, quantified disagreement of \(31/48\approx0.6458\) between two structurally motivated conventions — roughly five to six orders of magnitude outside the \(10^{-6}\) cross-route agreement threshold this program otherwise applies (compare the \(\le4\times10^{-14}\) sphere-calibration agreement of Section 3, and the explicit \(10^{-6}\) target named for R1's eventual two-route reconciliation). This is reported as-is: not resolved here, not averaged over, not dropped in favor of the more convenient route. Marked OPEN (structural) — the BRST-forced ghost multiplicity and sign ( \(-2\) ) is not in question; only the endomorphism convention feeding the trace is unsettled. Closing condition, stated explicitly: derive the Faddeev–Popov ghost Lagrangian directly from the de-Donder gauge condition to select the physically correct endomorphism uniquely, then reconverge Routes A and B to within \(10^{-6}\) .

 The \(124/315\) color-sector ratio. A related \(K_6\) scalar heat-kernel ratio, \(a_6/a_0=124/315\) , is reported as DERIVED-PENDING-INDEPENDENT-REPRODUCTION : it holds only at the specific value \(\mathrm{Scal}_{K_6}=7.5\) (the Levi-Civita-normalized scalar value corresponding to Killing-form \(\mathrm{Scal}=5/2\) ; under the pure Killing-norm evaluation the analogous number reads \(0.0252\) , i.e. this ratio is metric-selected, not normalization-invariant like \(23/75\) or \(1/6\) ). It is routed through the same engine that needed the R2 correction and has not been reproduced off a structurally independent engine. Not banked; feeds Route B of R1 if and when independently confirmed.

 None of R1, R3, R5, or the \(124/315\) ratio is a leg the CERTIFIED-IRREDUCIBLE terminal depends on. They are computation-debts owed to consumer gates — chiefly UQF-9 (the shared Clay-class strong-coupling wall) and Gap-01 — and are listed here with their actual current numbers, precisely so that residuals are shown in full rather than rolled into a hedge on the terminal itself.

 6. Negative controls: the wrong answers this construction could have produced, and does not

 A certificate is only as good as the negative controls that could have falsified it. Six are on record for this gate, each independently checkable by a reader:

 Curvature-ratio negative control. \(|\mathrm{Riem}|^2/\mathrm{Scal}^2\) must never read \(31/147=0.2108843537414966\) (the historically retired, Bianchi-violating value, residual \(1/7\) ) or \(60\) (the value for the round unit \(S^6\) — a different, locally symmetric manifold entirely; recall \(K_6\) has \(|\nabla\mathrm{Riem}|^2=1/4\ne0\) , so it is homogeneous but not symmetric, and getting \(60\) would signal that a reader has silently substituted the wrong space). Only \(23/75=0.3066666666666667\) , certified by the \(3.05\times10^{-16}\) Bianchi closure and by three independent routes, is consistent with the frozen \(K_6\) geometry at its chamber center.

 Fiber-dimension negative control. The graviton fiber dimension must read \(91=\binom{14}{2}\) , computed directly and only from \(D=13\) . The specific value \(67\) is on permanent record as a previously fabricated number caught by an earlier verification pass; it does not correspond to \(\mathrm{Sym}^2\) of any dimension count consistent with \(D=13\) , so its reappearance anywhere downstream is itself the signature of fabrication, not a rounding variant. Likewise the FP ghost fiber must read \(13=D\) , never \(11\) ; the net weight must read \(65=91-2\cdot13\) .

 Even/odd-dimension control on the dissolution mechanism. As derived in full in Section 4, the identical pole-location argument at adjacent even \(D=12\) or \(D=14\) produces a well-posed slot ( \(s=0\) or \(s=4\) respectively), while \(D=13\) does not ( \(s=7/2\) ). This is the sharpest test of the mechanism itself: the dissolution is not "heat kernels are generically messy," it is a checkable consequence specifically of \(D\) being odd. A reader who mistakenly truncated the geometry — e.g. dropping \(S^1_Y/\mathbb{Z}_2\) to land on \(D=12\) — would obtain a spuriously well-posed but physically wrong-arena test, exactly the "residual under a truncated object is an artifact" failure mode named in the framework's general discipline.

 Bianchi identity as a standing falsifier, not a one-time audit. Any future modification to the \(K_6\) curvature engine is required to be re-run against the first-Bianchi residual. A residual departing from machine zero — as the retired \(31/147\) value did, at \(1/7\) — is a decisive, target-blind signal of a computational error, independent of whether the resulting downstream number happens to look physically reasonable.

 The \(R_0\) /Planck-length non-identification control. \(R_0\equiv(2\pi M_U)^{-1}\) is constructed as a pure color/gauge object — fixed solely by the two-loop RG closure condition \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) with residual \(9.6\times10^{-11}\) , with zero gravitational input anywhere in its definition — and numerically \(R_0/\ell_{\rm Planck}\approx194\) . These two length scales are not numerically close, let alone identified. This is a targeted control against a specific illegitimate move: importing \(R_0\) as if it were a graviton-sector short-distance cutoff, to manufacture a spurious "finite-grain" resolution of the strong-coupling wall (5C, exported to UQF-9). Because \(R_0\) is demonstrably a color-sector quantity, an argument that quietly borrows it as a gravity cutoff would be cross-sector scale smuggling (false-flooring); the \(\approx194\times\) non-identification is precisely the numerical check that catches such a move before it is made. This control protects the boundary between this gate and the exported UQF-9 wall; it makes no claim about 5C itself.

 Topological negative control. \(\chi(K_6)=6=|S_3|\) (the order of the \(A_2\) Weyl group), \(\chi(S^2)=2\) , \(\chi(S^1_Y/\mathbb{Z}_2)=1\) are exact, frozen topological invariants that any curvature or index computation on this arena must remain consistent with; a computation returning an Euler characteristic inconsistent with these values (for instance, from a mis-specified holonomy projection) would be an immediate, cross-checkable sign of an error in bundle bookkeeping, independent of the curvature-ratio checks above.

 7. Step-by-step reproduction: rebuilding every number in this section from scratch

 A reader holding only the four irreducible anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) and the frozen geometric data quoted throughout this dossier can regenerate every claimed number above by the following explicit sequence.

 Step 1 — fix the arena and its parity. Set \(D=\dim\mathcal M_4+\dim K_6+\dim S^2+\dim S^1_Y=4+6+2+1=13\) . Confirm \(D\) is odd. This single fact is everything Section 4's mechanism computation needs; no curvature data is required to reach the pole-location conclusion.

 Step 2 — fix the \(K_6\) chamber witness and metric. Take \(K_6=SU(3)/T^2\) with \(SU(3)\) -invariant metric \(g(\vec u)=\sum_i u_i\langle\cdot,\cdot\rangle_{\mathfrak m_i}\) on the three root 2-planes of the \(A_2\) root system ( \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) ), and evaluate at the symmetric chamber center \(\vec u=(1,1,1)\) — the only witness admissible under Weyl-rigidity (the chamber \(\vec u\in[1/2,3/2]^3\) ; off-center points fail admissibility and are eliminated by the selector).

 Step 3 — compute the curvature ledger by at least two of the three independent routes, and check Bianchi before trusting anything downstream. Using the general-chamber Ricci formula
$$
\mathrm{Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12x_1x_2x_3}\quad(\text{cyclic for Ric}_2,\mathrm{Ric}_3),
$$
evaluate at \(x_1=x_2=x_3=1\) to get \(\mathrm{Ric}_i=5/12\) for all \(i\) , hence (with \(\dim\mathfrak m_i=2\) each) \(\mathrm{Scal}=\sum_i \dim(\mathfrak m_i)\,\mathrm{Ric}_i=2\cdot3\cdot(5/12)=5/2\) . Independently, run the Peter–Weyl spectral route over \(C_2(p,q)=(p^2+q^2+pq+3p+3q)/3\) and confirm the same scalar curvature emerges from the heat-trace short-time expansion. Either route should return \(|\mathrm{Riem}|^2=23/12\) and the ratio \(23/75\) . Before trusting any further downstream number, verify the first-Bianchi residual is at machine-zero level (below, say, \(10^{-14}\) ); a residual near \(10^{-1}\) signals a sign error in the naturally-reductive curvature formula — check the quarter-bracket sign convention against Besse Proposition 7.38 / Kobayashi–Nomizu Chapter II, exactly the fix that converted the historical \(1/7\) residual to \(3.05\times10^{-16}\) .

 Step 4 — build the Lichnerowicz endomorphism and diagonalize. Form \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) on the traceless symmetric 2-tensor sector \(\mathrm{Sym}^2_0\) (dimension 20, from \(\binom{7}{2}=21\) minus the 1-dimensional pure-trace mode over the 6-real-dimensional \(K_6\) tangent space). Diagonalizing against the \(A_2\) Weyl-class decomposition returns four eigenvalues, \(1/6,5/12,7/6,17/12\) , with multiplicities \(6,6,6,2\) (summing to \(6+6+6+2=20\) , the correct dimension count). Sum with multiplicity:
$$
\mathrm{tr}\,E_L=6\cdot\tfrac16+6\cdot\tfrac{5}{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12}=1+\tfrac52+7+\tfrac{17}{6}=\tfrac{40}{3},
$$
and the sum of squares:
$$
\mathrm{tr}\,E_L^2=6\left(\tfrac16\right)^2+6\left(\tfrac{5}{12}\right)^2+6\left(\tfrac76\right)^2+2\left(\tfrac{17}{12}\right)^2
=\tfrac16+\tfrac{25}{24}+\tfrac{49}{6}+\tfrac{289}{72}.
$$
Putting over the common denominator 72: \(\tfrac{12}{72}+\tfrac{75}{72}+\tfrac{588}{72}+\tfrac{289}{72}=\tfrac{964}{72}=\tfrac{241}{18}\) , confirming the stated trace of squares exactly.

 Step 5 — count the fibers. Compute \(\dim\mathrm{Sym}^2(T)=\binom{D+1}{2}\) with \(D=13\) : \(\binom{14}{2}=\tfrac{13\cdot14}{2}=91\) . Set the FP ghost fiber dimension equal to \(D=13\) . Apply the BRST-forced \(-2\) subtraction multiplicity: net weight \(=91-2\cdot13=91-26=65\) . This step needs only the integer \(D=13\) from Step 1 — no curvature data enters.

 Step 6 — run the mechanism computation. Using the pole-location formula \(s=D/2-k\) for the \(k\) -th heat-kernel coefficient \(a_{2k}\) , substitute \(D=13,k=3\) : \(s=13/2-3=7/2\) , a half-integer, so \(\zeta_L(0)\) is holomorphic and no anomaly/positivity slot exists at this order. Cross-check by repeating the identical substitution at adjacent even \(D=12\) ( \(s=6-3=3\) for \(k=3\) , an integer; at even \(D\) the \(s=0\) anomaly slot itself is reached at the integer order \(k=D/2=6\) — well-posed) and \(D=14\) ( \(s=7-3=4\) , also integer; its \(s=0\) slot at \(k=7\) ) to confirm the dissolution is specifically an odd- \(D\) phenomenon, not an artifact of the reader's own arithmetic.

 Step 7 — calibrate the machinery against the four exactly solvable sphere cases. Independently compute \(a_6/a_0\) for round \(S^2\) , \(S^4\) , \(S^6\) , and the conformally coupled case, both by direct spectral summation over the known sphere Laplacian eigenvalues and by the closed-form Gilkey/Seeley-DeWitt algebraic formula; confirm agreement at the \(10^{-14}\) level or better in every case. This validates that the same machinery which must eventually produce the \(K_6\) graviton \(a_6\) trace (R1, still open) behaves correctly on cases where the answer is independently known — without asserting this validates the not-yet-computed \(K_6\) value itself.

 Step 8 — attempt, and expect not to close, the R5 ghost-route reconciliation, honestly. Compute both Route A ( \(-251/504\) ) and Route B ( \(149/1008\) ) for the two candidate ghost-operator endomorphism conventions, and confirm the numerical gap \(-31/48\approx-0.6458\) found in Section 5. A reader reproducing this dossier should not expect this step to converge; obtaining the same persistent, quantified gap is itself the correct and honest reproduction outcome at the current state of the record.

 A reader who completes Steps 1–7 will have independently regenerated every certified number this gate's terminal actually depends on: the arena's odd parity, the curvature ledger ( \(5/12\) , \(5/2\) , \(23/75\) , \(1/6\) , \(6\) ), the \(E_L\) spectrum and its two traces ( \(40/3\) , \(241/18\) ), the fiber counts ( \(91\) , \(13\) , \(65\) ), the odd- \(D\) dissolution mechanism ( \(s=7/2\) vs. \(s=0,4\) at adjacent even \(D\) ), and the four sphere-calibration cross-checks — without taking any of them on faith. Step 8 independently confirms that one specific named residual (R5) is real, quantified, and currently unresolved, which is the intended and honest outcome of attempting to reproduce it. No step in this procedure touches, or is capable of touching, the CERTIFIED-IRREDUCIBLE / RESOLVED +0 terminal itself, because that terminal is reached on the arena's parity (Step 6) and the structural mode content (Steps 1–5), neither of which is contingent on R1, R3, or R5 ever closing.

 Open gaps & the specialist closure path

 UQF-5A/5B is CERTIFIED-IRREDUCIBLE / RESOLVED +0. That terminal is reached on named legs — the structural graviton mode, the fiber weights 91/13/65, the two-helicity IR limit, and the odd- \(D\) dissolution of the one-loop positivity certificate — none of which is contingent on any item below. Every hole in this section is a residual shown, not a leg that gates the terminal : several are finite compute-debts owed to consumer gates (UQF-9, Gap-01), one is a discipline to maintain rather than a hole to close, and one is a genuine shared frontier exported whole to UQF-9. Ordered by leverage, each is stated as a target-blind bet with an explicit success criterion and an explicit refutation criterion, so a specialist can pick one up without re-deriving the frame.

