/gates/ ledger is the closure-of-record. The analysis below is the conservative least-closed-residual attack vintage published as a mid-audit record, which records those same residuals as “OPEN.”What this is. The full per-gate dossier for Gap-01: the sixth Seeley–DeWitt heat-kernel coefficient, tr[a₆], of the de-Donder graviton-plus-ghost operator on the frozen 13D branch — the program's one-loop-finiteness / UV-completion-input probe. It expands the 30-second public brief (
articles/GATE_BRIEF_GAP01.md) into the rigorous treatment a working physicist can both check and build on, and it hands the open holes to the specialist who will close them.Binding discipline (carried verbatim). STATUS-UPGRADES:0. Frozen branch
dcc66f1b2685/ manifest metaa5b1e6f9d951is READ-ONLY. The dimensionless, scale-free content is genuinely DERIVED; the dimensionful magnitude is dissolved-as-ill-posed at odd D=13 and, where reported, is a labeled consistency coefficient only; the total value is NOT emitted; the positivity functional is UNMADE (no sign asserted). given-E ≠ derivation-of-E; anchored ≠ derived; dissolved ≠ solved; AXIOM-CLOSED ≠ proven; a captured terminal log ≠ an independent reproduction. This gate touches none of the famous walls — it is a finite-matching question, not a route across the Yang–Mills mass gap or the cosmological constant.
The exact, scale-free heart of the a₆ calculation is genuinely derived and triple-checked. Four independent exact-rational sphere cross-checks (S²=4/315, S⁴=74/63, S⁶=1139/63, conformal=5/63) match the actual S^n eigenvalue heat trace to 13–16 decimal places, computed on the spectral side independently of the Gilkey contraction. (The dimensionless K₆ color factor 124/315 is also pinned, but carries a caveat — it is metric-selected, holding at the normalization Scal_K6=7.5; the genuinely normalization-robust scale-free statements are the ratios a₄/a₂²=66/125 etc. See §3.2.) That is real, recovered physics: a clean, scheme-free piece of one-loop quantum gravity on the frozen geometry, computed two structurally different ways that agree.
And the gate stays honestly OPEN, because the dimensionful magnitude rides an injected scale and — at the level that actually decides the gate — two computation routes for the K₆ heat-kernel value disagree. We say both halves plainly. The win is the win; the gap is the gap; neither is dressed as the other.
OPEN — partial-derivation banked (scale-free sector DERIVED + dual-validated; bulk magnitude reported only as a labeled consistency coefficient; total value not emitted). Direction of travel: strengthened. No status was ever upgraded. The gate's status is set by its least-closed residual, and several residuals are non-terminal (a route inconsistency in the value, a positivity functional not built, a field-level sufficiency question shared with all of quantum gravity). One open piece ⇒ gate OPEN.
It establishes, rigorously and target-blind:
It does NOT establish — and explicitly does not claim:
This is a serious candidate / partial-unification result, not a validated closure.
No one in physics has a UV completion of quantum gravity. Perturbatively quantized general relativity is non-renormalizable: at two loops the pure-gravity effective action develops a divergence proportional to a cubic-curvature (mass-dimension-6) invariant, the Goroff–Sagnetti / van de Ven term, with no counterterm available in the classical action. The heat-kernel (Seeley–DeWitt / Gilkey) expansion is the standard organizing tool for exactly this structure: for a Laplace-type operator D on a d-dimensional manifold,
Tr e^{−tD} ~ (4π t)^{−d/2} Σ_{k≥0} t^k ∫ √g tr_V[a_{2k}(x)] (t → 0+),
where each a_{2k} is a local, universal polynomial in the curvature, the endomorphism E, and the bundle curvature Ω of D, of mass dimension 2k. The coefficient a₆ (the k=3, mass-dimension-6, cubic-curvature term) is the one that governs the two-loop divergence structure and the leading UV behavior of the graviton self-energy. Its finiteness and sign are the canonical one-loop UV diagnostic.
Gap-01 asks the bounded, well-posed version of that question on the program's frozen geometry: can tr[a₆] of the frozen 13D graviton+ghost operator be computed from the frozen spectrum, is it finite and well-behaved, and would such a value contribute toward a UV completion? The named object is MO-9, the assembled basis-projected tr[a₆] vector in the cubic-curvature basis; the adjudication objects are MO-10/MO-11 (a positivity functional and its decision-grade pass-semantics) and MO-12 (the necessary-not-sufficient / sufficiency question).
The Gilkey a₆ coefficient is known in closed form. The canonical references are Gilkey's Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem (Theorem 4.8.16, the universal a₆ polynomial) and Vassilevich's review Heat kernel expansion: user's manual (Phys. Rept. 388 (2003) 279; hep-th/0306138), whose equation (4.29) is the explicit a₆ functional ("a6nobou") this gate's engine contracts. The smooth-bulk coefficient is therefore available; the difficulties are (i) actually contracting it against a specific high-dimensional graviton+ghost spectrum, (ii) the boundary/defect contribution, and (iii) the physical interpretation.
On (ii), the boundary heat-kernel tower is the genuine frontier: the published mixed-boundary-condition coefficients run out at a₅ (Branson–Gilkey–Kirsten–Vassilevich, Nucl. Phys. B563 (1999) 603; hep-th/9906144). An order-6 mixed Neumann/Dirichlet boundary coefficient does not exist in the literature. This is the obstruction the gate's earlier framing called the "missing-literature object." Part of this gate's progress (§3.5) is showing that for this geometry the relevant object is not a boundary coefficient at all.
On (iii), gauge-fixing dependence is a known hazard: off-shell graviton heat-kernel coefficients are gauge- and parametrization-dependent and become invariant only on-shell (Einstein + Λ). This is documented in Bastianelli–Bonezzi–Melis, EPJC (2022), arXiv:2206.13287, corroborated by hep-th/9809169. This bears directly on whether a positivity verdict can even be read (§3.6).
s = 7/2 (equivalently the t-power t^(−7/2) per skeleton_check.py) — a power divergence that is scheme-dependent and vanishes in dim-reg. There is no log term and no conformal-anomaly slot (a_{d/2}=a_{6.5} is non-integer, hence absent). So the dimensionful value is not a clean scheme-independent number — it is ill-posed as stated. This is not a failure of effort; it is a property of odd D.The honest community position: the total a₆ for a realistic high-dimensional graviton+ghost spectrum has not been assembled by anyone. Gap-01's contribution is to compute the scale-free part exactly, to dissolve the false boundary wall into a tractable graded-bulk object, to locate the open value-leg precisely, and to keep every honest limit on the table.
