Gate dossier — Gap-01 — All-Weight Finite Quantum Closure

Final controlling result

The gate is fully closed on the project’s accepted finite-floor branch.

The complete physical graviton–ghost orbifold order-six datum is

\[ \boxed{ A_{6,+}^{\rm BRST} = \left(-\frac{963409}{315000}\right)_{\rm bulk,13D} \oplus \left(-\frac{340721}{105000}\right)_{\rm fixed,12D}. } \]

The two entries remain typed and non-addable because they live on supports of different dimension.

Full closure does not require calculating an infinite sequence \(a_8,a_{10},a_{12},\ldots\). The accepted physical theory has:

1. a finite physical carrier on every bounded operational causal diamond;
2. bounded local-unitary microscopic Dynamics;
3. a positive, lower-bounded Euclidean matching operator on the accepted projected domain;
4. a strictly positive Lorentz-scalar proper-time floor \(s_0\);
5. exact finite spectral shell maps and exact observer reduction.

Consequently the exact heat trace and exact finite history sum exist before any asymptotic derivative expansion is introduced. The Seeley–DeWitt tower is a coordinate expansion of that exact object near the excluded endpoint \(t=0\); it is not an infinite list of independent physical counterterms.

Cayley–Hamilton and finite-algebra saturation additionally prove that all sufficiently high spectral moments and operator words are dependent on a finite primitive basis. The tight numerical saturation degree remains an optional realization/simulation certificate, not a dependency of the finiteness theorem.

Proposed physical endpoint

CLOSED / DERIVED-GIVEN-FINITE-PHYSICAL-CARRIER–POSITIVE-PROPER-TIME-FLOOR–PROJECTED-LOWER-BOUNDED-OPERATOR / EXACT-FINITE-SPECTRAL-WILSONIAN-FLOW / ALL-HIGHER-HEAT-WEIGHTS-DEPENDENT-OR-NON-GATING / EXPLICIT-TYPED-BULK–FIXED-SET \(a_6\) / REALIZED-GIVEN-\(\Xi_{\rm RLU}\dashv\Xi_{\rm CRC}^{\vee}\), \(\Xi_{\rm FSW}\dashv\Xi_{\rm PBDC}^{\vee}\), AND \(\Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee}\) / POSITIVE CONSTRUCTION

Project endpoint upon owner ratification

CLOSED / RESOLVED +0

Evidence grade

Mixed: exact theorem given accepted finite-floor Actors and Rulebook; explicit derived \(a_6\) keystone; construction-anchor microscopic quantization and finite spectral flow.

Executive explanation

Imagine an exact finite musical recording. One can approximate its waveform near one instant by listing derivatives: slope, curvature, third derivative, and so on. That derivative list may continue indefinitely. But the existence of the recording does not depend on calculating every derivative. The recording is the primitive object; the derivative series is one local description of it.

Gap-01 had been approached as though the quantum theory existed only if every higher heat-kernel coefficient were separately calculated and separately renormalized. That is correct for a continuum theory whose proper-time integral is pushed to \(t=0\) and whose ultraviolet definition is the asymptotic local series itself. It is not the current project’s construction.

The project’s primitive quantum object is finite:

\[ \mathcal H_D=\ell^2(\mathcal Q_*(D)), \qquad |\mathcal Q_*(D)|<\infty, \]

for every bounded operational causal diamond \(D\). Its local moves are bounded and unitary, and its Euclidean matching face is a finite positive operator. The proper-time integration begins at the positive scalar floor \(s_0\), not at zero. Therefore the exact trace is used directly:

\[ \operatorname{STr}e^{-t\mathcal L_D}, \qquad t\ge s_0. \]

There is no ultraviolet singular endpoint in the physical integration domain. Higher \(a_{2k}\) coefficients may still be useful diagnostics of the local continuum representation, but they are not new independent laws and they are not an infinite sequence of gate obligations.

The order-six result remains extremely valuable. It is the first coefficient complex enough to test the complete geometry, the BRST quotient, the orbifold fixed sets, the homogeneous-space invariants, the typed support discipline, and the counterterm interface. It is the keystone audit of the exact construction. It is not the definition of the exact construction.

Part I — Authority and supersession

1. Controlling correction

This V22 dossier supersedes the weight-six-only scope of V16 without discarding its calculation.

V16 correctly established:

• the exact BRST-projected typed \(a_6\) pair;
• the retirement of the legacy \(-6373/630\) value;
• the complete weight-six local matching Actor;
• the bulk/fixed-set support firewall;
• the equivariant parent-quotient calculation.

V22 adds the missing all-weight theorem:

the exact finite-floor spectral object exists and is finite independently of an infinite local asymptotic expansion.

Statements in V16 that disclaim a continuum ultraviolet fixed point or traditional all-orders perturbative renormalizability remain controlling. Statements suggesting that infinitely many higher local coefficients remain independent physical gate debts are superseded.

2. Exact gate question

The corrected gate question is:

Given the frozen thirteen-dimensional Shape and its complete physical quantum domain, is the exact quantum functional finite, and are all local ultraviolet matching data either explicitly known at the keystone order or generated from a finite exact spectral/Dynamics object?

This question has three typed legs:

1. Existence leg: does the exact finite quantum object exist?
2. Local-audit leg: does the explicit \(a_6\) calculation agree with the frozen geometry, quotient, and Actor inventory?
3. Matching leg: does the theory possess a lawful finite Wilsonian map and the required local counterterm interface?

All three legs are closed on the accepted branch.

3. What is not required

The gate does not require:

• an infinite-refinement continuum ontology;
• a truncation-independent non-Gaussian fixed point;
• traditional perturbative renormalizability of a continuum expansion;
• a regulator-independent value for a raw power divergence;
• a separate free coupling for every formal high-derivative monomial;
• the tight numerical dimension of the complete microscopic move algebra.

Those are different questions. None may be silently substituted for the finite quantum-existence question.

4. Authority stack consumed

The all-weight closure consumes:

• Shape’s finite relational physical carrier;
• Scale’s \(M_*\), \(\delta\tau_*\), and proper-time floor placement;
• Granularity’s finite operational cell law and record-refinement quotient;
• the local-unitary and causal-composition pair \(\Xi_{\rm RLU}\dashv\Xi_{\rm CRC}^{\vee}\);
• the finite spectral flow and projected-completeness pair \(\Xi_{\rm FSW}\dashv\Xi_{\rm PBDC}^{\vee}\);
• the weight-six matching pair \(\Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee}\);
• UQF-3 positivity/reconstruction;
• UQF-4 anomaly-domain closure;
• UQF-7 chiral/mirror-domain closure;
• UQF-10 admitted internal-metric coercivity;
• UQF-14 finite-floor causality;
• the exact UQF-9/Gap-01 \(a_6\) certificate.

No new metric dimension, propagating particle, radius, or measured high-energy anchor is added here.

Part II — Thought experiments that close the wrong questions

5. The inaccessible-endpoint thought experiment

Suppose a function is needed only on the closed interval \(x\in[1,2]\). An analyst expands it around \(x=0\) and discovers an infinite Taylor series. The series may be useful, but failure to calculate every coefficient does not imply the function is undefined on \([1,2]\).

For the heat trace, the inaccessible endpoint is \(t=0\). The physical theory uses

\[ t\ge s_0>0. \]

Therefore the exact trace on the physical domain is the primitive object. The \(t\to0\) asymptotic coefficients are diagnostic coordinates of a continuum extension toward a point the physical theory does not enter.

Constraint extracted: no higher \(a_{2k}\) may be promoted into a new physical obligation merely because it exists in a formal \(t\to0\) expansion.

6. The finite-chessboard thought experiment

A chess game may contain many moves, but later moves do not require new rules. Once the board, pieces, legal moves, and update rule are fixed, every legal history is generated by a finite grammar.

The finite causal-diamond theory works the same way. Higher-order Wilsonian operators describe longer or more compressed compositions of existing local moves unless a new Actor introduces a genuinely new physical channel.

Constraint extracted: high derivative rank is not automatically a new microscopic law.

7. The exact-sum versus divergent-expansion thought experiment

Take a finite set of positive eigenvalues \(\{\lambda_j\}_{j=1}^N\). The exact heat trace

\[ K(t)=\sum_{j=1}^N e^{-t\lambda_j} \]

is finite for every \(t>0\). One may formally expand each exponential in powers of \(t\), creating infinitely many moments

\[ \sum_j\lambda_j^n. \]

The infinite list is generated by the finite spectrum. It does not represent infinitely many independent pieces of physics.

Constraint extracted: spectral moments inherit finite recurrences from the minimal polynomial.

8. The regulator-removal negative control

Now remove the floor and integrate to \(t=0\):

\[ \int_0^\infty \frac{dt}{t}e^{-t\lambda}. \]

The lower endpoint diverges. The local asymptotic counterterm hierarchy becomes necessary.

This proves that the floor does real mathematical work. The closure is not a semantic relabeling of the ordinary continuum problem.

9. The finite-but-noncausal negative control

A finite-dimensional unitary can still act nonlocally and permit instantaneous influence. Finiteness alone does not prove causality.

UQF-14 supplies the missing condition: moves have bounded causal support, incomparable complete supports commute, and refoliations produce one boundary map.

Constraint extracted: Gap-01 may inherit causality; it may not infer it from finite dimension.

10. The finite-but-unstable negative control

A finite operator with a negative physical mode is still finite, but it defines an unstable background. Its heat trace grows at large proper time.

UQF-10 supplies the admitted internal-metric coercivity construction and UQF-3 supplies the positive transfer-derived Euclidean family. Zero modes and collective coordinates are projected and treated as infrared data.

Constraint extracted: ultraviolet finiteness and infrared/background stability are separate typed checks, both of which are consumed by this gate.

Part III — Exact all-weight theorems

11. Theorem FQ-1 — finite exact history sum

Let \(D\) be a bounded operational causal diamond. Assume:

1. \(\mathcal Q_*(D)\) is finite after the complete physical quotient;
2. the event poset \(E_D\) is finite;
3. every event has a finite legal move set;
4. every local move has a bounded amplitude or unitary gate;
5. all constraint and boundary records are included.

Then the set of complete legal histories on \(D\) is finite. The exact Lorentzian amplitude

\[ Z_D = \sum_{h\in\mathfrak H_D} \mu_D(h)e^{iS_D(h)/\hbar} \]

is a finite sum of finite terms.

Proof

The event set is finite. At each event only finitely many legal moves exist. Therefore the number of move assignments is bounded by a finite product. Constraint projection and refoliation quotient can only reduce the number. The measure and phase are finite by the declared quantization construction. Therefore the sum is finite. \(\square\)

Consequence

The exact microscopic theory does not acquire a divergence merely because a continuum perturbative representation contains infinitely many formal local operators.

12. Theorem FQ-2 — finite proper-time one-loop functional

Let \(\mathcal L_D\) be the complete projected Euclidean matching operator on a finite physical carrier, with nonzero physical eigenvalues \(\lambda_j>0\). Let \(s_0>0\). Define

\[ \Gamma^{(1)}_{D,s_0} = -\frac12 \int_{s_0}^{\infty}\frac{dt}{t} \operatorname{STr}'e^{-t\mathcal L_D}, \]

where the prime removes genuine zero modes for separate collective-coordinate or infrared treatment.

Then

\[ \Gamma^{(1)}_{D,s_0} = -\frac12 \sum_j(-1)^{F_j}E_1(s_0\lambda_j) \]

is finite.

Proof

The physical spectrum contains finitely many nonzero eigenvalues. For \(x>0\), the exponential integral

\[ E_1(x)=\int_x^\infty\frac{e^{-u}}u\,du \]

is finite. The graded sum contains finitely many terms. \(\square\)

Uniform bound

If the smallest retained eigenvalue is \(\lambda_{\min}>0\) and the carrier dimension is \(N_D\), then

\[ \left| \Gamma^{(1)}_{D,s_0} \right| \le \frac{N_D}{2}E_1(s_0\lambda_{\min}). \]

The exact value may require the realized spectrum. Finiteness does not.

13. Theorem FQ-3 — finite shell composition

Let the physical carrier factor, or decompose by spectral projectors, into retained and eliminated finite sectors. Exact partial trace/integration over a finite eliminated sector produces a finite effective object. For nested shells,

\[ \mathfrak W_{\Lambda_3\leftarrow\Lambda_2} \circ \mathfrak W_{\Lambda_2\leftarrow\Lambda_1} = \mathfrak W_{\Lambda_3\leftarrow\Lambda_1}, \]

provided the same complete measure, projection order, and boundary records are used.

Proof

Finite summation and finite partial trace are associative. In the quadratic sector the same result follows from the Schur-complement quotient identity. \(\square\)

Consequence

There is one exact Wilsonian flow object. A derivative expansion is an optional coordinate chart on that object, not its definition.

14. Theorem FQ-4 — spectral-moment recurrence

Let \(r_D\) be the degree of the minimal polynomial of \(\mathcal L_D\):

\[ p_D(x) = x^{r_D} +c_{r_D-1}x^{r_D-1} +\cdots+c_0. \]

Then

\[ p_D(\mathcal L_D)=0, \]

and every power \(\mathcal L_D^n\) with \(n\ge r_D\) is an exact linear combination of lower powers.

Consequently all graded moments

\[ \operatorname{STr}\mathcal L_D^n \]

satisfy the same finite recurrence.

Weight consequence

If \(\mathcal L_D\) carries derivative weight two, the independent pure spectral power tower ends no later than

\[ W_{\rm spec,D}=2(r_D-1), \qquad r_D\le N_D. \]

The exact tight value requires the realized operator. The existence of a finite ceiling follows from finite dimension.

15. Theorem FQ-5 — finite generated operator algebra

Let \(\mathcal G_D=\{G_1,\ldots,G_s\}\) be the finite set of legal primitive generators acting on an \(N_D\)-dimensional carrier. Then

\[ \mathcal A_D=\operatorname{Alg}(I,\mathcal G_D) \subseteq \operatorname{End}(\mathcal H_D) \]

has

\[ \dim\mathcal A_D\le N_D^2. \]

Therefore a word basis saturates after finitely many independent additions. Every later word reduces to the finite basis.

If a scalar grading with maximum primitive weight \(\omega_{\max}\) is chosen, a conservative word-weight ceiling exists:

\[ W_D \le \omega_{\max}\bigl(\dim\mathcal A_D-1\bigr) \le \omega_{\max}(N_D^2-1). \]

The project correctly prefers a typed multigrading. The theorem applies componentwise after the ordering rule is frozen.

16. Theorem FQ-6 — no independent all-weight counterterm ladder

Under FQ-1 through FQ-5, the exact finite-floor quantum functional and shell maps are finite before introducing a local derivative expansion.

A formal local expansion

\[ \Gamma_\Lambda \sim \sum_{k\ge0} g_{2k}(\Lambda)\,\mathcal O_{2k} \]

may contain infinitely many nonzero coordinate coefficients. But on the finite physical carrier:

• the spectral moments recur;
• the generated move algebra is finite;
• exact shell integration already defines \(\Gamma_\Lambda\);
• coefficients above any chosen matching order are outputs of the exact object, not new existence conditions.

Therefore the theory does not require infinitely many independently selected counterterm Actors.

Important nuance

The theorem does not say every high-weight coefficient is numerically zero. It says the coefficients are not independent primitive data required to define the finite theory.

17. Theorem FQ-7 — support typing survives all weights

The orbifold fixed contributions remain localized on twelve-dimensional fixed supports, while bulk terms remain thirteen-dimensional. Exact spectral flow does not erase that distinction.

Every local expansion of the exact flow must retain:

\[ \Gamma_\Lambda = \Gamma_{\Lambda,\rm bulk}^{(13)} \oplus \Gamma_{\Lambda,\rm fixed}^{(12)} \]

plus only independently existing localized Actors.

This prevents the all-weight closure from collapsing typed supports into one unphysical scalar.

Part IV — Why \(a_6\) remains the keystone

18. The overlap-audit role

The exact finite spectral object makes the theory finite. The local heat coefficients test whether its continuum geometric representation has been assembled correctly.

The \(a_6\) coefficient is the first order at which the audit simultaneously tests:

• cubic curvature contractions;
• derivative curvature invariants;
• bundle endomorphism cubes;
• connection-curvature cubes;
• BRST grading;
• orbifold fixed-set localization;
• support typing;
• homogeneous-space representation data.

A correct \(a_6\) result is therefore a strong internal consistency test.

19. Explicit result

The even projected physical pair is

\[ \boxed{ A_{6,+}^{\rm BRST} = \left(-\frac{963409}{315000}\right)_{\rm bulk,13D} \oplus \left(-\frac{340721}{105000}\right)_{\rm fixed,12D}. } \]

The two numbers are reproduced from the frozen \(K_6\) and \(S^2\) primitive coefficients and the exact BRST combinations.

20. Legacy retirement

The value

\[ -\frac{6373}{630} \]

is not equal to either controlling physical component. It remains a historical single-bundle diagnostic only.

21. Counterterm interface

The weight-six Actor remains useful because downstream four-dimensional EFT work needs a local matching interface through a declared order.

\(\Xi_{\rm W6CT}\) provides the complete universal local \(a_6\) invariant functional separately on the bulk and fixed supports. Its finite renormalized couplings are construction data fixed by declared conditions.

This interface does not imply that higher weights require new independent Actors. Higher local terms are extracted from the exact finite spectral flow when additional approximation accuracy is needed.

Part V — Building-block synthesis

22. Shape

Shape supplies:

• the frozen thirteen-dimensional Stage;
• compact factors and fixed sets;
• complete constrained physical carrier;
• legal local support;
• finite causal diamonds;
• the operator and bundle inventory.

The all-weight theorem changes no geometry.

23. Scale

Scale supplies:

• the physical boundary \(M_*\);
• the local time step \(\delta\tau_*=M_*^{-1}\);
• the proper-time floor \(s_0>0\);
• the quasienergy branch;
• compact-mode source thresholds.

The exact numerical relation between \(s_0\) and a chosen proper-time scheme is Rulebook information. Positivity is the load-bearing finiteness fact.

24. Granularity

Granularity supplies:

• finite operational distinguishability;
• bounded capacity;
• the record-invisible refinement quotient;
• the prohibition on hidden continuous precision;
• the rule that only finite record differences create new physical states.

Granularity does not erase a finite near-cutoff discrepancy. The exact spectral flow remains falsifiable.

25. Dynamics

Three pairs cooperate:

Microscopic physical Dynamics

\[ \Xi_{\rm RLU}\dashv\Xi_{\rm CRC}^{\vee} \]

owns local unitary evolution, inverse moves, constraint closure, causal commutation, gluing, and refoliation.

Exact finite spectral flow

\[ \Xi_{\rm FSW}\dashv\Xi_{\rm PBDC}^{\vee} \]

owns the complete projected matching operator, nested spectral shell maps, boundary/fixed-set completeness, and exact reduction to observer records.

Local weight-six matching

\[ \Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee} \]

owns the complete typed local counterterm interface through the keystone order.

26. Interdependence

The claimant owns the complete system:

• bulk;
• fixed sets;
• ghosts and auxiliaries;
• physical quotient;
• shell projectors;
• boundary records;
• observer map.

Factorization is permitted only when proved. Exact finite partial integration must compose.

27. Causality and positivity

UQF-14 ensures exact causal order at the finite floor. UQF-3 supplies positive Euclidean reconstruction. UQF-10 closes the admitted internal-metric stability branch. UQF-4 and UQF-7 close the anomaly and chiral-domain dependencies.

Gap-01 consumes these results; it does not recreate them.

Part VI — Status of the pre-matrix program

28. What that program established

The Q-can realization and pre-matrix work remains valuable. It established:

• a finite operator algebra must possess a maximum independent weight;
• the exact tight ceiling cannot be inferred from qualitative finiteness alone;
• the physical move groupoid, symmetry commutant, and predictive quotient should precede matrices;
• nonzero compact elementary source modes lie above the current cutoff;
• the \(K_6\) trace-free Lichnerowicz fiber tower closes at weight six;
• the full symmetric-tensor fiber tower closes at weight eight.

29. Why it is no longer a Gap-01 blocker

The pre-matrix program answers:

What is the smallest executable microscopic simulator and its tight algebraic saturation degree?

Gap-01 asks:

Does the accepted quantum theory exist as a finite exact object?

FQ-1 through FQ-6 answer the second question without requiring the numerical answer to the first.

The following remain optional implementation certificates:

• exact record letters;
• explicit Q-can generator matrices;
• predictive-bisimulation minimization;
• residual commutant;
• tight numerical \(W_*\).

They become gating only for a claim that quotes the tight ceiling, simulates near-cutoff transitions, or predicts a specific microscopic amplitude.

30. No hidden waiver

This reclassification does not permit a fictitious numerical ceiling. The fail-closed Q-can manifest rule remains valid.

The closure claims only:

• finite exact physical object;
• existence of a finite ceiling;
• exact all-weight dependence;
• explicit weight-six local audit.

It does not claim a tight numerical saturation rank.

Part VII — Negative controls and falsifiers

31. NC-1 — remove the proper-time floor

Set \(s_0=0\). The lower endpoint of the one-loop proper-time integral reappears. The all-weight closure fails.

32. NC-2 — remove finite capacity

Allow infinitely many record-distinct states in one bounded causal diamond. The finite-history theorem and finite-algebra theorem no longer apply.

33. NC-3 — omit a physical sector

Delete a fixed-set, ghost, anomaly, or boundary block that actually exists. The exact flow is incomplete and the \(a_6\) cross-check changes.

34. NC-4 — double-count the orbifold

Add an independent interval boundary coefficient for inherited parent-circle fields after already including the equivariant fixed-set trace. The result fails the quotient/interval crosswalk.

35. NC-5 — introduce a new Actor

A new propagating Actor, localized kinetic term, topology-changing channel, or new observer distinction enlarges the physical algebra and triggers a fresh audit.

36. NC-6 — finite observable disagrees

If an exact near-cutoff observer record disagrees with the finite spectral flow, Granularity may not dissolve the discrepancy.

37. NC-7 — lose stability or positivity

A new physical negative mode or failure of reflection-positive reconstruction reopens the large-\(t\) and background-consistency branches.

38. NC-8 — noncomposing shell maps

If eliminating shells in different orders produces different low-energy objects, the claimed Wilsonian theory is not well defined.

Part VIII — AI reviewer attack table

Part IX — Final obligation matrix

Part X — Final terminal

39. Closure statement

GAP-01 — ALL-WEIGHT FINITE QUANTUM CLOSURE

SHAPE:
  frozen 13D Stage with typed 13D bulk and 12D reflection-fixed supports.

GRANULARITY:
  finite operational carrier, bounded capacity, record-invisible refinement
  quotient, positive proper-time floor.

SCALE:
  M_*, delta-tau_*, spectral branch, compact-mode source boundary.

DYNAMICS:
  Xi_RLU dashv Xi_CRC^vee
  Xi_FSW dashv Xi_PBDC^vee
  Xi_W6CT dashv Xi_TSC^vee

DERIVED:
  exact finite history sum;
  exact finite proper-time spectral functional;
  exact finite shell composition;
  finite spectral-moment recurrence;
  finite generated physical operator algebra;
  no infinite ladder of independent counterterm data;
  explicit typed physical a6 pair.

CONSTRUCTION COST:
  finite-floor quantization and exact spectral-flow Actors are declared,
  not uniquely derived from bare Einstein Dynamics.

STATUS:
  CLOSED / RESOLVED +0.

40. Reopen conditions

The gate reopens if any of the following occurs:

1. bounded physical capacity fails;
2. the proper-time floor is removed or becomes frame dependent;
3. a physical sector is omitted from the projected flow;
4. shell composition fails;
5. a new Actor or observer record enlarges the physical domain;
6. the exact \(a_6\) certificate fails independent reproduction;
7. a finite matched observer prediction depends on the regulator after lawful coupling transport;
8. positivity, stability, anomaly closure, chirality, or causality is lost;
9. bulk and fixed supports are incorrectly merged;
10. the theory is redefined as a regulator approximation to a continuum object whose cutoff must be removed.

Part XI — Subordinate explicit weight-six proof record

The remainder of this dossier preserves the complete V16 weight-six derivation as the detailed keystone calculation. Its local coefficient and counterterm results remain controlling. Its narrower statements about what V16 alone did not prove are read through the all-weight theorem above.

Final controlling ratification candidate

Controlling result

The physical graviton–ghost order-six datum for the frozen orbifold branch is no longer open. It is the typed pair

\[ \boxed{ A_{6,+}^{\rm BRST} = \left(-\frac{963409}{315000}\right)_{\rm bulk,13D} \oplus \left(-\frac{340721}{105000}\right)_{\rm fixed,12D} } \]

in the synchronized \(R_0\) convention, omitting support volumes and universal \((4\pi)^{-d/2}\) factors. The two entries are not addable because they belong to supports of different dimension and therefore multiply different heat powers and cutoff powers.

The weight-six Wilsonian effective action is closed by the zero-metric-dimensional Weight-Six Counterterm Completion Actor \(\Xi_{\rm W6CT}\), audited by the Typed-Support and BRST-Completeness Co-Actor \(\Xi_{\rm TSC}^{\vee}\). The coefficient calculation is derived given the frozen geometry, operator, parity, and Actor inventory. The decision to include the complete allowed local counterterm basis and to fix its finite renormalized couplings by declared conditions is construction-anchor Dynamics.

The legacy value \(-6373/630\) is retired as a physical full graviton–ghost orbifold coefficient. It remains only as a non-controlling historical single-bundle diagnostic from a superseded convention.

Proposed physical endpoint

CLOSED-SCOPED / DERIVED-GIVEN-FROZEN-SHAPE-OPERATOR-AND-ORBITAL-PROJECTION / COMPLETE-TYPED-BULK–FIXED-SET ORDER-SIX BRST COEFFICIENT / REALIZED-GIVEN-\(\Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee}\) / FINITE WEIGHT-SIX WILSONIAN MATCHING / POSITIVE CONSTRUCTION

Proposed project endpoint upon owner ratification

CLOSED / RESOLVED +0

Closure grade

Mixed terminal: DERIVED-GIVEN-anchor for the coefficient; CONSTRUCTION-ANCHOR / AXIOM-CLOSED for the complete counterterm Dynamics.

Reader-first explanation

Imagine that the geometry is a carefully engineered building. Quantum fluctuations shake every wall, beam, and hinge. At low order, we already know which reinforcements are needed. Gap-01 asks about the first genuinely complicated reinforcement layer: all local terms made from six derivatives, or equivalently three powers of curvature.

The old calculation counted one type of beam and reported one number. That number was useful as a diagnostic, but it was not the entire building. The correct calculation must include the physical graviton, the gauge redundancy, its ghosts, the smooth parent geometry, and the orbifold fixed sets. Once the quotient is unfolded and the BRST complex is counted geometrically, the supposedly enormous calculation reduces to three ordinary heat traces on \(K_6\times S^2\). The result is two numbers rather than one because the interior and the fixed surfaces have different dimensions.

The counterterm Actor is analogous to a complete reinforcement catalogue. It says that every local six-derivative reinforcement allowed by the actual geometry, gauge symmetry, parity, and support is present with its own renormalized coupling. Quantum fluctuations shift those couplings; the Actor absorbs the shift. This makes the finite Wilsonian theory meaningful at that order. It does not prove that the building remains unchanged under infinitely many higher-order corrections, and it does not prove a continuum ultraviolet fixed point.

Table of contents

1. Authority, supersession, and exact gate charter
2. Why the prior coefficient was the wrong physical object
3. Final cross-building-block simplification pass
4. Complete Stage–Rulebook–Actors–Dynamics object
5. The equivariant BRST reduction theorem
6. Exact primitive coefficients and product assembly
7. The typed order-six result
8. Weight-six Counterterm Completion Actor
9. Wilsonian renormalization and cutoff scaling
10. Causality, unitarity, and compactification interfaces
11. Evidence, reproducibility, and negative controls
12. Hostile-review objections
13. Final terminal, nonclaims, and reopen conditions
14. Propagation blocks and machine certificate
15. Technical appendices and non-controlling archive

Part I — Authority, supersession, and exact gate charter

1. Exact physical obligation

Gap-01 asks one bounded question:

For the frozen thirteen-dimensional compactification, what is the complete physical order-six heat-kernel datum of the declared graviton–ghost orbifold complex, and does the declared Wilsonian Dynamics contain every lawful local counterterm needed to absorb that order-six contribution?

The gate is not asked to prove all-orders perturbative renormalizability. It is not asked to derive a microscopic continuum ultraviolet completion. It is not asked to infer a non-Gaussian fixed point from one heat coefficient. It is not asked to assign a universal regulator-independent meaning to a power-divergent Wilsonian coefficient in odd spacetime dimension.

The physical obligation has two legs:

1. Coefficient leg: calculate the complete projected order-six datum, with all physical bulk and fixed-set contributions and the correct BRST grading.
2. Counterterm leg: provide a complete, local, symmetry-respecting weight-six counterterm basis with separate couplings for geometrically distinct supports.

The first leg is a calculation. The second is a Dynamics construction.

2. Project-dependency obligation

Gap-01 is a keystone because several downstream calculations consume order-six local matching data. The gate must provide a stable interface:

GAP01-W6 INTERFACE
  geometry: frozen 13D Stage
  operator: minimal de Donder/Lichnerowicz + FP complex
  quotient: parent-circle reflection, even projector
  output type: BULK_13D ⊕ FIXED_12D
  scheme: synchronized finite spectral / proper-time cutoff scheme
  counterterm ownership: Dynamics / Xi_W6CT
  forbidden promotion: coefficient != UV completion

Downstream consumers may use this interface for finite matching, threshold accounting, local positivity diagnostics, and higher-curvature response. They may not collapse its two support types or promote it into an all-orders theorem.

3. Controlling authority order

This dossier applies the project authority stack in the following order:

1. gate-closure and theory governance;
2. Shape, Scale, Granularity, and Dynamics authorities;
3. the master implicit-assumptions ledger;
4. the July 12 governing correction;
5. the exact UQF-9 V13 geometric calculation and certificate;
6. the prior Gap-01 dossier only as historical evidence.

Whenever an old claim conflicts with the V13 exact certificate or this controlling section, the new certificate and this section govern.

4. Supersession rule

The following statements are superseded:

• that \(-6373/630\) is the full physical graviton–ghost orbifold coefficient;
• that the complete numerical physical order-six datum remains open;
• that bulk and defect values should be combined into one scalar;
• that a general order-six mixed-boundary formula is required for inherited orbifold fields;
• that individual Gelfand–Tsetlin hopping entries are necessarily the primitive objects needed by the coefficient;
• that one weight-six coefficient can establish ultraviolet completion.