 R1 — the d=13 graviton-minus-ghost \(a_6\) trace (highest leverage, computation-debt)

 The precise open object. The physical sixth Seeley–DeWitt heat-kernel coefficient
$$
\mathrm{tr}[a_6]^{\rm phys} = \mathrm{tr}[a_6(\text{grav})] - 2\,\mathrm{tr}[a_6(\text{FP ghost})],
$$
evaluated on the frozen \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) background, for the Lichnerowicz operator \(L_{\rm grav}=-(\nabla^2+E_L)\) acting on the graviton TT sector \(\mathrm{Sym}^2_0\) ( \(E_L\) spectrum \(\tfrac16(\times6),\tfrac5{12}(\times6),\tfrac76(\times6),\tfrac{17}{12}(\times2)\) , giving \(\mathrm{tr}\,E_L=40/3\) , \(\mathrm{tr}\,E_L^2=241/18\) ), minus twice the Faddeev–Popov ghost trace on the vector bundle ( \(E=\mathrm{Ric}=(5/12)\,\mathrm{Id}\) , \(\mathrm{tr}\,E=5/2\) , \(\mathrm{tr}\,E^2=25/24\) ). This is not computed anywhere in the corpus. The dimensionless even-coefficient sign \(C\approx-6.39\) (a number that surfaces in the corpus's cross-checking apparatus as a target the eventual \(a_6\) must not be tuned to reproduce) is verified absent — no engine has produced it honestly — and it must never be fabricated or "recovered" by adjusting scheme choices until it appears.

 Why it is hard, and the specific traps. The universal \(a_6\) coefficient at general dimension is a \(\sim\) 46-term cubic-curvature functional (Gilkey's Theorem 3.3.1 template, specialized by Avramidi's Ch. 4 machinery to \(d=13\) ): terms built from \(\mathrm{tr}(R^3)\) -type contractions of the Riemann tensor, \(\mathrm{Ric}\cdot\mathrm{Riem}\cdot\mathrm{Riem}\) contractions, \((\nabla\mathrm{Riem})^2\) -type derivative terms, and \(E\) -curvature cross terms ( \(\mathrm{tr}(E\,\mathrm{Ric})\) , \(\mathrm{tr}(E^3)\) , \(\mathrm{tr}(E\,\Omega_{ab}\Omega^{ab})\) , and so on), each with a fixed rational Gilkey coefficient. The frozen geometry supplies exactly nine of the needed cubic/weight-6 invariants at full precision — \(\mathrm{Scal}^3=125/8\) , \(\mathrm{Scal}\,|\mathrm{Ric}|^2=125/48\) , \(\mathrm{Scal}\,|\mathrm{Riem}|^2=115/24\) , \(|\mathrm{Ric}|^3=125/288\) , \(\mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=125/288\) , \(\mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}=115/144\) , \(K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=-113/72\) , \(K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-5/72\) , \(|\nabla\mathrm{Riem}|^2=1/4\) — plus the \(E_L\) spectrum above. What is missing is not curvature data but operator data : the \(SU(3)\) Gelfand–Tsetlin off-diagonal matrix elements that gate the Lichnerowicz first-order (hopping) term on \(\mathrm{Sym}^2_0\) . The graviton bundle's fiber decomposes under the \(T^2\subset SU(3)\) isotropy into five Weyl-inequivalent weight classes, and the Lichnerowicz curvature term \(-2R_{acbd}h^{cd}\) mixes these classes off-diagonally — a mixing controlled by the standard \(SU(3)\) lowering-operator (Gelfand–Tsetlin ladder) formula, which is exact in principle but has not yet been enumerated for this specific representation content. Skipping this step and simply plugging the nine scalar invariants into the Gilkey template as if the operator were diagonal would silently drop the hopping contribution — that is the trap: a plausible-looking but wrong number, indistinguishable in form from the honest one, is worse than an open flag.

 A second, sharper trap is the κ³/π kill-test (R3) : because the \(a_6\) magnitude also carries a scheme/scale dependence (see R3 below), any calculation that is tuned — even unconsciously, by choosing a convention until the answer looks "reasonable" — to land on \(C\approx-6.39\) or on a previously-circulated bulk magnitude is true by construction and relocates the problem rather than closing it. The route must be run target-blind, with the sign and magnitude reported whatever they come out to be, before comparison to any external target.

 Exactly what closes it, target-blind, with success/refutation criteria. The closure path is:
1. Derive the \(SU(3)\) Gelfand–Tsetlin off-diagonal matrix elements coupling adjacent GT patterns on the \(\mathrm{Sym}^2_0\) representation content at the \(K_6=SU(3)/T^2\) isotropy, using the standard lowering-operator formula (this is Route A, the Lichnerowicz/graviton-native route).
2. With R2 fixed (see below — the curvature-engine sign bug must be corrected first, since R1 consumes the same engine), assemble \(\mathrm{tr}[a_6(\text{grav})]\) from the full off-diagonal-corrected operator through the Gilkey template.
3. Independently assemble \(\mathrm{tr}[a_6]^{\rm phys}\) via Route B (ghost+vector reconstruction: the certified vector \(E=\mathrm{Ric}\) trace plus the scalar backbone ratio \(a_6/a_2^3=7936/39375\) , banked across three or more independent engines).
4. Success criterion: Route A and Route B must agree to within \(10^{-6}\) relative, and the resulting scalar-sector \(a_6/a_0\) must reproduce all four independent sphere calibrations exactly ( \(S^2=4/315\) , \(S^4=74/63\) , \(S^6=1139/63\) , conformal \(=5/63\) , each already verified to \(\le4\times10^{-14}\) relative error against the exact spectral values) as a consistency check on the shared machinery, before the graviton-specific number is banked.
5. What a refuting/negative result looks like, and how it is still a valid close: if Route A and Route B disagree persistently beyond \(10^{-6}\) after the GT elements and the R2 fix are both in hand, that is a reportable structural obstruction (not a failure to hide) — it would mean the two heat-kernel constructions are not computing the same trace, which is itself new information about the operator's well-posedness. A computed value that fails to reproduce the sphere calibrations independently falsifies the specific implementation, not the target closure. Under no reading does either outcome touch the RESOLVED +0 terminal, since R1 is consumer-charged, not gate-defining.

 Leverage. This is the single highest-leverage residual in the whole gate: it directly supplies the number owed to UQF-9 (the strong-coupling/UV-completion wall) and to Gap-01 , and it is the prerequisite input for any future attempt to run the (dissolved-as-inapplicable-at-odd- \(D\) , but potentially informative-as-a-diagnostic) positivity functional at an even-dimensional companion construction. Closing R1 does not reopen 5B — the odd- \(D\) dissolution stands regardless of the trace's value — but it retires the largest named "OWED" tag in the entire graviton-sector ledger and unblocks two consumer gates simultaneously.

 R2 — the 31.23% Riemann-norm engine bug (prerequisite, disclosed-corrected)

 The precise open object. The corpus's \(a_6\) -computation engine, prior to correction, carried the curvature ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2 = 31/147 = 0.2108843537414966\) instead of the certified \(23/75 = 0.3066666666666667\) — a \(31.23\%\) deficit, localized specifically to the Riemann-norm sector (the companion Ricci ratio \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) was correct all along in the same engine, which is precisely what let the bug hide: a partial self-check passed while the full tensor was wrong).

 Why it is hard, and the specific trap. The trap that let this bug survive is exactly the one to guard against when reusing or re-deriving the engine: checking only a subset of the invariants (here, the Ricci-only self-consistency) creates false confidence in the whole computation. The wrong ratio is not a random numerical slip — it corresponds to a specific wrong sign convention in a curvature-formula bracket (two quarter-bracket signs need to flip to the Besse-7.38 / Kobayashi–Nomizu-II naturally-reductive form), so it silently produces a self-consistent-looking but Bianchi-violating tensor. The diagnostic that catches it is the first-Bianchi identity , which the wrong tensor fails at residual \(1/7\approx0.1429\) (a clean rational, not numerical noise — itself a fingerprint of a sign error rather than a truncation error) while the corrected tensor closes to \(3.05\times10^{-16}\) (machine zero, i.e. floating-point noise, i.e. genuinely exact).

 Exactly what closes it, and what a refutation looks like. This one is a completed, localized code correction , not an open derivation: flip the two quarter-bracket signs in the naturally-reductive curvature formula to the Besse-7.38 / Kobayashi–Nomizu-II form. Success criterion: the first-Bianchi residual drops to machine zero ( \(\sim10^{-16}\) ) and the ratio \(23/75\) is emitted on-disk by the corrected engine, both of which have already been demonstrated across three independent target-blind routes (explicit \(\mathfrak{su}(3)\) Gell-Mann structure-constant computation via the Nomizu/Koszul formula; a curvature-free spectral heat-trace check over the scalar-Laplacian Casimir spectrum; an independent Levi-Civita/full-Riemann build over the 3-parameter invariant metric). A refutation would be if a fourth independent route failed to reproduce \(23/75\) or failed the Bianchi test — that has not happened; all three completed routes agree. The retired value \(-2.817995812\times10^{94}\ \mathrm{GeV}^6\) , which rode the buggy \(31/147\) ratio, is a decision-grade negative result and must never be revived; the corrected-pending replacement \(-2.995681680\times10^{94}\ \mathrm{GeV}^6\) ( \(+6.30\%\) shift, sign preserved, tracking the \(\kappa=7/12\to5/12\) correction) is scheme-anchored and explicitly not banked as a derived result — it is a placeholder pending the scheme-object resolution in R3.

 Leverage. R2 is the gating prerequisite for R1: any \(a_6\) trace assembled on the uncorrected engine inherits the same Riemann-norm error. It also protects every other cubic-curvature invariant quoted in this dossier's own geometry ledger, since \(K_1\) , \(K_2\) , and the weight-6 invariants would all shift under the same sign flip if it were not already applied — the values quoted throughout this dossier ( \(K_1=-113/72\) , \(K_2=-5/72\) , and the weight-6 table) are the corrected , post-fix values.

 R3 — the dimensionful \(a_6\) magnitude is scheme-anchored, not measured (axiom-open)

 The precise open object. Even once R1 and R2 are both resolved, the resulting \(a_6\) will be a dimensionless rational (like the sphere-calibration ratios \(4/315\) , \(74/63\) , etc.) that must be converted to a physical, dimensionful one-loop contribution via an injected heat-kernel scheme object — effectively a choice of renormalization/regularization convention and an overall mass scale (entering through the \(M_*\) floor: either \(M_*\approx6.01\times10^{16}\) GeV from the UQF-9 handoff route or \(M_*=7.467050992135091\times10^{16}\) GeV from the Planck-normalization route, \(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})=4.023836152402511\times10^{185}\ \mathrm{GeV}^{11}\) with \(M_{\rm Pl}=1.220900000000000\times10^{19}\) GeV, ordinary not reduced). Additionally, Theorem L′ shows \(|\mathrm{Riem}|^2/\mathrm{Scal}^2\) (and hence, presumably, any cubic invariant built from it) is not a \(K_6\) invariant in the abstract — it is metric-modulus-dependent, equal to \(23/75\) only at the frozen witness \(\vec u=(1,1,1)\) , versus \(1/3\) at the Kähler–Einstein point \((1,1,2)\) and \(\approx0.3609\) at a generic point like \((1,2,3)\) — so the magnitude (not the ratio-structure) inherits a moduli dependence that must be tracked explicitly.

 Why it is hard, and the trap (κ³/π kill-test). No physical measurement fixes the heat-kernel scheme object directly — it is a convention, and any convention can be dialed to produce nearly any magnitude. This is exactly the decision family flagged by the corpus's κ³/π kill-test : a scheme object reverse-engineered so that the \(a_6\) magnitude lands on a wanted value (e.g. to match some target bulk-vacuum-energy magnitude, or the retired \(10^{94}\ \mathrm{GeV}^6\) -scale numbers) is true by construction and relocates the openness rather than resolving it — the same failure mode this program's discipline is built to catch elsewhere in the corpus. The honest move is to name the scheme dependence as what it is.

 Exactly what closes it, target-blind, with success/refutation criteria. Two honest outcomes are available, and both are legitimate closes: (a) find and verify, target-blind, that the shared one-loop heat-kernel scheme object used elsewhere in the program (i.e., already fixed by some other, independent physical requirement, not invented for this purpose) determines the \(a_6\) magnitude uniquely — success criterion: the same scheme object, applied without modification to at least one other already-closed heat-kernel computation in the corpus, reproduces that computation's banked value; or (b) explicitly name and verify a value-free axiom, AXIOM-HEATKERNEL-SCHEME-OBJECT , stating that the magnitude (not the ratio-structure) of \(a_6\) is a scheme choice, not a prediction — this is a legitimate ANCHORED-type closure (axiom, not derivation), provided it is stated as such rather than presented as measured. What refutes a claimed closure: any instance where the scheme object is adjusted between computations to produce a wanted number is an immediate disqualification and relocates the problem — this is the single sharpest trap in the entire graviton-sector residual list, because it is the easiest one to get away with informally.

 Leverage. R3 sits between R1 and the interacting/strong-coupling sector (R8/R9 below): it is exactly the same "scale-free ratio vs. scheme-anchored magnitude" split that recurs at UQF-9, so a clean, target-blind resolution of R3 (in either direction (a) or (b)) is directly reusable as the template for the analogous scheme-anchoring question at the shared UV wall.

 R4 — the positivity functional itself: dissolved under canon, but flagged for any future pursuit

 The precise open object. Classically, gate 5B was framed as requiring a positivity test \(P(a_6)\ge0\) on the physical trace, with (in the older framing) three competing candidate readings of what functional \(P\) should even be. Under the canonical, dossier-fixing resolution, this functional dissolves as ill-posed at the frozen total dimension \(D=13\) : \(a_6\) sits at the half-integer heat-kernel pole \(s=7/2\) , which is a power divergence, scheme-dependent and identically zero in dimensional regularization; the operator zeta function \(\zeta_L(0)\) is holomorphic for odd total dimension (no logarithmic/anomaly pole), so there is no even-dimensional one-loop positivity/anomaly slot to test at odd \(D\) . \(P(a_6)\ge0\) is therefore not a well-posed pass/fail question at \(D=13\) — this is the specific mechanism that makes the gate terminal (RESOLVED, not perpetually OPEN) rather than a claim that the test was run and passed.

 Why it is subtle, and the trap. The trap here runs in two directions simultaneously, and a specialist must avoid both. First: do not select or construct a version of \(P\) engineered so that some computed \(a_6\) magnitude "passes" — that would be axiomatizing a pass, the mirror image of the κ³/π trap. Second: do not treat the dissolution itself as requiring rescue — the correct reading is that the test does not apply , which is a positive structural fact about odd-dimensional heat kernels (a property of the mathematics, holding for any odd- \(D\) theory, not a special pleading for this framework), not an evasion.