Full construction lives in the corpus chapter
…/TOE/TOE_Self_close_chapters/ch_02_gap01_uv_gravity_a6.mdand the engine…/computational_runs_2026-06-23/a6_computation/(a6_compute.py,berger_symbolic.py,bianchi_test.py). This section is the rigorous recap, grounded line-by-line in the certificates listed in §5.
The frozen 13D branch is the product orbifold M₄ × K₆ × S² × S¹_Y/ℤ₂, with K₆ = SU(3)/T² the program's clean color carrier. The geometry supplies, concretely and load-bearingly (not as decoration):
a₆^phys = a₆[grav] − 2·a₆[ghost] + 0. (Source: certificate B1_…/03_OBJECT_IDENTITY.md, §A2; E_OMEGA_OPERATOR.md §2.5.)So the geometry makes the question well-posed and pins the scale-free ratios. That is genuine work. What the geometry does not do: certify that the coefficient closes the high-energy problem, fix the dimensionful magnitude (which rides one geometry-unfixed scheme object), or determine that the operator/spectrum is unique (given-E ≠ derivation-of-E).
This is the gate's earned win. The claim is that, given the frozen Laplace-type operator, the canonical one-loop a₆ object is forced — by locality, diffeomorphism/gauge covariance, elliptic consistency, dimensional homogeneity, and product/orbifold functoriality — to be the universal Gilkey Thm 4.8.16 / Vassilevich eq.(4.29) a₆ functional, and its scheme-independent invariants are uniquely determined.
The four exact-rational sphere cross-checks. Compute the a₆ heat-trace coefficient on round spheres two structurally independent ways and require agreement:
The two routes agree on:
| Object | Value | Route A vs Route B agreement |
|---|---|---|
| S² a₆ | 4/315 | rel. error ~1.3×10⁻¹⁴ |
| S⁴ a₆ | 74/63 | rel. error ~0 |
| S⁶ a₆ | 1139/63 | rel. error ~2×10⁻¹⁶ |
| S⁶ conformal | 5/63 | rel. error ~4×10⁻¹⁴ |
(Source: 01_DOSSIER.md §2.1/W1; CERT_SCALEFREE_SECTOR_2026-06-25.md; B1_…/08_TWO_ROUTE_AGREEMENT.md; re-run 2026-06-24.) These are exact rationals reproduced on the spectral side independently of the Gilkey contraction — the textbook definition of target-blind. A grep of the verifier confirms no 2.8/94/target tokens; the rationals are not back-fitted.
The color factor 124/315. This is the dimensionless K₆ color factor that the scale-free sector pins; it is dual-validated (witness W2). Honest caveat (carry it): 124/315 is metric-selected — it holds at the specific normalization Scal_K6 = 7.5; the corpus's own declared K₆ curvature gives a different decimal. So 124/315 is the clean scale-free invariant of the validated normalization, and must be reported with that caveat, not sold as "the one target-blind ratio that drops out of nothing." The genuinely normalization-robust scale-free statements are the ratios of §3.4 (a₄/a₂² = 66/125 etc.), which are exact independent of R₆.
Why this is the real recovered physics. The scale-free sector is a cost-0 theorem of Gilkey invariance theory plus standard QFT, given the frozen operator. It is the only earned, gate-relevant DERIVED win — and it is genuinely scheme-free.
The a₆ functional contains derivative-of-curvature monomials (e.g. |∇Riem|²), so the engine must differentiate the curvature correctly. A bianchi_test.py implementing both differentiation paths on the Berger 3-sphere (the squashed S³, a stress-test with known curvature) shows:
Three independent codes agree on 256·a²(a²−1)². Conclusion: the engine's ∇-machinery is correct; the prior retraction was a (1,3)-tensor differentiation bug, not a wrong closed form (witness W3; 01_DOSSIER.md §2.2). This validates the machinery on Berger — it does not by itself validate the K₆ magnitude against any published value (there is none).
A real error sits inside the corpus a₆ engine, and the certificate CERT_CURVATURE_DELTA_2026-06-25.md (cross-confirmed by B1_…/04_R3_K6_BIANCHI_FIX.md) pins it precisely. The K₆ = SU(3)/T² curvature builder a6_compute.py::k6_riem_gilkey::Rop() writes the two naturally-reductive quarter-terms as
+1/4 [[Y,Z]_m, X]_m − 1/4 [[X,Z]_m, Y]_m
whereas the correct naturally-reductive form (Besse 7.38 / Kobayashi–Nomizu) carries the opposite sign. The consequence is checkable target-blind, because the criterion is the first Bianchi identity — a theorem every Riemann tensor must satisfy — not any a₆ value:
| Quantity | engine as-written | sign-flipped (correct) |
|---|---|---|
| 1st-Bianchi max-residual | 0.142857 = 1/7 (VIOLATES) | 3.05×10⁻¹⁶ (satisfies) |
| |Riem|²/Scal² | 31/147 = 0.21088 | 23/75 = 0.30667 |
| Ricci eigenvalues | 0.5 ×6 (isotropic) | 0.5 ×6 (isotropic) |
| Scal/Ric_eig | 6.000000 | 6.000000 |
| Einstein constant κ | 7/12 | 5/12 |
| tr[a₆]_bulk (GeV⁶) | −2.817995812×10⁹⁴ | −2.995681680×10⁹⁴ |
| Δ | — | +6.305%, sign preserved (negative) |
(Source: B1_…/04_R3_K6_BIANCHI_FIX.md; CERT_CURVATURE_DELTA_2026-06-25.md.) The engine self-check missed the error because it tested only Ricci isotropy and Scal/Ric=6, both insensitive to the sign (the Ricci contraction is unchanged). Per-channel: the Ricci/scalar-only cubics (R³, R|Ric|², Ric³, RicRicRiem) are unchanged (ratio 1.0000); only the Riemann-cubics move, diluting the ~45% jump in |Riem|² down to +6.305% in the bulk.
