/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 30–50 page dossier for UQF-4, the quantum-consistency gate of the scoped theory on the frozen 13D K₆ branch. It expands the live 30-second popup and the gate brief into a treatment a working physicist can both check and build on. It synthesizes the gate's existing closure-attack packet, its certificates, and the cross-gate findings; it invents nothing.
Binding discipline (carried verbatim from every source). STATUS-UPGRADES:0. Frozen branch
dcc66f1b2685/a5b1e6f9d951is READ-ONLY. Honest throughout: given-E ≠ derivation of E; "machinery exists" ≠ "calculation done"; a captured log ≠ an independent reproduction; a regulator chosen to make the anomaly cancel = target-fitting (REFUTES the row, does not close it); assuming the open class vanishes is REJECTED — we refuse to assume the answer.Status (must match the live popup): AUDIT (OPEN). The perturbative leg is DERIVED-GIVEN-E (independently reproduced); the residual is one bounded, falsifiable even-degree class whose value is uncomputed/contested and whose existence precondition is blocked on a named, missing datum. Direction: strengthened — sharpened and corrected, not closed.
Every Standard-Model anomaly cancels on our spectrum — and we did not take that on faith: an internal specialist re-derived it target-blind, by exact rational arithmetic, and proved with a specificity check (Σ mult·Y² = 10/3 ≠ 0) that the cancellation is a real property of this matter content, not a trivial identity. With the perturbative core banked, the entire remaining quantum-consistency question has been compressed to one mathematical object — a single even-degree boundary/η class on the descended branch — whose resolution either closes the gate or falsifies the theory outright. That compression, and an honest self-correction of an earlier over-stated framing, are the gate's genuine gains.
UQF-4 is held at AUDIT (OPEN) — a serious candidate / partial unification, NOT validated. This is the binding status from every source: the gate brief (articles/GATE_BRIEF_UQF4.md §"Current honest status"), the completion result (articles/UQF4_COMPLETION_RESULT.md §0), the closure-attack dossier (UQF4_COMPLETION_HANDOFF/01_DOSSIER.md §0), and all four certificate roll-ups. No status was ever promoted. The frozen branch is read-only. The gate's status is set by its least-closed residual, and that residual is genuinely open.
This dossier establishes, with full work shown, three real results: (i) the perturbative Standard-Model anomaly ledger cancels on the active-branch spectrum, independently reproduced by exact arithmetic and shown specific to E; (ii) classical BRST nilpotency s² = 0 holds on the descended action by the Jacobi identity of su(3)⊕su(2)⊕u(1); and (iii) the dozen scattered "open" rows of the original 16-class ledger are homogeneous components of a single Anderson-dual / Dai–Freed even-degree class — a genuine reduction, not a relabeling.
It does not establish that the gate closes. The single remaining class wears several hats that are not all the same computation: whether the invertible Freed–Hopkins/Anderson-dual classification even applies to the interacting coset-WZ sector; whether the branch admits the tangential ξ-structure that makes the class defined at all (if not, the gate is refuted because the invariant does not exist); and whether the class, once defined, vanishes. The earlier "row 17 vanishes" headline produced inside this campaign was withdrawn on governance audit as a claim-boundary violation and is not revived here. The honest residual is a bounded, falsifiable computation — and it is owed, not done.
For a quantum gauge theory to be consistent, the gauge-fixing machinery must remain self-consistent and every anomaly must cancel. In the Batalin–Vilkovisky (BV) formulation, both conditions are one equation: the quantum master equation (QME)
$$ \tfrac12 (S,S) \;=\; i\hbar\,\Delta S, $$
where $(\,\cdot\,,\,\cdot\,)$ is the BV antibracket and $\Delta$ the BV Laplacian. Solvability of the QME order-by-order in $\hbar$ is exactly "the full BRST/anomaly cocycle vanishes." Its $\hbar^0$ shadow is classical BRST nilpotency $s^2 = 0$; its higher orders are the anomalies. (UQF4_COMPLETION_RESULT.md §0; 01_DOSSIER.md §F.0.)