 What would close it further, target-blind, and what refutation looks like. If pursued further by a specialist (this is explicitly optional — the gate's terminal does not depend on it): the honest outcomes are (i) an independent confirmation that the ill-posedness is generic to odd total dimension (this confirms the terminal, it does not change it) or (ii) the discovery of a genuinely new even-dimensional companion object — for instance, a boundary or defect contribution living on an even-dimensional locus within the odd- \(D\) bulk — that does carry a well-posed positivity slot. Outcome (ii), if it occurred, would have to be reported as a new object , not as a rescue of the original \(P(a_6)\ge0\) test on the bulk trace; conflating the two would be a category error. No outcome under (i) or (ii) reopens 5B to OPEN, because the terminal was reached by showing the original test inapplicable, and that showing is not contingent on what a specialist later finds about some different, new object.

 Leverage. Confirms (never threatens) the RESOLVED +0 terminal; a rigorous, fully-written-out proof of the odd- \(D\) holomorphy argument (beyond the structural statement already banked) would strengthen the certificate from "structurally clear" to "explicitly demonstrated," which is worth doing for completeness even though it changes no status.

 R5 — the ghost/BRST subtraction is well-posed in sign but unverified in route (structural, live disagreement)

 The precise open object. The Faddeev–Popov ghost subtraction's overall sign ( \(-2\) , i.e., minus twice the ghost trace) is forced by BRST nilpotency of the de-Donder-gauge-fixed action and is not in question. What is unverified is the route by which the ghost operator's own endomorphism \(E\) is chosen: two candidate physical prescriptions — Bochner ( \(E=0\) ) versus Lichnerowicz ( \(E=-\mathrm{Ric}\) ) — give ghost traces that disagree by
$$
\text{Route A} = -\frac{251}{504}=-0.4980158730158730,\qquad \text{Route B} = \frac{149}{1008}=0.1478174603174603,\qquad \Delta = -\frac{31}{48} \approx -0.6458333333333333,
$$
a discrepancy roughly six orders of magnitude outside the \(10^{-6}\) agreement tolerance used elsewhere in this dossier's cross-checks — not a small numerical wobble but a genuine unresolved choice-of-operator question.

 Why it is hard, and the trap. The trap is to treat this as "just pick the convention that makes downstream numbers nicer" — that would be the same relocate-not-close failure as R3's kill-test, applied to an operator choice instead of a scale. The two candidates are not notational variants of the same physics; \(E=0\) (Bochner Laplacian) and \(E=-\mathrm{Ric}\) (Lichnerowicz-type on the ghost's vector bundle) are genuinely different differential operators with different spectra, and the "right" one is fixed by which physical principle governs the ghost field's own kinetic operator under the de-Donder gauge-fixing — a question that must be settled by writing out the ghost action explicitly from the Faddeev–Popov procedure applied to the actual gauge-fixing condition, not by inspection of which answer looks better.

 Exactly what closes it, target-blind, with success/refutation criteria. Write out the Faddeev–Popov ghost Lagrangian explicitly from the de-Donder gauge-fixing condition \(\nabla^M h_{MN}=\tfrac12\nabla_N h\) applied to the linearized diffeomorphism transformation, identify unambiguously which curvature term (if any) the resulting ghost kinetic operator carries, and recompute both routes' traces from that single, explicit derivation rather than from the two candidate endomorphisms treated as free choices. Success criterion: the explicit derivation selects one of \(E=0\) or \(E=-\mathrm{Ric}\) (or some third possibility not yet considered) uniquely, and the two previously-competing traces collapse to agreement within \(10^{-6}\) once only the correct operator is used (i.e., the "disagreement" should turn out to be a symptom of comparing the wrong operator to itself under two names, not a genuine physical ambiguity). What a refuting/negative result looks like: if the explicit derivation is carried out correctly and a persistent disagreement remains — i.e., the ghost operator is genuinely still ambiguous even after writing out the gauge-fixing explicitly — that is itself a reportable structural obstruction , honestly stated as such, not resolved by fiat.

 Leverage. R5 feeds directly into R1: the ghost trace \(\mathrm{tr}[a_6(\text{FP})]\) is one of the two terms in \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{grav})]-2\,\mathrm{tr}[a_6(\text{FP})]\) , so R1 cannot be honestly closed while R5's \(31/48\) discrepancy stands unresolved — any R1 computation using one of the two ghost routes without addressing this is provisional at best.

 R6 — the \(\mathbb{Z}_2\) orbifold-defect order-6 boundary heat-kernel coefficient (blocked, literature gap)

 The precise open object. The bulk \(a_6\) discussed in R1 is only part of the story on an orbifold: the \(S^1_Y/\mathbb{Z}_2\) boundary at the two fixed points \(\theta=0,\pi\) contributes its own defect heat-kernel coefficients (the Donnelly equivariant construction already supplies the lower-order defect terms: per-fixed-point \(a_0\) defects of \(\pm1/4\) , from the reflection \(g\) -trace \(\sum 1/|1-dg| = 2\times\tfrac12=1\) and the orbifold-trace decomposition \(K^\pm=\tfrac12K_{\rm circle}\pm\tfrac12\) ). What is needed at order 6 — the mixed Neumann \(\oplus\) Dirichlet boundary coefficient analogous to the bulk \(a_6\) — does not exist in the published literature : the standard boundary heat-kernel tower (Branson–Gilkey–Kirsten–Vassilevich; Kirsten's monograph treatment) stops at \(a_5\) .

 Why it is hard, and the specific trap. This is not a computation the corpus's engines failed to run — it is a genuine gap in the published mathematical literature on mixed-boundary-condition heat kernels at this order. The trap is to treat "the literature stops at \(a_5\) " as license to extrapolate informally (e.g., by pattern-matching the \(a_0\) – \(a_5\) boundary coefficients and guessing the \(a_6\) form) — any such extrapolation would be exactly the kind of fabricated-coefficient risk this dossier is built to avoid, and must not be presented as derived.

 Exactly what closes it, target-blind, and what a refutation looks like. Two honest paths: (i) a genuine new specialist construction of the order-6 mixed Neumann \(\oplus\) Dirichlet boundary coefficient, built from first principles using the same Seeley–DeWitt recursive machinery that produced \(a_0\) through \(a_5\) in the existing literature — this would be a new, citable mathematical result in its own right, independent of this physics program; or (ii) a proof that the Donnelly equivariant construction's \(\tfrac12 c_3^\gamma\) -type equivariant-index term already supplies the needed order-6 data without requiring the ordinary (non-equivariant) boundary tower to be extended — i.e., that the equivariant route sidesteps the literature gap rather than filling it. Success criterion for either path: the resulting defect coefficient, combined with the corrected bulk \(a_6\) from R1, must satisfy the same Bianchi-type/positivity-type internal consistency checks used for the bulk sector, and must not be presented as a total (bulk+defect) \(a_6\) until both pieces are independently verified. Refutation: if neither path succeeds, R6 remains honestly BLOCKED — a specialist-level open mathematical problem, not a flaw in this framework, and the dossier must not fabricate the defect total or the combined total to manufacture an appearance of closure.

 Leverage. Genuinely modest for this gate's terminal (the bulk trace R1 is the dominant residual) but non-trivial for any future orbifold-boundary graviton computation in the broader program; a general order-6 mixed-boundary result would be reusable well beyond this specific gate.

 The 124/315 color ratio — derived-pending independent reproduction

 The precise open object. The scalar-Laplacian sixth coefficient ratio \(a_6/a_0 = 124/315\approx0.3937\) (referred to in the corpus as the "color ratio") is currently metric-selected : it holds only when \(\mathrm{Scal}_{K_6}=7.5\) (the \(R_6\) -scaled companion normalization, \(\mathrm{Scal}=15/2\) ), and is routed entirely through the same engine that R2 corrects. At the Killing-form normal metric used for the primary curvature ledger in this dossier ( \(\mathrm{Scal}=5/2\) ), the ratio the same procedure would naively give is \(0.0252\) , not \(124/315\) — the two do not match without accounting for the scale-dependence explicitly, which has not yet been done rigorously across a change of normalization.

 Why it is hard, and the trap. The trap is treating a ratio computed in one normalization as directly comparable, unscaled, to a ratio in another — the dossier's own discipline (never mix an absolute value across the Killing-form and \(R_6\) -scaled companion metrics) applies with special force here because \(a_6/a_0\) has mass-dimension \(-6\) relative to \(a_0\) under the \(R_6\) -scaled convention and is dimensionless in the Killing-form convention; conflating them without the explicit conversion factor is a normalization error masquerading as a physics discrepancy.

 Exactly what closes it, target-blind, with success/refutation criteria. Reproduce \(124/315\) from a structurally-independent engine — one that does not share the R2 code path — starting from the same \(\mathrm{Scal}=7.5\) normalization, and separately verify the conversion to the Killing-form normalization reproduces \(0.0252\) (or corrects it, if the naive conversion itself turns out to be the error) via the explicit scale factor connecting the two metrics. Success criterion: independent reproduction to the same \(\le4\times10^{-14}\) -class precision achieved for the four sphere calibrations. Refutation: if an independent engine does not reproduce \(124/315\) , the ratio must not be banked and reverts to OPEN with that specific engine's result reported honestly.

 Leverage. Directly informs R1's Route B scalar backbone; a clean independent reproduction would strengthen confidence in the shared scalar-sector machinery that both Route A and Route B partially depend on.

 R7 — the scope firewall " \(a_6 \ne\) UV completion" (a discipline to maintain, not a hole to close)

 The precise open object. There is no open object here in the technical sense — R7 names a boundary that must never be crossed: no single heat-kernel coefficient, however completely and correctly computed, can settle the strong-coupling/UV-completion question, because the relevant operator ladder \(a_6 < a_8 < a_{10} < \cdots\) is unbounded — each higher coefficient probes shorter distances and higher-order curvature invariants that \(a_6\) alone cannot constrain.

 Why it matters as a discipline. The trap this firewall exists to block is a one-sentence non-sequitur: "we computed \(a_6\) (R1 closes) \(\Rightarrow\) the strong-coupling graviton (5C) is under control." That inference is invalid regardless of how cleanly R1 closes, because \(a_6\) is a necessary but never sufficient piece of one-loop data — sufficiency for a genuine UV completion would require control of the full ladder, which is precisely the content of the shared UQF-9 wall.

 What "closes" it. Nothing does, structurally — this is maintained by vigilance in every future write-up, not resolved by computation. The discipline is: report R1's eventual value (once closed) as exactly what it is, a one-loop coefficient, and never let it be quoted as evidence for or against strong-coupling consistency.

 Leverage. Protects the boundary between this gate (RESOLVED) and UQF-9 (the genuine open frontier) in every future paper or dossier that cites the eventual R1 number — without this firewall stated explicitly, a future reader could easily over-claim on the program's behalf.

 R8/R9 (5C) — the interacting/strong-coupling graviton and above-cutoff unitarity: exported whole to UQF-9

 The precise open object. Beyond the linearized (free-field) graviton certified by 5A/5B, the interacting theory — loop corrections from graviton self-interactions, unitarity of graviton scattering above the compactification cutoff, and the existence of a consistent UV completion — is completely open, here and everywhere in the field. This is not a private weakness of this framework: no approach to quantum gravity (string theory, asymptotic safety, loop quantum gravity, or this constraint-first geometric construction) has a finished answer to strong-coupling graviton consistency. The gate does not attempt to resolve this; it is exported in full to UQF-9 as the shared Clay-class UV-completion wall (also touching UQF-14), carried under the flag bounded:false .

 Why it is hard. The technical obstruction is that graviton self-interactions are governed by an operator with negative mass dimension (in four large dimensions, Newton's constant \(G_N\) has dimension \(-2\) ; in the full 13-D theory the relevant coupling inherits dimension from \(M_*\) ), making the theory perturbatively non-renormalizable order by order — precisely the reason the field as a whole has spent decades on non-perturbative approaches (asymptotic safety's non-Gaussian fixed point, string theory's extended objects, loop quantum gravity's discretized geometry) rather than a perturbative fix.

 Exactly what closes it, target-blind, with success/refutation criteria. The only thing that closes 5C is closing UQF-9 itself: a target-blind, truncation-independent demonstration of a non-Gaussian renormalization-group fixed point (in the Functional Renormalization Group / Wetterich-equation sense) evaluated on the frozen chamber data of this exact geometry — \(M_U\approx1.0\times10^{16}\) GeV, \(R_0=(2\pi M_U)^{-1}=1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) , chamber center \(\vec u=(1,1,1)\) — obtained via two structurally different computational routes that agree. A clean no-go theorem is an equally valid close : a rigorous demonstration that no such fixed point exists on this geometry would be genuine, publishable progress, not a failure, provided it is reported as a no-go rather than buried. The nearest tractable sub-target inside this frontier is P0 (below). Refutation/failure mode to avoid: a truncation-dependent FRG result (i.e., one that changes materially when the truncation order of the effective-average-action ansatz is increased) does not count as a fixed point for this purpose — this is the standard, field-wide pitfall of FRG calculations, and the corpus's discipline of requiring two independent routes exists specifically to catch it.

 Leverage. This is the single largest piece of leverage in the entire graviton sector, precisely because it is not claimed here: correctly exporting it (rather than either quietly ignoring it or overclaiming a resolution) is what keeps UQF-5A/5B's RESOLVED +0 terminal honest. Any progress on UQF-9 automatically upgrades the graviton sector's interacting completeness without this gate's own terminal needing to move.

 P0 — does gravity own an intrinsic shortest length, or inherit the color scale \(R_0\) ? (cleanest UQF-9 sub-target)

 The precise open object. Is there a graviton-intrinsic finite-grain cutoff distinct from the compactification radius \(R_0\) , or does the gravitational sector simply inherit \(R_0\) (the same scale that sets the \(SU(3)_c\times SU(2)_L\times U(1)_Y\) unification radius) with no independent gravitational input?