The correction was reproduced four target-blind ways: (1) a Nomizu Levi-Civita rebuild R = ∇∇ − ∇∇ − ∇_[·,·]; (2) a covariant 2nd-Bianchi build with |∇Riem|²=0.25; (3) the corpus's own finite-difference reference k6_fd_nablaR.py (which outputs |Riem|²=1.91667 and Ricci eig 5/12 — its stale comment says 7/12); (4) sectional-curvature structure sums giving Scal=5/2. A from-scratch rebuild with a different inner product (−tr(XY) vs the corpus's −6·tr(XY)) gives Bianchi residual 0, |Riem|²/Scal²=23/75 and |Ric|²/Scal²=1/6 exactly — zero contact with the corpus answer. The fix is a one-line sign flip; it is documented as DISCLOSED-CORRECTED / computation-debt (a correction owed), not a closure, and it does not silently flip the bulk sign (corrected bulk is still negative). The scale-free sphere cross-checks (§3.2) still pass — the fix is K₆-local.
The companion derivative sector (+1.260×10⁹² GeV⁶, −0.447% of bulk) rests on the K₆ value |∇Riem|² which has no direct external published anchor; its trust comes only from its sub-percent weight plus transfer-of-confidence from the Berger validation (§3.3). It is honestly flagged as such.
This is one of the dossier's strongest structural results, and it changes the kind of object the "defect" is. The earlier framing (and the still-live public brief) treated the S¹_Y/ℤ₂ contribution as a missing order-6 mixed Neumann/Dirichlet boundary heat-kernel coefficient, absent from the literature because the boundary tower stops at a₅. The certificate CERT_Z2_DEFECT_STRUCTURE_2026-06-25.md (and B1_…/03_OBJECT_IDENTITY.md) shows that this is the wrong object.
A global isometric ℤ₂ involution on a closed manifold is not a manifold-with-boundary BVP. Its order-6 contribution is the fixed-point term of the equivariant (Lefschetz) heat trace — the Donnelly expansion (Math. Ann. 224 (1976) 161) — which exists to all orders, has integer t-powers, no extrinsic-curvature monomials, and is built from bulk local invariants on the fixed locus weighted by the normal action. On the frozen branch:
The decisive computation: the twisted trace Tr_σ(e^{−tD}) on S¹_R/ℤ₂ = 1 exactly, t-independent (single fixed mode n=0; cos modes +1 and sin modes −1 cancel per n). That is an integer-power t⁰ Donnelly/Lefschetz series with no 1/√t half-integer boundary tower — the structural signature that there is no boundary object here at all.
Hence the exact reduction:
Tr_{M/Z2}(e^{−tD}) = (1/2) Tr_M(e^{−tD}) ± (1/2) Tr_M(σ e^{−tD})
Tr_M(σ e^{−tD}) = Tr_base(γ e^{−tD_base}) (exact factorization)
tr[a6]^{Z2-defect} = (1/2) c3^γ
where c₃^γ is the t³ (order-6) Seeley–DeWitt coefficient of the same de-Donder graviton+ghost operator on the 12d base M₄ × K₆ × S², using the same eq.(4.29) functional, same E, Ω, same K₆+S² curvature, with the fibre trace tr_V replaced by the graded trace tr_V(γ·), γ being the reflection action on the fibre. The grading constants are pure linear algebra, forced by D=13 + the reflection (zero reference to any a₆ value):
γ = Sym²(diag(1₁₂, −1)): tr γ_grav = 67, tr γ_ghost = 11,
Block-A graded weight = 67 − 2·11 = 45 (vs. bulk 65).
This is REDUCE-not-RELOCATE: the subproblem (graded a₆ of a 12d base via the already-built, sphere-validated engine plus a trivial γ insertion) is strictly easier than deriving a new research-grade boundary BVP functional. The certificate verifies the reduction with six target-blind tests to machine precision — including the decisive T5: a spinning vector field on curved S² × (S¹/ℤ₂), whose direct orbifold spectrum equals the graded reduction through all Seeley–DeWitt orders (diff ~1×10⁻¹³), confirming the grading captures all orders through a₆, not just the leading one. An independent curved scalar check on S² × (S¹/ℤ₂) gives defect a₆ = 2/315 = (1/2)(4/315) to rel. 1.3×10⁻¹⁴ (B1_…/03_OBJECT_IDENTITY.md). All grading commutators [γ, E], [γ, Ω], [γ, trace-reversal] vanish exactly on the actual engine matrices (necessary for the factorization).
The honest residue. The structure is built and the false boundary wall is dissolved; but the numeric c₃^γ is not emitted — it inherits the dimensionful scheme problem (R1) and, more sharply, the still-open Levi-Civita correction (R4, §3.7). So "no total emitted" remains dominant — but for the correct reason (a runnable computation-debt), not a literature wall.
The decision object MO-10/MO-11 is a positivity functional Π that would read a UV verdict off the total a₆. The certificate CERT_POSITIVITY_FUNCTIONAL_2026-06-25.md does the disciplined thing: it fully specifies Π across six legs (operator basis; ghost signs; gauge-fixing dependence; zero modes; boundary terms; pass/fail semantics) before any sign is read, then shows Π has no valid realization on this branch — for four reasons, none of which uses sign(a₆):
Any one reason suffices. The functional is therefore OPEN/UNMADE, sharpened to BLOCKED-as-valid-certificate; no sign is asserted. Crucially, this construction makes the gate fail-to-build — it cannot manufacture a false PASS, the opposite of target-fitting.
The dimensionful magnitude has two compounding problems. First, the odd-D dissolution: there is no canonical finite dimensionful a₆ at odd n. a₆ sits at the half-integer ζ-pole s=7/2 (a power divergence, scheme-dependent, zero in dim-reg) and ζ_L(0) is holomorphic for odd n (no log, no anomaly slot). The banked GeV⁶ number is a local density × the prefactor (4π)^(−13/2) (reproduced 7.1637×10⁻⁸), which is neither the canonical A₆ (units GeV⁻⁷) nor an object that legitimately pairs with that prefactor. So the magnitude is dissolved-as-ill-posed, admissible only as a labeled consistency coefficient.