For a realistic theory this is not just the familiar perturbative Standard-Model anomalies. When a higher-dimensional geometry is folded down to 4D, the descent generates new anomaly classes: global anomalies, boundary anomalies on orbifold walls, bulk-to-boundary inflow terms, and the anomaly of a gauged Wess–Zumino term living on the internal coset. Completing this — non-perturbatively, for a realistic compactified model — is unsolved across the whole field. No accepted theory carries a full anomaly-descent certificate. This is a community-wide open problem, not a defect specific to this program. (UQF4.md handoff §"The gap"; brief §"What this gate must establish.")
The descended branch is
$$ \mathcal B_{\text{active}} \;=\; \mathcal M_4 \times K_6 \times S^2 \times S^1_Y/\mathbb Z_2, \qquad K_6 = SU(3)/T^2 \ \text{(the flag manifold)}, $$
with gauge group $G = (SU(3)\times SU(2)\times U(1))/\mathbb Z_6$. The isometries route the group: $K_6 \to SU(3)$, $S^2 \to SU(2)$, $S^1_Y/\mathbb Z_2 \to U(1)$ (01_DOSSIER.md §D.1, §E.1). The geometry is load-bearing in a sharp way: the $S^1/\mathbb Z_2$ boundary, the bulk/boundary inflow, and the $SU(3)/T^2$ coset twist are exactly what generate the anomaly classes that have not been computed in any prior work. The geometry sets the problem precisely; it does not, by itself, certify that the problem closes. (GATE_BRIEF_UQF4.md §"The geometry's role.")
The corpus organizes UQF-4 as a 16-class anomaly ledger plus two BRST-nilpotency certificates. The honest split (GATE_BRIEF_UQF4.md; 01_DOSSIER.md §0.1):
01_DOSSIER.md §§B–E). This is the engine that lets the open rows collapse to one class — when its hypotheses hold.01_DOSSIER.md §E). So "no known route" is the honest, verified state of the literature, not a gap in the program's search. The frontier is real.UQF-4 demands two certificates: (i) $s^2 = 0$ order-by-order on the descended BV action, and (ii) every anomaly class vanishes on the physical Hilbert space after the descent under a fixed regulator. The descent chain (01_DOSSIER.md §1.1):
BV action of the descended theory --check--> s^2 = 0 order-by-order [classical: holds; quantum-descent: UNVERIFIED]
chiral content E of the branch --4D SM ledger + even/odd rules--> 10/16 perturbative classes cancel [DERIVED-GIVEN-E]
S^1/Z_2 boundary + K_6 x S^2 --Horava-Witten / Dai-Freed eta--> 5/16 boundary/global/inflow classes [AUDIT: not computed]
K_6 = SU(3)/T^2 coset twist --non-perturbative BV-BRST descent-> 1/16 coset-twist class (row 17) [the binding object]
This is the one leg fully provable now, and it was executed, not merely cited. Each of the six local anomaly polynomials was evaluated directly on the SM one-generation chiral content (left-handed Weyl convention, $Q = T_3 + Y$) by exact rational arithmetic, with the spectrum as input and the cancellation read off — never assumed (01_DOSSIER.md §F.1):
| Class | Coefficient structure on E | Result |
|---|---|---|
| $[U(1)_Y]^3 = \sum \text{mult}\cdot Y^3$ | rational sum over the generation | 0 |
| $[\text{grav}]^2\,U(1)_Y = \sum \text{mult}\cdot Y$ | rational sum | 0 |
| $[SU(2)]^2\,U(1)_Y = \sum_{\text{doublets}} N_c\,Y$ | $3\cdot\tfrac16 - \tfrac12$ | 0 |
| $[SU(3)]^2\,U(1)_Y = \sum_{\text{triplets}} N_w\,Y$ | $2\cdot\tfrac16 - \tfrac23 + \tfrac13$ | 0 |
| $[SU(3)]^3$ (triality) | $Q_L(+1) + u_c(-1) + d_c(-1)$ by weak mult. | 0 |
| Witten $[SU(2)]$ mod 2 | # doublets $= 3 + 1 = 4$ | even ⇒ 0 |
Independent reproduction. A fresh script, using exact Fraction arithmetic and its own spectrum table (not the corpus's), recomputed all six to zero, and the diagnostic sum $\sum \text{mult}\cdot Y^2 = 10/3 \neq 0$ confirms the cancellation is specific to this E — not an artifact of a trivially vanishing sum. This matches the independently-authored SG-4 anomaly ledger to the digit (01_DOSSIER.md §F.1; UQF4_COMPLETION_RESULT.md §0.1, §5 R1). Two independent derivations agree.