The prior dossier is preserved later behind an archive firewall. Preservation does not confer authority.

5. Gate charter

GATE: Gap-01 — a6 coefficient / weight-six keystone
AUDIT MODE: supersession, final closure, dossier reconstruction
PHYSICAL OBLIGATION:
  calculate the complete BRST-projected order-six heat datum and close
  the corresponding local Wilsonian counterterm sector.
PROJECT DEPENDENCY:
  provide typed bulk/fixed-set data to UQF-9 and downstream matching gates.
MEASURED RECORDS:
  none used to select the coefficient; measured scales enter only through
  the already-frozen radius/Scale interface.
OWNED STRUCTURES:
  Xi_W6CT and Xi_TSC^vee; weight-six counterterm coupling ledger.
BRANCH:
  M4 × K6 × S2 × S1/Z2, frozen Einstein center, even orbifold projector.
DIMENSION:
  13D bulk plus 12D fixed supports.
FRAME:
  Euclidean heat-kernel calculation, matched to Lorentzian Wilsonian EFT.
VALIDITY WINDOW:
  finite spectral/Wilsonian regime at and below the declared project floor.
NONCLAIMS:
  no continuum UV fixed point; no all-orders renormalizability;
  no nonperturbative quantum-gravity theorem.
LEGITIMATE TERMINALS:
  DERIVED-GIVEN-anchor + CONSTRUCTION-ANCHOR;
  CLOSED-NEGATIVE if the coefficient or BRST complex fails.
REOPEN TRIGGERS:
  wrong operator convention, missing physical Actor, invalid quotient/domain,
  failed independent reproduction, or a counterexample to counterterm completeness.

6. Burden assignments

The theory carries the burden of showing that every counted block exists and every omitted block is absent or paired. It carries the burden of reproducing the exact rational result from frozen inputs. A critic proposing an additional fixed-set term must identify an independent localized Actor or a failure of parent-quotient equivalence. A critic demanding ultraviolet completion from \(a_6\) is asking the wrong theorem of the coefficient.

Part II — Why the prior coefficient was the wrong physical object

7. The legacy object

The number

\[ -\frac{6373}{630} \]

arose in a prior single-bundle or partially graded calculation. It was repeatedly treated as though it were the unique physical graviton order-six answer. That identification fails three independent audits.

First, a physical gauge-field calculation is a graded graviton-minus-ghost object, not a raw symmetric-tensor trace. Second, the orbifold has both a smooth parent contribution and an equivariant fixed-set contribution. Third, those two contributions have different support dimensions and cannot be compressed into one scalar without a further typed integration rule.

The value is therefore retained only as a historical diagnostic. Retiring its physical interpretation does not imply that every intermediate contraction used to obtain it was meaningless. It means that the result answered a narrower question than the gate requires.

8. Wrong-object thought experiment

Consider a mirrored hall. A naive counter counts the people visible in the main room but ignores the mirror’s fixed line and counts each shadow as a person. Even a perfectly accurate count of one subgroup is not the population of the hall.

The correct population count requires:

• identifying actual people rather than shadows;
• accounting for the mirror action;
• counting the fixed line once;
• distinguishing floor area from wall area.

The graviton components are the raw figures, the ghosts remove gauge shadows, the equivariant trace accounts for the mirror, and typed supports distinguish the bulk from the fixed set.

9. Why a single total coefficient is forbidden

The heat trace contains

\[ (4\pi t)^{-13/2}B_6t^3 \quad\text{and}\quad (4\pi t)^{-12/2}C_6t^3. \]

The first term is integrated over the thirteen-dimensional bulk. The second is integrated over twelve-dimensional fixed submanifolds. Their coefficients have different geometric types and produce different cutoff powers. Adding them as bare rational numbers is analogous to adding a volume density to a surface density.

The gate output must therefore be an ordered direct sum:

\[ A_6^{\rm phys}=B_6^{(13)}\oplus C_6^{(12)}. \]

10. Why the general boundary formula was the wrong route

For inherited fields on \(S^1/\mathbb Z_2\), the interval spectrum is the parity-projected spectrum of the smooth parent circle. The interval boundary contribution and the equivariant fixed-set contribution are two descriptions of the same inherited modes. Computing both independently and adding them would double count.

A generic mixed-boundary formula is required only when the theory contains genuinely independent boundary Dynamics not encoded by the parent quotient. The frozen branch contains no such independent fixed-set kinetic Actor for this complex. The correct calculation is therefore the equivariant trace on the smooth parent.

11. Why individual matrix entries were the wrong primitive

Heat coefficients depend on invariant traces such as \(\operatorname{tr}E^3\), \(\operatorname{tr}(E\Omega^2)\), and cubic connection-curvature contractions. They do not depend on an arbitrary choice of basis. The homogeneous-space structure constants and representation invariants determine these traces directly. Enumerating every off-diagonal hopping entry is an optional reconstruction, not the primitive calculation.

Part III — Final cross-building-block simplification pass

12. TECRAC: existence before counting

A term enters only if all of the following are true:

1. the support exists in the frozen Stage;
2. the field or bundle exists in the Actor inventory;
3. the operator domain and parity permit it;
4. it survives the BRST quotient with its correct sign;
5. it is not a duplicate representation of another contribution;
6. the corresponding invariant is nonzero on the actual geometry.

This removes corners, intersecting-boundary invariants, independent localized kinetic fields, and odd fixed-point zero modes before calculation.

13. Shape: unfold the quotient

Let \(g\) reflect the parent circle. For the even sector,

\[ P_+=\frac12(1+g), \qquad \operatorname{Tr}_{P_+}e^{-tL} =\frac12\operatorname{Tr}e^{-tL} +\frac12\operatorname{Tr}(ge^{-tL}). \]

The first trace is a smooth closed-manifold trace. The second localizes on the two fixed components. The quotient geometry supplies the decomposition automatically.

14. Interdependence: count the complete complex first

The primitive object is not a list of separately computed scalar, vector, ghost, bulk, and boundary numbers. It is the complete equivariant BRST supertrace. Factorization into simpler blocks is permitted only after proving that the direct-product connection, kinetic operator, reflection, and grading preserve the decomposition.

This ordering prevents a convenient decomposition from being mistaken for primitive independence.

15. Anomaly-descent source census

The anomaly-descent building block contributes a reusable rule: distinguish inherited sources from independent localized sources before evaluating any trace.

• inherited parent fields enter through the equivariant quotient;
• independent fixed-set Actors require their own local term;
• absent auxiliary fields are not promoted to physical blocks;
• a parity label does not create a zero mode;
• a local trace contribution does not prove the existence of a normalized state.

The current source census contains only the inherited graviton–ghost complex.

16. UQF-7 domain completeness

The accepted chiral/gauge domain fixes which projected fields and ghosts are legal. Gap-01 imports that domain; it does not invent a second parity table. This prevents a heat calculation from silently including mirror or forbidden sectors already removed by the physical domain.

17. Scale: typed support and cutoff powers

Scale supplies the decisive same-ruler rule. The bulk and fixed contributions are separately meaningful because each is paired with its own support measure and cutoff scaling. No physical observable consumes the bare sum of the two rationals.

18. Granularity: finite inventory, not a substitute for calculation

Granularity terminates the source inventory and removes a demand for an infinitely divisible physical regulator. It does not determine the coefficient. The exact rational must still be calculated from the frozen geometry.

19. Dynamics: perturbative counterterms, not new fundamental poles

The weight-six terms are Wilsonian insertions. They are not resummed into a new exact higher-derivative propagator. This rule is essential: a local higher-curvature counterterm used perturbatively need not introduce a new asymptotic ghost state below the cutoff, whereas treating a truncated derivative expansion as a fundamental all-frequency kinetic operator can manufacture spurious poles.

20. UQF-10 compactification interface

The calculation is performed on the accepted compactification branch and its admitted physical operator domain. A tachyonic or undefined background would invalidate the heat expansion. The current branch supplies the coercive internal-metric sector required for this local one-loop calculation.

21. UQF-14 causality interface

The counterterm Actor is local on its declared support and is used within the finite Wilsonian expansion. Its coefficients modify local matching data; they do not create an independent above-cutoff signal channel. The causal construction remains owned by UQF-14.

22. Time-synchronization audit

The heat parameter \(t\) is not physical clock time. It orders spectral suppression. Gap-01 makes no claim that small \(t\) is a literal elapsed time or that a coefficient directly measures signal speed.

23. Elegance result

After all building blocks are applied, the problem reduces to:

1. three primitive closed-manifold heat traces on \(Y_8=K_6\times S^2\);
2. two exact BRST combinations;
3. one orbifold projector;
4. one typed counterterm ledger.

No general boundary catalogue, infinite KK enumeration, or component-by-component graviton matrix is required.

Part IV — Complete physical object

24. Stage

\[ X_{13}=\mathcal M_4\times K_6\times S^2\times S^1_\chi/\mathbb Z_2, \qquad K_6=SU(3)/T^2. \]

Only the Stage carries metric dimension. The fixed components are two disjoint copies of

\[ F_{12}\cong\mathcal M_4\times K_6\times S^2. \]

The normal direction is one-dimensional. In the direct-product geometry the fixed components are totally geodesic, their normal bundle is trivial, and they do not intersect. Therefore corner and extrinsic-curvature sectors are absent.

25. Rulebook

The relevant Rulebook entries are:

• standard Euclidean Laplace-type sign convention;
• de Donder gauge for the metric fluctuation;
• Faddeev–Popov grading;
• even orbifold projector on the inherited complex;
• synchronized \(R_0\) normalization;
• no independent fixed-set kinetic Actor;
• perturbative Wilsonian treatment of weight-six operators;
• no addition of differently typed support coefficients.

26. Actor inventory

The counted one-loop complex is:

• unconstrained symmetric two-tensor metric fluctuation;
• Faddeev–Popov vector ghost with multiplier \(-2\) in the BRST heat trace;
• no second-order contribution from the ultralocal third ghost in the declared formulation.

The counterterm sector contains the new zero-dimensional Actor pair:

\[ \Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee}. \]

27. Dynamics

The primitive kinetic operators are the standard minimal Lichnerowicz operator on symmetric tensors and Hodge operator on vectors, evaluated on the frozen background. The one-loop Wilsonian action is represented by

\[ \Gamma_1=-\frac12\int_{\Lambda^{-2}}^\infty\frac{dt}{t} K_{\rm BRST}(t), \]

with the overall sign and factor frozen in the machine certificate. The counterterm Actor supplies the negative of the local cutoff-sensitive contribution plus independently declared finite renormalized couplings.

28. Complete object statement

The gate does not compute “geometry alone.” It computes

\[ (\text{Stage},\text{Rulebook},\text{Actors},\text{Dynamics}) \]

as one object. Changing the gauge complex, parity action, operator sign, radius conversion, or support inventory changes the coefficient and constitutes a new branch.

Part V — Equivariant BRST reduction theorem

29. Internal fixed geometry

Define

\[ Y_8=K_6\times S^2, \qquad Z_9=Y_8\times S^1, \qquad X_{13}=\mathcal M_4\times Z_9. \]

Let \(K_0\), \(K_1\), and \(K_2\) denote the scalar, one-form, and unconstrained symmetric-two-tensor heat traces on \(Y_8\).

30. Parent bulk decomposition

The direct-product tensor decomposition gives

\[ K_2(X_{13})-2K_1(X_{13}) = \bigl[K_2(Y_8)+3K_1(Y_8)+5K_0(Y_8)\bigr]K_0(S^1). \]

The rank check is

\[ 36+3\cdot8+5=65=\frac{13\cdot14}{2}-2\cdot13. \]

This is an exact component-count check on the graded complex.

31. Reflection-fixed decomposition

Reflection acts positively on tangential indices and negatively on the single normal index. The equivariant BRST trace becomes

\[ \boxed{ K_g^{\rm BRST}=K_2(Y_8)+K_1(Y_8)+K_0(Y_8) }. \]

The equivariant rank check is

\[ 36+8+1=45. \]

32. Theorem GAP01-EQBRST

Theorem. For the declared direct-product metric, standard minimal de Donder/Faddeev–Popov complex, reflection-compatible domain, and absence of independent fixed-set kinetic Actors, the complete inherited orbifold order-six heat datum is determined by the three ordinary closed-manifold heat traces \(K_0(Y_8)\), \(K_1(Y_8)\), and \(K_2(Y_8)\) through the two combinations above.

Proof sketch. The direct-product connection preserves the canonical tangent decomposition. The Laplace-type operators commute with the reflection and BRST grading. The parent-circle trace factorizes. The reflection trace on the circle equals one because only the zero Fourier mode contributes diagonally. Tangential/normal index characters give the displayed signs. No additional strata or localized Actors exist. Therefore no other contribution is available.

33. Consequence

The supposed missing general mixed-boundary coefficient is eliminated as a wrong object for this branch. A future independent boundary Actor would create a new term and reopen the source census, but none is present here.

Part VI — Exact primitive coefficients and product assembly

34. Frozen geometric invariants

At the \(K_6\) Einstein center in Killing normalization:

\[ \operatorname{Ric}=\frac5{12}g, \qquad R=\frac52, \]

\[ |\operatorname{Ric}|^2=\frac{25}{24}, \qquad |\operatorname{Riem}|^2=\frac{23}{12}, \qquad |\nabla\operatorname{Riem}|^2=\frac14, \]

\[ K_1=-\frac{113}{72}, \qquad K_2=-\frac5{72}. \]

The nonzero derivative invariant is essential. Homogeneity does not imply local symmetry.

35. Primitive \(K_6\) coefficients

In the common \(R_0\) normalization, the coefficient vectors \([a_0,a_2,a_4,a_6]\) are

\[ K_6\text{ scalar}: \left[1,\frac12,\frac{33}{250},\frac{992}{39375}\right], \]

\[ K_6\text{ vector}: \left[6,0,-\frac{47}{250},-\frac{251}{10500}\right], \]

\[ K_6\text{ symmetric tensor}: \left[21,-\frac{27}{2},\frac{1643}{250},-\frac{6613}{3750}\right]. \]

36. Primitive \(S^2\) coefficients

For the unit two-sphere:

\[ S^2\text{ scalar}: \left[1,\frac13,\frac1{15},\frac4{315}\right], \]

\[ S^2\text{ vector}: \left[2,-\frac43,\frac2{15},\frac8{315}\right], \]

\[ S^2\text{ symmetric tensor}: \left[3,-7,\frac{61}{5},-\frac{1256}{105}\right]. \]

These values pass exact spectral controls on the round sphere.

37. Product coefficients on \(Y_8\)

The product convolution rule gives

\[ Y_8\text{ scalar}: \left[1,\frac56,\frac{137}{375},\frac{4537}{39375}\right], \]

\[ Y_8\text{ vector}: \left[8,\frac53,-\frac{43}{750},-\frac{6919}{157500}\right], \]

\[ Y_8\text{ symmetric tensor}: \left[36,-20,\frac{1624}{125},-\frac{86116}{13125}\right]. \]

38. Parent bulk vector

Combining \(K_2+3K_1+5K_0\) gives

\[ \boxed{ \left[ 65,-\frac{65}{6},\frac{2197}{150},-\frac{963409}{157500} \right] }. \]

39. Reflection-fixed vector

Combining \(K_2+K_1+K_0\) gives

\[ \boxed{ \left[ 45,-\frac{35}{2},\frac{133}{10},-\frac{340721}{52500} \right] }. \]

40. Orbifold projection

The even projector contributes one half of each parent trace:

\[ B_6^{(13)}=-\frac{963409}{315000}, \qquad C_6^{(12)}=-\frac{340721}{105000}. \]

These are the exact trace coefficients used by the gate.

41. Effective-action convention

With

\[ \Gamma_1=-\frac12\int_{\Lambda^{-2}}^\infty\frac{dt}{t}K_{\rm BRST}(t), \]

the cutoff-sensitive weight-six multipliers per unit normalized support volume are

\[ \Gamma_{1,\rm bulk}^{(6)} \supset (4\pi)^{-13/2}\Lambda^7R_0^{-6} \frac{963409}{2205000}, \]

\[ \Gamma_{1,\rm fixed}^{(6)} \supset (4\pi)^{-6}\Lambda^6R_0^{-6} \frac{340721}{630000}. \]

The counterterm contribution has the opposite sign in this frozen convention. Finite renormalized pieces remain construction parameters fixed by declared matching conditions.

42. Why the cutoff powers differ

For the bulk term, \(t^3(4\pi t)^{-13/2}/t\) integrates as \(\Lambda^7\). For the fixed term, \(t^3(4\pi t)^{-6}/t\) integrates as \(\Lambda^6\). This independently confirms that the coefficients are different support types.

43. Why \(a_6\) is not a universal logarithmic anomaly here

A logarithmic heat-kernel divergence occurs when the coefficient order matches the support dimension. Weight six does not match thirteen or twelve. The present coefficient is a Wilsonian power-sensitive matching datum in the frozen finite spectral scheme. Its local operator basis is meaningful; its bare power-divergent numerical normalization is scheme-bound. This is exactly why the counterterm Actor and explicit scheme record are required.

Part VII — Weight-Six Counterterm Completion Actor

44. Definition of \(\Xi_{\rm W6CT}\)

The elegant primitive is not a hand-written list of dozens of monomials. It is the universal local order-six heat functional itself, evaluated on the declared BRST complex and split by support:

\[ \Xi_{\rm W6CT} = \mathscr A_{6,\rm bulk}\!\left[\Delta_2-2\Delta_1\right] \oplus \mathscr A_{6,g}\!\left[\Delta_2-2\Delta_1\right]. \]

Here \(\mathscr A_{6,\rm bulk}\) is the universal closed-manifold Gilkey functional and \(\mathscr A_{6,g}\) is its reflection-equivariant fixed-locus functional. Each is a local polynomial in the Riemann tensor, bundle endomorphism \(E\), connection curvature \(\Omega\), and their allowed derivatives. The invariance theorem guarantees that this functional is the complete local one-loop order-six object for the frozen Laplace-type operator. Expanding it into a chosen monomial basis is a representation of the Actor, not the Actor’s definition.

Its counterterm action is

\[ S_{\rm W6CT} = c_{\rm B}(\mu)\int_{X_{13}}\!\sqrt g\, \mathscr a_{6,\rm bulk}^{\rm BRST} + c_{\rm F}(\mu)\sum_{f\in\{0,\pi\}} \int_{F_f}\!\sqrt h\, \mathscr a_{6,g}^{\rm BRST}. \]

The divergent parts of \(c_{\rm B}\) and \(c_{\rm F}\) are fixed to the negative of the calculated cutoff-sensitive terms. Their finite renormalized parts are declared matching data. Bulk and fixed couplings are distinct because the supports have different dimensions and cutoff powers. Reflection equates the two fixed-component couplings but never equates them with the bulk coupling.

45. Why the functional definition is complete

For a fixed Laplace-type operator, the local heat-kernel theorem states that the order-six coefficient is a universal invariant polynomial. Therefore every one-loop local order-six contribution generated by the declared graviton–ghost complex is already contained in \(\mathscr A_6\). No additional monomial can be generated at this order without appearing in that universal functional, modulo Bianchi identities, integration by parts, and field-redefinition/equation-of-motion equivalence.

If a reviewer prefers an explicit basis, the functional may be expanded into classes including cubic curvature contractions, curvature–endomorphism terms, cubic endomorphism traces, connection-curvature contractions, and derivative terms. That expansion is finite and algorithmic. It is not required to establish completeness because completeness belongs to the invariant functional, not to one choice of coordinates on its operator space.

46. Fixed-set specialization

The equivariant functional is evaluated only on the actual reflection fixed locus. Terms requiring corners, intersecting fixed sets, nonzero extrinsic curvature, twisted normal-bundle curvature, or independent localized fields vanish or are absent before evaluation. Thus the fixed-set Actor is not the most general boundary functional in mathematics; it is the complete equivariant functional for the physical support that can actually exist in this branch.

47. Co-Actor \(\Xi_{\rm TSC}^{\vee}\)

The Co-Actor enforces:

1. support typing;
2. BRST closure;
3. parity compatibility;
4. basis completeness modulo identities;
5. no duplicate boundary/orbifold representation;
6. no absent Actor;
7. separate renormalization conditions for independent supports;
8. perturbative use of the derivative expansion;
9. certificate and hash consistency.

48. Why this is construction-anchor closure

The geometry determines the divergent combination, but it does not force the finite values of every allowed Wilsonian coupling. Including the complete counterterm basis and choosing renormalization conditions are lawful Dynamics choices. They are not derived from the original Einstein–Hilbert term alone.

The gate therefore does not mislabel the counterterm Actor as a prediction. It closes because a complete explicit construction exists and absorbs the computed local contribution.

49. No new metric dimensions

Both Actor and Co-Actor are zero-metric-dimensional. They modify the Dynamics and admissibility ledger, not the Stage dimension count.

50. No hidden fit to the coefficient

The basis is generated from local symmetry, support, and BRST cohomology before the numerical coefficient is loaded. The coefficient determines the required running/bare shift inside that predeclared basis. No operator is added merely because its contraction helps match the answer.

51. Perturbative-pole firewall

The truncated weight-six action is never declared fundamental at arbitrarily high frequency. It is expanded perturbatively around the accepted two-derivative kinetic operator within the Wilsonian window. Any pole appearing only after nonperturbative resummation of a truncated derivative series is excluded from the physical-state inventory unless the full microscopic Dynamics independently establishes it.

Part VIII — Wilsonian meaning and interfaces

52. What “survive as a finite quantum theory” means here

At this gate, survival means:

• the one-loop order-six local contribution is finite after the declared cutoff and exactly calculable;
• every local structure it generates exists in the counterterm Actor;
• gauge redundancy is handled by the BRST complex;
• quotient fixed-set terms are counted exactly once;
• renormalized couplings can be fixed without altering the frozen geometry;
• no finite contradiction or uncancelled local divergence remains at this order.

53. What it does not mean

It does not mean:

• that all \(a_{2k}\) are computed;
• that only finitely many higher-curvature operators exist at all orders;
• that gravity is perturbatively renormalizable in the traditional finite-parameter sense;
• that a UV fixed point exists;
• that nonlocal or nonperturbative effects are controlled by \(a_6\);
• that the regulator may be removed to infinite resolution.

54. UQF-9 relationship

UQF-9 owns the broader high-energy meaning of the finite spectral Wilsonian theory. Gap-01 supplies its exact weight-six local matching datum. Closing Gap-01 strengthens UQF-9 but does not erase UQF-9’s permanent nonclaim concerning a conventional continuum UV fixed point.

55. Compactification relationship

The local coefficient assumes the frozen compactification background and operator domain. If UQF-10’s accepted construction were withdrawn or a physical tachyon reintroduced into the counted sector, the heat expansion would require re-audit.

56. Causality relationship

Local Wilsonian counterterms preserve the support structure used by the causal construction. The perturbative-pole firewall prevents a finite derivative truncation from being mistaken for a new microscopic signal law.

57. Positivity relationship

The sign of the evaluated coefficient is not, by itself, a complete positivity theorem. Positivity bounds act on specific projected four-dimensional amplitudes after matching, subtraction, and field redefinition. Downstream gates must perform that observer-map calculation rather than reading causality from the bare sign.

58. Field-redefinition discipline

Operators proportional to lower-order equations of motion may be moved within the local basis by field redefinitions. The Co-Actor tracks equivalence classes rather than treating every written monomial as an independent observable. The branch-specific heat coefficient remains invariant as an integrated trace datum under lawful basis changes, while individual coupling labels may change.

Part IX — Evidence, reproducibility, and negative controls

59. Exact machine reproduction

The V13 source calculation and the new V16 Gap-01 certificate independently reproduce

\[ B_6^{(13)}=-\frac{963409}{315000}, \qquad C_6^{(12)}=-\frac{340721}{105000}. \]

All arithmetic is exact rational arithmetic.

60. Rank controls

The parent graded rank is 65 and the equivariant rank is 45. A missing ghost multiplier, incorrect normal-index character, or wrong tensor decomposition fails these checks immediately.

61. Curvature controls

The calculation reproduces the frozen exact values

\[ |\operatorname{Riem}|^2=\frac{23}{12}, \quad |\nabla\operatorname{Riem}|^2=\frac14, \quad K_1=-\frac{113}{72}, \quad K_2=-\frac5{72}. \]

The rejected values from contaminated curvature conventions remain permanent negative controls.

62. Sphere controls

The scalar, vector, and symmetric-tensor coefficients on \(S^2\) are checked against exact spectra. Sphere agreement credentials the universal local formula and sign convention but does not substitute for the \(K_6\) computation.

63. Pure-trace control

Under the standard Lichnerowicz convention, the pure trace has the required zero action on an Einstein background in the relevant algebraic check. A prior nonzero pure-trace assignment signals an incompatible operator convention and is not mixed into the controlling result.

64. Negative-control register

65. Independent reproduction contract

A second implementation must:

1. regenerate the primitive \(K_6\) coefficient vectors from the exact invariant core;
2. regenerate the \(S^2\) vectors from spectral sums or an independent local engine;
3. apply product convolution;
4. reproduce both BRST combinations;
5. apply the projector;
6. emit the typed pair without adding it;
7. match every rational exactly.

A failure reopens the numerical leg until reconciled.

Part X — Hostile-review objections

66. “You closed the gate by adding arbitrary counterterms.”

The counterterm basis is indeed a construction, and the dossier labels it as such. The non-arbitrary content is completeness: every term is selected before comparison by local symmetry, support, BRST cohomology, and the actual Actor inventory. Finite renormalized coefficients are not predicted; they are declared Dynamics data. This is legitimate construction-anchor closure, not from-nothing derivation.

67. “A theory with infinitely many possible counterterms is not UV complete.”

Correct—and this dossier does not call it UV complete in that sense. Gap-01 closes one finite weight-six Wilsonian sector. UQF-9 owns the broader finite-floor theory, and conventional continuum UV completion remains a separate nonclaim.

68. “Power divergences are scheme-dependent, so the coefficient is meaningless.”

The bare magnitude is scheme-bound, but the local operator classification and the exact coefficient in a frozen scheme are meaningful matching data. Wilsonian effective actions necessarily carry scheme and scale. The remedy is explicit scheme ownership, not pretending the coefficient is universal or refusing to calculate it.

69. “Why compute \(a_6\) if it is not logarithmic?”

Because it controls the first cubic-curvature/local six-derivative matching sector in the chosen expansion. It is a finite audit of the geometry and operator complex and supplies required higher-curvature couplings for downstream matching. It need not be a log anomaly to be physically useful.

70. “The fixed set should use boundary heat coefficients.”

That would be appropriate for an independently specified interval boundary problem. Here the inherited interval theory is the parity quotient of a smooth parent circle. The equivariant trace is spectrally exact and already contains the fixed contribution. An independent boundary term requires an independent boundary Actor.

71. “The two fixed points should give two extra copies.”

The reflection trace on the parent circle equals one: each fixed point contributes the normal determinant factor one half. The two together give one. The orbifold projector then supplies its additional factor one half. Counting two full copies would fail the equivariant trace identity.

72. “Unconstrained symmetric tensors are not physical gravitons.”

Correct in isolation. They are used inside the complete gauge-fixed BRST complex. The physical object is the graded combination, not the raw tensor trace. This is precisely why the legacy single-bundle interpretation was retired.

73. “Higher derivatives create ghosts.”

A finite Wilsonian derivative expansion used perturbatively does not define new exact asymptotic poles. Ghost conclusions drawn from resumming a truncated series are outside its validity. A genuine additional state would require the microscopic completion to establish a pole within the admitted physical spectrum.

74. “You have not enumerated every off-background coefficient separately.”

The Actor contains the complete cohomological basis, while the heat calculation evaluates the exact branch-specific linear combination. A full monomial expansion would strengthen the public exhibit but is not required for completeness: the universal invariant functional is the complete object, and any lawful basis expansion is only a coordinate representation of it.

75. “Why trust the standard Lichnerowicz convention?”

Because the operator is frozen explicitly, satisfies the pure-trace and sphere controls, and is the operator used by the exact certificate. A different convention is a different branch and must be translated rather than silently merged.

76. “This is only one loop.”

Yes. The gate charter is the weight-six one-loop/Wilsonian keystone. No all-loop statement is promoted from it.

77. “Does the negative sign imply instability?”

No. The sign is the coefficient of a local heat-trace combination in a specified Euclidean scheme. Stability requires the full quadratic action and physical projection at the relevant background and scale. It cannot be inferred from this sign alone.

78. “Can field redefinitions remove the result?”

They can redistribute coefficients among equivalent local operators, particularly equation-of-motion terms. They cannot erase the full integrated heat-trace datum or the need for a complete counterterm equivalence class.

Part XI — Final terminal and propagation

79. Subleg ledger

80. Final physical endpoint

The complete physical order-six graviton–ghost orbifold heat datum is derived as a typed bulk/fixed-set pair. Every local weight-six structure generated by that datum lies in an explicit complete BRST- and support-respecting counterterm Actor. The finite Wilsonian theory is therefore closed at weight six on the frozen branch.

81. Project endpoint

Upon owner acceptance of \(\Xi_{\rm W6CT}\dashv\Xi_{\rm TSC}^{\vee}\):

Gap-01 — CLOSED / RESOLVED +0.

82. Permanent nonclaims

• No claim of a continuum UV fixed point.
• No claim of finite-parameter perturbative renormalizability at all orders.
• No claim that power-divergent coefficients are regulator-independent observables.
• No claim that finite counterterm couplings are predicted by Shape.
• No claim that \(-6373/630\) is the physical total.
• No claim that bulk and fixed support coefficients can be added.
• No claim about topology-changing or nonperturbative branches outside the frozen Stage.

83. Reopen conditions

The gate reopens only on a named failure:

1. a reproducible error in the exact rational calculation;
2. an incorrect Lichnerowicz/ghost convention relative to the physical path integral;
3. a physical Actor or localized term omitted from the frozen source census;
4. failure of parent-quotient spectral equivalence;
5. failure of BRST compatibility with reflection;
6. proof that the declared counterterm cohomology basis is incomplete;
7. a measured contradiction attributable specifically to the weight-six sector;
8. withdrawal of the compactification background required by the operator calculation.

A request for a continuum UV fixed point does not reopen Gap-01; it belongs to the broader UV gate.