Second — and this is the binding open hole — the scale-free ratios that would survive at fixed scheme are blocked by a route inconsistency in the K₆ value. The K₆ coset is naturally reductive but NON-symmetric, so the Levi-Civita connection differs from the canonical (Peter–Weyl/Casimir) connection by Λ(X)Y = (1/2)[X,Y]_m. The Casimir/Peter–Weyl spectral method gives the canonical-connection spectrum, which is exact at a₀, a₂ but misses higher coefficients on the non-symmetric coset. The gap is located exactly (B1_…/05_LC_CORRECTION.md):
So the two routes for the graded value disagree because one (the spectral peel) silently uses the canonical connection and inherits the 1/24-class error, while the genuine object needs the Levi-Civita Bochner spectrum. The clearest partial result is in the ghost sector — but it is partial, not a reconciliation: a structurally-independent, Gilkey-free su(3) route earned the Bochner (E=0) ghost a₆/a₀ = 149/1008 (0.147817463; also via the corpus Gilkey engine with two curvature builds; exactly −16/315 + 143/720 = 149/1008), which dissolved the named su(3) root-shell blocker for the Bochner ghost only. It does NOT reconcile the physical ghost: the corpus's binding Faddeev–Popov ghost is the Lichnerowicz/Hodge operator with E = −Ric, whose Route-A (physical) value is −251/504, so against the Gilkey-free Route-B 149/1008 the two ghost routes DISAGREE by |−251/504 − 149/1008| = 31/48 ≈ 0.646 — ~6 orders outside the pre-fixed 1×10⁻⁶ tolerance. (An earlier draft claim "ghost sector reconciled / two routes agree exactly" was caught by the verifier as target-fitting: the apparent agreement at 149/1008 came only from silently replacing the physical E = −Ric ghost with the pure-Bochner E = 0 operator. The corpus binding status is R7 = FAIL_VALUE_MISMATCH for the ghost as well as the graviton.) The graviton sector is likewise open: canonical (Bochner) and Levi-Civita (Lichnerowicz) values differ (e.g. graviton a₄ Bochner 233/210 vs Lichnerowicz 893/210), Route A's K₆-bundle a₆ fails its own canonical anchor (−43/504 vs −16/315), and the corpus carries two graviton a₄ anchors (893/210 vs 1643/210, same 2/21 gap but different absolute value) so the a₆ sign is not yet pinned. (Source: handoff first-pass plug results; PLUG_RESULTS_FIRSTPASS_2026-06-29.md; B1_…/05_LC_CORRECTION.md, 06, 06b, 07, 08, 13_FINAL_ROLLUP.md.) The disagreement sits upstream of the scheme anchor, in the SU(3) Gelfand–Tsetlin off-diagonal matrix elements that gate the 5-class Levi-Civita Lichnerowicz graviton hopping on K₆ — and it is a Bochner-vs-Lichnerowicz operator-identity ambiguity in both the ghost and graviton sectors. The honest statement is therefore route-INCONSISTENT (R7 = FAIL_VALUE_MISMATCH) for the graded value, which is stronger than "uncomputed."
Consolidating the certificates and the B1 completion ledger:
| ID | Residual (named object) | Disposition |
|---|---|---|
| R1 | Dimensionful BULK magnitude rides an injected scale | DISSOLVED-as-ill-posed at odd D=13 (no GeV⁶ value; consistency coefficient only) |
| R2 | ℤ₂ orbifold-defect identity | STRUCTURE-BUILT — Donnelly (1/2)c₃^γ; the boundary-coefficient "wall" was a wrong-object artifact |
| R3 | ~31% curvature-input error | DERIVED-VERIFIED / DISCLOSED-CORRECTED (23/75 Bianchi-exact; engine 31/147 Bianchi-violating) |
| R4 | Levi-Civita-vs-canonical correction (LC machinery) | COMPUTED/VALIDATED (latest pass, 13_FINAL_ROLLUP) — vector a₄ gap = 1/24 exact, graviton Sym²(T) gap = 2/21 exact (canonical 291/70 or 541/70, LC 893/210 or 1643/210); Ω^LC = base Riemann to 1.1×10⁻¹⁶. (Earlier pass 02_RESIDUAL_LEDGER recorded this OPEN; rollup supersedes.) Not by itself gate-closing. |
| R5 | Route A graded c₃^γ value | ASSEMBLED candidate, ANCHOR-INCONSISTENT — single-engine cubic assembled, but its K₆-bundle a₆ fails the canonical anchor (−43/504 vs −16/315, MATCH=False). Binding. |
| R6 | Route B independent value | PARTIAL — Gilkey-free Bochner-ghost a₆/a₀ = 149/1008 earned (named su(3) blocker dissolved for the Bochner ghost only); graviton Sym²(T) LC a₆ still missing (does not inherit the collapse; brute peel diverges). Binding. |
| R7 | Two-route agreement for the VALUE | FAIL_VALUE_MISMATCH — sphere scale-free sector agrees 2 routes, but the B1 coefficient does not (ghost Route A −251/504 vs Route B 149/1008, |Δ|=31/48 ≈ 0.646, ~6 orders out). Binding. |
| R8 | BRST σ-evenness sufficiency | PROVED (Phase H, 2026-06-26) — reduces to finite σ-equivariance; no longer the binding wall |
| MO-12 / R5(field) | Sufficiency: a₆ is one term in unbounded a₈/a₁₀/… tower | OPEN (field-level) — necessary-not-sufficient, shared with all QG |
| R(positivity) | Positivity functional MO-10/MO-11 | OPEN/UNMADE — no valid realization on this off-shell odd-D branch; no sign asserted |
| R(given-E) | Spectrum given, not derived | DISCLOSED (given-E) / scoped |
(Sources: B1_…/02_RESIDUAL_LEDGER.md, 13_FINAL_ROLLUP.md; the four CERT_.md certificates; 01_DOSSIER.md §3.) Gate status = least-closed residual = OPEN.*
Three quantitative objects that looked derived from the frozen geometry — the gauge threshold vector (SG-7), the moduli loop coefficient c_loop (SG-6), and the a₆ magnitude (this gate) — do not pose three independent questions. They collapse onto one shared technical decision: which heat-kernel object is the finite one-loop coefficient (a scale-invariant object, which the blind runs use and which sends the color row toward zero, vs a dimension-laden object, which the corpus uses to reach its numbers). Settle that one decision and all three numbers move together. The sharpening from the 2026-06-28/29 audit is that the obstruction lives one level deeper still: in the SU(3) Gelfand–Tsetlin off-diagonal / Levi-Civita-hopping sector that R4 names. Reconcile the routes ONCE — fix the GT off-diagonal matrix elements (5-class LC Lichnerowicz hopping) — and the fix propagates to Gap-01 (R1/R4), SG-6 (its breathing-mode singlet / FRG-2 wall), and SG-7 (its threshold row-formula normalization). This is the gate's single highest-leverage move.
These are the moves that produced the progress, now shared at working-physicist depth so a reader can reproduce and extend them.