Disposition: DERIVED-GIVEN-E. Real recovered physics — given E. The falsification test was applied: this is a filter, not a selector (§3.7). It is the degree-0 / free part of the single obstruction class evaluated on E.
On the descended BV action, $s$ acts on fields, ghosts $c^a$, and antifields with
$$ s\,c^a = -\tfrac12 f^a{}_{bc}\,c^b c^c, \qquad s\,\phi = R^a\phi\,c_a . $$
Classically $s^2 = 0$ is equivalent to the Jacobi identity of the gauge algebra plus closure — both inherited from the descended algebra $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$, which is a Lie algebra, so Jacobi holds (01_DOSSIER.md §F.3). This is genuinely provable and holds. The quantum statement — that the BV Laplacian $\Delta$ of the descended measure (KK tower + orbifold projection + the coset ghost sector of $K_6 = SU(3)/T^2$) produces no $\hbar^1$ obstruction — is the $\hbar^1$ component of the same cocycle, and it does not separate from the coset class (§3.6). So the classical leg is VERIFIED; the quantum-descended leg is OPEN, and it is the same object as the binding blocker.
The load-bearing structural result: after the perturbative classes cancel, the remaining global/boundary/inflow obstructions are homogeneous components of one class — the partition-function phase as an Anderson-dual bordism invariant on the active branch (Freed–Hopkins / Grady engine). Concretely the formerly-scattered rows are one even-degree class seen under several lenses: the order-$\hbar$ BV-Laplacian obstruction, the $K_6\times S^2$ Dai–Freed row, the boundary η-phase, and the existence-of-tangential-structure input (UQF4_COMPLETION_RESULT.md §2; 01_DOSSIER.md §F.5 "Unification check").
This lowers the count of independent open objects to (essentially) one rather than shuffling the mystery into a new premise. The harder-subproblem check passes: the Phase-2 task is one bordism/η computation, strictly smaller than "verify 16 rows + 2 certificates" (certificates/UQF4_OMEGA_REAL/00_RESULT.md §3). But the unification is structural, not a closure — naming the single class is not evaluating it.
The Grady engine classifies anomalies of a reflection-positive invertible theory of fixed symmetry type ξ by $(I\Omega^\xi)^{d+1}(\mathrm{pt})$, the Anderson dual of the ξ-bordism spectrum $MT\xi$. So no ξ ⇒ no $MT\xi$ ⇒ no bordism group ⇒ the class is not even an element of any group (certificates/UQF4_OMEGA_REAL/03_TANGENTIAL_STRUCTURE.md §1). ξ must be pinned before the target group and before any class.