84. Canonical source-of-truth block

Gap-01 — a6 coefficient / weight-six Wilsonian keystone
Status: CLOSED-SCOPED / DERIVED exact typed coefficient pair
        + CONSTRUCTION-ANCHOR complete counterterm Actor / RESOLVED +0.
Exact trace output, common R0 convention:
  bulk 13D:  -963409/315000
  fixed 12D: -340721/105000
Rule: DO NOT ADD; distinct supports and cutoff powers.
Legacy -6373/630: retired as physical full coefficient; archive only.
Counterterm ownership:
  Xi_W6CT ⊣ Xi_TSC^vee, zero metric dimensions.
Scope:
  weight-six finite Wilsonian matching only;
  no continuum UV fixed-point or all-orders claim.

85. Shape and Dynamics propagation block

ADD TO ACTOR INDEX:
  Xi_W6CT — Weight-Six Counterterm Completion Actor.
  Xi_TSC^vee — Typed-Support and BRST-Completeness Co-Actor.
DIMENSION:
  both zero metric dimensions.
DYNAMICS RULE:
  maintain separate bulk-13D and fixed-12D local coupling ledgers;
  use counterterms perturbatively within the Wilsonian window;
  never infer new asymptotic poles from resumming a truncated series.
SOURCE RULE:
  inherited orbifold fields are counted through the parent equivariant trace;
  independent fixed-set terms require independent localized Actors.

86. Downstream propagation

• UQF-9 may replace “complete projected \(a_6\) open” with “computed as typed pair,” while retaining its no-UV-fixed-point boundary.
• Gap-13 and other consumers must remove references to a missing generic order-six boundary coefficient for this inherited branch.
• Any downstream scalar “total \(a_6\)” must be replaced by typed support data and the correct support integrals.
• The legacy \(-6373/630\) must be removed from public physical summaries.

87. Formal acceptance block

OWNER DECISION:
  [ ] ACCEPT Xi_W6CT and Xi_TSC^vee as the controlling weight-six Dynamics pair.
  [ ] RATIFY the exact typed coefficient pair.
  [ ] RETIRE -6373/630 as the physical full coefficient.
  [ ] PROPAGATE Gap-01 to CLOSED / RESOLVED +0.

Technical appendices

Appendix A — Exact one-loop cutoff integration

For a support of dimension \(d\), an order-six term contributes

\[ K(t)\supset(4\pi t)^{-d/2}a_6t^3. \]

Using the frozen effective-action convention,

\[ \Gamma_1^{(6)}=-\frac12(4\pi)^{-d/2}a_6 \int_{\Lambda^{-2}}^\infty dt\,t^{2-d/2}. \]

For \(d=13\), the divergent local part is proportional to \(\Lambda^7\). For \(d=12\), it is proportional to \(\Lambda^6\). The universal finite upper-limit contribution depends on the full spectrum and is not encoded by the asymptotic coefficient alone.

Appendix B — Counterterm-basis completion contract

A future explicit basis enumeration must satisfy:

1. Generate local ghost-number-zero BRST cohomology at weight six.
2. Split by bulk and fixed support.
3. Reduce with Bianchi identities and integration by parts.
4. Quotient equation-of-motion/field-redefinition redundancies.
5. Apply reflection parity and actual bundle representations.
6. Delete nonexistent supports and Actors.
7. Verify that the branch-specific heat density is in the span.
8. Freeze a basis transformation matrix and rank certificate.

The present Actor is defined by this complete cohomological class rather than by one arbitrary monomial basis.

Appendix C — Child-level recap

The geometry is a castle with an interior and two special mirrored walls. Quantum shaking tells us how much reinforcement is needed. The interior reinforcement and wall reinforcement are measured per different kinds of space, so we keep two numbers. We now know both numbers exactly. The counterterm Actor is the complete box of legal reinforcement pieces. It makes the castle repairable at this order, but it does not prove that no new kind of repair will ever be needed at higher orders.

Appendix D — Reviewer reconstruction algorithm

1. Verify the source hashes.
2. Run the V13 exact script.
3. Run the V16 Gap-01 certificate script.
4. Confirm all primitive vectors.
5. Confirm the bulk and fixed BRST combinations.
6. Confirm the one-half orbifold projector.
7. Confirm the cutoff exponents \(7\) and \(6\).
8. Confirm that the output is a typed pair.
9. Inspect the source census for independent localized Actors.
10. Audit the counterterm Actor against the BRST cohomology contract.

Appendix E — Reference and authority register

The controlling project sources are listed in the integrity manifest distributed with this dossier. External mathematical background is used only for standard heat-kernel, equivariant-trace, BRST, and Wilsonian-EFT machinery; the branch-specific numbers are generated by the project’s exact scripts.

Appendix F — Integrity hashes

Appendix G — Controlling V13 exact-calculation record

Authority note: This appendix is controlling for the exact geometric calculation and is incorporated without changing its numerical content. Where its provisional status language differs from this V16 dossier, the V16 terminal governs; its equations and certificate values remain controlling technical evidence.

UQF-9 — Final Building-Block Simplification Pass and Exact Order-Six Calculation

Version: V13 calculation amendment
Date: 2026-07-17
Status: Technical ratification candidate; not owner-ratified
Scope: The declared minimal de Donder/Lichnerowicz graviton plus Faddeev–Popov ghost complex on \(\mathcal M_4\times K_6\times S^2\times S^1_\chi/\mathbb Z_2\), with \(K_6=SU(3)/T^2\), no independent fixed-set kinetic Actor, and the synchronized Shape v2.10 radius convention.

Executive result

The final cross-building-block pass eliminates the supposed need for a general order-six mixed-boundary formula. The correct object is the equivariant heat trace on the smooth parent circle. After applying the complete BRST grading and decomposing the tangent bundle geometrically, the entire problem reduces to three primitive heat traces on the eight-dimensional fixed internal geometry

\[ Y_8=K_6\times S^2. \]

The exact parent-bulk and reflection-fixed-set combinations are

\[ K_{\rm bulk}^{\rm BRST}(t) =K_2(Y_8;t)+3K_1(Y_8;t)+5K_0(Y_8;t), \]

\[ K_{g}^{\rm BRST}(t) =K_2(Y_8;t)+K_1(Y_8;t)+K_0(Y_8;t), \]

where \(K_0,K_1,K_2\) denote the scalar, Hodge-vector, and unconstrained symmetric-two-tensor Lichnerowicz heat traces. This is exact for the declared direct-product operator.

In the common \(R_0\) normalization, omitting the universal volume and \((4\pi)^{-d/2}\) factors, the order-six coefficients are

\[ \boxed{b_3^{\rm bulk}=-\frac{963409}{157500}}, \qquad \boxed{c_3^{\gamma}=-\frac{340721}{52500}}. \]

The orbifold projector \(P_+=(1+g)/2\) therefore supplies

\[ \boxed{\frac12 b_3^{\rm bulk}=-\frac{963409}{315000}}, \qquad \boxed{\frac12 c_3^{\gamma}=-\frac{340721}{105000}}. \]

These two numbers must not be added. The bulk term and fixed-set term occupy different powers in the heat expansion because their supports have dimensions thirteen and twelve:

\[ K_{\rm orb}(t)\supset (4\pi t)^{-13/2}\left(\frac12 b_3^{\rm bulk}\right)t^3 + (4\pi t)^{-12/2}\left(\frac12 c_3^\gamma\right)t^3. \]

Thus the complete projected order-six datum is a typed pair, not one scalar “total \(a_6\).”

Part I — What the building blocks simplify

1. TECRAC: existence before counting

No term is admitted merely because it appears in a universal boundary formula. A block enters only if:

1. its geometric support exists;
2. its field or bundle exists in the frozen Actor inventory;
3. its domain and parity permit it;
4. it survives the BRST quotient with the correct sign;
5. it is not another representation of a block already counted.

This deletes corners, boundary intersections, and independent localized fields from the present problem. The two fixed components are disjoint and no independent fixed-set kinetic Actor is declared.

2. Shape: unfold the quotient

Write the interval as the quotient of the smooth parent circle by the reflection \(g:y\mapsto-y\). For every inherited field,

\[ \operatorname{Tr}_{P_+}e^{-tL} =\frac12\operatorname{Tr}e^{-tL} +\frac12\operatorname{Tr}\!\left(g e^{-tL}\right). \]

The parent term and the fixed-set term are therefore not independent boundary additions. The equivariant term already is the inherited quotient defect.

For the circle eigenbasis \(|n\rangle\), reflection sends \(|n\rangle\mapsto|-n\rangle\). Hence

\[ \operatorname{Tr}_{S^1}\!\left(g e^{t\partial_y^2}\right)=1 \]

exactly: only the \(n=0\) diagonal matrix element survives. In the local fixed-point language, the two fixed points each contribute the usual one-half normal determinant, and together give one.

3. Anomaly-descent and UQF-7: inherited versus localized sources

The anomaly-descent source-census rule transfers directly:

• inherited parent fields are counted through the equivariant trace;
• a separate fixed-set term is admitted only for an independent localized Actor;
• an orbifold-odd label is not evidence for a localized zero mode;
• absent regulator auxiliaries are not promoted into physical heat-kernel blocks.

Under the current Shape inventory, there is no additional fixed-set kinetic Actor. Therefore the defect is completely inherited from the parent graviton/ghost complex.

4. Interdependence: no primitive factorization

The primitive object is the complete parent BRST trace. Scalar, vector, tensor, bulk, and defect ledgers are derived decompositions. They may be separated only because the direct-product connection, the reflection, and the declared kinetic operator commute and preserve the displayed blocks.

This prevents adding independently calculated “boundary,” “orbifold,” and “ghost” answers that may encode the same states.

5. Dynamics: quotient before component counting

The counted object is

\[ K^{\rm BRST}(t)=K_{\rm grav}(t)-2K_{\rm FP}(t), \]

not the raw tensor-component count. The ultralocal third ghost has no second-order heat operator and contributes zero to the \(a_{2k>0}\) tower in the declared formulation.

6. Scale and same-ruler discipline

The Killing-normalized \(K_6\) metric has \(\mathrm{Scal}=5/2\). The project’s synchronized \(R_0\) normalization has \(\mathrm{Scal}=3/R_0^2\). Therefore

\[ g_{K_6,R_0}=\frac56 R_0^2 g_{K_6,\rm Killing}, \]

and a local coefficient of order \(2k\) scales by \((6/5)^kR_0^{-2k}\). This conversion is applied before combining \(K_6\) with the unit-normalized \(S^2\).

The bulk and defect supports have different dimensions. Same-ruler discipline therefore forbids adding their coefficients even though both carry the order label six.

7. Time synchronization: Euclidean proper time is not physical clock time

The parameter \(t\) in the heat trace is a spectral/proper-time bookkeeping variable. No BB-TS physical clock Actor is added to the coefficient. The relevant synchronization is instead the common operator, radius, quotient, and normalization tuple.

8. Granularity: finite inventory, not coefficient substitution

Granularity guarantees that the project does not owe an infinitely divisible physical continuum. It does not supply the numerical coefficient. Here its useful role is narrower: the Actor inventory and quotient grammar are finite, so the existence sieve terminates. The surviving coefficient is still calculated exactly.

Part II — The geometric reduction

9. Tangent decomposition

Let

\[ Y_8=K_6\times S^2, \qquad Z_9=Y_8\times S^1_\chi, \qquad X_{13}=\mathcal M_4\times Z_9. \]

On a direct product with flat factors, the standard Lichnerowicz and Hodge operators preserve the canonical bundle decompositions.

For the internal circle:

\[ K_0(Z_9)=K_0(Y_8)K_0(S^1), \]

\[ K_1(Z_9)=\big[K_1(Y_8)+K_0(Y_8)\big]K_0(S^1), \]

\[ K_2(Z_9)=\big[K_2(Y_8)+K_1(Y_8)+K_0(Y_8)\big]K_0(S^1). \]

For the flat four-dimensional factor:

\[ K_1(X_{13})=4K_0(Z_9)+K_1(Z_9), \]

\[ K_2(X_{13})=10K_0(Z_9)+4K_1(Z_9)+K_2(Z_9). \]

Consequently,

\[ K_2(X_{13})-2K_1(X_{13}) =\big[K_2(Y_8)+3K_1(Y_8)+5K_0(Y_8)\big]K_0(S^1). \]

The rank check is

\[ 36+3\cdot8+5\cdot1=65=91-2\cdot13. \]

10. Reflection-fixed trace

The circle reflection acts as \(+1\) on tangential indices and \(-1\) on a normal index. Thus on \(Y_8\times S^1\),

\[ K_0^g=K_0(Y_8), \quad K_1^g=K_1(Y_8)-K_0(Y_8), \quad K_2^g=K_2(Y_8)-K_1(Y_8)+K_0(Y_8). \]

After including the four flat tangential directions and the ghost multiplier,

\[ \boxed{K_g^{\rm BRST}=K_2(Y_8)+K_1(Y_8)+K_0(Y_8)}. \]

The equivariant rank is

\[ 36+8+1=45=67-2\cdot11. \]

This is the missing Donnelly coefficient reduced to ordinary closed-manifold coefficients on \(Y_8\). No general boundary tower remains.

Part III — Exact primitive coefficients

11. Universal formula and sign convention

The calculation uses the standard Laplace-type convention

\[ D=-\left(g^{ij}\nabla_i\nabla_j+E_V\right). \]

For a positive Weitzenböck operator \(\nabla^*\nabla+Q\), the coefficient formula therefore consumes

\[ E_V=-Q. \]

This sign is essential. It is independently checked on the unit two-sphere: the Hodge one-form coefficient comes out \(a_2=-4/3\), and the unconstrained symmetric-tensor Lichnerowicz coefficients reproduce the exact spectral identities below.

The full order-six formula includes the nonzero derivative sectors \(\nabla\Omega\) and \(\nabla E\). They cannot be deleted on \(K_6\), because

\[ \lVert\nabla\mathrm{Riem}\rVert^2=\frac14\neq0. \]

12. Exact \(K_6\) geometry

The script reconstructs \(SU(3)/T^2\) directly from the Gell-Mann structure constants and Nomizu’s normal-homogeneous connection. It returns

\[ \mathrm{Ric}=\frac5{12}g, \qquad \mathrm{Scal}=\frac52, \qquad \lVert\mathrm{Ric}\rVert^2=\frac{25}{24}, \]

\[ \lVert\mathrm{Riem}\rVert^2=\frac{23}{12}, \qquad \lVert\nabla\mathrm{Riem}\rVert^2=\frac14, \]

\[ K_1=-\frac{113}{72}, \qquad K_2=-\frac5{72}. \]

These reproduce the project’s strongest curvature negative controls.

13. Standard Lichnerowicz operator

With the curvature convention used by the reconstruction,

\[ R^{\rm std}_{acbd}=\langle e_a,R(e_b,e_d)e_c\rangle, \]

and the standard endomorphism is

\[ (Q_Lh)_{ab} =\mathrm{Ric}_a{}^c h_{cb} +\mathrm{Ric}_b{}^c h_{ac} -2R^{\rm std}_{acbd}h^{cd}. \]

A decisive negative control is

\[ Q_L(g)=0. \]

Any index placement giving a nonzero pure-trace eigenvalue is not this standard Lichnerowicz operator. The exact pointwise spectrum on \(\mathrm{Sym}^2(TK_6)\) is

\[ 0\;(\times1),\quad \frac14\;(\times2),\quad \frac12\;(\times6),\quad \frac54\;(\times6),\quad \frac32\;(\times6). \]

14. \(K_6\) coefficients in Killing normalization

The exact local coefficient vectors \([a_0,a_2,a_4,a_6]\) are

\[ K_0(K_6)= \left[1,\frac5{12},\frac{11}{120},\frac{124}{8505}\right], \]

\[ K_1(K_6)= \left[6,0,-\frac{47}{360},-\frac{251}{18144}\right], \]

\[ K_2(K_6)= \left[21,-\frac{45}{4},\frac{1643}{360},-\frac{6613}{6480}\right]. \]

The scalar result gives the established check

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

The physical six-dimensional graviton-minus-two-ghost order-six combination is

\[ \boxed{a_6^{K_6,\rm BRST} =-\frac{6613}{6480}-2\left(-\frac{251}{18144}\right) =-\frac{139}{140}}. \]

This value includes the full Levi-Civita derivative sectors. No Gelfand–Tsetlin state-by-state hopping enumeration is needed because the local coefficient depends only on basis-invariant contractions of the connection curvature and endomorphism.

15. Common \(R_0\) normalization

After the \((6/5)^k\) conversion,

\[ K_0(K_6)=\left[1,\frac12,\frac{33}{250},\frac{992}{39375}\right], \]

\[ K_1(K_6)=\left[6,0,-\frac{47}{250},-\frac{251}{10500}\right], \]

\[ K_2(K_6)=\left[21,-\frac{27}{2},\frac{1643}{250},-\frac{6613}{3750}\right]. \]

16. Unit \(S^2\) coefficients and spectral controls

For the unit sphere,

\[ K_0(S^2)=\left[1,\frac13,\frac1{15},\frac4{315}\right]. \]

The Hodge one-form spectrum gives the exact identity

\[ K_1(S^2;t)=2\big(K_0(S^2;t)-1\big), \]

hence

\[ K_1(S^2)=\left[2,-\frac43,\frac2{15},\frac8{315}\right]. \]

For the unconstrained standard Lichnerowicz operator,

\[ K_2(S^2;t)=3K_0(S^2;t)-2-6e^{-2t}, \]

which yields

\[ K_2(S^2)=\left[3,-7,\frac{61}{5},-\frac{1256}{105}\right]. \]

These exact spectral identities independently verify the operator sign and index placement used in the local coefficient engine.

Part IV — Product assembly

17. Primitive traces on \(Y_8=K_6\times S^2\)

Product convolution gives

\[ K_0(Y_8)= \left[1,\frac56,\frac{137}{375},\frac{4537}{39375}\right], \]

\[ K_1(Y_8)= \left[8,\frac53,-\frac{43}{750},-\frac{6919}{157500}\right], \]

\[ K_2(Y_8)= \left[36,-20,\frac{1624}{125},-\frac{86116}{13125}\right]. \]

18. Parent bulk

Applying \(K_2+3K_1+5K_0\),

\[ K_{\rm bulk}^{\rm BRST}= \left[ 65, -\frac{65}{6}, \frac{2197}{150}, -\frac{963409}{157500} \right]. \]

19. Fixed-set equivariant term

Applying \(K_2+K_1+K_0\),

\[ K_g^{\rm BRST}= \left[ 45, -\frac{35}{2}, \frac{133}{10}, -\frac{340721}{52500} \right]. \]

The orbifold projector supplies one-half of each stratum.

Part V — What was removed

The final calculation contains none of the following because the present construction does not support them:

• corner or edge-intersection coefficients;
• extrinsic-curvature invariants of an independently imposed interval boundary;
• a separate boundary coefficient added on top of the equivariant defect;
• orbifold-odd fixed-point zero modes;
• regulator mirror sectors absent from Shape v2.10;
• independent localized kinetic Actors not declared in the frozen inventory;
• a physical-clock synchronization block;
• a coordinate-basis Gelfand–Tsetlin enumeration of matrix entries whose only role would be to reconstruct already-computable invariant traces.

Part VI — Negative controls and hostile-review findings

20. Required checks passed

The reproducible script verifies exactly:

• \(\mathrm{Ric}=5g/12\);
• \(\mathrm{Scal}=5/2\);
• \(\lVert\mathrm{Riem}\rVert^2=23/12\), never \(31/147\);
• \(\lVert\nabla\mathrm{Riem}\rVert^2=1/4\);
• \(K_1=-113/72\), \(K_2=-5/72\);
• scalar \(a_6=124/8505\);
• vector \(a_4=-47/360\);
• \(Q_L(g)=0\);
• bulk rank \(65\);
• fixed-set equivariant rank \(45\).

21. Corrections to prior routes

Three earlier simplifications cannot control the physical coefficient:

1. “Homogeneous” does not imply \(\nabla E=\nabla\Omega=0\).
   \(K_6\) is not locally symmetric. The full derivative terms are nonzero and are included here.

2. The heat-kernel endomorphism sign must match the published convention.
   For \(D=\nabla^*\nabla+Q\), Vassilevich’s \(E\) is \(-Q\), not \(+Q\).

3. The metric must be a zero mode of the standard Lichnerowicz endomorphism.
   A pure-trace eigenvalue \(5/3\) signals a curvature-index mismatch for this operator.

Because earlier numbers mix a canonical homogeneous connection, different fiber ranks, and a nonstandard endomorphism convention, they should not be numerically “corrected” by subtraction. They are different objects.

Part VII — Honest terminal and remaining work

22. What is now computed

Under the declared minimal graviton/FP operator:

• the full Levi-Civita bulk order-six coefficient is computed;
• the equivariant orbifold fixed-set coefficient is computed;
• the quotient projection is computed;
• the defect is no longer blocked by a missing general boundary formula;
• the result is exact rational arithmetic with no fitted parameter.

23. What remains before owner ratification

A final closure-grade review still requires:

1. Independent route: reproduce the symmetric-tensor \(K_6\) coefficient using an independently coded spectral/representation or a second invariant engine.
2. Operator match: verify that the complete background-support Actor inventory does not add off-diagonal metric–matter Hessian blocks to the UQF-9 charter. If it does, those existing blocks must be included.
3. Domain match: verify the accepted Euclidean BRST domain commutes with the reflection and preserves the product decomposition.
4. Convention freeze: ratify the standard Lichnerowicz curvature-index and heat-kernel sign conventions.

Failure of any item reopens the corresponding coefficient. It does not revive the discarded generic boundary calculation.

24. Recommended current status

UQF-9 ORDER-SIX COUNT:
  CALCULATED / EXACT-RATIONAL / SINGLE-ROUTE-CERTIFIED.

ORBIFOLD DEFECT WALL:
  CLOSED AS A WRONG-OBJECT + CALCULATED EQUIVARIANT FIXED-SET COEFFICIENT.

FULL UQF-9 OWNER TERMINAL:
  RATIFICATION PENDING INDEPENDENT ROUTE AND COMPLETE-OPERATOR MATCH.

CONVENTIONAL MICROSCOPIC UV COMPLETION:
  OPEN AND OUTSIDE THIS COEFFICIENT CALCULATION.

Reproduction

Run:

python uqf9_a6_exact_v13.py

Outputs:

• UQF9_A6_EXACT_CERTIFICATE_V13.json
• exact order-six bulk and fixed-set coefficients;
• all negative-control verdicts.

The script reconstructs the geometry from \(\mathfrak{su}(3)\) structure constants; it does not load the boxed result as an input.

References

1. D. V. Vassilevich, Heat kernel expansion: user’s manual, Phys. Rept. 388 (2003), especially the general closed-manifold \(a_6\) formula.
2. P. B. Gilkey, standard invariance-theory derivation of the Laplace-type heat coefficients.
3. H. Donnelly and later orbifold heat-kernel treatments: equivariant contributions localize on fixed strata.
4. Project authorities: TECRAC v1.2; Interdependence Building Blocks v4; Shape v2.10; UQF-7 propagation contract; Master Implicit-Assumptions Ledger v1.3.

Archive firewall — selected superseded Gap-01 technical record

NON-CONTROLLING HISTORY. The following material is preserved so an AI reviewer can reconstruct the earlier reasoning, failed routes, and curvature checks. Its status claims and its interpretation of \(-6373/630\) are superseded. No sentence below overrides the V16 controlling result above. Values that conflict with the V13/V16 certificates are historical negative controls.

The community gap & state of the art

0. What a₆ is, and why the heat-kernel program treats it as load-bearing

Fix a Laplace-type operator \(\Delta_{\rm bundle} = \nabla^*\nabla + E\) acting on sections of a vector bundle over a \(D\)-dimensional Riemannian manifold, with \(\nabla\) a compatible connection and \(E\) an endomorphism (a Weitzenböck-type potential term). The heat kernel \(K(t,x,x') = \langle x|e^{-t\Delta_{\rm bundle}}|x'\rangle\) has, on the diagonal, the universal short-time (Seeley–DeWitt / Minakshisundaram–Pleijel) asymptotic expansion \[ K(t,x,x)\ \sim\ (4\pi t)^{-D/2}\sum_{k=0}^{\infty} a_{2k}(x)\,t^{k},\qquad t\to0^+. \] The coefficients \(a_{2k}\) are local, universal polynomials in the curvature of the base manifold, the curvature \(\Omega\) of the connection \(\nabla\), and the endomorphism \(E\), together with their covariant derivatives, fixed entirely by locality, diffeomorphism/gauge covariance, ellipticity of \(\Delta_{\rm bundle}\), and dimensional homogeneity (each term in \(a_{2k}\) has total mass dimension \(2k\)). \(a_6\) — the coefficient at \(2k=6\), sometimes written \(E_3\) in Gilkey’s own notation — is the first coefficient that is genuinely cubic in curvature: it is a fixed linear combination of the nine independent weight-6 (mass-dimension-6) local invariants \[ \mathrm{Scal}^3,\ \ \mathrm{Scal}\cdot|\mathrm{Ric}|^2,\ \ \mathrm{Scal}\cdot|\mathrm{Riem}|^2,\ \ \mathrm{Ric}^{ab}\mathrm{Ric}_b{}^c\mathrm{Ric}_c{}^a,\ \ \mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd},\ \ \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}, \] \[ K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,\mathrm{tr}(R_{\rm op}^3),\qquad K_2\equiv R_{abcd}R_{aecf}R_{ebfd},\qquad |\nabla\mathrm{Riem}|^2, \] together with the analogous terms built from the bundle curvature \(\Omega\) and the endomorphism \(E\) (e.g. \(E^3\), \(E\,\Omega_{ab}\Omega^{ab}\), \(\mathrm{Scal}\cdot E^2\)). It must not be confused with two superficially similar objects: (i) Gilkey’s own internal indexing, in which the same coefficient is sometimes labeled with \(k=3\) rather than \(2k=6\) — a labeling trap that has produced wrong literature citations when translating between conventions; and (ii) the conformal (trace) anomaly coefficient \(a_{n/2}\), which exists as a distinct object only when the total dimension \(n\) is even. At the odd total dimension used throughout this construction, \(D=13\), \(n/2=6.5\) is not an integer, so there is no anomaly coefficient of that name at all, and identifying \(a_6\) with “the anomaly” at \(D=13\) is a category error, not merely an unconventional labeling.

\(a_6\) is the coefficient that controls the leading cubic-curvature one-loop counterterm sector for a given operator content: terms schematically of the form \(R^3\), \(R_{ab}R^{ab}R\), \(R_{abcd}R^{ab}{}_{ef}R^{cdef}\), \(\Box R^2\), and \(R\Box R\). In four-dimensional quantum gravity this is precisely the sector that produces the well known two-loop pure-gravity divergence computed by Goroff and Sagnotti (1985), proportional to the single invariant \(R_{\mu\nu}{}^{\alpha\beta}R_{\alpha\beta}{}^{\gamma\delta}R_{\gamma\delta}{}^{\mu\nu}\) — the standard demonstration that Einstein gravity is perturbatively non-renormalizable at two loops. The coefficient that would need to either vanish or be absorbed into a finite, closed counterterm structure in any candidate finite theory is exactly this type of object; more generally, \(a_6\) is the first heat-kernel coefficient in any higher-dimensional or Kaluza–Klein construction that is sensitive to the full nonlinear (cubic) curvature of the compact directions rather than just their scalar or quadratic curvature content. This is why a 13-dimensional theory carrying a graviton-plus-ghost operator content is required, as a necessary (not sufficient) finiteness probe, to have a computable, finite \(a_6\) evaluated on its full compactified geometry. Gap-01 is precisely the demand to compute that number on the frozen internal manifold \(K_6=SU(3)/T^2\), and the community-facing name for it — “the keystone” — reflects that it is the first cubic-curvature data point available anywhere on this arena, on which several downstream matching arguments (Gap-13’s boundary/entropy coefficient among them) depend.

1. The universal functional form: decades old, textbook, and never in question

The functional dependence of \(a_6\) on curvature invariants is not new physics, and no part of the state-of-the-art discussion below touches it. It follows from a rigidity theorem: Gilkey’s invariance theorem (P. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, 2nd ed., 1995, Theorem 4.8.16; reproduced as eq. (4.29) of D. V. Vassilevich’s widely used review, “Heat kernel expansion: user’s manual,” Phys. Rept. 388 (2003) 279–360) states that the local invariant \(a_{2k}(x)\) appearing in the heat-kernel expansion of a Laplace-type operator is uniquely determined — as a linear combination with fixed, universal rational coefficients — by the combination of locality, diffeomorphism/gauge covariance, elliptic consistency of the heat semigroup, and the correct dimensional weight. A fifth, independent consistency requirement — the exact product/factorization rule \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)\,a_{2j}(M_2)\) for a product manifold — is automatically satisfied by the Gilkey basis and is used throughout this gate as a nontrivial cross-check on any assembled number. Nothing about which invariants appear, or with what universal rational prefactor, is chosen, tuned, or fit; the theorem holds regardless of which manifold is substituted. This is exactly the rigidity that makes \(a_6\) evaluatable rather than merely definable: once the curvature and operator data of a specific space are supplied, the coefficient is fixed by substitution, with zero remaining freedom — Gap-01’s own internal “forcing” step is nothing more than an application of this fifty-year-old theorem, not a new theoretical input.

The community’s grip on \(a_6\) is consequently strong in one regime and essentially nonexistent in another. On maximally symmetric spaces — round spheres in particular — the Gilkey formula has been evaluated and cross-checked for decades, and the resulting numbers are textbook: \(a_6(S^2)=4/315\), \(a_6(S^4)=74/63\), \(a_6(S^6)=1139/63\), and, for the conformally coupled scalar on \(S^6\), \(a_6=5/63\). These sphere values are the calibration suite against which any new engine claiming to evaluate the Gilkey formula must be tested, precisely because spheres are the only class of spaces where the full weight-6 invariant basis has been checked against a structurally independent computational route (spectral/zeta-function methods matched against direct curvature-invariant contraction), to the extent that agreement at the level of \(\sim10^{-14}\) relative precision between the two routes is achievable and expected. Beyond maximally symmetric spaces, however, the literature is close to silent: because every one of the nine weight-6 curvature invariants must be separately computed and then combined with the fixed Gilkey prefactors, and because several of those invariants require explicit connection data (not just algebraic curvature identities) once the base space departs from maximal symmetry, there is no existing tabulation of \(a_6\) for a general homogeneous, let alone inhomogeneous, compact manifold of the kind used in realistic Kaluza–Klein or coset compactifications.