(I) Separate the scale-free skeleton from the dimensionful flesh — and only claim the skeleton. The decisive discipline was to recognize that Gilkey invariance theory forces the dimensionless invariants regardless of any scale choice, while the dimensionful magnitude rides a scheme object the geometry does not fix. By computing the sphere cross-checks on the spectral side (direct eigenvalue heat trace, Vandermonde t³ peel) independently of the Gilkey contraction, the scale-free sector becomes a genuine two-route, target-blind theorem — and the magnitude is openly excluded from the "derived" set. This is the κ³/π discipline in action: write the object without knowing the target.
(II) Use a theorem, not the target, as the correctness criterion. The ~31% curvature error was caught and fixed not by matching a known a₆ (there is none) but by the first Bianchi identity — a law every Riemann tensor obeys. The buggy tensor fails it (1/7); the corrected one passes (3×10⁻¹⁶) and lands on the clean rational 23/75 simultaneously. One sign choice cannot fake both an exact rational and Bianchi-exactness — that double coincidence is what makes the fix trustworthy without any target.
(III) Identify the object, not just attack the wall. The biggest conceptual move was refusing to derive a "missing order-6 boundary coefficient" and instead asking what the S¹_Y/ℤ₂ fixed points actually are. A flat-circle reflection on a closed manifold is a Donnelly equivariant fixed-point problem with integer t-powers and a totally-geodesic fixed locus — emphatically not a boundary BVP and not a Cheeger cone. The diagnostic that settled it was the twisted trace being exactly 1 and t-independent (no 1/√t tower). This is the "is the problem itself the unicorn?" move (M5/M7 from the attack arsenal): the wall was an artifact of describing the wrong object. The payoff is a reduction (graded bulk a₆ via the existing engine), not a relocation to a harder problem.
(IV) Specify the decision functional before reading the answer. For positivity, the win was to freeze all six legs of Π before any sign was read, then let the functional's own well-posedness conditions render the verdict. Because the four kill-reasons are each independent of sign(a₆), the construction cannot manufacture a false PASS — it can only honestly report "not buildable here." This is anti-target-fitting by construction.
(V) Locate the gap to the exact rational, then name the machinery that closes it. Rather than declaring the value "uncomputed," the work pinned the open leg to a single exact number: the Levi-Civita-vs-canonical vector a₄ gap = 1/24, with scalars provably immune (which is why 66/125 is exact). That converts a vague "the value disagrees" into a concrete computation-debt — the −2·Σ_i Λ(e_i)∇^can correction via explicit su(3) Gelfand–Tsetlin matrix elements on Peter–Weyl harmonic sections — that a specialist can run.
(VI) Collapse three "independent" open numbers onto one decision. Recognizing that the SG-6 modulus, the SG-7 threshold vector, and the Gap-01 a₆ value all ride the same heat-kernel scheme object (and, deeper, the same GT/LC-hopping sector) turns three gates' worth of work into one load-bearing computation. This is leverage discovered by auditing what the numbers share, not by attacking each in isolation.
| W | Witness | What it shows | Status |
|---|---|---|---|
| W1 | a6_compute.py run_crosschecks() |
S²=4/315, S⁴=74/63, S⁶=1139/63, conf=5/63 — exact-rational, independent of the Gilkey contraction | PASS (re-run 2026-06-24) |
| W2 | Color factor 124/315 | Exact dimensionless ratio, dual-validated (metric-selected at Scal_K6=7.5 — carry the caveat) | DERIVED (caveated) |
| W3 | berger_symbolic.py + indep_berger.py + bianchi_test.py |
256·a²(a²−1)², null at a=1; retraction = (1,3)-tensor verifier bug | VERIFIED |
| W4 | Bulk value −2.817995812×10⁹⁴ GeV⁶ (corrected −2.995681680×10⁹⁴) | Reproduced exactly by a6_compute.py |
labeled consistency coefficient ONLY |
| W5 | ℤ₂ defect = (1/2)c₃^γ (Donnelly); curved test defect = 2/315 to 1.3×10⁻¹⁴ | Structure built; value not emitted | STRUCTURE-BUILT |
| W6 | R3 sign fix: 31/147 (Bianchi-violating) → 23/75 (Bianchi-exact); +6.305% bulk | A real engine error, corrected | DISCLOSED-CORRECTED |
| W7 | Bochner (E=0) ghost graded a₆/a₀ = 149/1008, earned Gilkey-free (su(3) peel diff 2.7×10⁻⁹; −16/315+143/720); dissolves the named su(3) root-shell blocker for the Bochner ghost | Partial: the physical (Lichnerowicz E=−Ric) ghost is −251/504, so ghost routes DISAGREE by 31/48 — NOT reconciled | REDUCE (Bochner blocker only); R7 = FAIL_VALUE_MISMATCH |
| W8 | LC-vs-canonical vector a₄ gap = 1/24 exact; scalars immune | The open value-leg, exactly located | OPEN/computation-debt |
| W9 | Π(a₆) six-leg spec + four sign-independent kill-reasons | Positivity functional non-realizable here | OPEN/UNMADE (no sign) |
| W10 | BRST σ-evenness (Phase H): [σ,Q_BRST]=0 all sectors, quartets σ-homogeneous, DeWitt measure σ-invariant | Sufficiency of the factorization grading | PROVED |
| W11 | Stale orphan a6_sphere_crosscheck.py (S²=−8/405, S⁶=299/27) |
A different Gilkey transcription that FAILS; superseded — do not invoke | housekeeping hazard |
Branch dcc66f1b2685 / manifest meta a5b1e6f9d951 (READ-ONLY). Engine .py files carry 2026-06-22 timestamps and were unmutated during all of this work; every correction (the R3 sign flip; the graded-γ variant) is a proposed drop-in run by monkeypatch in a scratchpad sandbox, REQUIRES-COUNTERSIGN before any corpus edit. The frozen geometry anchor is …/TOE/00_EXACT_GEOMETRY_ANCHOR.md and 03_FROZEN_GEOMETRY_CONTEXT.md.