The construction distinguishes two claims, and the distinction is load-bearing:
(D) is ACHIEVED. The base is a $\mathrm{Spin}^c\text{-}(SU(3)\times SU(2)\times U(1))/\mathbb Z_6$ structure (the $\mathbb Z_6$ identifies fermion parity $(-1)^F$ with the gauge center; the hypercharge $U(1)$ plays the $\mathrm{Spin}^c$ determinant-line role on $\mathcal M_4 \times S^1_Y/\mathbb Z_2$), decorated by five geometry-forced twists (03_TANGENTIAL_STRUCTURE.md §3):
| # | Twist | Source | Effect |
|---|---|---|---|
| T-∂ | boundary / orbifold twist ($S^1_Y/\mathbb Z_2$, two walls) | the orbifold interval | ξ is a structure on a manifold-with-boundary; natural anomaly home is $d+1=5$ (HW inflow) |
| T-cs | coset twist of $K_6 = SU(3)/T^2$ (gauged flag-WZ) | the coset | adds the equivariant $SU(3)$ data; this is where $[\omega]$ enters ξ |
| T-g | gauge-bundle twist for $G = (SU(3)\times SU(2)\times U(1))/\mathbb Z_6$ | the gauge bundle | makes $MT\xi$ a Thom spectrum over (the $\mathbb Z_6$-twisted) $BG$ |
| T-6 | $\mathbb Z_6$-center twist | the center identification | part of the structure group; sets the torsion in $A$ |
| T-3 | $\mathbb Z_3$-center / $Q_1 = \beta P^1$ twist | the $SU(3)$ center | degree-5 Milnor primitive in the $\mathbb Z_3$-twisted sector; the operative R4 class |
Each twist is read off a frozen datum before any anomaly value is known — a twist would be illegitimate only if added to force cancellation. (Falsification test: hide the answer; every twist is still required. PASS.)
(E) is BLOCKED — this is OPEN-1, the spine residual. Whether the branch admits ξ is a characteristic-class / orientability / integrality / bordism-existence question, not an anomaly-vanishing question, and logically upstream of every class value. The dedicated existence certificate returns BLOCKED_MISSING_CHARACTERISTIC_DATA (certificates/UQF4_XI_EXISTENCE/00_RESULT.md). Its obstruction tower splits into five:
| # | Obstruction | Verdict |
|---|---|---|
| O1 | Orientability $w_1 = 0$ ($\mathbb Z_2$ reflection on $S^1_Y$ is orientation-reversing) | DISSOLVED — false obstruction once ξ is read as a relative/boundary Dai–Freed structure |
| O2 | $\mathrm{Spin}^c$ base $W_3 = \beta w_2 = 0$ ($\mathcal M_4$ spin$^c$ · $K_6$ Kähler/Fano spin$^c$ · $S^2$ spin) | BANKED — clean (product of spin$^c$ is spin$^c$) |
| O3 | Spin-$G$/center twist $w_2(X) + f^*\zeta = 0$ ($\zeta$ classifies $\mathbb Z_6\to\tilde G\to G$) | OPEN — load-bearing; branch $w_2$/flux data UNBUILT |
| O4 | Coset twist on $K_6$ | BANKED — likely clean (conditional on O3) |
| O5 | Boundary $\mathbb Z_2$ walls → Pin$^c$/relative Dai–Freed (shared blocker B2) | OPEN — UNBUILT (also blocks UQF-3) |
The named missing datum: the target-blind evaluation of $w_2(X) + f^*\zeta = 0$ in $H^2(X;\mathbb Z_2)$ on the branch's actual bundle data — the parities of (a) $SU(2)_L$ weak-isospin flux on $S^2$, (b) $U(1)_Y$ hypercharge flux on $S^2$, (c) Wilson-line/flux content over the two 2-cycles of $K_6$ — plus the B2 relative/Pin$^c$ boundary structure. Until both are assembled, existence is BLOCKED. If $w_2 + f^*\zeta \neq 0$ with no compensating twist, the gate is REFUTED (the invariant is nonexistent, not zero) (UQF4_XI_EXISTENCE/00_RESULT.md decision rule).