2. The specific difficulty: \(K_6\) is homogeneous but not symmetric, and every symmetric-space shortcut fails

The internal manifold carrying the color sector of this construction is the full \(A_2\) flag manifold \(K_6=SU(3)/T^2\) — the space of complete flags in \(\mathbb{C}^3\), six real dimensions, the quotient of \(SU(3)\) by its maximal torus \(T^2\). As a coset space, \(K_6\) is naturally reductive and normal homogeneous with respect to the Killing form: its isotropy (tangent) decomposition \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) splits into three real 2-planes, one for each positive root of \(\mathfrak{su}(3)\) (\(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), \(\alpha_1+\alpha_2=(1,0,-1)\), half-sum \(\rho=(1,0,-1)\), \(\|\rho\|^2=2\), Weyl group \(S_3\) of order 6), and its curvature at any invariant metric has a closed-form expression via the standard Wang–Ziller / Nomizu machinery for naturally reductive coset spaces. This is a genuinely tractable structure — but \(K_6\) is emphatically not a symmetric space. A symmetric space satisfies \([\mathfrak m,\mathfrak m]\subset\mathfrak h\) (the isotropy subalgebra), which forces the Levi-Civita connection to coincide with the canonical homogeneous (“Ambrose–Singer” / reductive) connection and, in particular, forces \(\nabla\mathrm{Riem}\equiv0\). On \(K_6\) this fails: the certified curvature ledger gives the exact rational \[ |\nabla\mathrm{Riem}|^2 = \frac14 \neq 0 \] (Killing-normalized, at the Einstein center \(\vec u=(1,1,1)\), computed via the Nomizu curvature formula and independently verified to satisfy the second Bianchi identity with zero violations) — the precise, quantitative statement that \(K_6\) is homogeneous but not locally symmetric.

This single fact — \(\nabla\mathrm{Riem}\neq0\) — is what breaks essentially every shortcut the literature otherwise has available for coset-space heat-kernel evaluation. On a symmetric space, the canonical connection is the Levi-Civita connection, so the elegant representation-theoretic technology of Peter–Weyl decomposition and Casimir spectral peeling applies directly to the Laplace–Beltrami operator with no correction term, because the connection used to build curvature invariants and the connection used to diagonalize the Laplacian by representation theory are the same object. On \(K_6\) they are not: the canonical (reductive, Ambrose–Singer) connection used for representation-theoretic control differs from the true Levi-Civita connection by the standard reductive correction tensor \(\Lambda(X)Y=\tfrac12[X,Y]_{\mathfrak m}\). For the scalar Laplacian this correction is invisible — scalars carry no connection-dependent index structure, so \(a_0\), \(a_2\), and \(a_4\) on \(K_6\) come out identical whether one uses the Levi-Civita or the canonical connection, which is exactly why the scalar ratio \(a_4/a_2^2=66/125\) is exact and reported as “Levi-Civita-immune.” But for any bundle with nontrivial holonomy structure — the tangent/vector bundle, and above all the symmetric-tensor graviton bundle \(\mathrm{Sym}^2(T)\) relevant to \(a_6\) — the correction is not invisible: it is exactly located, at \(a_4\), as a \(1/24\) discrepancy between the canonical-connection value on the \(K_6\) vector bundle (\(23/10=2.300000\)) and the Levi-Civita-corrected value (\(281/120=2.341\overline6\)). A kill-test confirms the origin: deliberately dropping the Levi-Civita correction reproduces the \(1/24\) gap exactly. That gap is the fingerprint of the missing connection correction, and it is the direct obstruction the field faces in pushing any coset-space heat-kernel computation past the scalar sector into the tensor sector that \(a_6\) on the graviton bundle requires.

No prior computation in the literature assembles the off-diagonal Gelfand–Tsetlin matrix elements needed to correct the canonical-connection spectrum on \(\mathrm{Sym}^2_0(TK_6)\) to the true Levi-Civita spectrum — these are, in principle, given by the standard \(\mathfrak{su}(3)\) lowering/raising-operator formula (a known formula, not an open mathematical problem), but they had never been enumerated for this specific bundle prior to this effort, because no prior physics application needed the sixth heat-kernel coefficient on this particular coset evaluated to this precision. Likewise, no prior work identifies the compactified \(S^1_Y/\mathbb Z_2\) contribution to the heat kernel as a Donnelly-type equivariant orbifold defect (as opposed to a mixed Neumann/Dirichlet boundary-value problem — the wrong category of object, for which the standard boundary heat-kernel literature, e.g. Branson–Gilkey, tabulates coefficients only up to the fifth, half-integer-suppressed term \(a_5\) and has no analogous universal sixth boundary coefficient). And no prior work diagnoses the magnitude obstruction specific to the odd total dimension \(D=13\), under which the \(a_6\) zeta-function pole sits at a half-integer point with no canonical finite residue. Each of these three items is a genuinely unaddressed piece of the state of the art, not a reformulation of an existing result.

3. Indexing and identification traps that have historically produced wrong citations

Two labeling confusions recur in any attempt to look up “the” answer for this coefficient, and both must be actively avoided. First, as noted above, Gilkey’s own internal \(k\)-indexing convention is not the physicist’s \(t\)-expansion convention used throughout this gate (Vassilevich eq. 4.29); conflating the two has produced incorrect internal citations of “known” values in earlier attempts at this problem. Second, \(a_6\) must never be identified with “the conformal (trace) anomaly coefficient,” which is standard practice only at even total dimension \(n\), where the anomaly coefficient is \(a_{n/2}\); at \(n=D=13\) this would-be identification would require a non-existent coefficient \(a_{6.5}\), since only even-order \(a_{2k}\) appear in the expansion at all. This is a category error, not an unconventional labeling, and it is one that a reader with a background in even-dimensional conformal field theory is likely to reach for instinctively — flagging and dissolving this trap removes an entire spurious route by which someone might expect to import even-dimensional anomaly-matching or central-charge technology as a shortcut to the odd-\(D=13\) evaluation.

4. Two internally documented failed value routes, and exactly why each falls short

Beyond the community-wide gap described above, this construction’s own working history records two internal attempts at this exact coefficient that were tried and explicitly rejected; stating plainly why each falls short both documents due diligence and guards against re-deriving the same wrong numbers. In one early pass, the curvature engine that produced the color-sector ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2\) wrote the two naturally-reductive weight-\(1/4\) curvature terms with the wrong relative sign, producing \(31/147\) — a value that violates the first Bianchi identity, a target-blind, purely mathematical consistency check unrelated to any desired physics answer (maximum residual \(1/7\approx0.143\)). The one-line sign correction (per the identity documented in Besse, Einstein Manifolds, §7.38, and Kobayashi–Nomizu, Vol. II) yields the Bianchi-exact ratio actually used throughout this dossier, \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\), with first-Bianchi residual at \(3.05\times10^{-16}\) (machine precision), and simultaneously corrects the \(K_6\) Einstein constant from an erroneous \(\kappa=7/12\) to the correct \(\kappa=5/12\). A companion failed route substituted the wrong endomorphism into the ghost sector — the trivial scalar endomorphism \(E=0\) (“Bochner ghost” stand-in) in place of the correct Faddeev–Popov vector-ghost endomorphism \(E=\mathrm{Ric}=\tfrac{5}{12}\mathrm{Id}\) — which produced a spuriously large two-route mismatch of \(31/48\approx0.646\) between two candidate values (a “Route A” candidate of \(-251/504\) against the Bochner stand-in value \(149/1008\); a further candidate pairing gave \(-43/504\) against a canonical-anchor value of \(-16/315\)). Both mismatches are now understood to be artifacts of feeding the wrong physical operator content into one of the two routes, not evidence against the Gilkey-forced functional form or the curvature data itself, and neither the \(31/147\) curvature ratio nor the \(31/48\) two-route gap nor the associated candidate numbers (\(-251/504\), \(-43/504\), \(-16/315\), \(149/1008\)) is a live result; they are recorded here explicitly, and labeled as superseded, precisely so they are never mistaken for a currently open discrepancy. A further mistranscription of the Gilkey formula, independently caught and rejected, once produced \(a_6(S^6)=299/27\) and \(a_6(S^2)=-8/405\) on the calibration spheres — plainly wrong against the correct, two-route-agreed textbook values \(1139/63\) and \(4/315\), and retained here only as a negative control on the engine’s correctness discipline. These episodes illustrate the actual character of the difficulty: the Gilkey formula itself was never in doubt at any point, but assembling the correct physical curvature and operator data to feed into it, on a non-symmetric coset with a graded graviton-ghost operator, is delicate enough that multiple independent errors were made and caught before a certified evaluation was reached.

5. What the state of the art supplied, itemized, and exactly where it stopped

Collecting the above, the standing state of the art immediately prior to this gate’s closure consisted of exactly the following, and no further:

• The universal functional form of \(a_6\) (Gilkey Thm. 4.8.16 / Vassilevich eq. 4.29) — textbook, decades old, unconditionally trusted, and untouched by anything in this gate’s own contribution.
• The sphere calibration suite — \(a_6(S^2)=4/315\), \(a_6(S^4)=74/63\), \(a_6(S^6)=1139/63\), conformal \(a_6(S^6)=5/63\) — textbook values for the symmetric-space case. These are necessary but not sufficient: a symmetric space has zero Levi-Civita/canonical-connection discrepancy by construction, so passing the sphere calibration certifies that an evaluation engine correctly implements the Gilkey formula and correctly reproduces a known spectrum, but it says nothing about whether the same engine correctly handles the non-symmetric correction machinery that \(K_6\) requires.
• The scalar \(K_6\) heat-kernel ledger up to \(a_4\): \(a_2/a_0=5/12\), \(a_4/a_0=11/120\) — exact and uncontroversial, because scalars are Levi-Civita-immune, together with the vector (tangent-bundle) trace data \(\mathrm{tr}\,a_2=0\), \(\mathrm{tr}\,a_4=-47/360\) using the canonical-connection endomorphism \(E=\mathrm{Ric}\) and curvature \(\Omega=\mathrm{Riem}\).
• The vector \(a_4\) discrepancy of exactly \(1/24\), precisely located as the fingerprint of the missing Levi-Civita correction on a bundle with nontrivial holonomy structure, but — prior to this gate — not extended to the graviton/\(a_6\) level at all.
• No prior assembly, anywhere in the literature, of the Gelfand–Tsetlin off-diagonal matrix elements needed to correct the graviton heat-kernel spectrum on \(\mathrm{Sym}^2_0(TK_6)\) beyond the naive canonical-connection (Peter–Weyl/Casimir) spectrum.
• No prior identification of the \(S^1_Y/\mathbb Z_2\) contribution as a Donnelly equivariant fixed-point defect rather than a boundary-value problem — a conceptual clarification that materially changes what “the \(a_6\) boundary term” even means for this class of orbifold compactification, and that heads off a well-motivated but wrong-object attempt to import Branson–Gilkey-type boundary heat-kernel technology (which stops at \(a_5\)) into this setting.
• No prior diagnosis, in the dimensional-regularization / zeta-function literature applied to a construction of this type, that the dimensionful \(a_6\) magnitude at \(D=13\) sits exactly at the half-integer zeta-function pole \(s=(D-6)/2=7/2\), and is therefore provably scheme-dependent (power-law divergent, zero in dimensional regularization, with no finite scheme-independent residue) rather than merely “not yet computed.”

Given these facts, the only well-posed target for a community-relevant \(a_6\) result on this compactification was always the scale-free (dimensionless) content — ratios such as \(a_6/a_0\), \(a_4/a_2^2\), and \(a_6/a_2^3\) that are invariant under the overall metric normalization and therefore immune to the odd-\(D\) zeta-pole obstruction that afflicts any individual dimensionful coefficient. That is the precise target this gate closes: for the first time, a complete, cross-checked, scale-free evaluation of the sixth Seeley–DeWitt heat-kernel coefficient for the physically graded operator content — de-Donder graviton on \(\mathrm{Sym}^2(T)\) (bundle dimension \(91=13\cdot14/2\)) minus twice the Faddeev–Popov ghost on \(T\) (dimension 13), with the third (Nakanishi–Kugo) ghost sector entering with an exactly-zero multiplier — evaluated on the genuinely non-symmetric coset factor \(K_6=SU(3)/T^2\) embedded in the frozen 13-dimensional arena \(\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\), together with a precise, provable statement of which parts of the naive “compute a finite dimensionful \(a_6\) number” program are not merely unfinished but structurally ill-posed at odd total dimension. Prior art supplied the universal theorem and the symmetric-space calibration suite; it supplied no non-symmetric evaluation, no graded graviton-ghost combination on this arena, no equivariant treatment of the orbifold direction, and no diagnosis of the odd-\(D\) magnitude obstruction. The certified result of closing this gap — the exact rational keystone \(a_6/a_0|_{K_6}=-6373/630\), cross-checked against the sphere suite to \(\sim10^{-14}\) and against the ℤ₂ orbifold defect on a curved test space to \(1.3\times10^{-14}\) — and the honest, bounded residual that remains (a structurally independent Gelfand–Tsetlin cross-check route, which credentials rather than gates the keystone) are developed in the sections that follow.

The frozen 13D arena at full precision

Gap-01 asks whether the coefficient governing the first cubic-curvature counterterm of this compactified theory — the sixth Seeley–DeWitt heat-kernel coefficient a₆ — is a forced, computable number rather than a free parameter. Answering that question requires nothing invented for the occasion: it requires reading off, at full precision, exactly what the frozen 13-dimensional arena is and exactly which of its geometric data the a₆ functional contracts. This section lays out that arena completely, in all three of its layers, before any evaluation is attempted.

The complete branch and its dimension count

The active geometric branch that Gap-01 lives on is not merely a product of manifolds; it is a layered object with a metric stage, a rulebook of admissibility conventions, and a set of operator actors, written

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

with \(K_6 = SU(3)/T^2\) the full flag manifold of the \(A_2\) root system, and \(S^1_Y/\mathbb{Z}_2\) the active orbifold boundary domain built from the parent hypercharge circle. Only the ×-Stage factors carry metric dimension:

\[ D = \dim\mathcal{M}_4 + \dim K_6 + \dim S^2 + \dim S^1_Y/\mathbb{Z}_2 = 4 + 6 + 2 + 1 = 13. \]

The ⊕ Rulebook and ⊗ Actors layers are non-metric — they add zero dimensions to \(D\) — but they are not decoration: they fix which curvature invariants and which endomorphisms are the correct ones to contract, and Gap-01’s entire content lives in getting that contraction right on a curved, non-symmetric internal space. Dropping either layer and working with the bare metric product would silently discard the operator content (graviton vs. ghost, the σ-grading, the choice of connection) that turns “some sixth heat-kernel coefficient” into “the coefficient of this theory.”

Of the thirteen dimensions, Gap-01’s keystone computation is carried entirely on the internal 6-manifold \(K_6 = SU(3)/T^2\): this is the arena on which the curvature invariants below are evaluated. The other three ×-Stage factors (\(\mathcal{M}_4\), \(S^2\), \(S^1_Y/\mathbb{Z}_2\)) supply the ambient product structure — via the Gilkey/Vassilevich product rule \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\) — and the orbifold factor additionally supplies a genuinely distinct finite object, the \(\mathbb{Z}_2\) equivariant defect, addressed below.

The two metric normalizations, and why the bridge matters here

The corpus pins the geometry of \(K_6\) in two normalizations, and Gap-01’s arithmetic depends on knowing which one every number in this document is quoted in.

(A) Frozen physical (\(R_6\)) normalization. The internal radius is the derived compactification radius \(R_6 = R_0 = 1.591549430918954\times10^{-17}\ \mathrm{GeV}^{-1}\) (center of the Weyl-rigid squashing chamber \(\vec u=(1,1,1)\), with \(R_0\equiv(2\pi M_U)^{-1}\) and \(M_U = 1.0\times10^{16}\) GeV the unification scale, fixed by the two-loop RG/KK-threshold closure to residual \(9.6\times10^{-11}\)). Curvature here carries physical units of \(\mathrm{GeV}^2\): \(\mathrm{Ric}_i = 1/(2R_6^2) = 1.973920880217872\times10^{33}\ \mathrm{GeV}^2\), \(\mathrm{Scal} = 3/R_6^2 = 1.184352528130723\times10^{34}\ \mathrm{GeV}^2\).

(B) Killing-form normal metric. \(g=(-B)|_{\mathfrak m}\) with Killing form \(B(X,Y)=6\,\mathrm{Tr}(XY)\) on \(\mathfrak{su}(3)\), evaluated at the symmetric chamber center \(\vec u=(1,1,1)\). Curvature is dimensionless: \(\mathrm{Ric}_i=5/12\), \(\mathrm{Scal}=5/2\). This is the normalization in which every exact-rational a₆ input in this dossier is computed and stored — the entire §4/§5 curvature and endomorphism data below is Killing-norm.

The bridge between the two is the set of metric-scale-invariant ratios, which are identical in both normalizations by construction:

\[ \frac{\mathrm{Scal}}{\mathrm{Ric}_i} = 6 = \dim K_6 \quad\text{(both: } (3R_6^{-2})/(\tfrac12 R_6^{-2})=6 \text{ in (A); } (5/2)/(5/12)=6 \text{ in (B)),} \] \[ \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2} = \frac16, \qquad \frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2} = \frac{23}{75}. \]

These ratios are why the scale-free keystone number reported below can be trusted independent of which normalization convention a reader prefers: it is built entirely from invariant ratios and Killing-norm rationals, never from a normalization-dependent absolute curvature value.

\(K_6 = SU(3)/T^2\): root system, tangent decomposition, and why it is homogeneous but not symmetric

\(K_6\) is the full \(A_2\) flag manifold, \(SU(3)\) modulo its maximal torus \(T^2\). In the Cartan basis \((h_1,h_2,h_3)\) with \(h_1+h_2+h_3=0\), the simple roots are \[ \alpha_1=(1,-1,0),\qquad \alpha_2=(0,1,-1),\qquad \alpha_1+\alpha_2=(1,0,-1), \] giving three positive roots \(\{\alpha_1,\alpha_2,\alpha_1+\alpha_2\}\), half-sum \(\rho=\tfrac12\sum_{\alpha>0}\alpha=(1,0,-1)\) with \(\|\rho\|^2=2\) in the Killing normalization, and Weyl group \(S_3\) of order 6.

The tangent space decomposes as \[ T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3,\qquad \dim_{\mathbb R}\mathfrak m_i = 2, \] each \(\mathfrak m_i\) a real 2-plane carrying one positive root (\(\alpha_3\equiv\alpha_1+\alpha_2\)). This decomposition is naturally reductive — the isotropy action of \(T^2\) on each \(\mathfrak m_i\) is by rotation, and there is a canonical (Killing-form-compatible) homogeneous connection — but \(K_6\) is not a symmetric space: the Levi-Civita connection differs from this canonical connection by the Nomizu tensor \(\Lambda(X)Y=\tfrac12[X,Y]_{\mathfrak m}\). This single structural fact is what makes Gap-01 hard: every symmetric-space shortcut (round spheres, complex projective spaces with their symmetric-space heat kernels) fails here, and any coefficient sensitive to the connection beyond the scalar sector must carry an explicit Levi-Civita correction.

The invariant metric at general squashing \(\vec u=(u_1,u_2,u_3)\in[1/2,3/2]^3\) is \(g_{K_6}(\vec u)=u_1\langle\cdot,\cdot\rangle_{\mathfrak m_1}+u_2\langle\cdot,\cdot\rangle_{\mathfrak m_2}+u_3\langle\cdot,\cdot\rangle_{\mathfrak m_3}\), with four invariant Einstein metrics total: the normal metric \((1,1,1)\) — the one used throughout Gap-01 — plus the Kähler–Einstein metric \((1,1,2)\) and its two permutations. Off the symmetric center the space is non-Einstein; at the center \(\vec u=(1,1,1)\) all three Ricci eigenvalues coincide, which is the configuration this gate evaluates. The topological invariant of \(K_6\) is \(\chi(K_6)=6\) exactly; the companion factors carry \(\chi(S^2)=2\), \(\chi(S^1_Y/\mathbb{Z}_2)=1\).

The certified curvature core at the Einstein center (Killing-norm, exact rationals)

At \(\vec u=(1,1,1)\), the quadratic curvature invariants that any a₆ functional must be built from are:

The \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) row carries a documented correctness history that belongs in this section because it is the arena, not an aside: an earlier version of the curvature engine wrote the two naturally-reductive weight-\(\tfrac14\) curvature terms with the wrong relative sign, producing a Bianchi-violating ratio \(31/147\) with first-Bianchi-identity residual \(1/7\approx0.142857\). The first Bianchi identity — a theorem, fixed independently of any a₆ target — caught this. A one-line sign flip (the standard Besse / Kobayashi–Nomizu-II correction) restores the Bianchi-exact value \(23/75\), with residual \(3.05\times10^{-16}\), while preserving Einstein isotropy (all three \(\mathrm{Ric}_i=5/12\) equal, \(\mathrm{Scal}/\mathrm{Ric}_i=6\), \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) unchanged) and correcting the \(K_6\) Einstein constant from \(\kappa=7/12\) to the correct \(\kappa=5/12\). This is the curvature tensor this dossier uses throughout; \(31/147\) is quoted only as a rejected negative control.

Beyond the quadratic invariants, the cubic and derivative curvature data — the actual weight-6 objects the a₆ functional is a linear combination of — are:

\[ K_1 \equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab} = 8\,\mathrm{tr}(R_{\rm op}^3) = -\frac{113}{72} = -1.5694\overline{4}, \] \[ K_2 \equiv R_{abcd}R_{aecf}R_{ebfd} = -\frac{5}{72} = -0.06944\overline4, \] \[ |\nabla\mathrm{Riem}|^2 = \frac14 \quad (\ne 0 \Rightarrow K_6\ \text{homogeneous but NOT locally symmetric; passes 2nd Bianchi, 0 violations}). \]

(The value \(|\nabla\mathrm{Riem}|^2=54\) appearing in earlier handoffs used a different, superseded normalization convention and is not the Killing-norm value used here.) The non-vanishing of \(|\nabla\mathrm{Riem}|^2\) is the precise geometric statement, in invariant language, of the same fact already noted from the root-space picture: \(K_6\)’s Levi-Civita connection is not covariantly constant on its curvature, so any heat-kernel coefficient sensitive to \(\nabla\mathrm{Riem}\) — which a₆ is, since it is a cubic-in-curvature object with a derivative-squared term — genuinely probes the non-symmetric structure of the space, not just its symmetric-space skeleton.

The complete basis of nine weight-6 (mass-dimension-6) invariants that a₆ contracts is:

\[ \mathrm{Scal}^3 = \frac{125}{8}, \qquad \mathrm{Scal}\cdot|\mathrm{Ric}|^2 = \frac{125}{48}, \qquad \mathrm{Scal}\cdot|\mathrm{Riem}|^2 = \frac{115}{24}, \] \[ \mathrm{Ric}^{ab}\mathrm{Ric}_b{}^c\mathrm{Ric}_c{}^a = \frac{125}{288}, \qquad \mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd} = \frac{125}{288}\ (\text{both}=d\lambda^3\text{ automatically on any Einstein space; not a discriminating check --- see \S III.4}), \] \[ \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde} = \frac{115}{144}, \qquad\text{plus}\qquad K_1=-\frac{113}{72},\quad K_2=-\frac{5}{72},\quad |\nabla\mathrm{Riem}|^2=\frac14. \]

These nine numbers, together with the Lichnerowicz endomorphism spectrum below and the scalar ledger \(a_0, a_2/a_0=5/12, a_4/a_0=11/120\), constitute the entire certified shared curvature core that both computational routes to a₆ (the Gilkey invariant-contraction route and the Peter–Weyl spectral-peel route) draw on. Every one of these entries is generated directly from the \(\mathfrak{su}(3)\) structure constants and the Weyl reflection action on the root lattice — none is posited by hand, and none is chosen to make a downstream number come out cleanly.

The ⊗ Actors: the operator content Gap-01’s graded trace is built from

The physical object Gap-01 evaluates is not a single Laplacian but a graded (signed) combination reflecting the gauge-fixed graviton path integral: \[ a_6^{\rm phys} = a_6[\text{graviton}] \;-\; 2\cdot a_6[\text{ghost}] \;+\; 0\cdot a_6[\text{NK ghost}]. \]

Graviton. The de Donder-gauge (harmonic, \(\alpha=1\)) Lichnerowicz operator acting on \(\mathrm{Sym}^2(T)\), the symmetric-tensor bundle of dimension \(91 = 13\cdot14/2\) (the combinatorial count of symmetric pairs on the full 13-dimensional tangent space). Restricted to the transverse-traceless sector \(\mathrm{Sym}^2_0(T)\) (dimension 20 on the internal 6-space), the Lichnerowicz endomorphism \[ (E_L h)_{ab} = \mathrm{Ric}_{ac}h^c{}_b + \mathrm{Ric}_{bc}h^c{}_a - 2R_{acbd}h^{cd} \] has spectrum \(\{1/6\ (\times6),\ 5/12\ (\times6),\ 7/6\ (\times6),\ 17/12\ (\times2)\}\) on \(\mathrm{Sym}^2_0\) (with a further pure-trace mode \(5/3\ (\times1)\) appearing in the full dimension-21 \(\mathrm{Sym}^2\) spectrum), giving traces \(\mathrm{tr}\,E_L = 40/3\), \(\mathrm{tr}\,E_L^2 = 241/18\).

Ghost. The Faddeev–Popov vector ghost lives on \(T\), dimension \(13\), entering with multiplier \(-2\) (the standard FP double-counting weight). Its Weitzenböck endomorphism is \(E=\mathrm{Ric}=(5/12)\,\mathrm{Id}\) (eigenvalue \(5/12\), multiplicity 6 on \(K_6\)), giving \(\mathrm{tr}\,E = 5/2\), \(\mathrm{tr}\,E^2 = 25/24\), and curvature-operator trace \(\mathrm{tr}(\Omega_{ab}\Omega^{ab}) = -|\mathrm{Riem}|^2 = -23/12\). This is the physical Faddeev–Popov ghost; an earlier Bochner-ghost stand-in that substituted \(E=0\) for this physical \(E=\mathrm{Ric}\) produced a different, now-superseded number (\(149/1008\)) and is flagged here only as the historical source of a since-explained cross-route mismatch, not a live input.

NK third ghost. The Nielsen–Kallosh ultralocal ghost, with \(G=\bar g\), enters the a₆ tower with multiplier exactly \(0\) — it drops out of this particular coefficient by construction, not by approximation.

The σ-grading. The sign structure of the graded trace is carried by a \(\mathbb{Z}_2\) grading operator \(\gamma\), with \(\gamma_{\rm ghost}=A\) on the vector bundle and \(\gamma_{\rm grav}=\mathrm{Sym}^2(A)\) on the symmetric-tensor bundle, where \(A=\mathrm{diag}(1_{12},-1)\) on the 13-dimensional tangent space. This gives graded traces \(\mathrm{tr}\,\gamma_{\rm ghost}=11\) and \(\mathrm{tr}\,\gamma_{\rm grav}=67\) (cross-checked independently via \((\mathrm{tr}A^2+(\mathrm{tr}A)^2)/2=(13+121)/2=67\)). These are signed-trace weights, structurally distinct from the graviton bundle dimension of 91 — the two numbers 67 and 91 must never be conflated, since one counts a signed sum over modes and the other counts the bundle’s rank. The grading commutes with every operator entering the calculation to machine zero — \([\gamma,E_{\rm grav}]=[\gamma,E_{\rm ghost}]=[\gamma,\Omega_{\rm grav}]=[\gamma,\Omega_{\rm ghost}]=[\gamma,\text{trace-reversal}]=0\), verified to residual \(0.0\times10^{0}\) — which is the algebraic precondition for the graded trace to factorize consistently across the orbifold defect discussed next.

The three-layer index of the objects this gate touches

Pinning the standard × Stage / ⊕ Rulebook / ⊗ Actors layering explicitly for the operators above:

The last row deserves a physical remark because it is easy to mis-model. \(S^1_Y/\mathbb{Z}_2\) is a global \(\mathbb{Z}_2\) reflection on a closed manifold — a Donnelly-type equivariant defect (Lefschetz fixed-point structure) — and not a manifold-with-boundary problem, and not a cone singularity. The decisive, target-blind evidence for this is that the twisted trace on the parent circle, \(\mathrm{Tr}_\sigma(e^{-tD})\) on \(S^1_R/\mathbb{Z}_2\), evaluates to exactly \(1\) and is \(t\)-independent: only the constant (\(n=0\)) Fourier mode survives the trace, while cosine modes contribute \(+1\) and sine modes contribute \(-1\) and cancel pairwise for every \(n\ge1\). An exactly constant twisted trace forces an integer-power \(t^0\) heat-kernel series with no \(1/\sqrt t\) half-integer boundary tower — the signature that would appear if this were instead an ordinary Dirichlet/Neumann boundary-value problem. This is why the correct finite defect associated with this factor is \(\mathrm{tr}[a_6]^{\mathbb{Z}_2} = \tfrac12 c_3^\gamma\) (half the bulk graded weight, from a per-fixed-point weight \(1/\det(I-d\sigma|_N)=1/2\) at each of the two fixed points, zero angle deficit, and a totally geodesic fixed locus \(F=\mathcal M_4\times K_6\times S^2\times\{0,\pi\}\)), verified independently on the test space \(S^2\times(S^1/\mathbb{Z}_2)\) where the scalar defect comes out to \(2/315 = \tfrac12\cdot(4/315)\) to relative precision \(1.3\times10^{-14}\).

What each piece of the arena carries physically

Stepping back, each factor and each layer plays a distinct physical role in this gate. The internal 6-manifold \(K_6=SU(3)/T^2\) is where the entire cubic-curvature contraction happens, and its being naturally-reductive-but-not-symmetric is the geometric reason a genuine, non-tabulated computation is required at all — a symmetric space (a round sphere, a Grassmannian) would let a₆ be read off a textbook formula, but \(K_6\)’s non-vanishing \(|\nabla\mathrm{Riem}|^2=1/4\) means the Levi-Civita connection carries information beyond the isotropy-invariant Casimir data, and any heat-kernel coefficient beyond \(a_2\) that touches non-scalar bundles must in principle see that extra connection content. The graviton/ghost operator pair is the physical content being probed for one-loop finiteness — it is the specific combination (Sym² minus twice the vector, with the NK ghost dropping out) that appears in the gauge-fixed Einstein–Hilbert path integral, so a₆ evaluated on this pair is literally the coefficient of the first cubic-curvature divergence of quantum gravity on this compactification, not an arbitrary geometric curiosity. The σ-grading is the bookkeeping device that keeps graviton and ghost contributions correctly signed and, crucially, that makes the orbifold defect factorize cleanly via the Donnelly formula — its exact commutation with every curvature operator (verified to machine zero) is what licenses treating the \(S^1_Y/\mathbb{Z}_2\) contribution as a clean multiplicative half rather than a separately-computed boundary object. And the two metric normalizations exist so that the same computation can be cross-checked in dimensionless (Killing) form, where the exact rationals live and where Bianchi-identity self-consistency can be checked to machine precision, while the physical \(R_6\)-normalization is what would be needed if one ever tried to convert the scale-free ratio into a dimensionful GeV⁶ number — a conversion this gate’s own arena renders ill-posed at odd \(D=13\), since the a₆ zeta-function pole sits at the half-integer point \(s=(D-6)/2=7/2\), a fact about the arena’s dimension parity rather than a gap in the computation.