 Why it is hard, and the trap. The trap is assuming the answer either way without the T-DEEP result already banked: \(R_0\) is proved to be a pure color/gauge object — defined entirely by the condition \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) , with zero gravitational input in its definition (residual \(9.6\times10^{-11}\) on the inverse-coupling equality) — and numerically \(R_0/\ell_{\rm Planck}\approx194\) , meaning the color length and the Planck length are not even numerically close to identified. Consequently, any claimed "finite-grain dissolution" of the strong-coupling graviton problem (5C) that simply uses \(R_0\) as if it were the graviton's own UV cutoff is smuggling in an unproven cross-sector scale identification — a specific, nameable version of the "false-flooring" failure mode (treating an unjustified floor as if it were derived) — and such an argument relocates onto UQF-9 rather than closing it.

 Exactly what closes it, target-blind, with success/refutation criteria. Determine, from first principles and without presupposing the answer, whether the graviton sector's short-distance behavior is controlled by \(R_0\) or by some independent scale. "Inherits \(R_0\) " as an outcome would confirm that there is no finite-grain shortcut around the UQF-9 strong-coupling wall — consistent with, and reinforcing, the T-DEEP result — and would be reported as a confirming non-result , honestly, not spun as new progress. "Owns its own intrinsic scale" as an outcome would open a genuinely new route into the finite-grain question and would need to be reported as new physics content, with its own independent derivation of that scale (not simply asserted). Refutation of any claimed answer: back-solving to whichever answer makes some other calculation come out nicer is exactly the trap named above; a legitimate resolution derives the answer from the operator content (i.e., from whether the graviton's own action or endomorphism structure, as opposed to the gauge sector's, contains any scale-setting term) rather than by comparing final numbers.

 Leverage. P0 is explicitly named as the cleanest tractable entry point into the UQF-9 frontier from the graviton side — resolving it either way narrows the UQF-9 search space materially (either by closing off the "reuse \(R_0\) " shortcut definitively, or by handing UQF-9 a genuinely new scale to work with) without itself requiring the full FRG fixed-point computation.

 Summary ordering and what a specialist should pick up first

 For a specialist physicist entering this sector cold, the ordering by both tractability and leverage is: R2 is already done (verify it, do not redo it) and gates R1 , which is the highest-leverage open computation and the one that most directly retires an "OWED" tag against consumer gates — start with the Gelfand–Tsetlin off-diagonal matrix elements. R5 must be resolved in parallel (it is a direct input to R1's ghost-subtraction term) by writing out the Faddeev–Popov ghost Lagrangian explicitly rather than choosing between the two candidate endomorphisms by inspection. R3 is a scheme question that can be settled independently of R1's numerics, by hunting for an existing, independently-fixed scheme object elsewhere in the program. R6 and the 124/315 reproduction are lower-leverage, semi-independent side quests — R6 in particular may require a genuinely new piece of mathematics (the order-6 mixed-boundary heat-kernel coefficient) that no one has published. R4 requires no further action for this gate's terminal, though a fully written-out proof of the odd- \(D\) holomorphy argument would be a worthwhile strengthening exercise. R7 requires no computation, only discipline in every future citation of R1's eventual value. R8/R9 and P0 are, correctly, not this gate's problem to solve — they are the field-wide UQF-9 frontier, and the single most valuable thing a specialist can do for the graviton sector specifically, short of solving quantum gravity, is to resolve P0 cleanly, since it is the one sub-question of the UQF-9 wall that is posed entirely in terms of objects (the color radius \(R_0\) , the Planck-scale ratio \(\approx194\) , the T-DEEP zero-gravitational-input proof) already fixed and available on this gate's own geometry.

 Honest ceiling, scope & the endpoint

 1. What this gate does NOT claim (the firewall, stated plainly)

 The preceding sections established a real, checkable structural result: on the frozen thirteen-dimensional arena \(\mathfrak{B}_{\rm active} = \mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) ( \(K_6=SU(3)/T^2\) , \(D=4+6+2+1=13\) ), the de-Donder-gauge-fixed metric fluctuation \(h_{MN}\) produces a Lichnerowicz-type operator \(L_{\rm grav}=-(\nabla^2+E)\) whose four-dimensional massless spin-2 mode is a graviton with exactly two helicities, propagating at light speed, reproducing linearized Einstein gravity and the Newtonian \(1/r^2\) limit in the infrared. The fiber-weight bookkeeping is exact and target-blind: graviton fiber dimension \(91=\dim\mathrm{Sym}^2(T)=\binom{14}{2}=13\cdot14/2\) over the complete \(d=13\) tangent bundle, Faddeev–Popov ghost fiber dimension \(13\) (one vector ghost per dimension), ghost multiplicity \(-2\) forced by BRST nilpotency, net graded weight \(91-2\cdot13=65\) . None of this is dialed in by hand. It is exactly as strong as it sounds, and it is exactly this strong and no stronger. The purpose of this section is to draw that boundary with the same rigor used to build the result, because a rigorous physicist checking this dossier will look here first for where the authors might be overclaiming — and finding nothing hidden is itself part of what makes the CERTIFIED-IRREDUCIBLE terminal honest.

 Four non-claims must be stated explicitly, because each is a place where a careless reading of Sections 1–9 could be pushed past what was actually shown.

 Non-claim 1 — this is not a derivation of General Relativity, and it is not a derivation of \(E\) . The Lichnerowicz endomorphism \(E_L\) that enters \(L_{\rm grav}=-(\nabla^2+E_L)\) is built from the Ricci and Riemann tensors of the already-fixed background geometry: \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) , evaluated at the Killing-form normal metric on the symmetric chamber center \(\vec u=(1,1,1)\) with \(\mathrm{Ric}_i=5/12\) . This endomorphism is supplied by the frozen shape — it is not derived from some deeper principle that would explain why gravity has this form rather than some other second-order operator. The graviton reproduces the observed general-relativistic long-wavelength limit; it does not derive general relativity from first principles more basic than "assume a metric fluctuation on a Riemannian background and gauge-fix it." This is the precise sense of the working shorthand used throughout this program: given- \(E\) \(\neq\) derivation-of- \(E\) . The observed GR/Newton infrared limit is consumed here as an input against which the construction is checked, not manufactured as an output from something more primitive. Anyone reading "gravity falls out of the same shape" as "we derived Einstein's equations from nothing" is reading past what is claimed.

 Non-claim 2 — this is not a claim that quantum gravity is solved, consistent, or complete. Section 5B's terminal — the one-loop positivity certificate dissolving as ill-posed at odd total dimension \(D=13\) — closes exactly one well-defined mathematical question (does a particular even-dimensional-style positivity test apply here) and closes it in the negative-existence sense: the test does not exist as a meaningful pass/fail gate at odd \(D\) . It says nothing about whether the interacting, strongly-coupled graviton sector is unitary, renormalizable, or UV-complete. That question — called 5C in this program's numbering — is carried forward untouched, exported in full to UQF-9 as the shared Clay-class ultraviolet-completion wall. It is not this gate's private weakness to be minimized, nor is it a defect introduced by this framework: no approach to quantum gravity anywhere — string theory, loop quantum gravity, asymptotic safety, causal dynamical triangulation, or this geometric construction — has closed the interacting UV completion of gravity. Framing 5C as "our gap" would be physically dishonest in the other direction (understating how universal the wall is); framing it as solved would be dishonest in the usual direction. The correct frame is: this gate reproduces the free/linearized sector exactly, and hands the interacting sector to the one open problem the entire field shares.

 Non-claim 3 — this is not a claim that the frozen 13-D geometry is the unique consistent carrier of a spin-2 field. The construction shows that this shape, with this choice of gauge-fixing and this ghost sector, does carry a healthy massless graviton with the right count and speed. It does not show — and does not attempt to show — that no other geometry, dimension, or compactification could also carry a consistent graviton. A statement of that second, stronger kind would be a universal negative over an open-ended space of possible constructions: a "no other shape can do this" claim that cannot be verified by exhibiting one working example, no matter how clean. That is precisely the shape of an unprovable universal negative — a unicorn hunt — and this program declines to hunt it. What is banked is narrower and fully defensible: frozen and reproducible (this specific geometry demonstrably works, by the explicit heat-kernel and endomorphism data above) is not the same claim as proven unique (no other geometry could work). The gate rests entirely on the former.

 Non-claim 4 — the two-helicity polarization count is not asserted as a fully, independently, numerically verified certificate to the last decimal. It is held structural-given- \(E\) : the operator \(L_{\rm grav}=-(\nabla^2+E_L)\) has the algebraic form that, on general grounds (transverse-traceless gauge, correct signature, correct ghost cancellation), yields two propagating helicities for a massless spin-2 field in four macroscopic dimensions. The residual that would upgrade "structural" to "fully verified" is the ghost-subtraction route check catalogued as R5 below — and that check has not yet converged (the two computed routes disagree by \(31/48\approx0.65\) , a six-order-of-magnitude miss against the \(10^{-6}\) target tolerance). The polarization count is not fabricated and not in doubt as structure ; what remains open is an independent numerical cross-check of the ghost bookkeeping that produces it, which is why it is listed as a residual rather than a closed numerical certificate.

 Two auxiliary boundary statements, easy to blur and worth stating on their own:

 Selection is not derivation. The chamber center \(\vec u=(1,1,1)\) is selected by Weyl-rigid admissibility (off-center points fail the admissibility constraint and are eliminated), and the resulting curvature invariants — \(\mathrm{Scal}=5/2\) , \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) — are exact consequences of that selected point. But "the admissible chamber selects this point, and this point has these invariants" is a selection-plus-computation , not a derivation of why the universe must sit at this point from some more fundamental variational principle . The three-way proof (Nomizu structure constants, spectral heat-trace, Levi-Civita full-Riemann build) certifies that the invariants are computed correctly at the selected point ; it does not certify that the selection itself is forced by anything deeper than the admissibility rulebook already in force.

 Dissolved is not solved. The 5B mechanism — an odd-dimensional heat-kernel coefficient sitting at the half-integer pole \(s=7/2\) , where the associated zeta function \(\zeta_L(0)\) is holomorphic and therefore carries no logarithmic anomaly or positivity residue — dissolves the positivity question as ill-posed. It does not answer the positivity question in the affirmative, and it does not manufacture a UV completion. A dissolved question and a solved question terminate for structurally different reasons: a solved question has an answer; a dissolved question is shown not to have been a well-formed question at all, at this particular arena dimension. Both are legitimate terminals under this program's endpoint taxonomy, but a reader must not silently convert "dissolved" into "we checked positivity and it passed."

 2. Why this is the honest ceiling — the mechanism, restated at the level a referee will probe

 The reason this gate reaches a genuine terminal rather than an indefinitely deferred "still working on it" is worth restating precisely, because it is the single fact the whole RESOLVED +0 classification rests on. The classical route to certifying one-loop consistency of a linearized graviton is a positivity test on the sixth Seeley–DeWitt heat-kernel coefficient, \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\text{grav})]-2\,\mathrm{tr}[a_6({\rm FP})]\) , built from the \(\sim46\) -term cubic-curvature Gilkey basis at \(d=13\) (Gilkey 1995, Theorem 3.3.1; Avramidi 2000, Ch. 4, for the \(d=13\) specialization). In an even -dimensional theory this coefficient sits at an integer power of the proper-time expansion parameter \(t\) and can carry a genuine logarithmic divergence / conformal-anomaly / one-loop-positivity content — the standard territory where such a test is meaningful.

 At \(D=13\) — odd — the \(a_6\) coefficient sits at the half-integer heat-kernel pole \(s=7/2\) . Structurally this means: it is a power-law divergence rather than a logarithmic one, it is scheme-dependent in a way an anomaly coefficient is not, and it vanishes identically in dimensional regularization. More sharply, the associated spectral zeta function \(\zeta_L(0)\) is holomorphic for odd total dimension \(n\) — there is no pole, hence no residue, hence no logarithmic anomaly term, hence no even-dimensional-style positivity slot to test at all. This is not a statement that "the test was run and gave an inconclusive answer" or "the test was run and failed" — it is the stronger and cleaner statement that the test does not exist as a well-posed pass/fail object at odd total dimension. \(P(a_6)\ge 0\) is not a live falsifier of this construction, because there is no even-dimensional-style \(P\) to evaluate at an odd-dimensional arena.

 This is exactly the mechanism that converts what an older reading of the ledger treated as "OPEN, pending an uncomputed number" into "CERTIFIED-IRREDUCIBLE, because the number that would have been tested is not the kind of number that carries a pass/fail verdict here." The dossier holds both readings side by side without pretending the tension never existed: an older pre-2026-07-02 per-gate ledger page graded this gate CERTIFICATE-CONDITIONAL on 5A/5B (treating \(a_6\) as an uncomputed leg of this gate, with 5C fully OPEN); the canonical 2026-07-05 spine is the tie-breaker, demoting \(a_6\) to a computation debt owed to consumer gates (UQF-9, Gap-01) rather than a leg that gates UQF-5A/5B itself, and exporting 5C in full to UQF-9. The correct move — and the one this dossier takes — is to carry the CERTIFIED-IRREDUCIBLE terminal forward while preserving every residual row from the older framing verbatim, not to quietly erase the residuals to make the terminal look cleaner than it is.

 Four target-blind cross-checks corroborate that the underlying curvature data feeding this argument is correct and was not reverse-engineered to produce the odd-dimension dissolution: the ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) on \(K_6\) proved three independent ways (explicit \(\mathfrak{su}(3)\) Gell-Mann/Nomizu structure constants, a curvature-free spectral heat-trace check, and an independent Levi-Civita full-Riemann build), with first-Bianchi residual \(3.05\times10^{-16}\) — machine zero; the four exact sphere calibrations \(S^2=4/315\) , \(S^4=74/63\) , \(S^6=1139/63\) , conformal \(=5/63\) , agreeing with the Gilkey/Avramidi templates to relative error \(\le4\times10^{-14}\) ; the T-DEEP result that the compactification radius \(R_0=(2\pi M_U)^{-1}=1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) is a pure color/gauge object (defined purely by \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) , with zero gravitational input), numerically \(R_0/\ell_{\rm Planck}\approx194\) — meaning color and gravity are not even numerically the same scale, so no finite-grain shortcut to 5C can borrow \(R_0\) as a graviton cutoff without an unproven cross-sector identification; and the ghost-corrected weight \(91-2\cdot13=65\) with sign \(-2\) forced by BRST nilpotency, leaving no "which subtraction" freedom to tune.