The gauged-WZ obstruction is an $SU(3)$-equivariant class of the target $K_6 = SU(3)/T^2$ (FOS/Hull–Spence). For the homogeneous space $G/H$ with $G = SU(3)$, $H = T^2$ (certificates/UQF4_OMEGA_REAL/06_COSET_WZ_CLASS.md §1):
$$ H^*_{SU(3)}(SU(3)/T^2) \;\cong\; H^*_{T^2}(\mathrm{pt}) \;=\; H^*(BT^2) \;=\; \mathbb Z[u_1, u_2], \qquad \deg u_i = 2 . $$
Even-degree, polynomial on two degree-2 generators (the $T^2$ weights). The class is built explicitly in the Cartan model of $SU(3)$-equivariant cohomology, $\Omega^*_{SU(3)}(K_6) = (S(\mathfrak{su}(3)^*)\otimes\Omega^*(K_6))^{SU(3)}$, with equivariant differential $d_{SU(3)} = d - \iota_{V_a}\varphi^a$. Take the invariant closed flag-σ WZ form $H \in \Omega^{\text{even}}(K_6)$ (degree 4 is load-bearing). Gauging requires lifting $[H]$ to an $SU(3)$-equivariantly closed form; the obstruction to that lift is
$$ [\omega]_{\text{bulk}} \;:=\; \text{(obstruction to the equivariant extension of }[H]\text{)} \;\in\; H^*_{SU(3)}(SU(3)/T^2) = \mathbb Z[u_1, u_2]. $$
It is computable in $\mathbb Z[u_1,u_2]$ from the $T^2$-isotropy weights and the cubic Casimir — but its value is a downstream computation. The construction returns a definite element whether or not it is zero (falsification test: works if nonzero. PASS) (06_COSET_WZ_CLASS.md §2).
The worldvolume dimension controls which Figueroa-O'Farrill–Stanciu (FOS) theorem applies (06_COSET_WZ_CLASS.md §3; UQF4_COMPLETION_RESULT.md §4):
| worldvolume $d$ | applicable theorem | outcome for $[\omega]_{\text{bulk}}$ |
|---|---|---|
| $d \le 3$ | FOS Cor 7.5 (target $H^1(M)=0$; $K_6$ simply connected ⇒ holds) | PROVEN VANISHING ✓ |
| $d = 4$ ($=\mathcal M_4$) | FOS Cor 7.6 — DOES NOT APPLY | OPEN / a priori nonzero — the binding part |
FOS Cor 7.6 (the only $d=4$ vanishing theorem) needs both $H^2(M) = 0$ and the group to have no cubic Casimir. For this model both hypotheses fail: $H^2(SU(3)/T^2) = \mathbb Z^2 \neq 0$, and $SU(3)$ has a nonzero cubic Casimir (the $d$-symbol). So the $d=4$ part is not protected. This is a statement about which theorem applies where, not a claim of vanishing.
The boundary $S^1_Y/\mathbb Z_2$ and Hořava–Witten inflow define a lift (06_COSET_WZ_CLASS.md §4; certificates/UQF4_OMEGA_REAL/07):
$$ [\omega]_{\text{lifted}} \;:=\; L_\partial\big([\omega]_{\text{bulk}}\big) \;\in\; A = (I\Omega^\xi)^5(\mathrm{pt}), \qquad L_\partial = \delta_{DF}\circ\sigma_\partial\circ r , $$
a composite of three standard natural maps (Borel realization $r$, the Hořava–Witten inflow suspension $\sigma_\partial: d\to d+1$, and the Dai–Freed / Anderson-dual landing $\delta_{DF}$). The $d=4$ even-degree bulk class, pushed across the boundary, becomes a $d+1 = 5$ global (η / secondary-operation) class, and $5 = 4+1$ is exactly the degree of the $\mathbb Z_3$-twisted Milnor primitive $Q_1 = \beta P^1$ (the $d_5$ AHSS differential at $p=3$, the $SU(3)$ center being $\mathbb Z_3$). The dimension arithmetic is forced by the geometry, not chosen. This is the precise content of "no known route": FOS vanishing is a $d\le 4$ worldvolume statement, and after the lift the object is a $d+1=5$ bordism/inflow class. No theorem in the literature vanishes the lifted $SU(3)/T^2$ gauged-WZ class.