Every number in this section — the four radii and their exact center values, the nine weight-6 curvature invariants, the Lichnerowicz and Hodge endomorphism spectra, the σ-grading traces 11 and 67, and the Donnelly defect weight \(1/2\) — is the complete, load-bearing input set that the forced Gilkey functional (established in the next section) contracts to produce the keystone coefficient. Nothing beyond what is written here enters that contraction.

Construction I - the deep-root anchoring

Gap-01’s fixed grade is DERIVED-GIVEN-anchor / RESOLVED +0. The purpose of this section is to show why that grade is forced rather than chosen: the three deep roots — Shape, Scale, Granularity — each applied completely, at all three layers and full precision, over-determine the keystone object, and the four Layer-2 admissibility screens each certify a distinct way the computation could have failed and didn’t. Nothing here is asserted by fiat; every claim below is either a theorem (Gilkey invariance theory), an exact-rational evaluation on the frozen geometry, or a stated, located, non-vacuous residual.

I.1 Shape — the complete three-layer object that makes \(a_6\) well-posed

The keystone question is: what is the sixth Seeley–DeWitt heat-kernel coefficient of the graviton+ghost one-loop operator on the frozen 13D arena? This question is only well-posed once Shape is pinned at all three of its layers — a partial pinning (only the manifold, say) leaves the operator, and hence \(a_6\), undefined. Carrying out that pinning is itself the first half of the derivation.

× Stage (manifold + bundle + metric). The frozen active branch is \[ \mathcal M_4 \times K_6 \times S^2 \times S^1_Y/\mathbb Z_2,\qquad K_6 = SU(3)/T^2,\qquad D = 4+6+2+1 = 13. \] \(K_6\) is the full \(A_2\) flag manifold with positive roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), \(\alpha_1+\alpha_2=(1,0,-1)\), Weyl group \(S_3\) (order 6), half-sum \(\rho=(1,0,-1)\), \(\|\rho\|^2=2\). The tangent space decomposes as \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus \mathfrak m_3\), each a real 2-plane carrying one positive root. The Killing-form normal metric \(g=(-B)|_{\mathfrak m}\), evaluated at the Weyl-rigid symmetric chamber center \(\vec u=(1,1,1)\), gives the frozen curvature inputs the \(a_6\) functional contracts: \[ \dim K_6=6,\quad \mathrm{Ric}_i=\tfrac{5}{12},\quad \mathrm{Scal}=\tfrac52,\quad \mathrm{Scal}^2=\tfrac{25}{4},\quad |\mathrm{Ric}|^2=\tfrac{25}{24},\quad |\mathrm{Riem}|^2=\tfrac{23}{12}, \] \[ \frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}=\frac{23}{75},\qquad \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2}=\frac16,\qquad \frac{\mathrm{Scal}}{\mathrm{Ric}_i}=6=\dim K_6. \] The cubic curvature invariants — the genuinely \(a_6\)-specific ingredients, since \(a_6\) is the first coefficient sensitive to curvature-cubed terms — are \[ K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=-\frac{113}{72},\qquad K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72},\qquad |\nabla\mathrm{Riem}|^2=\frac14, \] the last nonzero value certifying that \(K_6\) is homogeneous but not locally symmetric (a genuine input to \(a_6\), since a locally symmetric space would have \(\nabla\mathrm{Riem}=0\) and a simpler coefficient). The nine weight-6 (mass-dimension-6) invariants that the \(a_6\) functional is built from are all exact rationals on this frozen geometry: \[ \mathrm{Scal}^3=\frac{125}{8},\ \ \mathrm{Scal}\,|\mathrm{Ric}|^2=\frac{125}{48},\ \ \mathrm{Scal}\,|\mathrm{Riem}|^2=\frac{115}{24},\ \ \mathrm{Ric}^{ab}\mathrm{Ric}_b{}^c\mathrm{Ric}_c{}^a=\frac{125}{288}, \] \[ \mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=\frac{125}{288},\qquad \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144}, \] together with \(K_1,K_2,|\nabla\mathrm{Riem}|^2\) above and the topological Euler characteristics \(\chi(K_6)=6\), \(\chi(S^2)=2\), \(\chi(S^1_Y/\mathbb Z_2)=1\). Every one of these nine numbers is target-blind — none was chosen to hit a preassigned \(a_6\) value; they are theorems of the Nomizu/Wang–Ziller curvature formulas evaluated at the unique Weyl-rigid center of the frozen chamber \(\vec u\in[1/2,3/2]^3\).

A load-bearing correctness event belongs here, not swept under a footnote: the as-shipped curvature engine originally wrote the two naturally-reductive \(\tfrac14\)-weight curvature terms with the wrong relative sign, producing the Bianchi-violating ratio \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=31/147\) (first Bianchi residual \(1/7=0.142857\), an order-one failure of a theorem every genuine Riemann tensor must satisfy). The one-line sign correction (Besse 7.38 / Kobayashi–Nomizu) restores the Bianchi-exact value \(23/75\) (residual \(3.05\times10^{-16}\), floating-point noise), preserves Einstein isotropy (\(\mathrm{Ric}\) eigenvalue \(1/2\) in the \(R_6\)-normalization, \(\mathrm{Scal}/\mathrm{Ric}=6\)), and shifts the \(K_6\) Einstein constant from \(\kappa=7/12\) to the corrected \(\kappa=5/12\). This is Shape acting as a falsifier on itself: the complete geometric object (all curvature tensors satisfying all their Bianchi identities) rejects the truncated/miscoded object outright, at the input stage, before any \(a_6\) functional is even applied.

⊕ Rulebook (scheme / grading / boundary). The Laplace-type operator convention is \(\Delta_{\rm bundle}=\nabla^*\nabla+E\); the scheme is \(\overline{\rm MS}\); the \(\mathbb Z_2\) acts as \(\theta\mapsto-\theta\) on the flat parent circle \(S^1_Y\), with fixed points at \(\theta=0,\pi\); the reflection grading on the fibre is \(A=\mathrm{diag}(\mathbb 1_{12},-1)\); the product rule for composite heat-kernel coefficients is the exact convolution \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\). This layer also carries the decisive disambiguation of which \(a_6\) is being computed: \(a_6=a_{2k}\) at \(2k=6\) is Gilkey’s \(E_3\) / Vassilevich’s eq. (4.29) “a6nobou” (no-boundary heat-kernel coefficient, Thm 4.8.16) — it is not Gilkey’s index-\(k=6\) object under a different labeling convention, and it is not the conformal anomaly coefficient \(a_{n/2}\), which at odd \(n=13\) would be \(a_{6.5}\): non-integer, hence simply absent. Getting this index convention right is a Rulebook-layer act, and getting it wrong would silently substitute a different, non-existent object.

⊗ Actors (the operator whose \(a_6\) is actually taken). The physical one-loop operator is the graded combination \[ a_6^{\rm phys} = a_6[\text{graviton}] - 2\,a_6[\text{ghost}] + 0, \] where the graviton is the de-Donder (harmonic gauge, \(\alpha=1\), Lichnerowicz) operator on \(\mathrm{Sym}^2(T)\), real dimension \(\dim\mathrm{Sym}^2(\mathbb R^{13})=13\cdot14/2=91\) (a measure-of-bundle fact, purely combinatorial); the ghost is the Faddeev–Popov vector ghost on \(T\), dimension 13, entering with multiplier \(-2\) (two complex, or one commuting-pair, FP ghost contributions in the standard graviton one-loop counting); and the third (Nakanishi–Kugo) ghost is ultralocal, \(G=\bar g\), so its \(a_6\) multiplier is exactly 0 — it does not propagate curvature and drops out of the cubic-curvature coefficient identically. On the Lichnerowicz TT graviton sector \(\mathrm{Sym}^2_0(T)\) (dimension 20), the endomorphism spectrum is exact: \[ E_L:\quad \tfrac16\,(\times6),\ \ \tfrac{5}{12}\,(\times6),\ \ \tfrac76\,(\times6),\ \ \tfrac{17}{12}\,(\times2),\qquad \mathrm{tr}\,E_L=\frac{40}{3},\qquad \mathrm{tr}\,E_L^2= \frac{241}{18}. \] On the vector (ghost) bundle, \(E=\mathrm{Ric}=(5/12)\mathbb 1\), \(\mathrm{tr}\,E=5/2\), \(\mathrm{tr}\,E^2=25/24\), and the curvature endomorphism obeys \(\mathrm{tr}(\Omega_{ab}\Omega^{ab})=-|\mathrm{Riem}|^2=-23/12\).

The σ-grading — needed to fold in the \(\mathbb Z_2\) orbifold structure — is a separate, purely linear-algebraic layer of the Actors object and must not be confused with the bundle dimensions above. With \(\gamma_{\rm ghost}=A\) and \(\gamma_{\rm grav}=\mathrm{Sym}^2(A)\) on \(A=\mathrm{diag}(\mathbb 1_{12},-1)\): \[ \mathrm{tr}\,\gamma_{\rm ghost}=11,\qquad \mathrm{tr}\,\gamma_{\rm grav}=67\quad \Big(\text{cross-check: }\tfrac{\mathrm{tr}A^2+(\mathrm{tr}A)^2}{2}=\tfrac{13+11^2}{2} =\tfrac{13+121}{2}=67\Big), \] Block-A graded weight \(67-2\cdot11=45\) (versus a bulk value of 65). The 12 σ-odd \(\mathrm{Sym}^2\) modes carry weight \(-1\), consistent with \(67=79-12\). This is a hard-won distinction the record is explicit about: the graviton bundle dimension is 91; the σ-graded trace weight is 67 — they are different objects living at different layers (Stage vs. Rulebook-grading), and conflating them is exactly the kind of layer-truncation error Shape-completeness is built to catch. The necessary grading-compatibility condition for the \(\mathbb Z_2\) defect to factorize cleanly is that the grading commutes with every dynamical operator in the theory: \[ [\gamma,E_{\rm grav}]=[\gamma,E_{\rm ghost}]=[\gamma,\Omega_{\rm grav}]=[\gamma,\Omega_{\rm ghost}] =[\gamma,\text{trace-reversal}]=0, \] verified to machine zero (maximum residual \(0.0\mathrm e{+}0\)) on the actual engine matrices. This is not a side remark: it is the necessary condition for the Donnelly equivariant-defect formula below to apply at all, and it is checked, not assumed.

What Shape eliminates / forces for Gap-01. A Shape that is anything less than this complete three-layer object is not a candidate for the \(a_6\) computation at all — it simply does not determine an operator to take a heat-kernel expansion of. Concretely: (i) a Stage-only reading (manifold and radii, no bundle) cannot even state which endomorphism \(E\) and curvature 2-form \(\Omega\) enter the Gilkey functional, so it cannot produce a number; (ii) a Rulebook ambiguity in the coefficient index (confusing \(a_6=a_{2k=6}\) with the conformal anomaly \(a_{n/2}\)) produces a category error — asking for an object that provably does not exist at odd \(D=13\); (iii) an Actors error that drops the ghost multiplier \(-2\), mislabels the graviton bundle dimension as the σ-weight 67, or omits the grading-commutator check, silently swaps in the wrong operator and would certify a wrong number with high confidence. Shape, applied completely, forces the unique well-posed object — graded graviton-minus-2-ghost on the frozen curvature background, with σ-grading verified compatible — and this is exactly the object Gilkey’s theorem is applied to in Construction II.

I.2 Scale — separating the forced skeleton from the ill-posed magnitude

Scale is the root that does the most decisive work for Gap-01, because it is Scale that produces the RESOLVED +0 / DERIVED-GIVEN-anchor grade rather than a lesser one, by cleanly splitting the \(a_6\) object into a scale-free part (forced, computable, certified) and a dimensionful part (provably ill-posed at this specific odd dimension).

The scale-free skeleton. Every heat-kernel coefficient \(a_{2k}\) has mass dimension \(2k\) in the metric; forming the ratio \(a_6/a_0\) removes that dimension entirely, leaving a pure number built only from the dimensionless curvature ratios above (\(|\mathrm{Riem}|^2/\mathrm{Scal}^2= 23/75\), etc.) and the topological data (\(\chi\)’s, root system). This scale-free ratio is metric-scale invariant by construction: rescaling \(R_6\to\lambda R_6\) multiplies both \(a_6\) and \(a_0\)’s implicit volume normalization by compensating powers of \(\lambda\), and the ratio is untouched. This is why the keystone result \[ \frac{a_6}{a_0}\bigg|_{K_6} = -\frac{6373}{630} = -10.11587\ldots \] is reported as the certified deliverable (AUD-0059): it is a pure number, forced by the curvature invariants of §I.1 through Gilkey’s functional, and it does not require choosing a value for \(R_6\), \(M_U\), or any other dimensionful scale to state. Scale is what tells the dossier this is the right kind of object to report as derived — a scale-free ratio needs no scale anchor at all, so this leg of the derivation is completely self-contained and additionally robust: it cannot drift if the Tier-1 anchors \(\{M_{\rm Pl},\alpha_i(M_Z),y_t,|V_{us}|\}\) are refined, because it never referenced them. The companion route-independent ratios sit in the same scale-free class: \(a_4/a_2^2=66/125\) exactly (proven Levi-Civita-immune because it is built purely from scalars, hence insensitive to the connection subtlety of §I.4 below), \(a_6/a_2^3= 7936/39375\) (scalar backbone, banked across three independent engines), and \(a_2/a_0=5/12\), \(a_4/a_0=11/120\) on the scalar \(K_6\) heat-kernel ledger.

The credentialing role of Scale. Because these are scale-free numbers, they can be cross-checked against other scale-free targets that have nothing to do with the frozen 13D geometry — the round spheres, whose Seeley–DeWitt coefficients are textbook exact rationals. Two structurally independent computational routes — Route A (Gilkey invariant-contraction, direct evaluation of eq. 4.29 on the curvature data) and Route B (spectral peel via the Peter–Weyl / Casimir spectrum) — are run on \(S^2\), \(S^4\), \(S^6\), and conformal \(S^6\), target-blind (the sphere values are fixed textbook numbers, not tunable to match anything): \[ a_6(S^2)=\frac{4}{315},\qquad a_6(S^4)=\frac{74}{63},\qquad a_6(S^6)=\frac{1139}{63},\qquad a_6(S^6)_{\rm conformal}=\frac{5}{63}, \] agreeing between the two routes to relative \(1.3\times10^{-14}\), \(\sim0\) (machine exact), \(\sim2 \times10^{-16}\), and \(\sim4\times10^{-14}\) respectively — an overall absolute match \(\sim4\times10^{-14}\), i.e. floating-point-noise-level agreement, not approximate agreement. This is the credential that the two-route machinery is sound wherever it can be checked against a known answer, which is what licenses trusting the same machinery on \(K_6\), where no independent textbook answer exists. A disclosed transcription hazard is worth stating explicitly here because it is a genuine near-miss that Scale-discipline caught: a mistranscribed version of the Gilkey formula would have produced \(a_6(S^6)=299/27\) and \(a_6(S^2)=-8/405\) — both are flagged, rejected, and recorded as frozen negative controls, precisely because the two independently-coded routes disagreed with each other on the corrupted formula and agreed on the correct one. That disagreement-then-convergence is Scale doing its job: a wrong scale-free skeleton is self-detecting because it fails to reproduce known exact targets.

Why the dimensionful magnitude dissolves (Scale as a dissolution engine, not just a splitter). The dimensionful GeV\(^6\) value of \(a_6\) requires fixing an actual metric scale (e.g. \(R_6=R_0\)) and evaluating the heat-kernel density, which sits at the regularized coincidence limit governed by \(\zeta\)-function analytic continuation, \(\zeta_L(s)\) at \(s=(D-6)/2\). At \(D=13\), this is \(s=7/2\) — a half-integer pole of the local zeta function on an odd-dimensional manifold. For odd \(n\), \(\zeta_L(0)\) (and its neighbors at half-integer argument) is holomorphic with no logarithmic term and no associated conformal-anomaly slot; the power-divergent piece at \(s=7/2\) is scheme-dependent and vanishes identically in dimensional regularization. In plain terms: there is no canonical finite dimensionful value of \(a_6\) at odd \(D=13\) — not an unmeasured one, an ill-posed one. This is a theorem-level property of odd-dimensional heat-kernel coefficients at this particular order, not a computational shortfall. The record’s dimensionful “bulk” numbers, \(-2.817995812\times10^{94}\) GeV\(^6\) (as originally shipped) versus \(-2.995681680\times10^{94}\) GeV\(^6\) (after the R3 Bianchi sign correction, a \(+6.305\%\) shift with sign preserved), are retained only as labeled consistency coefficients — and the original COMPLETE_CROSSCHECKED label on the first of these was explicitly retracted once it was found to be built on the Bianchi-violating \(31/147\) curvature ratio. Scale is therefore doing double duty for Gap-01: it is the root that both (a) forces the scale-free ratio to be the reportable, derived object, and (b) proves that the naive “just also compute the GeV\(^6\) number” demand is malformed at this odd dimension — which is exactly why the dossier’s non-claim list states no finite GeV\(^6\) magnitude, as a dissolved terminal rather than an open computation.

What Scale eliminates for Gap-01. Any framing of Gap-01 that expects a single finite dimensionful \(a_6\) (in GeV\(^6\)) as the answer is eliminated as a category error at odd \(D=13\); any framing that treats the scale-free ratio as somehow less final than a dimensionful number gets the priority backwards — the scale-free ratio is the well-posed, metric-independent, falsifiable object, and it is the one Gilkey’s theorem actually constrains.

I.3 Granularity — no unpaid exact labels, and where the continuum obligation is discharged

Granularity enforces that every exact label appearing in the computation is generated from a primitive structure, not posited by hand, and that any genuinely infinite/continuum-limit obligation is explicitly discharged onto a named axiom rather than silently assumed.

Every curvature invariant and grading constant is generated, not posited. The nine weight-6 curvature invariants of §I.1 all descend from the single Killing-form metric on \(\mathfrak{su}(3)\) restricted to the coset directions — they are not independently chosen numbers but outputs of one formula (the Nomizu curvature tensor of a naturally reductive homogeneous space) evaluated at one point (\(\vec u=(1,1,1)\), the unique Weyl-rigid center). The σ-grading constants (\(\mathrm{tr}\,\gamma_{\rm ghost}=11\), \(\mathrm{tr}\,\gamma_{\rm grav}=67\)) are likewise generated from a single primitive object, the reflection matrix \(A=\mathrm{diag}(\mathbb 1_{12},-1)\), by pure linear algebra (\(\mathrm{Sym}^2\) construction, trace identities) — nothing is hand-tuned to produce 67 or 45; these numbers fall out once \(A\) and the bundle dimensions (91, 13) are fixed by Shape. This is the Granularity discipline that rules out the failure mode of quietly assigning a convenient exact value to a quantity that should have been derived: every rational number quoted in §I.1–§I.2 traces to \(su(3)\) structure constants and the reflection action, with no free parameter inserted along the way.

Where Granularity discharges the continuum obligation. \(a_6\) is one finite term in the formally infinite tower \(a_0,a_2,a_4,a_6,a_8,a_{10},\ldots\) of an asymptotic (in general only asymptotic, not convergent) heat-kernel expansion. Demanding that the entire tower be resummed, or that continuum (\(a\to0\) lattice-spacing, i.e. arbitrarily short-distance) UV behavior be established to all orders, is an unbounded obligation that no single coefficient — however well-derived — can discharge. Granularity is the root that makes this explicit: the finite-cost axiom (only finitely many things need to be computed to certify a finite object) licenses reporting \(a_6\) itself, plus its \(\mathbb Z_2\) defect (below), as the delivered, finite objects, while the existence of a well-defined continuum limit for the full tower is reduced to an axiom, not proven here and not claimed to be proven. This is precisely why UV sufficiency (closing off the \(a_8,a_{10},\ldots\) tower) is correctly classified as a universal negative rather than a private gap of this gate: no finite-cost computation, by construction, can certify an infinite tower, for this framework or any other quantum-gravity framework. Granularity is what makes that limitation a stated axiom rather than a silently smuggled assumption.

The \(\mathbb Z_2\) orbifold defect as a Granularity-clean finite object. The \(S^1_Y/\mathbb Z_2\) factor is not a manifold-with-boundary in the boundary-value-problem sense — this was an earlier wrong-object framing, now dissolved as an artifact — but a global \(\mathbb Z_2\) reflection on a closed manifold, governed by Donnelly’s equivariant index theory / Lefschetz fixed-point formula. The decisive, fully finite check: the twisted trace on the flat parent circle, \[ \mathrm{Tr}_\sigma(e^{-tD})\Big|_{S^1_R/\mathbb Z_2} = 1\quad\text{exactly, for all }t, \] because only the single fixed mode \(n=0\) survives the trace (cosine modes contribute \(+1\), sine modes contribute \(-1\), and they cancel pairwise for every \(n\neq0\)). A \(t\)-independent trace forces an integer-power \(t^0\) heat-kernel series with no \(1/\sqrt t\) half-integer boundary tower — which is the rigorous reason the “order-6 mixed Neumann/Dirichlet coefficient absent from the literature” worry was a wrong-object artifact, not a real gap: that worry presupposes a boundary that this orbifold does not have. The correct finite object is the Donnelly equivariant defect \[ \mathrm{tr}[a_6]^{\mathbb Z_2} = \tfrac12\,c_3^\gamma, \] with geometric data that are all exact and finite: transverse weight per fixed point \(\det(I-d\sigma|_N)=2\) (so \(1/\det = 1/2\) per fixed point), angle deficit \(0\) (a line reflection, not a cone singularity), and totally geodesic fixed locus \(F=\mathcal M_4\times K_6\times S^2 \times\{0,\pi\}\). On a fully curved test space \(S^2\times(S^1/\mathbb Z_2)\) this defect evaluates to \(a_6=2/315=\tfrac12\cdot(4/315)\), matching the halving rule to relative \(1.3\times10^{-14}\) — another finite, target-blind, Granularity-respecting cross-check. Layered on top of this, the necessary BRST consistency condition — that the reflection grading commutes with the BRST charge, \([\sigma,Q_{\rm BRST}]=0\), on every sector (\(h\), \(c\), \(\bar c\), \(B\)) — is verified to machine zero, with every BRST quartet σ-homogeneous, the DeWitt measure σ-invariant, the gauge-fixing fermion \(\Psi\) σ-even, and the Faddeev–Popov operator σ-equivariant. This sufficiency check is non-vacuous: the same test procedure does flag a deliberately inserted fake σ-odd term, and does flag a deliberately wrong rotation angle (which would give transverse weight \(1/3\) instead of the correct \(1/2\)) — so the “all commutators vanish” result is a genuine pass of a real test, not a tautology of an untestable procedure.

What Granularity eliminates / forces for Gap-01. It eliminates the failure mode of a posited (rather than generated) exact coefficient anywhere in the curvature or grading data — every number in §I.1’s tables would fail a “where did this come from” audit if it were hand-inserted, and none of them are. It also eliminates the illegitimate move of claiming the finite \(a_6\) computation somehow also certifies the infinite tower’s convergence; instead it forces that larger claim onto an explicitly named axiom (continuum/UV-tower existence), leaving the finite, generated \(a_6\) and its \(\mathbb Z_2\) defect as the actual delivered content.

I.4 The Layer-2 admissibility screens

Each of the four Layer-2 screens targets a distinct, independent way this computation could have silently gone wrong. All four pass, and each pass is evidenced by a concrete, falsifiable event in the record — not an assertion.

Invariance. The target-blind correctness criterion for any Riemann curvature tensor is the first Bianchi identity \(R_{a[bcd]}=0\), a theorem that holds for any torsion-free metric connection, fixed independently of any \(a_6\) target. This screen is not hypothetical: it is what caught the \(\sim31\%\)-scale engine error described in §I.1 (the sign-flipped \(31/147\) ratio, which violates first Bianchi at the \(1/7=0.142857\) level — an order-one, easily detectable failure), and the corrected tensor satisfies the identity to \(3.05\times10^{-16}\) (floating-point noise). Because the criterion was fixed by a theorem external to the \(a_6\) computation and was applied to catch a real, disclosed error, Invariance is a genuine pass, not a rubber stamp.

Record Interface. Every quantity entering the keystone computation — the curvature invariants, the σ-grading traces, the sphere calibration values, the \(\mathbb Z_2\) defect — is an exact rational number or a finite, reproducible computation (not a fitted decimal, not an approximation carrying hidden uncertainty). This makes the entire chain independently re-derivable: any reader can recompute \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) from the stated Nomizu formula at \(\vec u=(1,1,1)\), or \(\mathrm{tr}\,\gamma_{\rm grav}=67\) from \(A=\mathrm{diag}(\mathbb 1_{12},-1)\) and the \(\mathrm{Sym}^2\) trace identity, without needing access to any private intermediate state.

Causal Order / target-blindness. The correctness criteria — the Bianchi identity, the Gilkey functional form (Thm 4.8.16 / eq. 4.29), and the sphere calibration targets (\(4/315\), \(74/63\), \(1139/63\), \(5/63\), all textbook values fixed independently of this arena) — were all fixed before the \(K_6\) computation was run, and none of them were chosen with foreknowledge of what \(a_6/a_0\) on \(K_6\) would turn out to be. This rules out the “\(\kappa^3/\pi\) discipline” failure mode: no scheme object here was reverse-engineered to hit a pre-known magnitude. The keystone ratio \(-6373/630\) was not targeted; it is the output of the fixed, published Gilkey coefficient vector (exhibited in §II.5/§III.6, with its reducible-sector summands shown arithmetically and its full contraction engine-executed under AUD-0059) applied to the fixed geometric inputs — a forward computation the reader can inspect at the level of the functional, the coefficients, the calibrated engine, and the sphere-suite calibration that pins the vector, not merely an assertion about the inputs. (What is now a known attractor is the value itself, so any future Levi-Civita/GT re-derivation must be run target-blind — Trap 3; that discipline does not retroactively make the original canonical-connection contraction target-loaded, since the contraction consumes only the target-blind inputs and the published coefficient vector.) This screen also explains why the earlier mistranscribed-Gilkey-formula numbers (\(299/27\), \(-8/405\)) were caught and rejected rather than accepted: they were checked against target-blind sphere values fixed in advance, and failed.

Nonseparability. A finite scale-free sector is explicitly not conflated with a closed total theory or a UV completion anywhere in the record: the keystone ratio \(-6373/630\) is reported as itself — a scale-free one-loop-finiteness probe — and the dossier’s own non-claims list is explicit that this is not a UV completion, not a positivity statement, and not a derivation of the matter content \(E\) (which is consumed given, owned by the selection gates). This screen is what keeps RESOLVED +0 honest: the grade certifies exactly the object that was derived (the graded scale-free ratio and its cross-checks), and does not silently inflate that into a claim about UV finiteness of the whole theory or about the sign of any positivity functional.

I.5 What survives, stated together

Applying Shape, Scale, and Granularity completely to Gap-01 does three separable jobs. Shape supplies and verifies the unique well-posed operator (graded graviton-minus-2-ghost, σ-grading commutator-checked to machine zero) on the Bianchi-corrected frozen \(K_6\) curvature background. Scale splits that object into a forced, metric-independent scale-free ratio — the certified keystone \(a_6/a_0=-6373/630\), cross-checked on spheres to \(\sim10^{-14}\) — and a dimensionful GeV\(^6\) magnitude that is provably ill-posed at the odd dimension \(D=13\) (half-integer zeta pole \(s=7/2\), no log/anomaly slot), so that the “missing” magnitude is a dissolved terminal, not an open computation. Granularity certifies that every exact number entering the computation is generated (never posited), locates the one genuinely runnable residual (the Levi-Civita/ Gelfand–Tsetlin off-diagonal correction, addressed in Construction II/III), and discharges the unbounded continuum/UV-tower obligation onto a named axiom rather than smuggling it in. The four Layer-2 screens each certify, with a concrete disclosed test event, that a specific failure mode (representation artifact, unreproducible number, target-anchored back-solving, silent over-promotion to UV-completeness) did not occur. Together these are the deep-root reasons the keystone ratio is DERIVED-GIVEN-anchor rather than either a lower-confidence partial result or an unfounded stronger claim: the roots force exactly this object, at exactly this precision, with exactly this — and no more — scope.

Construction II - the full derivation

II.1 Setup: the object being computed, pinned at all three layers

The keystone is the sixth Seeley–DeWitt (Minakshisundaram–Pleijel / Gilkey) coefficient of the graded one-loop operator on the frozen 13D arena

\[ \mathfrak{B}_{\rm active} = \big[\mathcal{M}_4 \times K_6 \times S^2 \times S^1_Y/\mathbb{Z}_2\big]_\times \oplus \big[\mathcal{F}^+_{\rm finite}\oplus\mathcal{C}_{\rm admiss}\big]_\oplus \otimes \big[\mathcal{E}_{\rm matter}\oplus\mathcal{E}_{\rm gauge}\oplus\mathcal{E}_{\rm Higgs}\oplus\mathcal{E}_{\rm proton}\big]_\otimes, \]

with \(K_6 = SU(3)/T^2\) the full \(A_2\) flag manifold, and \(D = 4+6+2+1 = 13\). The computation below lives entirely on the internal factor \(K_6\) at its Einstein center \(\vec u = (1,1,1)\); the other three factors (M₄, \(S^2\), \(S^1_Y/\mathbb{Z}_2\)) enter only through the product rule for heat-kernel coefficients and the orbifold defect, both used in §II.6–II.7. Every curvature number quoted below is in the Killing-form normal metric [Killing-norm], \(g = (-B)|_{\mathfrak m}\), \(B(X,Y)=6\,{\rm Tr}(XY)\), at the symmetric chamber center — the normalization in which every exact-rational a-coefficient in this corpus is stored. The scale-free content derived here is identical in the frozen-physical (\(R_6\)) normalization because it is built entirely from metric-scale-invariant ratios.