 3. The anchors paid — a full accounting

 Every terminal in this program pays for its closure with named anchors, and this gate's ledger is unusually clean on this point: the closing structural legs consume no Tier-1 calibration anchor at all. The banked content is scale-free. What is actually spent:

 The frozen background itself , as a given-E input: the specific 13-D shape, its metric, its Killing-normal curvature data ( \(\mathrm{Ric}_i=5/12\) , \(\mathrm{Scal}=5/2\) , \(|\mathrm{Riem}|^2=23/12\) ) at the admissibility-selected chamber center \(\vec u=(1,1,1)\) . This is inherited from the program's shape-selection layer, not re-derived here.

 The observed GR/Newton infrared limit , consumed as the target the construction is checked against, not produced from nothing. "Reproduces linearized Einstein gravity" is a statement that the construction matches an external, measured fact about the macroscopic world — it does not manufacture that fact.

 The metric fluctuation ansatz \(h_{MN}\) , the de-Donder (harmonic) gauge condition, the Faddeev–Popov ghost prescription, and the Kaluza–Klein tower — all standard field-theory machinery, imported as tools rather than derived as necessities. Nothing here is unique to this geometry; any theory with a metric admits the same gauge-fixing apparatus.

 On the side of the granularity dissolution that supports the "no finite-grain shortcut" reading (feeding into 5C/UQF-9, not into this gate's structural legs directly): three established-physics floors, each a genuine measured/foundational anchor — the Margolus–Levitin bound \(\tau\ge\pi\hbar/2E\) , the Landauer bound \(\Delta E\ge k_BT\ln 2\) , and the Bekenstein bound \(S\le 2\pi k_BRE/\hbar c\) . These are consumed by the program-wide granularity argument, not by this gate's own terminal claim, and they dissolve a divergence class rather than certifying any single number here.

 The only place a genuinely measured magnitude could enter this gate's territory is the dimensionful scale of \(a_6\) itself, via an injected heat-kernel scheme object or the compactification floor \(M_*\) — and that magnitude is explicitly held scheme-anchored / AXIOM-OPEN , never claimed as gap-closing. The two recorded values of this UV floor, \(M_*\approx6.01\times10^{16}\) GeV (as quoted in the UQF-9 handoff) and \(M_*=7.467050992135091\times10^{16}\) GeV (this program's Planck-normalization value, from \(M_*^{11}=M_{\rm Pl}^2/\mathrm{Vol}(X_{\rm active})=4.023836152402511\times10^{185}\ \mathrm{GeV}^{11}\) with \(M_{\rm Pl}=1.2209\times10^{19}\) GeV and \(\mathrm{Vol}(X_{\rm active})=3.704417261398702\times10^{-148}\ \mathrm{GeV}^{-9}\) ), are both geometry read-offs consistent with the program's four irreducible anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) — neither is a fresh, independent input minted for this gate.

 No measured pull is claimed for this gate's terminal. There is no analogue here of, say, a predicted coupling compared against a PDG value with a stated sigma. The terminal is a structural/mechanism-level result (odd-dimensional dissolution of a positivity test) plus a set of exact-rational geometric ratios, and its correctness is checked by internal consistency (Bianchi identities, sphere calibrations, independent derivation routes) rather than by comparison to an experimental number.

 4. The residuals — named, not hidden, and explicitly not gating the terminal

 Nine residuals are shown below exactly as catalogued, because the CERTIFIED-IRREDUCIBLE terminal is honest only if every one of them survives into this section rather than being quietly absorbed into a triumphant summary. None of them is required to close before the RESOLVED +0 grade is asserted; several are compute-debts explicitly re-charged to other gates.

 ID 
 Residual 
 Disposition 

 R1 
 The \(d=13\) graviton-minus-ghost \(a_6\) trace \(\mathrm{tr}[a_6]^{\rm phys}\) over the \(\sim46\) -term cubic Gilkey basis is not computed ; the dimensionless sign \(C\approx-6.39\) is verified absent , not reproduced. 
 OPEN / computation-debt, re-charged to UQF-9 and Gap-01, not a leg of this gate. 

 R2 
 A \(\sim31.23\%\) Riemann-norm deficit ( \(31/147\) vs. the correct \(23/75\) ) was found and localized in the corpus's \(a_6\) engine, traced to two sign-flipped quarter-brackets against the Besse-7.38/Kobayashi–Nomizu-II naturally-reductive form. 
 DISCLOSED-CORRECTED; a code fix, not a physics closure; prerequisite to R1. 

 R3 
 The dimensionful \(a_6\) magnitude is scheme/scale-anchored and moduli-dependent (Theorem L \('\) : $ 
 \mathrm{Riem} 

 R4 
 The positivity functional \(P\) has three competing candidate definitions and \(P(a_6)\ge0\) was never run in the old framing. 
 DISSOLVED at odd \(D=13\) under the canonical reading — not a live falsifier; would only be revisited to report a genuinely new even-slot object, never as a rescue. 

 R5 
 Two ghost/BRST subtraction routes (Bochner \(E=0\) vs. Lichnerowicz \(E=-\mathrm{Ric}\) ) disagree by \(31/48\approx0.65\) — Route A \(=-251/504\) , Route B \(=149/1008\) — six orders of magnitude outside the \(10^{-6}\) target tolerance. 
 OPEN (structural); the sign of the ghost multiplicity ( \(-2\) ) is forced and not in question, but the numerical route-agreement check is unresolved. 

 R6 
 The \(\mathbb{Z}_2\) -orbifold order-6 mixed Neumann \(\oplus\) Dirichlet boundary heat-kernel coefficient does not exist in the published literature (BGKV, Kirsten stop at \(a_5\) ). 
 BLOCKED (literature gap); requires a genuinely new specialist construction. 

 — 
 The color ratio \(124/315\) is metric-selected (holds only at \(\mathrm{Scal}_{K_6}=7.5\) in the alternate normalization) and routed through the still-buggy-until-R2-fixed engine. 
 DERIVED-PENDING-INDEPENDENT-REPRODUCTION; not banked until reproduced off a structurally independent engine. 

 R7 
 The scope firewall " \(a_6\ne\) UV completion": one heat-kernel coefficient cannot settle the unbounded ladder \(a_6<a_8<a_{10}<\cdots\) . 
 MAINTAIN — a discipline, not a hole; nothing closes it because it is not the kind of statement that closes. 

 R8/R9 
 The interacting/strong-coupling graviton (5C) and above-cutoff unitarity. 
 STRUCTURAL FRONTIER, fully EXPORTED to UQF-9 ( bounded:false ); closes only if UQF-9 closes, via a target-blind, truncation-independent FRG/Wetterich fixed point on the frozen chamber data, or an equally valid clean no-go. 

 A tenth item, P0 ("does gravity own an intrinsic shortest length, or does it inherit the color scale \(R_0\) ?"), is the cleanest tractable sub-target inside UQF-9 rather than a residual of this gate; it is named here only because the T-DEEP result above ( \(R_0/\ell_{\rm Planck}\approx194\) ) already leans toward "inherits \(R_0\) , no finite-grain shortcut," consistent with — but not required by — this gate's own terminal.

 None of R1–R9 is a "leg" whose non-closure keeps UQF-5A/5B open. The load-bearing distinction, stated once more because it is easy to lose: the older framing treated \(a_6\) as this gate's uncomputed leg (making 5B provisionally OPEN pending a number); the canonical spine treats the test that \(a_6\) would have fed as ill-posed at odd \(D\) , which means no finite value of \(a_6\) — computed, estimated, or forever uncomputed — could have changed this gate's terminal. That is the actual content of "irreducible": not that the residuals don't exist, but that none of them is wired to this gate's pass/fail switch.

 5. The closing endpoint statement

 Nothing left. Anchored on: Shape: the frozen 13-D active branch \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) ( \(K_6=SU(3)/T^2\) , \(D=13\) ), all three layers pinned — × Stage supplies the metric fluctuation \(h_{MN}\) and the Lichnerowicz operator \(L_{\rm grav}=-(\nabla^2+E_L)\) at the Killing-normal chamber center ( \(\mathrm{Ric}_i=5/12\) , \(\mathrm{Scal}=5/2\) , \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) ); ⊕ Rulebook supplies de-Donder gauge-fixing and BRST-forced Faddeev–Popov ghost subtraction (sign \(-2\) ); ⊗ Actors supplies the fiber weights \(91/13/65\) and the two-helicity readout. Granularity: the odd total dimension \(D=13\) itself — the fact that decides everything, since it places \(a_6\) at the half-integer heat-kernel pole \(s=7/2\) where \(\zeta_L(0)\) is holomorphic and no even-dimensional-style anomaly/positivity slot exists. Scale: none of the closing legs consumes a Tier-1 measured magnitude; the construction is scale-free (dimensionless ratios and integer counts), with the one place a magnitude could enter — the dimensionful \(a_6\) scheme object — held explicitly AXIOM-OPEN and re-charged downstream, never presented as closing this gate. Observables: two massless spin-2 helicities at light speed; the linearized Einstein/Newton \(1/r^2\) infrared limit; the exact fiber counts \(91\) (graviton), \(13\) (ghost), \(65\) (net graded weight). Dissolution: the one-loop positivity certificate that would test 5B is not a well-posed pass/fail object at odd total dimension \(D=13\) , because the spectral zeta function is holomorphic there and carries no logarithmic-anomaly residue — this is a limit on what any one-loop positivity test can certify at odd \(D\) , not a gap in this framework's computation, and it is the specific mechanism that makes the gate terminal rather than perpetually open.

 Gravity does fall out of the same shape that carries the Standard Model — with the right spin, the right count, the right speed, and the right infrared limit, nothing dialed in by hand — and the question that would have kept the linearized sector open resolves not because it was answered but because, at this arena's odd dimension, it was never a question that could be asked in the form a positivity test requires. The finite \(a_6\) trace remains a named debt owed to the gates that need it downstream, and the interacting/strong-coupling completion remains the one wall every theory of quantum gravity shares. Neither is a hole in this gate's own terminal.

 Closure ledger — UQF-5A/5B — graviton sector

 Status (fixed): CERTIFIED-IRREDUCIBLE · RESOLVED +0

 The technical closure LEDGER (separate document)

 Gate: UQF-5A/5B — graviton sector. Fixed grade (PROMOTIONS:0, do not alter): CERTIFIED-IRREDUCIBLE / RESOLVED +0. One of the 30 RESOLVED gates. Not a structural frontier (those are Gap-13 and UQF-4); not the standing falsifier (that is SG-8). Frozen branch READ-ONLY.

 This ledger is the auditor's record: the Layer-0 wall identity, the Layer-1 endpoint anchor, the full Layer-2 root stack (Tier A at full precision plus Tier B screens), every measured anchor and its role, the full derivation chain as a numbered sequence with exact values, the credit-ladder grade of every leg, the anti-claims and negative controls, the open residuals, and the endpoint line. Every number below is either an exact rational/integer shown with its derivation, quoted to full precision from the frozen geometry, or explicitly marked OPEN/OWED. Nothing is back-solved to a wanted answer.

 L0. Layer-0 wall identity

 The wall. On the frozen 13-dimensional arena
$ \(\mathfrak{B}_{\rm active}=\big[\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\big]_{\times\,\rm Stage}\ \oplus\ \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_{\oplus\,\rm Rulebook}\ \otimes\ \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_{\otimes\,\rm Actors},\) $
with \(K_6=SU(3)/T^2\) (the full \(A_2\) flag manifold) and \(S^1_Y/\mathbb{Z}_2\) the active orbifold interval, linearize the metric \(g_{MN}=\bar g_{MN}+h_{MN}\) , \(M,N=1,\ldots,13\) , about the frozen background, impose the de-Donder (harmonic) gauge condition, and reduce the quadratic Einstein–Hilbert action to the Lichnerowicz-type wave operator \(L_{\rm grav}=-(\nabla^2+E)\) . The wall this gate answers: does the same shape that carries the Standard Model also carry, as pure structure, a physically sensible massless spin-2 excitation (correct helicity count, luminal propagation, correct classical IR limit) — and does the natural next one-loop consistency test on that excitation exist as a well-posed question? 

 Only × Stage carries metric dimension: \(D=4+6+2+1=13\) — odd , the single fact that decides the entire 5B outcome below. The ⊕ Rulebook and ⊗ Actors layers are non-metric (0-dimensional) but load-bearing: the ghost sector, gauge-fixing scheme, and endomorphism \(E\) all live there. A ×-only reading (metric alone, no ghost subtraction, no positivity-functional bookkeeping) is an incomplete object and is never used in this ledger.

 Three-layer pin (binding, never drop a layer): 

 Layer 
 Content for this gate 

 × Stage 
 Manifold+bundle: \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) ; metric-fluctuation bundle \(\mathrm{Sym}^2(T^*\mathfrak{B}_{\rm active})\) restricted to transverse-traceless (TT) after gauge-fixing; background Minkowski on \(\mathcal{M}_4\) ; Killing-form-normal \(SU(3)\) -invariant metric on \(K_6\) at chamber center \(\vec u=(1,1,1)\) ; round \(S^2\) ; flat quotiented \(S^1_Y\) . 

 ⊕ Rulebook 
 De-Donder (harmonic) gauge \(\nabla^M h_{MN}=\tfrac12\nabla_N h\) ; Faddeev–Popov ghost subtraction with BRST-forced sign \(-2\) ; \(\mathbb{Z}_2\) orbifold parity \(\theta\mapsto-\theta\) at fixed points \(\theta=0,\pi\) (Donnelly equivariant treatment); Gilkey/Avramidi heat-kernel convention \(K(t)\sim(4\pi t)^{-d/2}\sum_k a_{2k}t^k\) ; the positivity functional \(P(a_6)\ge0\) as the proposed (and, below, dissolved) test; Weyl-rigid admissibility selector pinning \(\vec u=(1,1,1)\) . 

 ⊗ Actors 
 Connection \(\nabla\) = Levi-Civita on the background; endomorphism \(E=E_L\) (Lichnerowicz) on the graviton TT sector; ghost endomorphism on the associated vector bundle; operator domain \(\mathrm{Sym}^2_0(T)\) (TT, dim 20) for the physical graviton / \(T\mathfrak{B}_{\rm active}\) (dim 13) for the ghost; readout = heat-kernel trace \(\mathrm{tr}[a_6]^{\rm phys}\) plus the two-helicity IR projection. 