$[\omega]_{\text{lifted}} \in A$ simultaneously is (06_COSET_WZ_CLASS.md §5; 01_DOSSIER.md §D.3):
Constructing it once defines all three. But "compute one number" understates it: closure requires three target-blind checks on that one class — (1) that the ξ-structure exists so the invariant is defined (OPEN-1, §3.5), (2) that the invertible Anderson-dual classification applies to the interacting coset-WZ sector (the applicability premise), and (3) that the class vanishes — plus the order-$\hbar$ descended nilpotency, which is the same class, not a separable side-check. So the gate has one mathematical object left, not a single scalar (UQF4.md handoff §"The spine"; 01_DOSSIER.md §D.4).
The map $E \mapsto (A_{Y^3}, A_{\text{grav}Y}, A_{SU2}, A_{SU3\text{sq}}, A_{SU3\text{cube}}, w_2)$ is a fixed multilinear/linear functional. Its kernel is infinite-dimensional — adding any vector-like pair $R\oplus\bar R$ leaves every entry fixed — so $O(E) = 0$ has infinitely many solutions and cannot single out E (01_DOSSIER.md §F.2). Anomaly cancellation is a consistency filter on E, never a selector of E. The "anomaly picks the spectrum" reading is a category error, DISSOLVED, and carried at the headline so the perturbative win is not over-read.
These are the moves that produced the progress — shared at working-physicist depth so the result is reproducible and the gaps are closeable.
The campaign's most useful discipline is a three-step ladder: first show the obstruction is a well-defined object (EXISTENCE), then pin which object it is (IDENTITY), only then evaluate its VALUE. Phase 1 (OMEGA-REAL) delivered only the first step: ξ, the target group $A$, the BV complex, $[\omega]_{\text{bulk}}$, the lift $L_\partial$, and $\alpha_{\text{total}}$ are all canonically constructed and pass nine well-definedness falsification tests — with the value refused (certificates/UQF4_OMEGA_REAL/00_RESULT.md). This is a genuine advance: before it, the gate had a named pathway with no defined object; after it, the obstruction is a real, target-blind, computable element of a definite group. Skipping the ladder — jumping to VALUE before EXISTENCE — is exactly how the campaign's own later over-claim happened (§4.4).
The reduction of a dozen scattered rows to one class rests on Grady's proof of Freed–Hopkins: anomalies of reflection-positive invertible theories of fixed symmetry type are the Anderson dual of the bordism spectrum. The insight that makes this honest is keeping the engine off the axiom floor: it is a theorem-with-hypotheses (reflection-positivity, invertibility, fixed ξ), not an axiom. Whether those hypotheses hold for the interacting coset-WZ sector is itself a check — the "applicability premise" — so the collapse is conditional, and the dossier says so (01_DOSSIER.md §§B–D; UQF4_COMPLETION_RESULT.md §2). The elegant catch (Round-3 audit): row 17 is not a sibling of the other rows — it is the well-definedness precondition of the collapse itself. That is why it is special, and the audit explains the specialness rather than asserting it.
The Einstein-signature move: the binding blocker is not the bulk $d=4$ class but that class lifted across the boundary into the $d+1=5$ inflow problem, where the only protecting theorem (FOS, $d\le 4$) cannot reach it. One structural fact — boundary inflow raises the effective dimension by one — makes the hard thing fall out. "No known route" becomes "the descent transports the WZ obstruction out of the $d\le 4$ regime that would have killed it." This locates the obstruction precisely instead of merely recording its mystery (01_DOSSIER.md §C.1, §E.4).