The certificates name runnable scripts (in the session scratchpad, target-blind, no corpus mutation):
a6_compute.py run_crosschecks() → the four sphere rationals; skeleton_check.py → (4π)^(−13/2)=7.1637×10⁻⁸ and the odd-D pole structure (CERT_SCALEFREE_SECTOR_2026-06-25.md).R3_ground_truth.py (prints 31/147 with Bianchi 1/7 vs sign-flipped 23/75 with Bianchi 3×10⁻¹⁶); R3_corrected_bulk.py (monkeypatches k6_riem_gilkey, re-runs main(), emits corrected bulk +6.305%); k6_independent_recompute.py / k6_verify_route1.py (four-method verification); k6_fd_nablaR.py (corpus's own FD reference) — see CERT_CURVATURE_DELTA_2026-06-25.md.A1_A2_object_real.py (det(I−A)=2, twisted trace=1 t-independent), R2_factorization.py (curved defect 2/315), A2_commutators_killtest.py (grading commutators = 0), test1..test6_*.py (the six validations) — CERT_Z2_DEFECT_STRUCTURE_2026-06-25.md, B1_…/03_OBJECT_IDENTITY.md.R4_scalefree_and_LCgap.py (vector a₄ gap = 1/24 exact; scalars immune; sphere peel); the Bochner-ghost reproducers (149/1008 via two curvature builds + Gilkey-free su(3) peel — not the physical Lichnerowicz ghost) — B1_…/05_LC_CORRECTION.md, 07_ROUTE_B_SPECTRAL_C3.md, 13_FINAL_ROLLUP.md.stage5_positivity_functional.md (six-leg spec + leg-by-leg falsification tests) — CERT_POSITIVITY_FUNCTIONAL_2026-06-25.md.Do NOT invoke the stale orphan a6_sphere_crosscheck.py (a different, failing Gilkey transcription; superseded — W11).
This is the most load-bearing section. Each open hole is a work-package: precise statement, why it's hard / traps to avoid, exactly what closes it (target-blind, with the refuting result spelled out), machinery & inputs (with paths), and leverage. Physics only; firewall the device applications. Every closure must hold κ³/π discipline — written without knowing the target — or it relocates rather than closes.
(a) Precise statement. Compute the first-order Levi-Civita correction to the graviton graded coefficient c₃^γ on the non-symmetric coset K₆ = SU(3)/T². The correction is −2 Σ_i Λ(e_i) ∇^can_{e_i} with Λ(X)Y = (1/2)[X,Y]m, evaluated on the tensor bundle Sym²(T) (graviton) — and settle the same correction for the physical (Lichnerowicz E=−Ric) ghost bundle T, where only the Bochner (E=0) variant is so far earned Gilkey-free. Concretely: derive the SU(3) Gelfand–Tsetlin off-diagonal matrix elements of the su(3) generators on Peter–Weyl harmonic sections (explicit V tensor model), which gate the 5-class Levi-Civita Lichnerowicz graviton "hopping" between curvature classes on K₆. The missing object is section-level matrix elements, beyond Casimir/weight bookkeeping.
(b) Why it's hard / prior-attempt lessons. The canonical (Peter–Weyl/Casimir) spectral method gives the wrong connection's spectrum on a non-symmetric coset — exact at a₀/a₂, but off by exactly 1/24 at vector a₄ (and worse for the graviton). The first-pass plug earned the Bochner (E=0) ghost value Gilkey-free (149/1008) — but this is not a reconciliation of the physical ghost: the corpus's binding Faddeev–Popov ghost is Lichnerowicz (E=−Ric, Route-A value −251/504), so the physical ghost routes DISAGREE by 31/48, exactly like the graviton. For the graviton the route fails for a sharper reason: the section trace moments T_j^LC(p,q) are Kostant quasi-polynomial — there are multiplicity deficits on the triangular-tip strata where fibre weights |w|² up to 8 exit the SU(3) weight hexagon. Global-polynomial resummation poisons even a₂; a brute-force peel is cost-bounded (reachable max Casimir ~124; the convergence needs thousands of irreps). Traps the verifier already caught here — do not repeat them: (1) TARGET-FITTING — an attempt to back-solve the GT elements toward a known a₆ was flagged; the elements must be derived structurally. (2) OVERCLAIM — a draft claimed "ghost sector reconciled ⇒ two routes AGREE EXACTLY ⇒ R7 MET"; that is FALSE against the frozen corpus (R7 = FAIL_VALUE_MISMATCH for both sectors; the apparent ghost "agreement" at 149/1008 came only from silently swapping the physical E=−Ric ghost for the pure-Bochner E=0 operator, and pairing the (1/2)c₃^γ reduction's own spectral anchor with the spectral peel is the same route twice, not two independent routes). Keep the earned Bochner-ghost blocker-dissolution strictly separate from the still-open physical-ghost and graviton value legs.
(c) Exactly what closes it. Derive the GT off-diagonal matrix elements target-blind; assemble the LC Lichnerowicz graviton graded c₃^γ; build a genuinely structurally independent second numeric route (not a second spectral peel — e.g. a non-spectral Gilkey contraction leg vs a spectral leg, distinct the way the sphere cross-checks are distinct); then re-run the two-route agreement against the pre-fixed 1×10⁻⁶ tolerance. Success: the two routes agree to tolerance on the graded graviton value, with no number fed in. Refuting result (also a valid close): if the routes cannot be reconciled without back-solving to a known number, report that the value is not target-blind-derivable and reduce the magnitude leg to a named, value-free axiom (AXIOM-HEATKERNEL-SCHEME-OBJECT) — a negative is a legitimate terminal.
(d) Machinery & inputs. Explicit su(3) V_{(p,q)} tensor model / Gelfand–Tsetlin basis; Peter–Weyl harmonic-section expansion; the Lichnerowicz/Bochner split on naturally-reductive cosets (Λ(X)Y=(1/2)[X,Y]_m). Start from: B1_…/05_LC_CORRECTION.md (the gap = 1/24, the correction formula), 06_ROUTE_A_GRADED_C3.md, 07_ROUTE_B_SPECTRAL_C3.md, 08_TWO_ROUTE_AGREEMENT.md; the corpus a6_compute.py (Gilkey contraction leg) and graviton_fiber.py (graviton fibre data in hand); R4_scalefree_and_LCgap.py. The handoff first-pass scratch (recon scripts: k6_curv2.py, nabla_riem.py, ghost_a6_mycurv.py, RECON_ghost_a6.py, PR21_clean_pins.py) reproduces the ghost leg and the failing graviton leg.
(e) Leverage. This single fix is load-bearing for three gates: Gap-01 (R1/R4 here), SG-6 (its R5 breathing-mode singlet / 13D FRG-2 wall), and SG-7 (its R2 threshold row-formula normalization). It is ONE object, not three — reconcile the routes once and the fix propagates to all three. Highest-value work-package in the gate.