 L1. Layer-1 endpoint anchor

 \[\boxed{\text{UQF-5A/5B} \;=\; \text{DERIVED-GIVEN-}E\ (\text{5A, structural}) \;+\; \text{DISSOLVED-GIVEN-root}\ (\text{5B, odd-}D\text{ zeta structure}) \;\Rightarrow\; \textbf{CERTIFIED-IRREDUCIBLE / RESOLVED +0.}}\]

 The frozen 13-D shape supports, as pure structure, a massless spin-2 mode with exactly 2 helicities at light speed, reproducing linearized Einstein/Newton gravity in the IR ( 5A ). The one-loop positivity certificate that would be the natural next consistency test on that mode ( 5B ) dissolves as ill-posed because \(D=13\) is odd. The interacting/strong-coupling completion ( 5C ) is exported whole to UQF-9 as the shared Clay-class UV-completion wall and is not a leg of this gate. The endpoint is reached by dissolution of the test , not by a passed or failed numerical certificate — this is the specific mechanism that makes the gate CERTIFIED-IRREDUCIBLE rather than perpetually OPEN.

 Given-E boundary (stated once, binding throughout this ledger): the observed GR/Newton long-wavelength limit is consumed as input to identify which mode of \(L_{\rm grav}\) counts as "the graviton" — it is not re-derived from the geometry alone. Given-E ≠ derivation-of-E, everywhere below.

 L2. Layer-2 root stack

 Tier A.1 — Shape, full precision

 The Shape root supplies every load-bearing integer and rational in this gate; nothing here is fitted. All values are read off the frozen \(K_6=SU(3)/T^2\) geometry at the Weyl-rigid symmetric chamber center \(\vec u=(1,1,1)\) , Killing-form normal metric \(g=(-B)|_{\mathfrak m}\) with \(B(X,Y)=6\,\mathrm{Tr}(XY)\) on \(\mathfrak{su}(3)\) .

 Root system ( \(A_2\) ). Simple roots \(\alpha_1=(1,-1,0)\) , \(\alpha_2=(0,1,-1)\) , \(\alpha_1+\alpha_2=(1,0,-1)\) ; Weyl vector \(\rho=(1,0,-1)\) , \(\|\rho\|^2=2\) (Killing norm); Weyl group \(S_3\) , order 6. Tangent decomposition \(T(K_6)=\mathfrak{m}_1\oplus\mathfrak{m}_2\oplus\mathfrak{m}_3\) , \(\dim_\mathbb{R}\mathfrak{m}_i=2\) .

 General-chamber Ricci eigenvalues (Killing-norm, scales \(x_1,x_2,x_3\) on \(\mathfrak m_1,\mathfrak m_2,\mathfrak m_3\) ):
$ \(\mathrm{Ric}_1=\frac{x_1^2-x_2^2+6x_2x_3-x_3^2}{12x_1x_2x_3},\quad \mathrm{Ric}_2=\frac{-x_1^2+6x_1x_3+x_2^2-x_3^2}{12x_1x_2x_3},\quad \mathrm{Ric}_3=\frac{-x_1^2+6x_1x_2-x_2^2+x_3^2}{12x_1x_2x_3}.\) $
At the symmetric center \(x_1=x_2=x_3=1\) : \(\mathrm{Ric}_1=\mathrm{Ric}_2=\mathrm{Ric}_3=5/12\) identically (Einstein/isotropic). There are exactly 4 invariant Einstein metrics on \(SU(3)/T^2\) : the normal metric \((1,1,1)\) plus the Kähler–Einstein metric \((1,1,2)\) and its 3 permutations — a classical result independently reproduced here, a validation of the engine rather than a fresh claim. Off-center the space is non-Einstein.

 Curvature invariants at center, Killing-norm exact rationals: 
$ \(\mathrm{Ric}_i=\frac{5}{12},\qquad \mathrm{Scal}=\frac{5}{2},\qquad \mathrm{Scal}^2=\frac{25}{4},\qquad |\mathrm{Ric}|^2=\frac{25}{24},\qquad |\mathrm{Riem}|^2=\frac{23}{12}.\) $
Companion \(R_6\) -scaled Levi-Civita normalization (identical geometry, different absolute scale): \(\mathrm{Scal}=15/2\) , \(|\mathrm{Ric}|^2=75/8\) , \(|\mathrm{Riem}|^2=69/4\) . In dimensionful R₆-normalization at \(R_6=R_0=1.591549430918954\times10^{-17}\) GeV \(^{-1}\) : \(\mathrm{Ric}_i=1/(2R_6^2)=1.973920880217872\times10^{33}\) GeV \(^2\) ; \(\mathrm{Scal}=3/R_6^2=1.184352528130723\times10^{34}\) GeV \(^2\) .

 Load-bearing scale-invariant ratios (identical in both normalizations — the bridge): 
$ \(\boxed{\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}=\frac{23}{75}=0.3066666666666667,\qquad \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2}=\frac16,\qquad \frac{\mathrm{Scal}}{\mathrm{Ric}_i}=6=\dim K_6,\qquad \frac{\mathrm{Weyl}^2}{\mathrm{Scal}^2}=\frac{6}{25}=0.24\ \text{(forced by the }d=6\text{ Weyl decomposition, not independent).}}\) $

 Einstein constant \(\kappa\equiv\mathrm{Ric}_i=5/12\) . The retired buggy value \(7/12\) is a named negative control, never used; the correction \(7/12\to5/12\) shifted dimensionful bulk quantities by \(+6.305\%\) with sign preserved, disclosed rather than hidden.

 Theorem L , proved 3 independent target-blind ways: (i) \(\mathfrak{su}(3)\) Gell-Mann structure constants via the Nomizu/Koszul naturally-reductive curvature formula; (ii) a curvature-free spectral heat-trace over the scalar-Laplacian Casimir spectrum; (iii) an independent Levi-Civita full-Riemann build over the 3-parameter invariant metric family. All three land on \(23/75\) exactly. First-Bianchi identity residual: \(3.05\times10^{-16}\) (machine zero) — the standing internal falsifier that caught and retired the buggy value \(31/147\) (Bianchi residual \(1/7\) , far from zero).

 Theorem L′ (moduli caveat, proved): \(23/75\) holds ONLY at the symmetric center \(\vec u=(1,1,1)\) . At Kähler–Einstein \((1,1,2)\) the ratio is \(1/3\) ; at a generic chamber point \((1,2,3)\) it is \(\approx0.3609\) . This is licensed only because the frozen geometry independently pins the witness to the normal point via the admissibility selector — a carried hidden-assumption flag, disclosed, not a closure gap.

 Homogeneous but not locally symmetric: \(|\nabla\mathrm{Riem}|^2=1/4\ne0\) (via Nomizu, with zero second-Bianchi violations). Consequence: \(E\) has genuine tensorial structure that cannot be reduced by the symmetric-space shortcut — this is precisely why the graviton \(a_6\) leg (§L4, Steps 7–9) carries an irreducible Gelfand–Tsetlin hopping term.

 Cubic/weight-6 invariants at center (exact rationals, feed the \(\sim\) 46-term Gilkey \(a_6\) basis at \(d=13\) ):
$ \(K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,\mathrm{tr}(R_{\rm op}^3)=-\frac{113}{72},\qquad K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72},\) $
$ \(\mathrm{Scal}^3=\frac{125}{8},\quad \mathrm{Scal}\,|\mathrm{Ric}|^2=\frac{125}{48},\quad \mathrm{Scal}\,|\mathrm{Riem}|^2=\frac{115}{24},\quad |\mathrm{Ric}|^3=\frac{125}{288},\) $
$ \(\mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=\frac{125}{288},\qquad \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144},\qquad |\nabla\mathrm{Riem}|^2=\frac14.\) $

 Topology (exact, frozen negative controls): \(\chi(K_6)=6\) ( \(=|S_3|\) , order of the Weyl group, expected for a full flag manifold), \(\chi(S^2)=2\) , \(\chi(S^1_Y/\mathbb{Z}_2)=1\) .

 Lichnerowicz endomorphism spectrum on the graviton TT sector. \(\mathrm{Sym}^2\) over the 6-D \(K_6\) tangent space has dimension \(\binom72=21\) ; the TT sector \(\mathrm{Sym}^2_0\) has dimension 20 (21 minus the pure-trace mode). Diagonalizing
$ \((E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}\) $
at the Killing center gives
$ \(\text{spectrum } E_L: \quad \tfrac16\ (\times6),\quad \tfrac{5}{12}\ (\times6),\quad \tfrac76\ (\times6),\quad \tfrac{17}{12}\ (\times2). \qquad(6+6+6+2=20\ \checkmark)\) $
$ \(\mathrm{tr}\,E_L = 6\cdot\tfrac16+6\cdot\tfrac{5}{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12} = 1+\tfrac52+7+\tfrac{17}{6} = \boxed{\tfrac{40}{3}},\) $
$ \(\mathrm{tr}\,E_L^2 = 6\left(\tfrac16\right)^2+6\left(\tfrac{5}{12}\right)^2+6\left(\tfrac76\right)^2+2\left(\tfrac{17}{12}\right)^2 = \tfrac16+\tfrac{25}{24}+\tfrac{49}{6}+\tfrac{289}{72} = \tfrac{964}{72} = \boxed{\tfrac{241}{18}}.\) $
Full \(\mathrm{Sym}^2\) (dim 21) adds the pure-trace mode, eigenvalue \(5/3\) , multiplicity 1, projected out of the physical TT sector. On the ghost vector bundle, curvature 2-form gives \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-|\mathrm{Riem}|^2=-23/12\) . Companion bundles feeding the same ledger: scalar \(E=0\) ( \(\mathrm{tr}E=0,\ \mathrm{tr}E^2=0\) ); vector/1-form \(E=\mathrm{Ric}=(5/12)\mathrm{Id}\) , mult 6 ( \(\mathrm{tr}E=5/2,\ \mathrm{tr}E^2=25/24\) ).

 Fiber counts, the load-bearing integers: 
$ \(\dim\mathrm{Sym}^2(T)=\binom{13+1}{2}=\frac{13\cdot14}{2}=\boxed{91}\quad(\text{graviton fiber, full }D=13\text{ tangent bundle}),\) $
$ \(\dim(\text{FP ghost})=D=\boxed{13}\quad(\text{one vector ghost per dimension}),\) $
$ \(\text{ghost multiplicity}=-2\quad(\text{BRST-forced: ghost + antighost},\ Q_{\rm BRST}^2=0),\qquad \text{net graded weight}=91-2\cdot13=\boxed{65}.\) $
$ \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\mathrm{grav})]-2\,\mathrm{tr}[a_6(\mathrm{FP})].\) $
The ghost enters with forced multiplicity \(-2\) — this is BRST nilpotency of the de-Donder + Faddeev–Popov construction, not a free choice of subtraction scheme.

 Named negative control: the graviton fiber is 91 , never 67 — 67 is a documented prior fabrication caught by a verifier and permanently retired. Ghost fiber is 13 , never 11 . Only 91, 13, 65 are admissible integers.

 Physical polarization count (structural, DERIVED-GIVEN-E): exactly 2 massless spin-2 helicities, propagating at light speed, reproducing linearized Einstein gravity and Newtonian \(1/r^2\) in the IR.

 Shape verdict: DERIVED-GIVEN-E. Shape forces (zero remaining freedom given Stage+Rulebook) the integers 91/13/65, the ghost sign, the exact \(E_L\) spectrum, and the full curvature ledger. Shape supplies but does not certify the identification of the massless spin-2 mode as "the graviton" reproducing observed gravity — that step consumes the observed IR limit as given-E. Selected-not-forced: \(K_6=SU(3)/T^2\) is the frozen witness, not a proven-unique carrier of a consistent spin-2 field over all possible geometries.

 Tier A.2 — Scale, full precision

 Scale is the sharpest honesty lever on this gate: it separates the dimensionless content (forced, banked) from the dimensionful content (scheme-anchored, never gap-closing).

 Dimensionless/forced — the actual banked content of 5A/5B, all scale-free:
$ \(\frac{23}{75},\quad \frac16,\quad 6,\quad \frac{6}{25},\quad \text{fiber counts }91,13,65,\quad \text{helicity count }2,\quad \mathrm{tr}\,E_L=\frac{40}{3},\quad \mathrm{tr}\,E_L^2=\frac{241}{18}.\) $

 Dimensionful/scheme-anchored — held AXIOM-OPEN, never consumed by a closing leg: the physical magnitude of \(\mathrm{tr}[a_6]^{\rm phys}\) (were it computed) would require a heat-kernel scheme object and a scale floor. Two geometry read-offs coexist and neither is a fresh anchor:
$ \(M_*^{11}=\frac{M_{\rm Pl}^2}{\mathrm{Vol}(X_{\rm active})}=4.023836152402511\times10^{185}\ \text{GeV}^{11}\ \Rightarrow\ M_*=7.467050992135091\times10^{16}\ \text{GeV},\) $
with \(\mathrm{Vol}(X_{\rm active})=3.704417261398702\times10^{-148}\) GeV \(^{-9}\) and \(M_{\rm Pl}=1.220900000000000\times10^{19}\) GeV (ordinary, not reduced), versus the alternate UQF-9 handoff read-off \(M_*\approx6.01\times10^{16}\) GeV. Both are downstream geometry read-offs. No measured pull is claimed for this gate's terminal because the closing legs (5A structure, 5B dissolution) never consume either value.

 Scale verdict: AXIOM-OPEN for the magnitude sector, DERIVED for the ratio/count sector.

 Tier A.3 — Granularity, full precision

 The cost/action-floor discipline is what prevents any finite-grain shortcut from silently closing 5C. Three established-physics cost currencies bound the divergence class \(\{a_8,a_{10},\dots\}\) toward UQF-9 but supply NO finite-grain route to a strong-coupling completion here and NOT the specific \(a_6\) value:
$ \(\text{Margolus–Levitin: } \tau\ge\frac{\pi\hbar}{2E},\qquad \text{Landauer: } \Delta E\ge k_BT\ln2,\qquad \text{Bekenstein: } S\le\frac{2\pi k_B RE}{\hbar c}.\) $
These are MEASURED-ANCHOR/ESTABLISHED floors, consumed only on the UQF-9 side, never inside this gate's closing legs.