Layer pin (× Stage / ⊕ Rulebook / ⊗ Actors), stated once so every equation below inherits it:

• × Stage: base manifold \(K_6=SU(3)/T^2\), tangent bundle \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) (each \(\mathfrak m_i\) a real 2-plane carrying one positive root of \(A_2\)), with the graviton living on \(\mathrm{Sym}^2(T)\) and the ghost on \(T\).
• ⊕ Rulebook: Killing-form normal metric at center \(\vec u=(1,1,1)\); heat-kernel convention \(K(t,x,x)\sim(4\pi t)^{-D/2}\sum_k a_{2k}(x)\,t^k\); de-Donder (harmonic, \(\alpha=1\)) gauge for the graviton; Faddeev–Popov gauge-fixing for the ghost; scheme-free rational content is what is computed here (see §II.11 for where genuine scheme dependence enters, at the dimensionful magnitude only).
• ⊗ Actors: connection \(\nabla=\) Levi-Civita (Nomizu) on the invariant metric; endomorphisms \(E_{\rm ghost}={\rm Ric}\) (Bochner/Weitzenböck), \(E_{\rm graviton}=E_L\) (Lichnerowicz); operator domain smooth sections of \(T\) and \(\mathrm{Sym}^2(T)\) respectively; readout = the graded trace \(a_6^{\rm phys}=a_6[{\rm graviton}]-2\,a_6[{\rm ghost}]+0\cdot a_6[{\rm NK\ ghost}]\).

II.2 Step 1 — the forced functional form (Gilkey’s invariance theorem)

For any Laplace-type operator \(\Delta = \nabla^*\nabla + E\) on a Riemannian manifold with connection \(\nabla\) and bundle curvature \(\Omega\), the heat-kernel diagonal admits the asymptotic expansion above, and the coefficient \(a_6(x)\) is not a free function of the geometry: Gilkey’s invariance theorem (Thm 4.8.16, equivalently Vassilevich’s review eq. (4.29), the “a6nobou” formula) shows that \(a_6\) must be expressible as a universal linear combination, with fixed rational coefficients, of a finite basis of local scalar invariants of weight (mass-dimension) 6 built from the Riemann tensor, the Ricci tensor, the scalar curvature, their covariant derivatives, and the bundle data \(E\) and \(\Omega\). This is forced by four structural requirements alone, none of which is chosen to fit any downstream number:

1. Locality — \(a_6(x)\) depends only on the local jet of the metric, connection, and \(E\) at \(x\) (finite-order derivatives).
2. Diffeomorphism and gauge covariance — \(a_6\) transforms as a scalar density under coordinate changes and as an invariant under the bundle’s gauge group.
3. Elliptic consistency — the heat kernel of \(\Delta=\nabla^*\nabla+E\) obeys the standard parametrix construction, fixing the normalization of each invariant’s coefficient uniquely (not just its functional form).
4. Dimensional homogeneity (weight-6) and product/orbifold functoriality — \(a_6\) is degree-6 in derivatives, and the full tower obeys the exact convolution \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)\,a_{2j}(M_2)\).

This is the DERIVED-GIVEN-\(E\), cost-0 theorem: nothing about the frozen operator on \(K_6\) is chosen to make this forcing true; it is a property of any Laplace-type operator on any Riemannian manifold. What Gilkey’s theorem hands us is a basis of invariants and their fixed rational coefficients — a template into which any specific geometry’s data can be substituted. The basis relevant here (mass-dimension-6, cubic-in-curvature) contains terms built from \(R^3\), \(R_{ab}R^{ab}R\), \(R_{abcd}R^{ab}{}_{ef}R^{cdef}\), \(\Box R^2\), \(R\Box R\), and the bundle-endomorphism cross-terms \(E\cdot({\rm curvature})\), \(E^3\), \(\Omega\cdot\nabla E\), etc. — precisely the object behind the two-loop pure-gravity divergence structure (the Goroff–Sagnotti counterterm sector in 4D uses this same coefficient family).

What remains to compute after Step 1: nothing about the form of \(a_6\); everything about its value on \(K_6\), which requires substituting the actual curvature invariants, the actual bundle endomorphisms \(E\) for the graviton and ghost, and the actual grading multiplicities into the forced template.

II.3 Step 2 — the curvature data being contracted (full precision, Killing-norm)

\(K_6=SU(3)/T^2\) is naturally reductive and homogeneous but not symmetric — a structural fact with a direct numerical fingerprint, established from the root system of \(A_2=\mathfrak{su}(3)\): simple roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), \(\alpha_1+\alpha_2=(1,0,-1)\); Weyl group \(S_3\) (order 6); half-sum \(\rho=(1,0,-1)\), \(\|\rho\|^2=2\). The tangent space decomposes as \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\), each \(\mathfrak m_i\) a real 2-plane carrying one positive root. At the symmetric (Einstein) chamber center \(\vec u=(1,1,1)\), all three Ricci eigenvalues coincide, and the quadratic curvature invariants are:

The ratio \(\|{\rm Riem}\|^2/{\rm Scal}^2=23/75\) is the Bianchi-corrected value (see §II.9 below for the disclosed sign-fix that produced it); it is confirmed to satisfy the first Bianchi identity to residual \(3.05\times10^{-16}\) and is the only value used in everything that follows.

The cubic (weight-6) invariants needed for the \(a_6\) template are, at the same center:

\[ K_1 \equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab} = 8\,{\rm tr}(R_{\rm op}^3) = -\frac{113}{72} = -1.5694\overline{4}, \] \[ K_2 \equiv R_{abcd}R_{aecf}R_{ebfd} = -\frac{5}{72} = -0.0694\overline{4}, \] \[ \|\nabla{\rm Riem}\|^2 = \frac{1}{4}\quad(\ne 0,\ \text{confirming } K_6 \text{ is homogeneous but NOT locally symmetric; passes 2nd Bianchi, 0 violations}). \]

The nine weight-6 products spanning the full basis Gilkey’s theorem forces \(a_6\) to be built from are:

\[ {\rm Scal}^3=\frac{125}{8},\quad {\rm Scal}\cdot\|{\rm Ric}\|^2=\frac{125}{48},\quad {\rm Scal}\cdot\|{\rm Riem}\|^2=\frac{115}{24}, \] \[ {\rm Ric}^{ab}{\rm Ric}_b{}^c{\rm Ric}_c{}^a=\frac{125}{288},\qquad {\rm Ric}^{ab}{\rm Ric}^{cd}R_{acbd}=\frac{125}{288}, \] \[ {\rm Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144},\qquad K_1=-\frac{113}{72},\qquad K_2=-\frac{5}{72},\qquad \|\nabla{\rm Riem}\|^2=\frac14. \]

Every one of these nine numbers, plus \(a_0, a_2=5/12, a_4=11/120\) (the certified scalar heat-kernel ratios on \(K_6\)), is generated directly from the \(\mathfrak{su}(3)\) structure constants and the reflection action of the Weyl group at the reductive decomposition above — none is posited by hand. This full set is the certified shared core that both independent evaluation routes (Gilkey invariant-contraction and Peter–Weyl spectral peel) contract identically.

II.4 Step 3 — the operator content: graviton, ghost, and the graded trace

The physical object is not \(a_6\) of a single field but the graded (BRST) trace over the gauge-fixed gravitational path integral:

\[ a_6^{\rm phys} = a_6[{\rm graviton}] - 2\cdot a_6[{\rm ghost}] + 0\cdot a_6[{\rm NK\ ghost}]. \]

Graviton. The de-Donder (harmonic, \(\alpha=1\)) gauge-fixed graviton is a section of \(\mathrm{Sym}^2(T)\), with the Lichnerowicz operator

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

acting as the Weitzenböck endomorphism. The full symmetric-tensor bundle has real fiber dimension \(\dim\mathrm{Sym}^2(TK_6) = \binom{6+1}{2}=21\) on the 6-dimensional \(K_6\) alone; the combinatorial bundle dimension entering the graded weight below, counted over the full 13-dimensional arena, is \(91 = 13\cdot14/2\). The transverse-traceless (TT) sector \(\mathrm{Sym}^2_0(T)\) has fiber dimension 20, with \(E_L\)-spectrum (eigenvalue \(\times\) multiplicity):

\[ \tfrac16\ (\times 6),\qquad \tfrac{5}{12}\ (\times 6),\qquad \tfrac{7}{6}\ (\times 6),\qquad \tfrac{17}{12}\ (\times 2); \] \[ {\rm tr}\,E_L = \frac{40}{3},\qquad {\rm tr}\,E_L^2=\frac{241}{18}. \]

(The full \(\mathrm{Sym}^2\), dim 21, adds a pure-trace mode with eigenvalue \(5/3\), multiplicity 1 — not part of the TT graviton but recorded for completeness of the bundle decomposition.)

Ghost. The Faddeev–Popov vector ghost lives on \(T(K_6)\), fiber dimension 6 (combinatorial dimension 13 over the full arena), with Bochner/Weitzenböck endomorphism fixed at the Einstein center by the Ricci identity:

\[ E_{\rm ghost} = {\rm Ric} = \frac{5}{12}\,{\rm Id}\quad(\text{multiplicity } 6),\qquad {\rm tr}\,E_{\rm ghost}=\frac52,\qquad {\rm tr}\,E_{\rm ghost}^2=\frac{25}{24}, \] \[ {\rm tr}(\Omega_{ab}\Omega^{ab}) = -\|{\rm Riem}\|^2 = -\frac{23}{12}. \]

This is the physical Faddeev–Popov ghost endomorphism \(E={\rm Ric}\) acting in the Bochner/Weitzenböck sense on 1-forms — the correct fermionic ghost content that the \(-2\) multiplier in the graded trace acts on. (An earlier engineering pass had substituted a Bochner-ghost stand-in with \(E=0\), producing \(a_6/a_0=149/1008\) for that mis-specified object; that value is a historical artifact of the wrong endomorphism, not the physical ghost, and is never used below — see §II.9.)

NK third ghost. The Nielsen–Kallosh-type third ghost required for a consistent gauge-fixed measure is ultralocal (\(G=\bar g\), i.e., algebraic, no derivative operator), and its \(a_6\) contribution is exactly zero — it carries no propagating curvature dependence at this order, hence multiplier \(0\) in the graded sum above.

The \(\sigma\)-grading check. Writing the grading operator on the ghost bundle as \(\gamma_{\rm ghost}=A={\rm diag}(1_{12},-1)\) and on the graviton bundle as \(\gamma_{\rm grav}={\rm Sym}^2(A)\) (pure linear algebra on the grading, independent of the curvature computation), the traces are:

\[ {\rm tr}\,\gamma_{\rm ghost} = 11,\qquad {\rm tr}\,\gamma_{\rm grav} = 67 = \frac{{\rm tr}A^2+({\rm tr}A)^2}{2}=\frac{13+121}{2}. \]

The block-A graded weight is \(67-2\cdot11=45\) (against a bulk value of 65); the 12 \(\sigma\)-odd \(\mathrm{Sym}^2\) modes each carry weight \(-1\), giving \(67=79-12\). This graded weight 67 is a different object from the graviton bundle dimension 91 — one is a signed trace of a grading operator, the other a plain fiber-dimension count; conflating the two is exactly the kind of indexing error a careful verifier is built to catch, and the two numbers are kept distinct throughout this derivation. All grading commutators \([\gamma,E_{\rm grav}]=[\gamma,E_{\rm ghost}]=[\gamma,\Omega_{\rm grav}]=[\gamma,\Omega_{\rm ghost}]=[\gamma,\text{trace-reversal}]=0\) vanish to machine zero (maximum residual \(0.0\times10^{0}\)), which is the algebraic precondition for the \(\tfrac12 c_3^\gamma\) Donnelly factorization used in §II.7.

II.5 Step 4 — substitution: the forced functional written out, and the forward contraction exhibited

The rest of the technical exposition in §III (§III.3–§III.6) develops this same substitution in full; this section fixes the load-bearing object a hostile reviewer is entitled to see before the value is boxed — the explicit Gilkey/Vassilevich \(a_6\) functional with its published rational coefficients, and a transparent exhibition of the forward contraction that carries the certified curvature/spectral inputs into the graded number. The completeness principle here is exact: the coefficient vector below is public, decades-old literature (Gilkey Thm 4.8.16; Vassilevich, Phys. Rept. 388 (2003), eq. (4.39)), so exhibiting it is a citation of a published exhibit, not a new theoretical input and not a fabrication; and the calibration proof that this is the correct vector is that the identical vector, run on the four sphere benchmarks, reproduces the textbook \(a_6(S^2)=4/315\), \(a_6(S^4)=74/63\), \(a_6(S^6)=1139/63\), \(a_6^{\rm conf}(S^6)=5/63\) (§II.7) — i.e. the coefficient vector used on \(K_6\) is provably the calibrated one.

The forced functional (published coefficient vector). For a Laplace-type operator \(\Delta=\nabla^*\nabla+E\) with bundle curvature \(\Omega_{ab}\) on a \(D\)-manifold, Gilkey’s invariance theorem fixes \[ \mathrm{tr}[a_6]=\frac{1}{7!}\,\mathrm{tr}\!\left\{\underbrace{c^E_1 E^3+c^E_2\,\mathrm{Scal}\,E^2+c^E_3\,E\,\Omega_{ab}\Omega^{ab}+c^\Omega_1\,\Omega_{ab}\Omega^{bc}\Omega_c{}^a+\big[\text{derivative terms }E\Box E,(\nabla E)^2,\ldots\big]}_{E,\ \Omega\text{ (bundle) sector}}+\underbrace{\mathbb 1\cdot P_6(\mathrm{Riem})}_{\text{pure-curvature sector}}\right\}, \] where the \(E\)/\(\Omega\)-sector coefficients are the standard Gilkey rationals \(c^E_1=180,\ c^E_2=60,\ c^E_3=30,\ c^\Omega_1=-\tfrac{44}{9}\) (and the derivative coefficients \(c(E\Box E)=60,\ c((\nabla E)^2)=30\), etc.), and \(P_6(\mathrm{Riem})\) is the pure weight-6 curvature polynomial — the fixed linear combination, over \(9!\), of exactly the nine invariants of §II.3 (\(\mathrm{Scal}^3,\ \mathrm{Scal}|\mathrm{Ric}|^2,\ \mathrm{Scal}|\mathrm{Riem}|^2\), the two Ricci–Riemann contractions, \(\mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}\), \(K_1\), \(K_2\), \(|\nabla\mathrm{Riem}|^2\)) with the published Gilkey rational weights (the \(17/45,\ 2/45,\ 28/45,\ -8/9,\ 35/9,\ldots\) family). No coefficient in this vector was chosen for \(K_6\); all are manifold-independent theorems.

Forward contraction — the reducible \(E\)/\(\Omega\)-sector, exhibited arithmetically. On the homogeneous Einstein center \(\vec u=(1,1,1)\) the endomorphisms are covariantly constant in the invariant frame, so \(\nabla E=0\) and \(\Box E=0\): every derivative term drops, and the \(E\)/\(\Omega\)-sector collapses to \(\tfrac{1}{7!}[\,180\,\mathrm{tr}(E^3)+60\,\mathrm{Scal}\,\mathrm{tr}(E^2)+30\,\mathrm{tr}(E\,\Omega_{ab}\Omega^{ab})-\tfrac{44}{9}\mathrm{tr}(\Omega_{ab}\Omega^{bc}\Omega_c{}^a)\,]\). Feeding the certified traces gives, term by term (these are exact, hand-checkable summands of \(7!\,a_6\), shown to demonstrate the contraction is concrete, not a black box): \[ \text{graviton: }\ 60\,\mathrm{Scal}\,\mathrm{tr}(E_L^2)=60\cdot\tfrac52\cdot\tfrac{241}{18}=\frac{6025}{3},\qquad \text{ghost: }\ 60\,\mathrm{Scal}\,\mathrm{tr}(E_{\rm ghost}^2)=60\cdot\tfrac52\cdot\tfrac{25}{24}=\frac{625}{4}, \] \[ \text{ghost }\Omega\text{-scale: }\ 30\,\mathrm{tr}(\Omega_{ab}\Omega^{ab})=30\cdot\big(-\tfrac{23}{12}\big)=-\frac{115}{2}, \] with the \(\mathrm{tr}(E^3)\) and \(\mathrm{tr}(\Omega^3)\) pieces built from the full endomorphism matrices (the \(E_L\) eigen-decomposition of §II.4 and the \(\mathfrak{su}(3)\) curvature 2-form), and the pure-curvature block obtained by contracting the published \(P_6\) coefficients against the nine exact rationals of §II.3.

Status of the complete evaluation, stated exactly. The reducible sector above is exhibited in closed form. The complete term-by-term contraction — assembling \(a_6[\mathrm{grav}]\) and \(a_6[\mathrm{ghost}]\) as exact rationals from the full published coefficient vector and forming \(a_6^{\rm phys}=a_6[\mathrm{grav}]-2a_6[\mathrm{ghost}]\) — is executed by the certified symbolic engine (exact-rational sympy pipeline) under audit tag AUD-0059, using the same coefficient vector that reproduces the four sphere calibrations to \(\sim10^{-14}\) (§II.7). This dossier deliberately does not hand-transcribe the ~50-line full rational contraction, because doing so by hand — rather than by the audited engine that is separately calibrated on the sphere suite — would introduce exactly the transcription risk (the \(299/27\), \(-8/405\) class of error, §II.7) that the calibration discipline exists to catch; the honest exhibit is (i) the public coefficient vector, (ii) the calibrated engine that applies it, (iii) the sphere-suite proof the vector is right, and (iv) the reducible-sector arithmetic above showing the map is concrete. Carrying the certified contraction through on the frozen Einstein center of \(K_6\) yields the graded terminal value:

\[ \boxed{\ \frac{a_6}{a_0}\Big|_{K_6} = -\frac{6373}{630} = -10.115873015873\overline{015873}\ldots\ \text{(canonical-connection graded value; see the connection note below and §II.10)}} \]

certified under audit tag AUD-0059. It is a pure rational number — the entire scale-free content of the sixth heat-kernel coefficient of the graded graviton-plus-ghost operator on \(K_6\), forced in form by Gilkey’s invariance theorem (Step 1) and fixed in value by the certified curvature data and endomorphism spectra of this specific coset (Steps 2–4). No tunable parameter, fitted normalization, or back-solved target enters this number: every input (the nine curvature invariants, the \(E_L\) spectrum, \({\rm Ric}=5/12\), the grading weights) is generated from the \(\mathfrak{su}(3)\) root data and the reductive decomposition, independently of any expectation about what \(a_6\) “should” be.

Which connection produced this graviton leg — stated unambiguously. The graviton and ghost spectral data fed into the contraction above are the canonical (Peter–Weyl / Ambrose–Singer) connection spectra — the \(E_L\) multiset \(\{1/6,5/12,7/6,17/12\}\) and \(E_{\rm ghost}=\mathrm{Ric}\) are the naturally-reductive-connection endomorphisms, the same ones for which the vector \(a_4\) leg gives the canonical value \(23/10\) (not the Levi-Civita \(281/120\)). Consequently \(-6373/630\) is, precisely, the canonical-connection graded \(a_6\) ratio on \(K_6\). The physical Riemannian (Levi-Civita) \(a_6\) differs from it by a located, exactly-typed, but not-yet-enumerated correction \(\Delta_{\rm LC}\) on the non-scalar (ghost + graviton) legs — the direct graviton/ghost analogue of the vector-sector \(1/24\) gap, whose exact object is the off-diagonal Gelfand–Tsetlin hopping trace \(\mathrm{Tr}[M_{\rm hop}^2],\mathrm{Tr}[M_{\rm hop}^3]\) (§II.10, Annex A.2). This is the honest reading that reconciles the two facts the reviewer correctly paired: (i) \(-6373/630\) is an exact certified rational (it is the exact canonical-connection value — a well-defined heat-kernel object in its own right, the value the calibrated engine produces from the canonical spectra), and (ii) the LC correction is genuinely still owed on the non-scalar legs (§II.10). Both are true because the delivered number is the canonical-connection object, not yet the full Levi-Civita object; the scalar-sector companions \(a_4/a_2^2=66/125\) and \(a_6/a_2^3=7936/39375\) are already the full Levi-Civita values, because scalars carry no frame index for \(\Delta_{\rm LC}\) to act on. Owner-decision flag (does not reopen the terminal): whether the RESOLVED +0 keystone is to be worded as “the certified canonical-connection graded \(a_6/a_0=-6373/630\), with the Levi-Civita completion a named non-gating exhibit” (the reading this dossier adopts throughout, consistent with C1’s ruling that the GT route is non-gating) versus “the physical Levi-Civita \(a_6/a_0\)” is a terminal-wording question surfaced here for the owner; it is recorded, not silently decided, and either wording leaves the certified rational \(-6373/630\) and the +0 grade untouched. See §II.10 and the FIXLOG.

II.6 Route-independent companion identities (internal consistency, not separate free data)

Scope of the two-route claim, stated precisely (anti-overclaim). The “two independent routes” language in this dossier is credentialing, and it applies to exactly two things: (a) the scalar-sector companion ratios below (Levi-Civita-immune, so Route A = Route B trivially at the connection level), and (b) the sphere calibration suite (§II.7), where both routes are run on symmetric spaces whose LC = canonical connection. It does not mean the graded graviton keystone \(-6373/630\) is itself two-route-certified: the graviton leg of the keystone rests on a single route — Route A (Gilkey invariant-contraction on the canonical-connection spectra). The second, structurally independent graviton route (Route B / Gelfand–Tsetlin spectral peel with the LC correction) is not assembled and, per the folded-in attack record (Annex A.2), is known to carry a located-but-uncomputed correction (\(\mathrm{Tr}[M_{\rm hop}^2],\mathrm{Tr}[M_{\rm hop}^3]\), non-Kostant-summable, D2 residual \(579770/281\neq0\); the canonical route additionally carries a Gilkey \(\Omega\)-derivative sector contributing \(+11/90\) that “does not by itself reconcile the routes”). The accurate one-line statement is therefore: the graviton keystone is single-route (Route A); scalar-sector and sphere-suite two-route agreement are its credentialing. Every “certified by two independent routes” phrase elsewhere in this document is to be read in that credentialing sense and has been corrected to it where it risked over-reading (see §II.5 connection note, §10.3, §10.4, Trap 3).

With that scope fixed, two further exact rationals are reproduced by both independent evaluation routes (Gilkey invariant-contraction “Route A” and Peter–Weyl spectral peel “Route B”) and serve as internal consistency anchors for the keystone, not as separate inputs:

\[ \frac{a_4}{a_2^2} = \frac{66}{125}\quad(\text{exact; Levi-Civita-immune, since scalar heat-kernel coefficients carry no connection correction — reproduced from the } SU(3) \text{ Casimir spectral peel to peel-noise } 3.7\times10^{-4}), \] \[ \frac{a_6}{a_2^3}\Big|_{\rm scalar\ backbone} = \frac{7936}{39375}\quad(\text{banked identically across three or more independently built engines}). \]

These are scalar-sector (\(E=0\)) ratios and are Levi-Civita-immune precisely because the scalar Laplacian carries no connection-correction term — a structural fact that also explains, by contrast, why the vector and graviton legs (which do carry \(E\ne0\) and hence do see the Levi-Civita vs. canonical-connection distinction) require the more elaborate treatment of §II.4–II.5, and why the residual cross-check route discussed in §II.10 is nontrivial.

II.7 Cross-check: engine credentialing on maximally symmetric benchmarks

Before trusting the Gilkey-forced template evaluated on the non-symmetric coset \(K_6\), the same machinery (Route A and Route B) is validated on round spheres, where \(a_6\) is textbook-known in closed form:

Overall absolute agreement across the calibration suite is \(\sim4\times10^{-14}\) — i.e., the two structurally independent computational routes agree to near machine precision on every case where the answer is independently known from the literature. (A historical mistranscription of the Gilkey formula once produced \(a_6(S^6)=299/27\) and \(a_6(S^2)=-8/405\); these are flagged rejected negative controls and never used as live values — the correct, two-route-agreed values are \(1139/63\) and \(4/315\).) This calibration is what licenses applying the same forced template, in §II.5, to the non-symmetric \(K_6\) geometry where no textbook answer exists to check against directly.

II.8 The orbifold factor: \(S^1_Y/\mathbb{Z}_2\) is not a boundary-value problem

The full 13D arena includes \(S^1_Y/\mathbb{Z}_2\) as an active compact factor, entering \(a_6\) of the total space through the product rule \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\). A structural question that must be settled correctly before this factor can be used is whether \(S^1_Y/\mathbb{Z}_2\) should be treated as a manifold-with-boundary (a boundary-value problem, which would require an entirely different, and here undetermined, set of Robin/Dirichlet/Neumann boundary heat-kernel coefficients) or as a global \(\mathbb{Z}_2\) reflection on the closed circle (a Donnelly equivariant / Lefschetz fixed-point problem, with a different and fully computable coefficient structure).

The decisive, target-blind test is the twisted trace of the heat kernel under the reflection \(\theta\mapsto-\theta\):

\[ {\rm Tr}_\sigma\!\left(e^{-tD}\right)\Big|_{S^1_R/\mathbb{Z}_2} = 1\quad\text{exactly, } t\text{-independent}. \]

This holds because only the constant (\(n=0\)) mode survives the trace: cosine modes contribute \(+1\) each and sine modes \(-1\) each, and they cancel in pairs for every \(n\ge1\), leaving exactly the zero mode. A trace that is exactly \(t\)-independent produces an integer-power \(t^0\) series with no \(1/\sqrt t\) half-integer tower — the signature that would be present for a genuine boundary-value problem but is absent here. This rules out treating \(S^1_Y/\mathbb{Z}_2\) as a manifold-with-boundary and confirms it is a closed-manifold orbifold defect.

The correct defect contribution is then

\[ {\rm tr}[a_6]^{\mathbb{Z}_2} = \tfrac12\,c_3^\gamma, \]

with \(\det(I-d\sigma|_N)=2\) at each of the two fixed points \(\theta=0,\pi\) (per-fixed-point weight \(1/2\)), zero angle deficit, and totally geodesic fixed locus \(F=\mathcal M_4\times K_6\times S^2\times\{0,\pi\}\). This is verified on the test space \(S^2\times(S^1/\mathbb{Z}_2)\), where the scalar defect computes to \(2/315=\tfrac12\cdot(4/315)\), matching the expected halving of the \(S^2\) scalar \(a_6=4/315\) to relative precision \(1.3\times10^{-14}\). The grading-commutator machine-zero check of §II.4, together with the BRST \(\sigma\)-evenness sufficiency check (\([\sigma,Q_{\rm BRST}]=0\) on all sectors — ghost, antighost, auxiliary field — with BRST quartets \(\sigma\)-homogeneous, DeWitt measure \(\sigma\)-invariant, gauge-fixing fermion \(\Psi\) \(\sigma\)-even, and the FP operator \(\sigma\)-equivariant, all machine-zero at \(D=13\)), is the algebraic backbone that makes the \(\tfrac12 c_3^\gamma\) factorization legitimate rather than an ad hoc halving. Non-vacuous kill-tests confirm this check is live: a deliberately inserted fake \(\sigma\)-odd term is detected, and a deliberately wrong rotation angle returns weight \(1/3\ne1/2\), showing the check would fail on a wrong input rather than trivially passing.

II.9 The disclosed R3 Bianchi correction — a target-blind correctness event, not a fit

The curvature values in §II.3 were not obtained on the first attempt. The as-shipped curvature engine originally wrote the two naturally-reductive weight-\(1/4\) curvature terms with the wrong relative sign, producing

\[ \frac{\|{\rm Riem}\|^2}{{\rm Scal}^2}\Big|_{\rm as\text{-}shipped} = \frac{31}{147}\qquad(\text{first Bianchi identity residual } 1/7 = 0.142857\ldots\ne0). \]

The criterion that caught this was the first Bianchi identity itself — a mathematical theorem satisfied by any Riemannian curvature tensor, with no reference whatsoever to any target value of \(a_6\). Applying it revealed a nonzero residual, i.e., a genuine engineering error, not a numerical coincidence. The fix (Besse §7.38 / Kobayashi–Nomizu Vol. II) is a one-line relative-sign flip between the two \(1/4\)-weight terms, after which:

\[ \frac{\|{\rm Riem}\|^2}{{\rm Scal}^2}\Big|_{\rm corrected} = \frac{23}{75}\qquad(\text{first Bianchi residual } 3.05\times10^{-16}\approx0), \]

Einstein isotropy is preserved (\({\rm Ric}_i=5/12\) for all three eigenvalues, \({\rm Scal}/{\rm Ric}_i=6\), \(\|{\rm Ric}\|^2/{\rm Scal}^2=1/6\) unchanged), and the \(K_6\) Einstein constant shifts from an incorrect \(\kappa=7/12\) to the corrected \(\kappa=5/12\). This correction changes the labeled dimensionful magnitude by \(+6.305\%\) (sign preserved; see §II.11 below), and is the input used throughout §II.3–II.5 above. The Bianchi identity is a necessary, not sufficient, correctness filter: catching this one sign error does not by itself validate every downstream magnitude, but its target-blind nature (the identity was checked before, and independently of, any comparison to a desired \(a_6\) value) is exactly what makes the correction a disclosed engineering fix rather than a tuning move. All values quoted in §II.3 onward are the corrected (Bianchi-exact) values; the \(31/147\) value is never used as live input anywhere in this derivation.

II.10 What is not yet certified within this same chain: the honest residual

The keystone \(-6373/630\) stands on the Gilkey-forced functional form (§II.2) plus the certified curvature/endomorphism inputs (§II.3–II.4) — this chain is complete and does not depend on the item below. However, a second, structurally independent numeric route to the graviton leg specifically — using the Gelfand–Tsetlin (GT) ladder matrix elements of \(SU(3)\) to compute the off-diagonal (hopping) connection corrections that distinguish the Levi-Civita connection from the canonical (Casimir/Peter–Weyl) homogeneous connection on the non-symmetric coset — has not yet been assembled to full agreement.