(a) Precise statement. The dimensionful bulk a₆ terminates on no measured invariant; it rides one shared, geometry-unfixed heat-kernel scheme/scale object (the SCHEME-ANCHORED cascade, §3.9). At odd D=13 there is moreover no canonical finite local t⁰ term, so the magnitude is dissolved-as-ill-posed.
(b) Why it's hard / traps. At odd n, a₆ sits at the half-integer pole s=7/2 (scheme-dependent power divergence, zero in dim-reg); there is no log/anomaly slot. Trap: the banked GeV⁶ figure (−2.817995812×10⁹⁴, corrected −2.995681680×10⁹⁴) is a local density × (4π)^(−13/2), not the canonical A₆ (GeV⁻⁷) — do not promote it to "the a₆ value." It is a labeled consistency coefficient and never gap-closing. Any object reverse-engineered to a known magnitude RELOCATES, it does not close.
(c) Exactly what closes it. Fix the shared one-loop scheme object target-blind — the κ³/π falsification test: it must be writable without knowing the target magnitude. If fixed and the magnitude lands → DERIVED-GIVEN-E. If not → Reduce to the named value-free axiom AXIOM-HEATKERNEL-SCHEME-OBJECT (AXIOM-CLOSED-pending-verification = OPEN unless the named object is verified to yield the value). Because the odd-D dissolution already shows no canonical finite value exists, the realistic terminal is the named axiom plus the dimensionless ratios from Hole 1.
(d) Machinery & inputs. CERT_SCALEFREE_SECTOR_2026-06-25.md (odd-D pole structure, skeleton_check.py); 01_DOSSIER.md §3.2 (the cascade); the SG-6/SG-7 dossiers (the shared object's other two faces).
(e) Leverage. Same shared object as Hole 1; settling the scheme object moves the SG-7 threshold vector + the SG-6 c_loop + a₆ together.
(a) Precise statement. The corpus engine's K₆ curvature builder writes a first-Bianchi-violating sign (|Riem|²/Scal²=31/147); the correct value is 23/75.
(b) Why it's hard / traps. It is not hard — it is a one-line sign flip — but it is owed, not closed. Trap: do not let R3's clean correctness inflate into gate-closure; correcting an input is necessary, not sufficient. The corrected bulk (−2.995681680×10⁹⁴) is still only a scheme-anchored consistency coefficient.
(c) Exactly what closes it. Apply the sign flip in k6_riem_gilkey::Rop() (and verify_k6_curvature.py::Rop()), re-run downstream, update GAP01_A6_FINAL.md with a disclosed-corrected note — REQUIRES-COUNTERSIGN. Success criterion (already met in sandbox): first Bianchi → 3×10⁻¹⁶, ratio → 23/75, bulk +6.305% sign preserved. Terminal as DISCLOSED-CORRECTED / computation-debt.
(d) Machinery & inputs. CERT_CURVATURE_DELTA_2026-06-25.md; B1_…/04_R3_K6_BIANCHI_FIX.md; scratch drop-ins k6_corrected_builder.py, run_corrected_engine.py, independent_bulk_recompute.py; corpus FD reference k6_fd_nablaR.py.
(e) Leverage. Removes one contamination source from the value computation (Hole 1) and from SG-6/SG-7.
(a) Precise statement. Emit the numeric graded c₃^γ and assemble the total a₆ = bulk + (1/2)c₃^γ. The structure is built (Donnelly reduction, §3.5); only the number is missing.
(b) Why it's hard / traps. Major correction to the old framing — do not repeat it: this is NOT a task to "derive a missing order-6 mixed Neumann/Dirichlet boundary coefficient." That object is a dissolved wrong-object artifact (the twisted trace is integer-power t⁰, no boundary tower; det(I−dσ|_N)=2, totally-geodesic, angle deficit 0). The public brief still uses the older "boundary coefficient, tower stops at a₅" language; the current corpus state (CERT_Z2 + B1 rollup) supersedes it. The real blocker is that the graded value inherits Hole 1 (the LC correction) — it is a single engine, so the binding two-route rule is UNMET.
(c) Exactly what closes it. After Hole 1 (LC correction) is computed: add a γ-graded variant of a6_density_operator() (Block A scalar × tr(γ); Block B fibre-traced with γ inserted), evaluate on the 12d base M₄×K₆×S², multiply by 1/2. Success: the graded value emitted with a structurally independent second route agreeing to tolerance. Refuting result: if even after Hole 1 the routes disagree, the defect value is terminal-BLOCKED and exported. No total is to be fabricated; "no total emitted" stays dominant until the value is genuinely two-route-clean.
(d) Machinery & inputs. CERT_Z2_DEFECT_STRUCTURE_2026-06-25.md (the reduction + six tests); B1_…/03_OBJECT_IDENTITY.md, 06_ROUTE_A_GRADED_C3.md; grading constants tr γ_grav=67, tr γ_ghost=11, weight 45.
(e) Leverage. Closing the total is what would let the positivity functional (Hole 5) even have an admissible input.
(a) Precise statement. Construct a decision-grade positivity functional Π(a₆_total) with pass-semantics and falsifier wiring, and evaluate it once the total exists. Until then, assert no sign.
(b) Why it's hard / traps. The functional is specified but has no valid realization on this branch for four sign-independent reasons (no admissible input; fatal off-shell gauge dependence; wrong odd-D predicate; object/predicate mismatch). Trap: do not read positivity off the bulk sign alone — that is not a verdict on the whole object, and off-shell gauge dependence means the bulk sign is not even invariant. Do not target-fit the conventions so a known number "passes."
(c) Exactly what closes it. Either (i) obtain an on-shell background without mutating the frozen geometry (likely impossible — flat × curved factors cannot be Einstein for any single Λ), plus the total value (Hole 4) and the unmade ghost-sign path-integral re-derivation, plus a theorem linking sign(a₆) to a positivity property in odd-D off-shell (nonexistent); or (ii) accept the certified verdict that Π is non-realizable here and keep it OPEN/UNMADE. A refuting result (Π provably cannot decide) is the realistic terminal.
(d) Machinery & inputs. CERT_POSITIVITY_FUNCTIONAL_2026-06-25.md (six-leg spec + falsification tests); arXiv:2206.13287 (off-shell gauge dependence); CONVENTIONS.md (residuals R-1 α, R-3 flat-vs-curved M₄); E_OMEGA_OPERATOR.md §0 (ghost-sign gating).
(e) Leverage. Gated on Hole 4; closes the MO-10/MO-11 adjudication leg if and only if a valid Π can exist.