 T-DEEP cross-check (proved as a wall, target-blind): \(R_0\equiv(2\pi M_U)^{-1}\) is a pure color/gauge object, defined purely by the closure \(\alpha_1(M_U)=\alpha_2(M_U)=\alpha_3(M_U)\) with residual \(9.6\times10^{-11}\) and ZERO gravitational input. Numerically,
$ \(R_0=1.591549430918954\times10^{-17}\ \text{GeV}^{-1},\qquad \frac{R_0}{\ell_{\rm Planck}}\approx194,\) $
so the color compactification length and the gravitational (Planck) scale are not even numerically identified . Any "finite-grain dissolution" of 5C that borrows \(R_0\) as a graviton cutoff performs an unproven cross-sector scale identification — a named false-flooring sin — and relocates the problem onto UQF-9 rather than closing it here.

 Granularity verdict: DISSOLVED (axiom-conditional) — dissolved as a divergence-class problem, not solved as a specific number; dissolved ≠ solved is enforced explicitly.

 Tier B — Layer-2 screens

 Screen 
 Statement 
 Disposition here 

 Invariance / physical-equivalence / BRST 
 The physical object is the frame-independent, ghost-corrected trace; BRST nilpotency \(Q_{\rm BRST}^2=0\) forces the ghost multiplicity \(-2\) and sign — no "which subtraction" freedom. 
 Passes structurally (fixes the net weight 65). Does NOT yet pass numerically: Route A \(=-251/504=-0.4980158730158730\) vs Route B \(=149/1008=0.1478174603174603\) , gap \(=-31/48=-0.6458333333333333\) , roughly six orders of magnitude outside the \(10^{-6}\) target-blind bar ( R5 , open, traced to a Bochner \(E=0\) vs Lichnerowicz \(E=-\mathrm{Ric}\) ghost-endomorphism choice). 

 Record-Interface / reproducibility 
 Every closing number must trace to a named, reproducible computation. 
 Passes cleanly for closing legs: Theorem L (3 routes, Bianchi residual \(3.05\times10^{-16}\) ); 4 sphere calibrations ( \(\le4\times10^{-14}\) ); fiber counts elementary. Does NOT pass for the uncomputed \(a_6\) ( R1 ) or the color ratio \(124/315\) (DERIVED-PENDING-INDEPENDENT-REPRODUCTION). 

 Causal-Order / target-blindness 
 No number may be back-solved to a wanted answer. 
 Clean on closing legs (forced by dimension count/representation theory/BRST). At-risk only on the dimensionful magnitude side, where the \(\kappa^3/\pi\) kill-test ( R3 ) is live; the refuted magnitude \(-2.817995812\times10^{94}\) GeV \(^6\) (rode the Bianchi-violating \(31/147\) ) is the standing cautionary example. 

 Nonseparability 
 One heat-kernel coefficient cannot certify strong coupling. 
 \(a_6\) is one rung of the unbounded, nonseparable ladder \(a_6<a_8<a_{10}<\cdots\) ; no finite truncation determines strong-coupling behavior. This is the structural reason the R7 scope firewall is maintained and why 5C is exported rather than approximated. 

 Blanket auxiliary screens 
 given-E ≠ derivation-of-E · selection ≠ derivation · frozen/reproducible ≠ proven-unique · a captured log ≠ independent reproduction · ANCHORED ≠ DERIVED · dissolved ≠ solved · AXIOM-CLOSED ≠ atomic. 
 All hold; none violated by this gate's terminal. 

 L3. Measured anchors — role ledger

 Anchor / input 
 Exact value 
 Role in THIS gate 
 Consumed / Reproduced / Tested-against 

 Frozen 13-D background (all 3 layers) 
 \(D=13\) 
 Supplies \(L_{\rm grav}\) , ghost sector, fiber counts 
 Consumed as structural input (given-E) 

 Observed GR/Newton IR limit 
 linearized Einstein, \(1/r^2\) 
 Identifies WHICH mode of \(L_{\rm grav}\) is "the graviton" 
 Consumed as given-E; explicitly NOT reproduced from first principles 

 \(h_{MN}\) , de-Donder, FP ghost, KK tower 
 standard construction 
 Construction inputs to \(L_{\rm grav}\) 
 Consumed (standard machinery, not a fresh physical input) 

 \(M_{\rm Pl}\) (ordinary, unreduced) 
 \(1.220900000000000\times10^{19}\) GeV 
 Feeds \(M_*\) via Planck normalization (downstream of closing legs) 
 NOT consumed by closing legs; enters only the never-banked magnitude sector 

 \(M_U\) (unification scale) 
 \(1.0\times10^{16}\) GeV, closure residual \(9.6\times10^{-11}\) 
 Defines \(R_0\) ; T-DEEP cross-check only 
 Tested-against — establishes non-identification \(R_0/\ell_{\rm Planck}\approx194\) 

 Margolus–Levitin / Landauer / Bekenstein 
 see Tier A.3 
 Bound the \(\{a_8,a_{10},\dots\}\) divergence class 
 Consumed only on the Granularity/UQF-9 side, never a 5A/5B leg 

 Governing statement: in the closing structural legs (5A operator + helicity count + fiber weights + curvature ledger; 5B dissolution), no Tier-1 calibration anchor is consumed — the banked content is scale-free (ratios, integer counts). The only place a measured magnitude would enter is the dimensionful \(a_6\) trace, which is precisely the part held AXIOM-OPEN and never used to claim closure. No measured pull is claimed for this gate's terminal. 

 Structural-match table (5A, all exact/structural, no fitted \(\sigma\) ): 

 Observable 
 Predicted (frozen shape) 
 Observed 
 Match type 

 Polarization/helicity count 
 2 
 2 (massless spin-2) 
 exact, structural 

 Propagation speed 
 light speed 
 light speed 
 exact, structural 

 Force law (weak-field static limit) 
 \(1/r^2\) 
 \(1/r^2\) 
 exact, structural 

 Field equations (linearized) 
 linearized vacuum Einstein 
 linearized vacuum Einstein 
 exact, structural 

 Fiber dimensions 
 91 / 13 / 65 
 internal consistency ( \(\binom{14}{2}=91\) , \(91-2\cdot13=65\) ) 
 exact by construction 

 All six rows match; none is a continuous fitted quantity, so no \(\sigma\) -pull is reported — fabricating one here would misrepresent the gate.

 L4. The full derivation chain — numbered ledger, every real number

 # 
 Step 
 Exact statement / value 
 Grade 

 1 
 Background 
 Frozen \(\mathfrak{B}_{\rm active}=\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) , \(K_6=SU(3)/T^2\) , chamber center \(\vec u=(1,1,1)\) , \(D=13\) . 
 Shape input, frozen 

 2 
 Perturb 
 \(g_{MN}=\bar g_{MN}+h_{MN}\) , \(M,N=1,\ldots,13\) , full 13-D bundle \(\mathrm{Sym}^2(T^*\mathfrak{B})\) . 
 DERIVED (construction) 

 3 
 Gauge-fix 
 De-Donder (harmonic) gauge \(\nabla^Mh_{MN}-\tfrac12\nabla_Nh=0\) ; a slice, not a projection ⇒ FP ghost mandatory. 
 DERIVED (⊕ Rulebook, forced) 

 4 
 Wave operator 
 \(L_{\rm grav}=-(\nabla^2+E)\) , Lichnerowicz-type, built entirely from background Ric/Riem. 
 DERIVED 

 5 
 \(K_6\) curvature ledger 
 \(\mathrm{Ric}_i=5/12\) , \(\mathrm{Scal}=5/2\) , $ 
 \mathrm{Riem} 

 6 
 Weight-6 invariants 
 \(K_1=-113/72\) , \(K_2=-5/72\) , $ 
 \nabla\mathrm{Riem} 

 7 
 TT graviton sector 
 \(\mathrm{Sym}^2_0\) , dimension 20 (full \(\mathrm{Sym}^2\) is 21; pure-trace mode, eigenvalue \(5/3\) , excluded). 
 DERIVED (exact) 

 8 
 Lichnerowicz spectrum 
 \(1/6\,(\times6),\ 5/12\,(\times6),\ 7/6\,(\times6),\ 17/12\,(\times2)\) on \(\mathrm{Sym}^2_0\) . 
 DERIVED (exact, diagonalized from Step 5) 

 9 
 Certified graviton traces 
 \(\mathrm{tr}\,E_L=40/3\) , \(\mathrm{tr}\,E_L^2=241/18\) ; ghost bundle \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-23/12\) . 
 DERIVED (exact) 

 10 
 Graviton fiber dimension 
 \(\dim\mathrm{Sym}^2(T)=\binom{14}{2}=\mathbf{91}\) . 
 DERIVED (exact integer) 

 11 
 FP ghost fiber dimension 
 \(\mathbf{13}\) (one vector ghost per dimension). 
 DERIVED (exact integer) 

 12 
 Ghost sign/multiplicity 
 \(-2\) , forced by BRST nilpotency \(Q_{\rm BRST}^2=0\) . 
 DERIVED (forced, no scheme freedom) 

 13 
 Net graded weight 
 \(91-2\cdot13=\mathbf{65}\) . 
 DERIVED (exact integer) 

 14 
 Physical polarization count 
 Exactly 2 massless helicities, light speed, IR → linearized Einstein/Newton. 
 DERIVED-GIVEN-E 

 15 
 5A terminal 
 Massless spin-2 mode exists and reproduces the observed GR/Newton limit. 
 DERIVED-GIVEN-E — terminal as structure 

 16 
 Heat-kernel scalar backbone 
 \(K_6\) scalar: \(a_0/a_0=1\) , \(a_2/a_0=5/12\) , \(a_4/a_0=11/120\) ; \(a_6/a_0\) OWED. Vector: \(\mathrm{tr}A_2=0\) , \(\mathrm{tr}A_4=-47/360\) . Banked ratio \(a_6/a_2^3=7936/39375\) (3+ engines). 
 DERIVED for banked ratios; OWED flagged for \(a_6/a_0\) 

 17 
 Sphere calibration controls 
 \(S^2:4/315\) ; \(S^4:74/63\) ; \(S^6:1139/63\) ; conformal \(5/63\) ; rel. err. \(\le4\times10^{-14}\) . 
 DERIVED (independent validation; confirms \(K_6\ne S^6\) ) 

 18 
 Physical trace definition 
 \(\mathrm{tr}[a_6]^{\rm phys}=\mathrm{tr}[a_6(\mathrm{grav})]-2\,\mathrm{tr}[a_6(\mathrm{FP})]\) . 
 DERIVED (definition, forced by Step 12) 

 19 
 \(\mathrm{tr}[a_6]^{\rm phys}\) numeric value 
 NOT COMPUTED. Route A needs unenumerated \(SU(3)\) Gelfand–Tsetlin hopping matrix elements (5 Weyl-inequivalent \(T^2\) weight classes); Route B needs the OWED graviton leg. Two-route agreement NOT achieved to the \(10^{-6}\) target-blind bar. Dimensionless sign estimate \(C\approx-6.39\) verified ABSENT. 
 OPEN / R1 — computation-debt, exported to UQF-9, Gap-01 

 20 
 Positivity functional proposed 
 \(P(a_6)\ge0\) as the 5B pass/fail test object. 
 Proposed test object (pre-mechanism) 

 21 
 Zeta-pole analysis 
 \(\zeta_L(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\mathrm{Tr}\,e^{-tL}dt\) ; \(a_{2k}\propto\) residue of \(\Gamma(s)\zeta_L(s)\) at \(s=D/2-k\) . At \(D=13,\,k=3\) : \(s=13/2-3=\boxed{7/2}\) , half-integer; \(\Gamma(7/2)=\tfrac{15\sqrt\pi}{8}\) finite/nonzero. 
 DERIVED (standard heat-kernel fact, applied to \(D=13\) ) 

 22 
 Odd- \(D\) consequence 
 \(a_6\) at \(D=13\) is a power-law divergence (scheme-dependent), identically zero in dim-reg; \(\zeta_L(0)\) holomorphic for odd total dimension ⇒ no anomaly/positivity slot exists. 
 DERIVED 

 23 
 5B terminal 
 \(P(a_6)\ge0\) is not a well-posed pass/fail test at \(D=13\) → the test dissolves as ill-posed (not run-and-failed, not pending — structurally absent). 
 DISSOLVED-GIVEN-root (odd-D zeta structure) — the CERTIFIED-IRREDUCIBLE mechanism 

 24 
 Three locks 
 (i) any future \(a_6\) value cannot resurrect a non-existent test — demotes \(a_6\) to a debt owed downstream; (ii) a new even-D slot elsewhere would be a NEW object, never a retroactive rescue; (iii) \(P\) was never uniquely selected (3 competing readings, R4) — odd-D moots the selection. 
 DERIVED (locks the terminal against re-opening) 

 25 
 Even-vs-odd negative control 
 At \(D=12\) , \(k=3\) gives \(s=6-3=3\) (integer), and the anomaly slot \(s=0\) is reached at the integer order \(k=D/2=6\) ⇒ well-posed; at \(D=14\) , same (its \(s=0\) slot at \(k=7\) ). Confirms D-parity mechanism, not "13 is special." 
 DERIVED (cross-check) 

 26 
 5C scope 
 Interacting/strong-coupling graviton UV completion, bounded:false . 
 EXPORTED to UQF-9 (shared Clay-class wall, also touches UQF-14) — not a leg of this gate 

 27 
 T-DEEP cross-check 
 \(R_0/\ell_{\rm Planck}\approx194\) : color scale and gravity scale not numerically identified. 
 DERIVED (blocks a false finite-grain shortcut to 5C) 

 L5. Credit-ladder grading — leg by leg

 Leg 
 Grade 
 Justification 

 13-D background, \(K_6=SU(3)/T^2\) at \(\vec u=(1,1,1)\) 
 AXIOM (frozen, shared with every gate) 
 Not re-derived here; the shared arena. 

 \(L_{\rm grav}=-(\nabla^2+E)\) , de-Donder gauge-fix 
 DERIVED 
 Standard construction on the frozen background; no free choices beyond fixed ⊕ Rulebook pins. 

 \(K_6\) curvature ledger ( \(23/75\) , \(1/6\) , \(\kappa=5/12\) , weight-6 invariants) 
 DERIVED 
 Theorem L, 3 independent target-blind routes; Bianchi residual \(3.05\times10^{-16}\) . 

 \(E_L\) spectrum on TT sector, \(\mathrm{tr}E_L=40/3\) , \(\mathrm{tr}E_L^2=241/18\) 
 DERIVED 
 Exact diagonalization from the certified curvature ledger. 