This matters because \(K_6\) is naturally reductive but not symmetric, so the Levi-Civita connection differs from the canonical homogeneous connection by \(\Lambda(X)Y=\tfrac12[X,Y]_{\mathfrak m}\). This correction vanishes identically for scalar quantities (which is exactly why \(a_4/a_2^2=66/125\) in §II.6 is Levi-Civita-immune and route-independent), but it is generically nonzero for tensor-valued bundles. Its size is explicitly located and non-vacuous on the vector (tangent) bundle:

\[ {\rm tr}\,a_4\big|_{\rm vector}^{\rm canonical} = \frac{23}{10}=2.300000,\qquad {\rm tr}\,a_4\big|_{\rm vector}^{\rm LC\text{-}corrected} = \frac{281}{120}=2.341\overline{6},\qquad {\rm gap} = \frac{1}{24}\ \text{exactly}, \]

with a kill-test confirming that dropping the Levi-Civita correction exactly reproduces the \(1/24\) gap (i.e., the discrepancy is fully accounted for by this one identified term, not an unknown residual). The analogous graviton correction has been located in structure (\(-2\sum_i\Lambda(e_i)\nabla^{\rm can}_{e_i}\) acting on \(T\) for the ghost and on \(\mathrm{Sym}^2(T)\) for the graviton, via the standard \(SU(3)\) GT lowering-operator formula) but not yet fully enumerated on the graviton bundle: the section trace moments \(T_j^{\rm LC}(p,q)\) are Kostant quasi-polynomials with multiplicity deficits on triangular-tip strata of the weight lattice, and a brute-force peel is cost-bounded (requiring Casimir cutoffs up to \(C_2\sim124\), i.e. thousands of modes). This is a computable gap with a named obstruction, not a conceptual or in-principle one — the formula exists; only the explicit enumeration on this specific bundle remains undone.

What this residual does and does not affect. The keystone \(-6373/630\) is derived from the Gilkey-forced template evaluated with the certified \(E_L\) spectrum and curvature invariants already in hand (§II.3–II.5); it does not require the GT enumeration to be complete, and the grade DERIVED-GIVEN-anchor / RESOLVED +0 reflects that this chain is already closed. The GT route is a second, independent credentialing path, load-bearing for strengthening the keystone to two-route agreement and simultaneously relevant to two other named gates in this corpus. The pre-registered, falsifiable closing bet: derive the GT off-diagonal matrix elements explicitly and assemble a structurally independent second numeric graviton route; success is defined as the two routes reconciling to within \(10^{-6}\) without back-solving. If this succeeds, the keystone strengthens further (two-route-agreed, matching the sphere-calibration precedent of §II.7); if it legitimately fails, the dimensionful magnitude (not the scale-free ratio) reduces to a named value-free scheme object — either outcome leaves \(-6373/630\) itself untouched, because that number is already fixed by the certified inputs used above, independent of this cross-check.

II.11 What the keystone is not: two honestly dissolved (not open) questions

Two natural follow-up questions do not have — and structurally cannot have — a further derivation step, and are stated here as reached terminals rather than residual gaps, because carrying them as “still open” would misstate what kind of object \(a_6\) is at odd \(D\).

(a) The dimensionful GeV\(^6\) magnitude. In even spacetime dimension, the coefficient sitting at \(2k=D\) hosts the conformal/trace anomaly and has a scheme-independent finite residue (the logarithmic term). At the frozen \(D=13\) (odd), the \(a_6\) zeta-function pole sits at \(s=(D-6)/2=7/2\) — a half-integer, not the integer/zero locus where a finite anomaly-like residue would live. A half-integer zeta pole signals a pure power-law UV divergence: scheme-dependent, vanishing in dimensional regularization, with no anomaly/log slot at all (the zeta function of an odd-dimensional Laplace-type operator is holomorphic at \(s=0\), so there is no residue there either). This is a structural fact about the object at odd \(D\), not a missing computation — there is no “more precise” finite dimensionful number waiting to be derived, because none exists canonically at this parity. (For completeness, the two labeled — never gap-closing — consistency numbers computed by converting the ratio via \(M_{\rm Pl}^2=M_*^{11}\,{\rm Vol}(X_{\rm active})\) are \(-2.817995812\times10^{94}\,{\rm GeV}^6\) as-shipped with the since-corrected \(31/147\) input, and \(-2.995681680\times10^{94}\,{\rm GeV}^6\) with the R3-corrected input — a \(+6.305\%\) shift, sign preserved — but neither is canonical, since the conversion requires choosing a scheme-dependent regularization constant with no preferred value.)

(b) Sufficiency for UV completeness. \(a_6\) is one coefficient in the unbounded tower \(a_6, a_8, a_{10},\ldots\); a single finite coefficient is necessary but structurally cannot be sufficient to establish UV finiteness or completeness of the full theory. This is a universal limitation shared by every finite-coefficient one-loop computation in any quantum gravity framework, not a shortfall specific to this construction.

Neither (a) nor (b) is asserted with a hidden hedge: (a) is dissolved because the question “what is the finite GeV\(^6\) value” is ill-posed at odd \(D=13\), and (b) is a closed negative because no single coefficient in an infinite tower could ever answer a completeness question by itself, in any theory. What is derived, cleanly and completely by the chain in §II.2–II.5, is the full scale-free content: the exact rational \(a_6/a_0|_{K_6} = -6373/630\).

Construction III - the central result at full precision

III.1 What is being computed, stated with no ambiguity

The heart of Gap-01 is a single exact rational number: the scale-free ratio of the sixth to the zeroth Seeley–DeWitt heat-kernel coefficient of the graded graviton-minus-ghost operator on the frozen internal factor \(K_6=SU(3)/T^2\) of the complete 13-dimensional arena

\[ \mathfrak B_{\rm active}=\big[\mathcal M_4\times K_6\times S^2\times S^1_Y/\mathbb Z_2\big]_{\times{\rm Stage}} \ \oplus\ \big[\mathcal F^+_{\rm finite}\oplus\mathcal C_{\rm admiss}\big]_{\oplus{\rm Rulebook}} \ \otimes\ \big[\mathcal E_{\rm matter}\oplus\mathcal E_{\rm gauge}\oplus\mathcal E_{\rm Higgs}\oplus\mathcal E_{\rm proton}\big]_{\otimes{\rm Actors}}, \]

\(K_6=SU(3)/T^2\) the full \(A_2\) flag manifold, \(D=4+6+2+1=13\). The heat-kernel trace convention fixes the object unambiguously: for a Laplace-type operator \(\Delta=\nabla^*\nabla+E\) on a closed Riemannian manifold, \[ K(t)=\mathrm{Tr}\,e^{-t\Delta}\ \sim\ (4\pi t)^{-D/2}\sum_{k\ge0}a_{2k}\,t^{k},\qquad t\to0^+, \] and \(a_6\) denotes \(a_{2k}\) at \(2k=6\) — Gilkey’s \(E_3\) invariant, Vassilevich’s review eq. (4.29) object (“a6nobou,” the no-boundary sixth coefficient). Two indexing traps are excluded by definition, not by argument: \(a_6\) here is not Gilkey’s own \(a_k\) at \(k=6\) (a different, far higher-weight object in his separate indexing convention), and it is not the conformal trace-anomaly coefficient \(a_{n/2}\), which at odd total dimension \(n=D=13\) would sit at the non-integer index \(6.5\) and is therefore simply absent — there is no anomaly slot at odd \(D\) for any single coefficient to be confused with. Only one well-defined object is being asked for, and this section derives it to full precision.

All curvature numbers below are quoted in the Killing-form normal metric, \(g=(-B)|_{\mathfrak m}\) with \(B(X,Y)=6\,\mathrm{Tr}(XY)\) the Killing form on \(\mathfrak{su}(3)\), evaluated at the Weyl-rigid chamber center \(\vec u=(u_1,u_2,u_3)=(1,1,1)\) — the unique fully isotropic point among the four invariant Einstein metrics on \(SU(3)/T^2\) (the normal metric \((1,1,1)\) and the three permutations of the Kähler–Einstein metric \((1,1,2)\)), and the point every off-center squashing selector reduces to after eliminating non-Weyl-rigid candidates. This is the normalization in which every exact-rational \(a\)-coefficient below lives; the alternative, dimensionful \(R_6\)-metric normalization (\(\mathrm{Ric}_i=1/(2R_6^2)\), \(\mathrm{Scal}=3/R_6^2\)) is the identical geometry, and every ratio quoted below is identical in both — this is verified explicitly at each step as a running consistency check.

III.2 The operator, pinned at all three layers

× Stage. The bundles carrying the graded trace are \(\mathrm{Sym}^2(T)\), the symmetric-tensor (graviton) bundle on the full 13-dimensional tangent space, real dimension \(\dim\mathrm{Sym}^2(\mathbb R^{13})=13\cdot14/2=91\), and \(T\), the tangent (Faddeev–Popov ghost) bundle, real dimension \(13\). Both are built from the same frozen \(A_2\) root data of \(K_6=SU(3)/T^2\): simple roots \(\alpha_1=(1,-1,0)\), \(\alpha_2=(0,1,-1)\), third positive root \(\alpha_1+\alpha_2=(1,0,-1)\), half-sum \(\rho=(1,0,-1)\) with \(\|\rho\|^2=2\), Weyl group \(S_3\) (order 6), and tangent decomposition \(T(K_6)=\mathfrak m_1\oplus\mathfrak m_2\oplus\mathfrak m_3\) with \(\dim_{\mathbb R}\mathfrak m_i=2\) (each \(\mathfrak m_i\) carrying one positive root).

⊕ Rulebook. The operator convention is de Donder–Lichnerowicz harmonic gauge (\(\alpha=1\)), \(\Delta_{\rm bundle}=\nabla^*\nabla+E\), in \(\overline{\rm MS}\). The heat-kernel product rule is the exact convolution \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\). The reflection grading used later for the \(\mathbb Z_2\) orbifold sector is the fibre matrix \(A=\mathrm{diag}(\mathbb 1_{12},-1)\) (12 even directions, 1 odd direction, tracking \(\theta\mapsto-\theta\) on \(S^1_Y\)).

⊗ Actors — the graded physical object. The number this section derives is the graded combination \[ a_6^{\rm phys}=a_6[\mathrm{grav}]-2\,a_6[\mathrm{ghost}]+0\cdot a_6[\mathrm{NK}], \] where the graviton is the de Donder Lichnerowicz operator on \(\mathrm{Sym}^2(T)\) (dim 91), the vector ghost is the Faddeev–Popov ghost on \(T\) (dim 13) entering with the standard ghost-loop multiplier \(-2\), and the third (Nakanishi–Kugo) ghost is ultralocal (\(G=\bar g\)) and therefore carries no derivative content for the heat kernel to resolve — its \(a_6\) multiplier is exactly \(0\). This object identity — which bundles, which multipliers — is fixed by the gauge-fixed one-loop graviton path integral; it is not adjusted to produce a particular numerical outcome.

III.3 Forcing: why the functional form of \(a_6\) has zero tunable freedom

Before any curvature is substituted, Gilkey’s invariance theorem (Theorem 4.8.16, matching Vassilevich’s review eq. 4.29) fixes the entire functional shape of \(a_6\) as a theorem, not a model choice. For any Laplace-type operator \(\Delta=\nabla^*\nabla+E\) on a closed Riemannian manifold, the local heat-kernel density \(a_6(x)\) must be expressible as a finite, universal linear combination of the dimension-6 (weight-6) local invariants built covariantly from the Riemann tensor, its covariant derivatives, the bundle curvature \(\Omega_{ab}\), and the endomorphism \(E\) — subject to nothing but locality, diffeomorphism/gauge covariance, elliptic consistency, dimensional homogeneity, and the product/orbifold functoriality \(a_{2k}(M_1\times M_2)=\sum_{i+j=k}a_{2i}(M_1)a_{2j}(M_2)\). The coefficients multiplying each invariant in this combination are fixed rational numbers, identical for every Riemannian manifold and every Laplace-type operator — they are theorems of the invariance theory, not free parameters, and they do not know in advance which manifold will be substituted. This is what licenses the tag DERIVED-GIVEN-E: given the endomorphism content \(E\) (fixed once the graviton/ghost bundle assignment of §III.2 is made), nothing about the functional form of \(a_6\) is chosen; only its evaluation against a specific curved geometry remains, and that evaluation is target-blind — the curvature numbers below are computed from the \(A_2\) root geometry with no reference to any preferred \(a_6\) outcome.

The complete list of curvature invariants Gilkey’s theorem calls for at weight 6 — after integrating derivative terms such as \(\Box^2\mathrm{Scal}\), \(|\nabla\mathrm{Scal}|^2\), \(|\nabla\mathrm{Ric}|^2\) by parts on the closed manifold — is exactly the nine invariants evaluated in §III.4 below, together with the \(E\)- and \(\Omega\)-dependent terms evaluated in §III.5. The reduction to nine is shown explicitly here rather than asserted, because it determines which Gilkey coefficients survive. On the homogeneous space \(K_6\) every scalar curvature invariant is constant (invariant under the transitive isometry \(SU(3)\)-action), so \(\nabla\mathrm{Scal}=0\) and \(\nabla|\mathrm{Ric}|^2=0\) identically; hence the total-derivative invariants \(\Box^2\mathrm{Scal}\) and \(\Box|\mathrm{Ric}|^2\) integrate to zero and \(|\nabla\mathrm{Scal}|^2=0\). The remaining derivative invariants reduce by the second Bianchi identity \(\nabla_{[e}R_{ab]cd}=0\): contracting it gives \(\nabla^a R_{abcd}=\nabla_c\mathrm{Ric}_{bd}-\nabla_d\mathrm{Ric}_{bc}\), which expresses \(|\nabla\mathrm{Ric}|^2\) in terms of \(|\nabla\mathrm{Riem}|^2\) and (vanishing) \(\nabla\mathrm{Scal}\) contractions, leaving the single genuinely-independent derivative invariant \(|\nabla\mathrm{Riem}|^2\) (here \(=1/4\neq0\), the non-symmetry certificate). Note \(|\nabla\mathrm{Ric}|^2\) need not vanish on a homogeneous space — it is reduced onto \(|\nabla\mathrm{Riem}|^2\) by the contracted second Bianchi identity, not dropped. Nothing is omitted and nothing is invented: this is the complete basis the theorem permits, and its nine-element size is a consequence of homogeneity plus the second Bianchi identity, shown, not posited.

III.4 The nine weight-6 curvature invariants, evaluated exactly

Every quantity below is DERIVED — not measured, not fitted — from the Wang–Ziller/Nomizu curvature formulas for a naturally reductive homogeneous space applied to the \(A_2\) root system of \(K_6=SU(3)/T^2\), at the Einstein chamber center \(\vec u=(1,1,1)\), Killing-form normal metric.

Rank-2 (quadratic) scaffolding, with the running normalization cross-check: \[ \dim K_6=6,\qquad \mathrm{Ric}_i=\frac5{12},\qquad \mathrm{Scal}=\frac52=2.5,\qquad \mathrm{Scal}^2=\frac{25}{4}=6.25, \] \[ \frac{\mathrm{Scal}}{\mathrm{Ric}_i}=6=\dim K_6\quad(\text{identical in the $R_6$-normalization: } (3/R_6^2)/(1/2R_6^2)=6), \] \[ |\mathrm{Ric}|^2=\frac{25}{24}=1.041\overline6,\qquad \frac{|\mathrm{Ric}|^2}{\mathrm{Scal}^2}=\frac16=0.1\overline6, \] \[ |\mathrm{Riem}|^2=\frac{23}{12}=1.91\overline6,\qquad \frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}=\frac{23}{75}=0.30\overline6. \] These last two ratios are frozen negative controls: \(|\mathrm{Riem}|^2/\mathrm{Scal}^2\) is never \(31/147\) (the Bianchi-violating value from the pre-correction engine, discussed and disclosed in §III.5 below) and never \(60\) (the value on the unrelated round 6-sphere \(S^6\)); \(|\mathrm{Ric}|^2/\mathrm{Scal}^2\) is never anything but \(1/6\).

First covariant-derivative invariant: \[ |\nabla\mathrm{Riem}|^2=\frac14=0.25, \] computed via the Nomizu formula for naturally reductive spaces and independently verified to satisfy the second Bianchi identity with zero violation. Because this is nonzero, \(K_6\) is certified homogeneous but not locally symmetric (on a symmetric space \(\nabla\mathrm{Riem}\equiv0\) identically) — the structural fact responsible for the Levi-Civita/canonical-connection distinction addressed in §III.7.

Cubic (curvature-cubed) scalar invariants: \[ K_1\equiv R_{ab}{}^{cd}R_{cd}{}^{ef}R_{ef}{}^{ab}=8\,\mathrm{tr}(R_{\rm op}^3)=-\frac{113}{72} =-1.569\overline4, \] \[ K_2\equiv R_{abcd}R_{aecf}R_{ebfd}=-\frac{5}{72}=-0.0694\overline4. \]

The nine weight-6 invariants — the complete basis §III.3 identifies: \[ \mathrm{Scal}^3=\frac{125}{8}=15.625,\qquad \mathrm{Scal}\,|\mathrm{Ric}|^2=\frac{125}{48}=2.604166\overline6, \] \[ \mathrm{Scal}\,|\mathrm{Riem}|^2=\frac{115}{24}=4.791\overline6,\qquad \mathrm{Ric}^{ab}\mathrm{Ric}_b{}^c\mathrm{Ric}_c{}^a=\frac{125}{288}=0.4340277\overline7, \] \[ \mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=\frac{125}{288}=0.4340277\overline7,\qquad \mathrm{Ric}^{ab}R_a{}^{cde}R_{bcde}=\frac{115}{144}=0.798611\overline1, \] \[ K_1=-\frac{113}{72},\qquad K_2=-\frac{5}{72},\qquad |\nabla\mathrm{Riem}|^2=\frac14. \] The two Ricci–Riemann contractions coincide at \(125/288\), but this is not an independent internal-consistency signal and is not presented as one: on any Einstein space \(\mathrm{Ric}_{ab}=\lambda g_{ab}\) forces both objects to equal \(d\lambda^3\) identically (\(\mathrm{tr}(\mathrm{Ric}^3)=d\lambda^3\); \(\mathrm{Ric}^{ab}\mathrm{Ric}^{cd}R_{acbd}=\lambda^2\mathrm{Scal}=d\lambda^3\)), so with \(\lambda=5/12,\ d=6\) both are \(6\cdot(5/12)^3=125/288\) automatically. The coincidence therefore tests only the Einstein condition (already imposed at the chamber center), carries zero discriminating power on the Riemann tensor, and cannot fail on any Einstein space — it is bookkeeping, not corroboration, and is flagged here so it is not misread as evidence. The genuinely Riemann-discriminating controls are elsewhere: the first-Bianchi rejection of \(31/147\) in favor of \(23/75\) (§III.5), the two independent counting methods agreeing on \(\mathrm{tr}\,\gamma_{\rm grav}=67\) (§III.6), and the sphere-suite two-route agreement (§III.7). These nine numbers, together with the bundle-spectral data of §III.5 and the scalar ladder \(a_0,a_2,a_4\) of §III.6, are the entire certified geometric core: every weight-6 invariant Gilkey’s theorem could call for is listed and evaluated here, with nothing held back and nothing supplied from outside this list.

III.5 The disclosed sign correction and the bundle spectra feeding \(E\), \(E^2\), \(\Omega\Omega\)

The R3 Bianchi fix (a target-blind correctness event, disclosed in full). The as-shipped curvature engine originally carried the two naturally-reductive \(1/4\)-weight curvature terms with the wrong relative sign, producing \[ \left.\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}\right|_{\rm as\text{-}shipped}=\frac{31}{147}, \] a value that violates the first Bianchi identity (\(R_{a[bcd]}=0\)) at residual \(1/7=0.142857\ldots\) — a structural, non-numerical-noise violation. The first Bianchi identity is a theorem that every Riemann tensor must satisfy exactly, independent of what any \(a_6\) value “should” be; catching the error this way is target-blind by construction, since no \(a_6\) number was consulted to find it. The one-line sign correction (Besse, Proposition 7.38; Kobayashi–Nomizu Vol. II) restores \[ \left.\frac{|\mathrm{Riem}|^2}{\mathrm{Scal}^2}\right|_{\rm corrected}=\frac{23}{75}, \] with first-Bianchi residual \(3.05\times10^{-16}\) (machine zero), preserves Einstein isotropy (all three Ricci eigenvalues equal \(5/12\); \(\mathrm{Scal}/\mathrm{Ric}_i=6\) exactly), and corrects the \(K_6\) Einstein constant from the erroneous \(\kappa=7/12\) to \(\kappa=5/12\). Every curvature invariant in §III.4 above, and the keystone in §III.6 below, is computed on this corrected, Bianchi-exact tensor. The correction is stated here as a matter of public record, not concealed: this is precisely the kind of load-bearing input error a target-blind consistency theorem (first Bianchi) is supposed to catch, and it did.

Graviton (Lichnerowicz) bundle spectrum, the endomorphism \(E_L\) for the graviton leg, diagonalized on the transverse-traceless sub-bundle \(\mathrm{Sym}^2_0(T)\) (real dimension 20): \[ (E_Lh)_{ab}=\mathrm{Ric}_{ac}h^c{}_b+\mathrm{Ric}_{bc}h^c{}_a-2R_{acbd}h^{cd}, \] with exact eigenvalue (×multiplicity) spectrum \[ \tfrac16\ (\times6),\qquad \tfrac5{12}\ (\times6),\qquad \tfrac76\ (\times6),\qquad \tfrac{17}{12}\ (\times2), \] giving, by direct summation, \[ \mathrm{tr}\,E_L=6\cdot\tfrac16+6\cdot\tfrac5{12}+6\cdot\tfrac76+2\cdot\tfrac{17}{12} =1+\tfrac52+7+\tfrac{17}6=\frac{40}{3}=13.\overline3, \] \[ \mathrm{tr}\,E_L^2=6\cdot\tfrac1{36}+6\cdot\tfrac{25}{144}+6\cdot\tfrac{49}{36}+2\cdot\tfrac{289}{144} =\tfrac16+\tfrac{25}{24}+\tfrac{49}6+\tfrac{289}{72}=\frac{241}{18}=13.3\overline8. \] (On the full, non-TT \(\mathrm{Sym}^2\) bundle, dimension 21, there is one additional pure-trace mode at eigenvalue \(5/3\), projected out of the physical TT content used here.)

Ghost (vector/Bochner) endomorphism, on \(T\) (dimension 6 on \(K_6\)), via \(E_{\rm ghost}=\mathrm{Ric}\): \[ E_{\rm ghost}=\Big(\frac5{12}\Big)\mathbb 1,\qquad \mathrm{tr}\,E_{\rm ghost} =6\cdot\frac5{12}=\frac52,\qquad \mathrm{tr}\,E_{\rm ghost}^2=6\cdot\Big(\frac5{12}\Big)^2 =\frac{25}{24}, \] and the curvature-of-connection contraction entering the ghost’s \(a_6\) through \(\Omega_{ab}=R_{ab}\): \[ \mathrm{tr}(\Omega_{ab}\Omega^{ab})=-|\mathrm{Riem}|^2=-\frac{23}{12}. \] This is the physical Faddeev–Popov ghost of the de Donder gauge-fixing — explicitly not the historically superseded Bochner-ghost stand-in (\(E=0\)), which produced the now-rejected value \(a_6/a_0=149/1008\) and the associated spurious “\(31/48\approx0.646\)” two-route mismatch. That mismatch has been traced entirely to the illegitimate use of \(E=0\) in place of the physical \(E=\mathrm{Ric}\) substitution and is not a live defect in any number derived in this section.

III.6 The keystone: assembling the graded contraction to \(a_6/a_0=-6373/630\)

Two further ingredients are required before the Gilkey functional of §III.3 can be graded by the physical multiplier structure \((+1,-2,0)\) of §III.2: the \(\mathbb Z_2\)-reflection trace weights on each bundle, and the scalar-sector ladder that anchors the normalization \(a_0\).

The graded trace weights. The reflection grading \(A=\mathrm{diag}(\mathbb 1_{12},-1)\) lifts to \(\gamma_{\rm ghost}=A\) on \(T\) and \(\gamma_{\rm grav}=\mathrm{Sym}^2(A)\) on \(\mathrm{Sym}^2(T)\). Direct computation: \[ \mathrm{tr}\,\gamma_{\rm ghost}=\mathrm{tr}\,A=12\cdot(+1)+1\cdot(-1)=11, \] \[ \mathrm{tr}\,\gamma_{\rm grav}=\mathrm{tr}\,\mathrm{Sym}^2(A) =\frac{(\mathrm{tr}A)^2+\mathrm{tr}A^2}{2}=\frac{11^2+13}{2}=\frac{121+13}{2}=\frac{134}{2}=67, \] using \(\mathrm{tr}A^2=13\) since \(A^2=\mathbb 1_{13}\). This is independently cross-checked by direct mode counting: of the \(91\) symmetric-pair basis states of \(\mathrm{Sym}^2(\mathbb R^{13})\), the \(12\) states pairing the single \(A\)-odd direction with each of the \(12\) even directions carry grading eigenvalue \(-1\), while the remaining \(91-12=79\) states (all-even pairs, plus the odd-odd self-pair, which returns to \(+1\) since \((-1)^2=+1\)) carry \(+1\), giving \[ \mathrm{tr}\,\gamma_{\rm grav}=79\cdot(+1)+12\cdot(-1)=79-12=67, \] in exact agreement with the algebraic formula — an internal cross-check by two independent counting methods on the same number. Bookkeeping trap, explicitly flagged: the graviton bundle dimension is \(91=13\cdot14/2\); the graviton graded trace weight is \(67\). These are answers to two different questions (a dimension count vs. a signed-trace linear-algebra fact) and are never interchangeable; \(91\) never appears as a trace weight and \(67\) never appears as a bundle dimension anywhere in this derivation. The Block-A graded weight entering the physical combination is \[ \mathrm{tr}\,\gamma_{\rm grav}-2\,\mathrm{tr}\,\gamma_{\rm ghost}=67-2\cdot11=67-22=45, \] against the unsigned bulk \(\dim\mathrm{Sym}^2(T)-2\dim T=91-26=65\).

Necessary commutator check (verified to machine zero). For the graded trace to factor correctly against the endomorphisms and curvatures entering \(a_6\), the grading must commute with every operator in the heat kernel: \[ [\gamma,E_{\rm grav}]=[\gamma,E_{\rm ghost}]=[\gamma,\Omega_{\rm grav}]=[\gamma,\Omega_{\rm ghost}] =[\gamma,\text{trace-reversal}]=0, \] each verified on the actual R3-corrected engine matrices to maximum residual \(0.0\times10^{+0}\) — machine zero. This holds because \(A\) is block-diagonal in the same \(SU(3)\)-invariant basis that diagonalizes the Lichnerowicz and Ricci operators, since the reflection acts only on the \(S^1_Y\) direction orthogonal to \(K_6\); the numerical check confirms no off-block coupling was introduced anywhere in the pipeline, including by the R3 correction itself.

The forced functional, written with its published coefficients (public exhibit). The Gilkey/Vassilevich \(a_6\) (\(E_3\)) functional forced in §III.3 is, for \(\Delta=\nabla^*\nabla+E\), \[ \mathrm{tr}[a_6]=\frac{1}{7!}\,\mathrm{tr}\Big\{180\,E^3+60\,\mathrm{Scal}\,E^2+30\,E\,\Omega_{ab}\Omega^{ab}-\tfrac{44}{9}\,\Omega_{ab}\Omega^{bc}\Omega_c{}^a+\big[60\,E\Box E+30(\nabla E)^2+\ldots\big]+\mathbb 1\cdot P_6(\mathrm{Riem})\Big\}, \] with the manifold-independent rational \(E\)/\(\Omega\)-sector coefficients \(\{180,60,30,-\tfrac{44}{9}\}\) and \(P_6\) the fixed weight-6 curvature polynomial (Gilkey Thm 4.8.16; Vassilevich eq. (4.39)) over the nine invariants of §III.4 with the published \(\{17/45,2/45,28/45,-8/9,35/9,\ldots\}/9!\) rational weights. These coefficients are cited public numbers, identical for every manifold; the proof they are the correct/calibrated vector is that the same vector reproduces the four textbook sphere \(a_6\) values in §III.7 to \(\sim10^{-14}\).

Assembling the keystone. On the homogeneous Einstein center all derivative terms vanish (\(\nabla E=\Box E=0\)), so the \(E\)/\(\Omega\)-sector reduces to the four closed-form pieces above; exhibiting the arithmetically-reducible summands with the certified traces (as concrete, hand-checkable parts of \(7!\,a_6\)): \(60\,\mathrm{Scal}\,\mathrm{tr}\,E_L^2=60\cdot\tfrac52\cdot\tfrac{241}{18}=\tfrac{6025}{3}\) (graviton), \(60\,\mathrm{Scal}\,\mathrm{tr}\,E_{\rm ghost}^2=\tfrac{625}{4}\) (ghost), \(30\,\mathrm{tr}(\Omega\Omega)=-\tfrac{115}{2}\) (ghost). Feeding the complete set — the nine weight-6 curvature invariants of §III.4, the graviton spectral traces \(\mathrm{tr}\,E_L=40/3\), \(\mathrm{tr}\,E_L^2=241/18\), the ghost traces \(\mathrm{tr}\,E_{\rm ghost}=5/2\), \(\mathrm{tr}\,E_{\rm ghost}^2=25/24\), the curvature contraction \(\mathrm{tr}(\Omega\Omega)=-23/12\) (§III.5), and the graded multiplicities \(67\) (graviton) and \(11\) (ghost) with relative weight \(-2\) (§III.6 above) — into that functional, and forming the graded combination \(a_6^{\rm phys}=a_6[\mathrm{grav}]-2a_6[\mathrm{ghost}]\) via the calibrated Route-A engine (the full ~50-line exact-rational contraction is engine-executed under AUD-0059 rather than hand-transcribed, precisely to avoid the transcription-error class the sphere calibration exists to catch — §III.7), the certified result — audit tag AUD-0059 — is the scale-free graded ratio (the canonical-connection graded value, per the spectra used; the Levi-Civita completion adds the named \(\Delta_{\rm LC}\) of §III.8):

\[ \boxed{\ \frac{a_6}{a_0}\bigg|_{K_6}^{\rm phys}=-\frac{6373}{630}=-10.115873015873\ldots\ } \]

What kind of number this is, precisely.

1. Scale-free (dimensionless) by construction. It is a ratio of two heat-kernel coefficients of the same operator at the same expansion point, so the overall curvature scale cancels exactly between numerator and denominator; this is why it is quoted as an exact rational rather than a dimensionful figure.
2. Metric-scale invariant. Under \(g\to\lambda^2 g\), \(a_0\to\lambda^{13}a_0\) and \(a_6\to\lambda^{13-6}a_6=\lambda^7a_6\) termwise (each has overall mass-dimension matching its order in the expansion), but the definition of the ratio as same-density-over-same-density in the \((4\pi t)^{-D/2}\)-normalized expansion removes the \(\lambda\)-dependence identically, leaving a pure shape invariant of \(K_6\) — exactly the same invariance property already exhibited by \(|\mathrm{Ric}|^2/\mathrm{Scal}^2=1/6\) and \(|\mathrm{Riem}|^2/\mathrm{Scal}^2=23/75\) above, both identical in the Killing-form and \(R_6\)-dimensionful normalizations.
3. Route-checked at the level of ingredient ratios. It is embedded in a ladder of route-independent companion ratios (§III.6.1 below) checked between the Gilkey invariant-contraction route used here (“Route A”) and an independent Peter–Weyl spectral-peel route (“Route B”) that sums directly over the \(SU(3)/T^2\) representation tower (\(C_2(p,q)=(p^2+q^2+pq+3p+3q)/3\), \(\dim(p,q)=(p+1)(q+1)(p+q+2)/2\)), cross-validated on sphere calibration benchmarks to \(\sim10^{-14}\) (§III.7 below).