(a) Precise statement. Whether a finite, positive total a₆ constitutes a UV completion.
(b) Why it's hard / why it is a dissolved unicorn, not an open weakness. a₆ is one term in the unbounded a₈/a₁₀/… Seeley–DeWitt tower; demanding that one coefficient certify the whole tower is a universal negative over an open-ended domain — unprovable in principle for any framework, shared with all of quantum gravity. This is dissolved as a shared ceiling, not a gap in our work. Trap: never frame necessary-not-sufficient as a hedge or a hidden weakness; it is the honest top. Never claim a finite a₆ "is" a UV completion.
(c) Exactly what closes it. No short route here; closing the field-level question is a quantum-gravity-wide problem. The bounded, honest claim — a₆ is a NECESSARY one-loop consistency check — is the ceiling. Keep necessary-not-sufficient binding.
(d) Machinery & inputs. 01_DOSSIER.md §1.2, §3.3 (R5); the continuum a₈/a₁₀ tower is separately REDUCED-TO-AXIOM (axiom-conditional on granularity / cost-floor) in the master ledger — but the finite a₆ + ℤ₂ defect survives as the real, bounded hole (Holes 1/4).
(e) Leverage. None gained by attacking it; do not spend the specialist's time here beyond keeping the scope honest.
(a) Precise statement. The operator/bundles are read off the frozen branch, not proven to be the unique forced spectrum.
(b)/(c)/(d)/(e). This is a scope export, not an a₆ hole: it is the shared geometry-selection unicorn (given-E ≠ derived-E), a universal negative over all possible spectra, exported to the selection gates. Terminal as DISCLOSED (given-E) / scoped. Do not attempt to "derive" the spectrum inside Gap-01.
What is claimed. The scale-free sector of a₆ on the frozen 13D branch is DERIVED — the color factor 124/315 (caveated as metric-selected) and the four exact-rational sphere cross-checks are forced by Gilkey invariance theory + standard QFT and reproduced two structurally independent ways to ~10⁻¹⁴; the geometric signs are derived; the ∇-machinery is Berger-validated; the ~31% curvature error is located and corrected target-blind; and and the "missing boundary coefficient" wall is dissolved into a tractable Donnelly graded-bulk object (1/2)c₃^γ. A partial value-leg result is banked but not sold as closure: the Bochner (E=0) ghost a₆/a₀ = 149/1008 was earned Gilkey-free, dissolving the named su(3) blocker for that object only — the physical (Lichnerowicz E=−Ric) ghost and the graviton remain route-inconsistent.
What is NOT claimed. No finite dimensionful tr[a₆] value (ill-posed at odd D=13; any GeV⁶ figure is a labeled consistency coefficient, never gap-closing). No total value (the graviton Levi-Civita value-leg is route-inconsistent; the binding two-route rule is UNMET). No sign of any positivity functional (unmade; non-realizable here). No claim that a finite a₆ constitutes a UV completion (MO-12: necessary-not-sufficient, a shared ceiling). No claim that the spectrum is derived (given-E).
The dissolutions are ceilings, not solutions. dissolved ≠ solved. The odd-D ill-posedness of the magnitude and the MO-12 sufficiency question are universal negatives — limits on what any framework could establish, shared with all of quantum gravity — reframed honestly as the top of the achievable, not as private gaps. The continuum a₈/a₁₀/… existence obligation is REDUCED-TO-AXIOM (axiom-conditional on granularity); only the finite a₆ + ℤ₂ defect survives as a real, bounded, plug-able hole.
Anchors paid. No measured invariant is consumed by this gate. The scale-free content terminates on the geometry + standard heat-kernel physics (genuinely derived); the dimensionful magnitude would, if pursued, terminate on one named, value-free scheme object (AXIOM-HEATKERNEL-SCHEME-OBJECT) shared across Gap-01 / SG-6 / SG-7.
The bounded, falsifiable bet. Stated exactly as it stands in the frozen corpus: the two value routes for the K₆ graded a₆ currently DISAGREE grossly — R7 = FAIL_VALUE_MISMATCH, ~6 orders outside the pre-fixed 1×10⁻⁶ tolerance — in both the physical (Lichnerowicz E=−Ric) ghost sector (Route A −251/504 vs Route B 149/1008, |Δ|=31/48) and the graviton sector. The plug-able task is to (a) deliver the missing graviton Sym²(T) Levi-Civita c₃^γ on K₆ via the SU(3) Gelfand–Tsetlin off-diagonal matrix elements, (b) settle the Bochner-vs-Lichnerowicz operator identity so the physical ghost and graviton routes are compared on the same operator (the Bochner-ghost 149/1008 is earned, but is not the physical ghost), (c) fix the route's K₆-bundle derivative/LC sector against a structurally independent route, and (d) re-run the two-route agreement within the pre-fixed tolerance — all target-blind. If the routes cannot be reconciled without back-solving to a known number, we will say so, and reduce the magnitude leg to a named value-free axiom. That same graviton/route fix is load-bearing for SG-6 and SG-7.
Final grade. Serious candidate, NOT validated — a finite-matching diagnostic that recovers the scale-free ratios and signs and dissolves a false boundary wall, not a closed UV completion. It touches none of the famous walls (the Yang–Mills mass gap stays Precisely-OPEN; Λ stays Weinberg-OPEN). STATUS-UPGRADES:0; frozen branch READ-ONLY; given-E ≠ derivation-of-E; a captured log ≠ an independent reproduction; ANCHORED ≠ DERIVED; AXIOM-CLOSED ≠ atomic; MO-12 necessary-not-sufficient is binding.
Dossier assembled from: the Gap-01 completion-handoff dossier (PER_GATE_DOSSIERS/GAP01_COMPLETION_HANDOFF/01_DOSSIER.md); the four certificates CERT_SCALEFREE_SECTOR_2026-06-25.md, CERT_CURVATURE_DELTA_2026-06-25.md, CERT_Z2_DEFECT_STRUCTURE_2026-06-25.md, CERT_POSITIVITY_FUNCTIONAL_2026-06-25.md; the B1 ℤ₂-defect completion ledger (certificates/B1_Z2_DEFECT_COMPLETION/02..13); the public brief GATE_BRIEF_GAP01.md; and the build handoff 30pagedoc/handoffs/GAP01.md. Body is rigorous recap + closure-path work plan; the corpus controls all common material. STATUS-UPGRADES:0.