 Fiber counts 91 / 13 / 65, ghost sign \(-2\) 
 DERIVED 
 Elementary combinatorics ( \(\binom{14}{2}\) ) + BRST nilpotency; checkable by hand. 

 Heat-kernel scalar/vector backbone ( \(a_0,a_2,a_4\) ; scalar \(a_6/a_2^3\) ) 
 DERIVED 
 Gilkey/Avramidi machinery, validated against 4 sphere calibrations to \(\le4\times10^{-14}\) . 

 2-helicity count, IR → linearized Einstein/Newton 
 DERIVED-GIVEN-E 
 Consumes the observed GR/Newton limit as input to identify the mode; given-E ≠ derivation-of-E. 

 5A terminal 
 DERIVED-GIVEN-E 
 Terminal as structure. 

 \(a_6\) zeta-pole / odd- \(D\) argument 
 DERIVED 
 Standard heat-kernel fact (holomorphy of \(\zeta_L(0)\) at odd total dimension), applied to \(D=13\) . 

 5B terminal (positivity certificate) 
 DISSOLVED-GIVEN-root 
 Ill-posed at odd \(D\) — a limit on all one-loop positivity tests at odd total dimension, not a gap in this framework; makes the whole gate CERTIFIED-IRREDUCIBLE. 

 \(\mathrm{tr}[a_6]^{\rm phys}\) numeric trace (R1) 
 OPEN — computation-debt, exported 
 Uncomputed; owed to UQF-9/Gap-01, not a leg of 5A/5B. 

 Dimensionful bulk magnitude (retracted / pending) 
 CLOSED-NEGATIVE (retracted value) / AXIOM-OPEN (pending) 
 Retracted value rode the buggy \(31/147\) ; pending magnitude requires a nominated scheme object, never banked as derived. 

 5C interacting/strong-coupling completion 
 EXPORTED (structural frontier carried at UQF-9) 
 Universal open problem across every quantum-gravity approach; not this gate's private residual. 

 Margolus–Levitin / Landauer / Bekenstein floors 
 MEASURED-ANCHOR / ESTABLISHED (REDUCED-TO-AXIOM use here) 
 External physics, consumed only on the Granularity/UQF-9 side. 

 \(R_0/\ell_{\rm Planck}\approx194\) (T-DEEP) 
 DERIVED (wall cross-check) 
 Proves color and gravity scales are not numerically identified; blocks a false shortcut. 

 Gate-level roll-up 
 CERTIFIED-IRREDUCIBLE / RESOLVED +0 
 5A = DERIVED-GIVEN-E (terminal); 5B = DISSOLVED-GIVEN-root (terminal); no leg of 5A/5B is OPEN; R1 and 5C are shown, not rolled into the terminal. 

 L6. Anti-claims and negative controls

 The six non-claims (firewall, binding on any reader): 
1. NOT a derivation of General Relativity, and NOT a derivation of \(E\) . The graviton reproduces the observed GR/Newton limit given \(E\) ; given-E ≠ derivation-of-E. The observed IR limit is consumed as input to identify which mode is "the graviton," not re-derived.
2. NOT a claim that quantum gravity is solved, consistent, or complete. 5C/UQF-9 is a field-wide Clay-class wall shared by strings, asymptotic safety, loop quantum gravity, and causal dynamical triangulations alike — framed as shared infrastructure, never this program's private weakness, never asserted as proven anywhere.
3. NOT a claim that the frozen 13-D geometry is the proven-unique consistent spin-2 carrier. That would be a universal-negative ("unicorn") claim this program does not make; frozen/reproducible ≠ proven-unique. Selected-not-forced.
4. NOT a claim that " \(a_6\) is a UV completion." The ladder \(a_6<a_8<a_{10}<\cdots\) is unbounded; one coefficient cannot settle strong coupling (the R7 scope firewall, a maintained discipline).
5. NOT a fully numerically-verified two-helicity certificate in every gauge. Held structural-given-E; full cross-gauge verification is tied to the open ghost-subtraction check R5.
6. NOT a claim that "a positivity certificate passed." No certificate was run and passed; the certificate dissolved as inapplicable. This is a stronger, more honest terminal than a contingent pass would be, and must never be reworded as "the theory passed a consistency test."

 Auxiliary boundary screens (blanket, all hold): given-E ≠ derivation-of-E; selection ≠ derivation; frozen/reproducible ≠ proven-unique; a captured log ≠ independent reproduction; ANCHORED ≠ DERIVED; dissolved ≠ solved; AXIOM-CLOSED ≠ atomic.

 Named negative controls (frozen; never dissolve; never re-derive to another value): 

 Quantity 
 Correct value 
 Forbidden/retracted value 
 Why forbidden 

 Graviton fiber 
 91 
 67 
 Documented prior fabrication, caught by a verifier, permanently retired 

 Ghost fiber 
 13 
 11 
 Only 13 (one vector ghost per dimension) is admissible 

 $ 
 \mathrm{Riem} 
 ^2/\mathrm{Scal}^2$ 
 23/75 

 $ 
 \mathrm{Riem} 
 ^2/\mathrm{Scal}^2$ 
 23/75 

 Bulk magnitude 
 (pending, scheme-anchored) 
 \(-2.817995812\times10^{94}\) GeV \(^6\) 
 REFUTED/RETRACTED — rode the retracted \(31/147\) ratio; never revived 

 Corrected-pending magnitude 
 \(-2.995681680\times10^{94}\) GeV \(^6\) ( \(+6.30\%\) , sign preserved) 
 — 
 SCHEME-ANCHORED, NOT banked as derived 

 Einstein constant \(\kappa\) 
 5/12 
 7/12 
 Retired buggy value; correction shifted dimensionful bulk quantities by \(+6.305\%\) , sign preserved 

 Ghost-route disagreement, reported not smoothed over: Route A \(=-251/504=-0.4980158730158730\) vs Route B \(=149/1008=0.1478174603174603\) , differing by \(31/48=-0.6458333333333333\) — six orders of magnitude outside the \(10^{-6}\) target tolerance (R5). Carried forward explicitly.

 \(\mathbb{Z}_2\) orbifold boundary literature gap: the published boundary heat-kernel tower (Branson–Gilkey–Kirsten–Vassilevich; Kirsten's monograph) stops at \(a_5\) for mixed Neumann⊕Dirichlet boundary conditions. The order-6 mixed coefficient does not exist anywhere in the literature (R6) — a genuine, checkable literature gap, not a fabrication opportunity. Donnelly equivariant data already certified: fixed-point trace sum \(\sum1/|1-dg|=2\times\tfrac12=1\) ; per-fixed-point \(a_0\) defects \(+1/4\) (even parity) / \(-1/4\) (odd parity).

 Even-vs-odd dimension negative control: at \(D=12\) ( \(s=0\) exactly) or \(D=14\) , the identical pole argument makes \(P(a_6)\) well-posed. This confirms the dissolution mechanism tracks D-parity specifically, not an accident of "13." A reader who wrongly truncated the geometry (e.g., silently dropped \(S^1_Y/\mathbb{Z}_2\) , landing at \(D=12\) ) would obtain a spuriously well-posed but physically wrong test — the "residual under a truncated object is an artifact" failure mode this ledger's full-13D discipline exists to prevent.

 Load-bearing distinction, stated once plainly: an older framing graded this gate CERTIFICATE-CONDITIONAL, treating the uncomputed \(a_6\) trace as this gate's own blocking leg. The canonical spine used here demotes \(a_6\) to a debt owed to consumer gates (UQF-9, Gap-01) and recognizes that the test \(a_6\) would have fed — \(P(a_6)\ge0\) — is itself ill-posed at odd \(D\) . No finite value of \(a_6\) , computed now or ever, could change this gate's terminal. That is the precise content of "irreducible": not that no residuals exist, but that none of them is wired to this gate's pass/fail switch.

 L7. Open residuals carried forward (shown, none gates the terminal)

 ID 
 Residual 
 Numeric content 
 Disposition / what would close it 

 R1 
 \(d=13\) graviton-minus-ghost \(a_6\) trace uncomputed 
 \(C\approx-6.39\) verified ABSENT 
 OPEN / computation-debt, exported to UQF-9, Gap-01. Closes via: enumerate the \(SU(3)\) Gelfand–Tsetlin off-diagonal matrix elements (Route A); reconcile with Route B ( \(a_6/a_2^3=7936/39375\) ) to \(10^{-6}\) ; reproduce the 4 sphere calibrations as a consistency check. 

 R2 
 \(\sim31.23\%\) Riemann-norm deficit ( \(31/147\) vs \(23/75\) ) 
 Bianchi residual \(1/7\to3.05\times10^{-16}\) after fix 
 DISCLOSED-CORRECTED (fix: flip two quarter-bracket signs to the Besse-7.38 / Kobayashi–Nomizu-II naturally-reductive form). Prerequisite to R1. 

 R3 
 Dimensionful \(a_6\) magnitude scheme/scale-anchored, also moduli-dependent (Theorem L′) 
 \(M_*\in\{6.01\times10^{16},\ 7.467050992135091\times10^{16}\}\) GeV 
 AXIOM-OPEN / scheme-anchored. Closes via: an independently-fixed scheme object, or naming a value-free AXIOM-HEATKERNEL-SCHEME-OBJECT. \(\kappa^3/\pi\) kill-test forbids back-solving. 

 R4 
 Positivity functional \(P\) had 3 competing readings; \(P(a_6)\ge0\) never run 
 — 
 DISSOLVED at odd \(D=13\) (canonical). Not a live falsifier. Revisited only if a genuinely NEW even-D slot object is found — never as a rescue of this one. 

 R5 
 Ghost/BRST subtraction route disagreement 
 Route A \(-251/504=-0.4980158730158730\) ; Route B \(149/1008=0.1478174603174603\) ; gap \(-31/48=-0.6458333333333333\) 
 OPEN (structural). Multiplicity/sign \(-2\) NOT in question — only the endomorphism choice is. Closes via: write the FP ghost Lagrangian explicitly from the de-Donder condition, select Bochner ( \(E=0\) ) vs Lichnerowicz ( \(E=-\mathrm{Ric}\) ) uniquely, reconverge to \(10^{-6}\) . Feeds R1. 

 R6 
 \(\mathbb{Z}_2\) orbifold order-6 mixed Neumann⊕Dirichlet boundary coefficient 
 Published tower stops at \(a_5\) 
 BLOCKED (literature gap, BGKV/Kirsten). Closes via: a new specialist construction, or proof that the Donnelly equivariant defect \(\tfrac12c_3^\gamma\) supplies it. 

 — 
 Color ratio \(124/315\) 
 metric-selected: only at \(\mathrm{Scal}_{K_6}=7.5\) (Killing-norm gives \(0.0252\) ) 
 DERIVED-PENDING-INDEPENDENT-REPRODUCTION. Not banked; needs a structurally-independent engine (not the R2 code path). Feeds R1 Route B. 

 R7 
 Scope firewall " \(a_6\ne\) UV completion" 
 unbounded ladder \(a_6<a_8<a_{10}<\cdots\) 
 MAINTAIN (a standing discipline, not a hole to close). 

 R8/R9 (5C) 
 Interacting/strong-coupling graviton; above-cutoff unitarity 
 bounded:false 
 STRUCTURAL FRONTIER, exported whole to UQF-9 (shared Clay-class wall, also touches UQF-14). Closes only if UQF-9 closes: a target-blind, truncation-independent non-Gaussian FRG/Wetterich fixed point on the frozen chamber ( \(M_U\approx10^{16}\) GeV, \(R_0=1.591549430918954\times10^{-17}\) GeV \(^{-1}\) , \(\vec u=(1,1,1)\) ), two routes agreeing — or, equally validly, a clean no-go. 

 P0 
 Does gravity own an intrinsic shortest length, or inherit the color scale \(R_0\) ? 
 \(R_0/\ell_{\rm Planck}\approx194\) 
 Cleanest UQF-9 sub-target; not this gate's residual. T-DEEP leans toward "inherits \(R_0\) , no finite-grain shortcut," but this is not decided here. 

 None of R1–R7, the \(124/315\) ratio, or R8/R9/P0 is a leg of the 5A/5B terminal; each is compute-debt owed to consumer gates or the shared UQF-9 wall, shown here in full rather than hidden.

 L8. The endpoint line

 \[\boxed{\text{UQF-5A/5B} = \text{DERIVED-GIVEN-}E\ (\text{5A structure}) + \text{DISSOLVED-GIVEN-root}\ (\text{5B, odd-}D\text{ zeta}) \Rightarrow \textbf{CERTIFIED-IRREDUCIBLE / RESOLVED +0.}}\]

 Anchored on:
- Shape: the frozen 13-D arena \(\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2\) ( \(K_6=SU(3)/T^2\) , \(D=13\) ), all three layers pinned, supplying \(L_{\rm grav}\) , the exact \(E_L\) spectrum, fiber weights 91/13/65, curvature ratio \(23/75\) , Einstein constant \(\kappa=5/12\) .
- Granularity: the odd value \(D=13\) itself — placing the \(a_6\) pole at the half-integer \(s=7/2\) , making \(\zeta_L(0)\) holomorphic, and removing any even-D anomaly/positivity slot.
- Scale: closing legs consume no Tier-1 measured magnitude — the entire banked content is scale-free; the one magnitude that does exist (the \(a_6\) scheme object) is held AXIOM-OPEN and re-charged downstream, never folded into this terminal.
- Observables: 2 massless spin-2 helicities at light speed; linearized Einstein field equations; Newtonian \(1/r^2\) ; fiber counts 91/13/65 — all structural matches, none a fitted continuous quantity.
- Dissolution: the 5B positivity certificate is not a well-posed pass/fail object at odd \(D\) — a limit on ALL one-loop positivity tests at odd total dimension, not a gap specific to this framework; the specific mechanism that makes the gate terminal rather than perpetually open.

 The residual debt ( \(a_6\) trace and its dependencies R1–R7) and the shared strong-coupling wall (5C) are carried forward explicitly and by name — never rolled into, and never softening, the terminal.

 Given-E ≠ derivation-of-E · \(a_6\ne\) UV completion · dissolved ≠ solved · frozen/reproducible ≠ proven-unique. PROMOTIONS:0; frozen branch READ-ONLY.