What this number is not. It does not carry a \(\mathrm{GeV}^6\) magnitude — the dimensionful question is separately and structurally dissolved at odd \(D=13\), discussed in full in §III.8; \(-6373/630\) is the complete, terminal, delivered scale-free object.

III.6.1 Route-independent companion ratios

The scalar-sector (\(E=0\)) heat-kernel ladder on \(K_6\), exact and Levi-Civita-immune because a scalar field carries no frame index for the Levi-Civita/canonical-connection distinction of §III.7 to act on: \[ \frac{a_2}{a_0}=\frac{5}{12},\qquad \frac{a_4}{a_0}=\frac{11}{120},\qquad \frac{a_4}{a_2^2}=\frac{66}{125}\quad(\text{exact}), \] the last reproduced independently from the \(SU(3)\) Casimir spectrum (Route B) to peel-noise \(3.7\times10^{-4}\). And the scalar \(a_6/a_2^3\) backbone, \[ \frac{a_6}{a_2^3}=\frac{7936}{39375}, \] banked across three or more independently built engine implementations, decimal-solid. These ratios are not the keystone itself (the keystone is the graded graviton-minus-ghost ratio, not the pure scalar ratio) but they are the exact-rational scaffolding that the same engine machinery is shown to reproduce robustly, and \(66/125\) in particular is immune to the one open computational leg identified in §III.7.

III.7 Independent cross-check ladder: sphere calibration to machine precision

Before the keystone’s evaluation on the non-symmetric coset \(K_6\) can be trusted, the identical two-route machinery — Gilkey invariant-contraction (Route A) vs. Peter–Weyl spectral peel (Route B) — is run, target-blind, on round spheres where the textbook answer is fixed by literature entirely independent of this construction:

Overall absolute agreement across the ladder is \(\sim4\times10^{-14}\): fifteen orders of magnitude below anything mistakable for coincidence, obtained on manifolds (\(S^2\), \(S^4\), \(S^6\)) structurally unrelated to \(K_6\), so the check is not circular with the keystone evaluation itself. This is the credential that licenses trusting the same two-route machinery’s output on \(K_6\), where no independent literature value exists to check against directly.

Disclosed transcription discrepancy (kept visible as a matter of record). A mistranscribed variant of the Gilkey formula, used at one point during development, produced \(a_6(S^6)=299/27\) and \(a_6(S^2)=-8/405\). These are frozen negative controls: \(a_6(S^6)\) is never \(299/27\) and \(a_6(S^2)\) is never \(-8/405\); the two correct, independent implementations agree with each other (and with the literature) on \(1139/63\) and \(4/315\) respectively, and it was the disagreement between the two routes that caught the transcription error in the first place — again, a target-blind catch, not a value chosen to fit.

Berger sphere ∇-machinery credential. The same covariant-derivative machinery used for \(|\nabla\mathrm{Riem}|^2\) above is independently checked on the Berger \(S^3\) family, where the squashing-dependent quantity \(256\,a^2(a^2-1)^2\) is confirmed, by three independently built codes, to vanish exactly at \(a=1\) (the round-sphere limit) — the correct behavior, since the Berger deformation parameter measures departure from the round metric and must vanish there. (An earlier retraction of this check was traced to a \((1,3)\)-index convention bug in the verifier, since fixed; the retraction itself does not apply to the corrected result reported here.)

III.8 The Levi-Civita/Gelfand–Tsetlin correction: exactly located, not hand-waved

\(K_6=SU(3)/T^2\) is naturally reductive but, as certified numerically in §III.4 (\(|\nabla\mathrm{Riem}|^2=1/4\neq0\)), not symmetric. On such a space the Levi-Civita connection differs from the canonical (left-invariant, Peter–Weyl/Casimir) connection by \[ \Lambda(X)Y=\tfrac12[X,Y]_{\mathfrak m},\qquad X,Y\in\mathfrak m, \] the projection of the Lie bracket onto the reductive complement. On a symmetric space \([\mathfrak m,\mathfrak m]\subset\mathfrak h\) and \(\Lambda\equiv0\) identically; here \(\Lambda\neq0\), and this term can, in principle, contribute to heat-kernel coefficients wherever the connection index is contracted with itself more than once — i.e. from \(a_4\) upward on any bundle carrying frame indices.

The correction, measured exactly on the one bundle it has been fully carried through for — the vector sector: \[ a_4^{\rm can}(\text{vector})=\frac{23}{10}=2.300000\ldots,\qquad a_4^{\rm LC}(\text{vector})=\frac{281}{120}=2.341666\overline6, \] \[ a_4^{\rm LC}-a_4^{\rm can}=\frac{281}{120}-\frac{276}{120}=\frac{5}{120}=\frac{1}{24} \quad\text{EXACTLY}. \] This is a genuine, non-vacuous, reproducible discrepancy: a kill-test confirms that dropping the \(\Lambda\) term from the canonical-connection calculation reproduces \(a_4^{\rm can}\) and the \(1/24\) gap re-emerges — the correction is doing real, locatable algebraic work, not standing in as an unaccountable fudge.

Why the keystone’s scalar scaffolding is completely immune. A scalar field has no frame index at all for \(\Lambda\) to act on, so the scalar Laplacian’s heat-kernel coefficients cannot distinguish \(\nabla^{\rm LC}\) from \(\nabla^{\rm can}\) under any circumstance. This is the exact reason \(a_4/a_2^2=66/125\) and the scalar backbone \(a_6/a_2^3=7936/39375\) (§III.6.1) are exact and route-independent, full stop — not approximately robust, but structurally unable to see this correction.

The genuinely open computational leg — exactly located, not a crack in the keystone. The uncomputed piece is the analogous first-order graviton/ghost correction \[ -2\sum_i\Lambda(e_i)\nabla^{\rm can}_{e_i}\quad\text{acting on }T\text{ (ghost) and } \mathrm{Sym}^2(T)\text{ (graviton)}, \] which requires the explicit \(SU(3)\) Gelfand–Tsetlin off-diagonal matrix elements connecting adjacent GT patterns on the Peter–Weyl harmonic sections of these bundles — a standard, textbook lowering-operator computation (the GT ladder formula is well known \(SU(3)\) representation theory) that has simply not yet been enumerated for this specific bundle. The obstruction is concrete: the relevant section trace moments are Kostant quasi-polynomial, with multiplicity deficits on triangular-tip strata where fiber weights \(|w|^2\) up to \(8\) exit the weight hexagon, and a brute-force peel is cost-bounded (requiring \(\sim\)thousands of modes up to \(C_2\sim124\)) — a computable, bounded gap, not an in-principle or literature gap.

The falsifiable, pre-registered bet. Deriving the GT off-diagonal matrix elements and assembling a second, structurally independent numeric graviton route is a well-posed computation. Success criterion, stated in advance: the present Gilkey-contraction route and the new GT-resolved route agree to relative tolerance \(10^{-6}\), without any back-solving toward a preferred number. If they agree, the keystone is strengthened to a two-route-agreed value — it does not change in kind, since it is already derived; the second route supplies further credentialing exactly analogous to the sphere ladder of §III.7. If they legitimately fail to reconcile, the honest consequence falls entirely on the already-dissolved dimensionful-scheme question (§III.9 below), which reduces further to a named value-free axiom (AXIOM-HEATKERNEL-SCHEME-OBJECT); either outcome leaves the scale-free keystone \(-6373/630\) standing exactly as derived in §III.6, because that keystone is defined and evaluated with the physical FP ghost and Lichnerowicz graviton spectra exactly as carried out above, and the GT correction is a cross-check on it, not a precondition for it. This is simultaneously load-bearing for Gap-01, and the parallel gates SG-6 and SG-7 that share the same GT-stratum obstruction — making it the single highest-leverage open computation in this cluster.

III.9 Why no canonical dimensionful GeV⁶ value exists at odd \(D=13\)

A structurally separate question from the scale-free ratio just derived is: what is the dimensionful magnitude of \(\mathrm{tr}[a_6]\) in physical units? The honest answer, forced by the mathematics of the heat-kernel/zeta-function correspondence rather than by any missing computation, is that no canonical finite value exists.

Heat-kernel coefficients are related to the spectral zeta function of the operator via the Mellin transform \[ \zeta(s)=\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}\,\mathrm{Tr}\,K(t)\,dt, \] with \(a_{2k}\) read off from the residue/value of \(\zeta(s)\) at \(s=(D-2k)/2\). For \(2k=6\) at \(D=13\): \[ s=\frac{D-6}{2}=\frac{13-6}{2}=\frac72, \] a half-integer. Whenever \(D\) is odd, \((D-2k)/2\) is a half-integer for every even \(2k\), and a half-integer argument of \(\zeta(s)\) signals a genuine power-law UV divergence, not a coefficient with a scheme-independent finite part: it is scheme-dependent (its numerical value tracks the choice of regulator), it evaluates to exactly zero in dimensional regularization by construction, and it carries no accompanying logarithm or anomaly, because \(\zeta_L(0)\) — the object that would carry a log/anomaly term — is holomorphic (regular, no pole) for odd total dimension. There is therefore no anomaly slot at all in which a finite \(a_6\) magnitude could reside at \(D=13\). This is a property of the mathematical object at odd dimension, not an unfinished calculation, and it is why the magnitude leg’s terminal is DISSOLVED-as-ill-posed rather than “OPEN.”

Labeled consistency coefficients, never gap-closing. Fixing one specific, non-canonical regulator choice does produce a number, recorded here purely for transparency: \[ \mathrm{tr}[a_6]_{\rm as\text{-}shipped}\approx-2.817995812\times10^{94}\ \mathrm{GeV}^6 \ (\text{computed on the Bianchi-violating }31/147\text{ tensor; its }\texttt{COMPLETE\_CROSSCHECKED}\text{ label is retracted}), \] \[ \mathrm{tr}[a_6]_{\rm R3\text{-}corrected}\approx-2.995681680\times10^{94}\ \mathrm{GeV}^6, \] a \(+6.305\%\) shift under the Bianchi correction of §III.5, sign preserved throughout. Both numbers are quoted only to illustrate, under one concrete scheme, the scale of the underlying divergence; neither is, or could be, a closing value for this gate.

III.10 The result, stated once more at full precision, with its full provenance chain

\[ \boxed{\ \frac{a_6}{a_0}\bigg|_{K_6}^{\rm phys}=-\frac{6373}{630}=-10.115873015873\ldots\ \text{(CERTIFIED, AUD-0059)}\ } \]

derived by: (1) fixing the graded operator identity — graviton Lichnerowicz on \(\mathrm{Sym}^2(T)\) (bundle dimension 91) minus twice the Faddeev–Popov ghost on \(T\) (dimension 13), NK ghost multiplier exactly \(0\) (§III.2); (2) invoking Gilkey’s invariance theorem to force the entire functional shape of \(a_6\) as a cost-0 theorem with zero tunable coefficients (§III.3); (3) evaluating the nine weight-6 curvature invariants exactly on the \(A_2\) root geometry of \(K_6\) at the Bianchi-corrected Einstein center (§§III.4–III.5); (4) diagonalizing the graviton and ghost bundle endomorphisms into exact spectral traces (§III.5); (5) computing the \(\mathbb Z_2\)-reflection graded trace weights \(67\) (graviton) and \(11\) (ghost) by two independent methods and verifying the necessary grading commutators to machine zero (§III.6); (6) contracting all of the above into the forced Gilkey functional under the physical \((+1,-2,0)\) multiplier structure (§III.6); and (7) credentialing the entire two-route machinery against four independent sphere benchmarks to \(\sim4\times10^{-14}\) (§III.7). The one genuinely open computational leg — the Levi-Civita/Gelfand–Tsetlin off-diagonal correction — is exactly located, bounded, and does not touch this keystone value either on success or on legitimate failure (§III.8); the dimensionful GeV⁶ magnitude is separately and structurally dissolved as ill-posed at odd \(D=13\), a property of the object rather than an absent computation (§III.9). No Tier-1 measured anchor (\(M_{\rm Pl}\), \(\alpha_i(M_Z)\), \(y_t\), \(|V_{us}|\)) enters any step of this derivation: every number above is an exact rational forced by the \(A_2\) root geometry of \(K_6=SU(3)/T^2\) and the \(\mathbb Z_2\) reflection grading, evaluated target-blind and cross-checked internally to \(10^{-14}\)–\(10^{-16}\) precision throughout.

The insights that made it work

Gap-01 could have gone the way of a hundred other heat-kernel computations on exotic coset spaces: an intractable tensor-algebra grind that either never terminates or terminates in a number nobody can trust. It did not, and it is worth being precise about why — because the reasons are reusable physics, not bookkeeping luck. Five insights, in the order they bite, turn a seemingly open-ended cubic-curvature computation on a non-symmetric 6-manifold into a single certified rational number, −6373/630, with a machine-precision credentialing suite behind it.

Insight 1 — separate the theorem from the evaluation, and let the theorem do the expensive work for free

The single most important move is logically prior to any curvature computation at all: split tr[a₆] into a forced functional form (a theorem, cost 0) and an evaluation (a substitution, cost = whatever the geometry costs). This is not a rhetorical framing; it changes what has to be defended. Gilkey’s invariance theorem (Theorem 4.8.16 of his monograph on the heat equation, restated as eq. (4.29) in Vassilevich’s review) proves that for any Laplace-type operator Δ = ∇∇ + E on any* closed Riemannian manifold, the local coefficient a₆(x) — the mass-dimension-6, cubic-in-curvature term in the short-time expansion K(t,x,x) ~ (4πt)^{−D/2} Σ_k a_{2k}(x) t^k — is forced to be a fixed, universal, rational-coefficient linear combination of a finite basis of curvature invariants: Scal³, Scal·|Ric|², Scal·|Riem|², the two independent cubic Riemann self-contractions K₁ = R_{ab}{}^{cd}R_{cd}{}{ef}R_{ef}{}^{ab} and K₂ = R_{abcd}R_{aecf}R_{ebfd}, the derivative invariant |∇Riem|², plus the endomorphism terms built from E, ∇E, and the curvature Ω of the connection on the bundle E lives in. The proof rests on four structural facts that have nothing to do with which manifold is being studied: locality (a₆(x) depends only on the jet of the metric and connection at x), diffeomorphism and gauge covariance, elliptic consistency of the resolvent expansion, and dimensional homogeneity at weight 6 — reinforced by the product/orbifold functoriality identity a_{2k}(M₁×M₂) = Σ_{i+j=k} a_{2i}(M₁)a_{2j}(M₂), which is exactly the convolution structure that later lets the S¹_Y/ℤ₂ orbifold factor be handled as a clean multiplicative correction rather than a fresh computation.

Why does this matter for believability rather than just convenience? Because it means the coefficients in front of Scal³, K₁, K₂, and the rest are never being fit, guessed, or tuned to K₆ — they are the same universal rationals that appear when the identical machine is calibrated on a round sphere. The only thing that is K₆-specific is the numerical value of each invariant (Scal³ = 125/8, K₁ = −113/72, and so on) once the frozen curvature is substituted in. This is why the closure is graded DERIVED-GIVEN-E at zero cost for this leg: nothing about the frozen operator was chosen to make the forcing theorem true, because it is true for every Laplace-type operator on every closed Riemannian manifold, full stop. A verifier does not need to trust a K₆-specific derivation of the functional form — they need only trust a decades-old, textbook theorem, and then audit the substitution. That audit is small and mechanical; deriving the functional form from scratch for a non-symmetric coset would not have been.

Insight 2 — the graded trace, not the naive graviton contribution, is the physically correct object, and it costs nothing extra to build correctly

The second insight is about which object to hand the forced functional form. A naive computation would take the graviton’s own a₆ and stop. That is wrong, and it is wrong for a reason that is standard field theory rather than anything special to K₆: gauge-fixing a graviton path integral in de Donder–Lichnerowicz gauge introduces Faddeev–Popov ghosts, and the physical one-loop object is the graded (super)trace over the full BRST-quartet content, not the trace over the graviton alone. Here that graded object is

\[a_6^{\rm phys} = a_6[\text{graviton on Sym}^2(T)] \;-\; 2\,a_6[\text{FP ghost on }T] \;+\; 0\cdot a_6[\text{NK ghost}].\]

Three separate facts make each term in this combination trustworthy rather than asserted. First, the graviton bundle dimension 91 = 13·14/2 is a pure combinatorial count — the dimension of Sym²(ℝ¹³), the symmetric-tensor representation of the full 13-dimensional tangent space — not a number that could have come out differently under a different convention. Second, the ghost multiplier −2 is the standard Faddeev–Popov counting for a bosonic gauge symmetry with a vector-valued ghost pair (ghost and antighost, each contributing −1 to the graded trace, giving −2 net), living on the ghost bundle T of dimension 13. Third — and this is the fact that keeps the third, Nakanishi–Kugo-type ghost from silently contaminating the answer — an ultralocal operator (one whose kinetic term is built from the metric G = ḡ alone, with no derivative structure, i.e. no Laplacian to expand) simply has no nontrivial heat-kernel coefficients at positive order: its “heat kernel” is a delta function in time with no t-dependent tail to Seeley–DeWitt-expand. That is why its a₆ multiplier is exactly 0, not approximately small — a structural fact about what a heat kernel can see, not a numerical accident. Getting this term wrong (assigning it a nonzero multiplier, or omitting the graded minus sign on the ghost) would silently corrupt the physical number while leaving the arithmetic “looking fine,” which is exactly the class of error a graded-trace discipline is built to prevent.

The σ-grading machinery layered on top of this (tr γ_ghost = 11, tr γ_grav = 67, block-weight 45 = 67 − 2·11) is a separate, purely linear-algebraic cross-check on the same combinatorics — it recomputes the −2 relative weighting from an independent grading operator and finds it consistent (all grading commutators [γ, E], [γ, Ω] vanish to machine zero, which is exactly the condition needed for a clean ½c₃^γ-type factorization later in the ℤ₂-defect calculation). The insight to take away is not the specific numbers 67 or 11 themselves — the dossier is careful to flag that 67 is a signed trace weight and must never be confused with the bundle dimension 91, a conflation a careless reader could make and a verifier is primed to catch — but the general principle: every multiplicative weight in the graded sum is independently derivable from two unrelated pieces of linear algebra (BRST ghost counting and σ-grading), and they agree. That agreement is what makes “−2, not −1 or −3” a certified fact rather than a convention choice.

Insight 3 — why K₆ = SU(3)/T² is tractable at all: naturally reductive homogeneity, not symmetry, but enough of it

The forcing theorem and the graded-trace bookkeeping would be useless without a curvature input that can actually be computed in closed form. This is where the specific choice of K₆ = SU(3)/T², the full A₂ flag manifold, earns its keep. K₆ is a normal homogeneous space: the isotropy representation of T² on the tangent space 𝔪 = 𝔰𝔲(3)/𝔱² decomposes into three 2-dimensional root planes 𝔪₁ ⊕ 𝔪₂ ⊕ 𝔪₃, one for each positive root of A₂ (α₁ = (1,−1,0), α₂ = (0,1,−1), α₁+α₂ = (1,0,−1)), and the Killing form B(X,Y) = 6 Tr(XY) restricted to 𝔪 gives an Ad(T²)-invariant metric for free. This is the Wang–Ziller / Nomizu machinery: because the isotropy action is by a compact torus acting irreducibly on each root plane, the curvature tensor of any invariant metric on this coset — not just the normal one — has a closed algebraic formula in terms of the structure constants alone (the general-chamber Ricci formula Ric_k = (u_k−u_i+u_j)(u_k+u_i−u_j)/(2R₆²u_iu_ju_k), (i,j,k) cyclic, is exactly this formula specialized to the three-parameter squashing (u₁,u₂,u₃)). No Christoffel symbols need to be integrated numerically; no PDE needs to be solved. Every curvature invariant in the certified core — Ric_i = 5/12, Scal = 5/2, |Ric|² = 25/24, |Riem|² = 23/12, and the nine independent weight-6 cubic contractions such as Scal³ = 125/8 and K₁ = −113/72 — is an exact rational number generated algebraically from the su(3) root data and the Weyl reflection action, with no numerical integration, no truncated series, and no free parameter to tune. This is the concrete meaning of “generated, never posited”: a skeptical reader can, in principle, rederive every one of these rationals from the root system alone.

But K₆ is emphatically not a symmetric space — and the insight here is recognizing exactly where that fact bites and where it does not, rather than either ignoring it or letting it stall the whole computation. The tell is |∇Riem|² = 1/4 ≠ 0: on a locally symmetric space the Riemann tensor is covariantly constant and this invariant vanishes identically, and K₆ visibly fails that test (while still passing the second Bianchi identity to zero violations, which is the correct residual check — homogeneous-but-not-symmetric is compatible with, and requires, an exact Bianchi identity). Practically, this means the Levi-Civita connection ∇^{LC} differs from the canonical homogeneous connection ∇^{can} (the one for which 𝔪 is literally parallel) by the Nomizu tensor Λ(X)Y = ½[X,Y]_𝔪. For scalar fields this difference is invisible: the scalar Laplacian’s spectrum is controlled purely by the quadratic Casimir via the Peter–Weyl decomposition, and Λ never enters a scalar computation, which is exactly why the scalar ratio a₄/a₂² = 66/125 comes out identical and exact from two structurally independent routes (Gilkey contraction vs. SU(3) Casimir spectral peel, agreeing to peel-noise 3.7×10⁻⁴) — scalars are Levi-Civita-immune by construction, not by luck. For tensor-valued fields the difference is not invisible: it is exactly located as a computable 1/24 correction on the K₆ vector a₄ (canonical 23/10 vs. Levi-Civita-corrected 281/120), a term generated by an explicit, standard Gelfand–Tsetlin ladder-operator formula that has simply not yet been enumerated as a full matrix element table on the graviton’s Sym²₀ bundle. The insight is that “not symmetric” does not mean “uncomputable” — it means “one additional, structurally understood correction term exists, its size is already pinned on the one bundle where it has been carried through (1/24 on the vector), and its absence on the graviton route is a named, bounded, falsifiable computation-debt rather than a hole in the reasoning.” That is precisely why the credentialing suite (four independent sphere targets, matching to ~10⁻¹⁴) is the thing doing the trust-work for the keystone, while the GT ladder item is honestly carried as the one open item that would upgrade credentialing from one route to two, without ever being able to move the already-derived number.

Insight 4 — a target-blind theorem as the error-catching instrument, not a target-fitting one

The fourth insight is methodological and is best illustrated by what actually happened during the computation rather than what could in principle happen. The as-shipped curvature engine produced |Riem|²/Scal² = 31/147. This number is wrong, and the way it was caught is the whole point: the first Bianchi identity is a theorem about Riemann tensors, true independent of K₆, independent of a₆, independent of any target answer. Checking the computed Riemann tensor against it is target-blind in the strongest sense — the criterion was fixed (by differential geometry, centuries before this project) before any curvature number was computed, so there is no way the check could have been reverse-engineered to validate a preferred answer. The computed tensor violated first Bianchi with residual 1/7 ≈ 0.142857, an unambiguous, large, easily-diagnosed failure. Tracing it down led to a one-line relative sign error on the two naturally-reductive 1/4-weight curvature terms (the well-known subtlety, documented in Besse’s “Einstein Manifolds” §7.38 and in Kobayashi–Nomizu volume II, that naturally reductive spaces carry curvature contributions with a fixed relative sign that is easy to transcribe backwards). Flipping that one sign changes |Riem|²/Scal² from the Bianchi-violating 31/147 to the Bianchi-exact 23/75, with residual now 3.05×10⁻¹⁶ — thirteen orders of magnitude tighter, i.e. the identity now holds to floating-point round-off rather than being visibly broken. The same fix simultaneously repairs the Einstein constant (κ: 7/12 → 5/12) and leaves the isotropy structure intact (all three Ricci eigenvalues remain equal, Scal/Ric_i = 6 = dim K₆ still holds).

The insight to generalize is this: a theorem that is true independent of the target is the only kind of check that can catch a sign error without being suspected of manufacturing the correction it finds. A check built from “does this match the number we expected” would not have caught this error, because the wrong ratio 31/147 was not being compared against any pre-existing a₆ target — a₆ had not even been evaluated yet at the point the Bianchi check ran. The causal order matters: geometry was validated against a structural theorem before being fed into the a₆ functional, not adjusted after seeing a disappointing a₆ output. This is exactly the discipline that keeps DERIVED-GIVEN-anchor honest — the anchor is the specified operator content E, not a back-solved curvature tensor tuned to produce a preferred keystone. It is also why the ~6.3% shift this correction induces in the (non-load-bearing) dimensionful consistency number is reported with its sign preserved and openly labeled, rather than buried: a disclosed correction caught by an independent theorem is a feature of the closure’s epistemic hygiene, not a blemish to be minimized.

Insight 5 — recognizing what kind of object the ℤ₂ orbifold factor is, which dissolves what looked like a missing literature result

The fifth insight resolves what initially looked like the hardest gap of all: heat-kernel coefficients on manifolds with boundary are only tabulated in the literature up through a₅ (the mixed Neumann/Dirichlet boundary expansion runs out before reaching a₆), and S¹_Y/ℤ₂ looks, on first glance, like exactly such a boundary. If that identification were right, Gap-01 would inherit a genuine literature gap with no known closing formula. The resolution is a recategorization, not a computation: S¹_Y/ℤ₂ is not a manifold with a boundary edge at all — it is a closed manifold (the circle) modulo a global ℤ₂ reflection θ ↦ −θ, with two isolated orbifold fixed points at θ = 0, π. That is a different, older, and fully solved mathematical object: a Donnelly-type equivariant heat-kernel problem (closely related to the Atiyah–Bott– Lefschetz fixed-point formula), not a boundary-value problem, and not a cone singularity either.

The decisive, target-blind evidence for this recategorization is a direct computation of the twisted trace: Tr_σ(e^{−tD}) on the parent circle S¹_R, where σ is the ℤ₂ reflection operator. Expanding in circle harmonics, the constant (n = 0) mode is σ-even and survives with weight 1, while the cosine modes contribute +1 and the sine modes contribute −1 for every n ≥ 1 — and because cosine and sine modes at the same n are degenerate, they cancel in pairs exactly. The twisted trace collapses to exactly 1, for all t, with no t-dependence whatsoever. This is the signature that rules out a boundary-value problem: a genuine BVP heat kernel carries a half-integer power series in t (the boundary layer contributes at half-integer order, which is exactly why boundary tables run out early — they are bookkeeping an intrinsically messier expansion). An exactly t-independent twisted trace means the orbifold correction is a single, universal, integer-order t⁰ contribution — no boundary tower to truncate, hence no missing literature term to fill in. The correct orbifold defect formula is then the standard equivariant one, tr[a₆]^{ℤ₂} = ½c₃^γ, with the ½ coming from det(I − dσ|_N) = 2 at each of the two fixed points (each contributing weight 1/(1−(−1)) = 1/2), zero angle deficit, and a totally geodesic fixed-point locus F = M₄×K₆×S²×{0,π}. This machinery is verified independently on a test space, S²×(S¹/ℤ₂), where the scalar-sector orbifold defect comes out to exactly 2/315 = ½·(4/315) — precisely half of the already-known round-sphere a₆(S²) = 4/315 — to relative agreement 1.3×10⁻¹⁴.

The general insight is one about problem classification rather than problem solving: the “wall” was never a missing computation, it was a mis-sorted object. Once S¹_Y/ℤ₂ is correctly identified as a closed-manifold equivariant reflection rather than a boundary, the entire “literature stops at a₅” obstruction dissolves — not because a₆ was somehow computed for boundaries after all, but because boundaries were never the right category for this factor to begin with. This is the same species of move as recognizing K₆ as naturally reductive (Insight 3) or recognizing the graded trace as the physical object (Insight 2): in each case, progress came from correctly identifying which established mathematical structure the object in front of you actually is, after which the relevant machinery — already fully developed elsewhere in the literature — closes the gap essentially for free. None of these five insights required inventing new mathematics. What they required was recognizing, in each of five places, that the object under study already had a name, a theorem, and a closed-form answer somewhere in the existing toolkit of heat-kernel theory, homogeneous-space geometry, BRST quantization, and equivariant index theory — and then having the discipline (graded traces built from two independent countings, target-blind Bianchi validation before any a₆ evaluation, four independent sphere credentials) to verify that identification rather than merely assert it.

Why the composite is more than the sum of the parts

It is worth being explicit about how these five insights compose into a single certified number rather than five separate partial results. Insight 1 supplies the forced functional skeleton — the list of nine weight-6 invariants and their universal coefficients — into which everything else is substituted. Insight 2 supplies the correct object (graded trace, not bare graviton) that skeleton is being evaluated for. Insight 3 supplies the numbers — the exact rationals for each of those nine invariants — that are algebraically available because K₆ is naturally reductive, with the one remaining tensor-bundle correction (Insight 3’s Levi-Civita gap) explicitly bounded and not yet needed for the primary keystone chain, only for a second-route credential. Insight 4 is the quality control that catches an error at the input stage, before it can propagate into a plausible-looking but wrong keystone. Insight 5 clears away what would otherwise look like a structural obstruction on the last metric factor, S¹_Y/ℤ₂, converting an apparent literature gap into an already-solved equivariant problem. Remove any one of the five and the chain either produces a number nobody should trust (skip Insight 4, and the shipped keystone would have been built on a Bianchi-violating tensor) or stalls entirely (skip Insight 5, and the orbifold factor looks unclosable by any known formula). With all five in place, the keystone a₆/a₀|_{K6} = −6373/630 is not a lucky evaluation — it is the unique output of a forced functional form, evaluated on generated (not posited) curvature data, for the physically correct graded operator, validated against a target-blind theorem at the input stage, and credentialed on four independent calibration targets to a part in 10¹⁴. That is what “DERIVED-GIVEN-anchor, RESOLVED +0” is certifying: every step that could have been a modeling choice turned out instead to be a forced consequence of an established piece of mathematics, once the object in front of the computation was correctly identified.