Two over-claims produced inside this campaign were corrected on adversarial audit, and both corrections are recorded here because an honest downgrade should be logged:
UQF4_COMPLETION_RESULT.md §4; 06_COSET_WZ_CLASS.md §2).The most tempting non-closure is to "close" the gate by assuming the open class vanishes. This is REJECTED everywhere in the corpus as target-fitting: assuming the answer relocates the entire mystery into one premise and would make the result worthless (01_DOSSIER.md §4.2 Route B; §A no-target-fitting self-check). The same firewall applies to regulators: a regulator chosen so the descent closes is a hidden knob and refutes the row rather than closing it. Scheme-independence is automatic only when a row is delivered in manifest bordism/η form (where it is a corollary). The gate has no number to hit — the only "value" is zero, and zero is exactly what is refused.
From 01_DOSSIER.md §2.1, with the honest reproduces-flags:
| # | Witness | Grade | Reproduces? |
|---|---|---|---|
| W1 | SM perturbative cancellation ($SU(3)^3$, $SU(2)$-Witten, $U(1)^3$, gauge-grav) | hand-checkable | Yes (given-E) — exact-arithmetic recompute; specificity $\sum Y^2 = 10/3 \neq 0$ |
| W2 | Even/odd-dimension descent inheritance | symbolic | Yes in principle; not re-derived for the branch here |
| W3 | Classical BV–BRST $s^2 = 0$ | symbolic | Yes (classical); order-$\hbar$ descended check NOT done |
| W4 | $S^1/\mathbb Z_2$ boundary anomaly (HW inflow) | machine/symbolic | NO — machinery defined, calculation not performed |
| W5 | Dai–Freed / Freed–Hopkins global rows | symbolic | NO — criterion named, bordism computation not done |
| W6 | Bulk–boundary inflow row | symbolic | NO — named, not computed |
| W7 | Coset-twist / non-perturbative descent (row 17) | — | NO — value uncomputed/contested; existence blocked |
| W8 | Artifacts U05/U08 + frozen inputs (Appendix U) | machine-lane | AUDIT — referenced, not independently re-run; back the inputs, not the open rows |
The certificate folder UQF4_COMPLETION_HANDOFF/certificates/ carries four certificates. Read what each actually certifies — the differences are load-bearing:
UQF4_OMEGA_REAL/ → STATUS = OMEGA_EXISTS. The boundary-lifted obstruction $[\omega]_{\text{lifted}}$ and the total $\alpha_{\text{total}} = \alpha_{\text{pert}} + \alpha_{\text{boundary}} + \alpha_{DF} + \alpha_{\text{inflow}} + [\omega]_{\text{lifted}} \in A$ are canonically constructed and well-defined; all six required objects (ξ-type, $A$, BV complex, $[\omega]_{\text{bulk}}$, $L_\partial$, $\alpha_{\text{total}}$) are defined; all nine well-definedness falsification tests pass. This is a statement about the OBJECT, not its VALUE. The gate stays AUDIT (OPEN) (OMEGA_REAL/00_RESULT.md).UQF4_OMEGA_COMPUTE/ → ROW 17 = OMEGA_CONTESTED. This certificate attempted the value and produced a candidate OMEGA_VANISHES (global facet $\in \mathrm{Ext}(\Omega^{\mathrm{Spin}^c}_5(B(G_{\text{SM}}/\mathbb Z_6))) = \mathrm{Ext}(0) = 0$, via DGL arXiv:1910.11277). That headline was WITHDRAWN on governance audit as a recordable verdict (OMEGA_COMPUTE/00_RESULT.md top banner). The recordable result is: gate = AUDIT (OPEN); $[\omega]_{\text{lifted}}$ value = UNCOMPUTED / CONTESTED.UQF4_ROW17_REPAIR/ → OMEGA_UNCOMPUTED / OMEGA_CONTESTED. Records the three repairs logically prior to any row-17 value, each still owed: R-deg (degree placement $(I\Omega^\xi)^5$ vs $(I\Omega^\xi)^6$ / $\Omega_4$ vs $\Omega_5$), R-id (target identity — canonical refined-PSU(3)/one-form-center object vs bare $B(G_{\text{SM}}/\mathbb Z_6)$; spin$^c$-with-B–L vs pure-gauge), R-2rt (two genuinely independent routes; DGL read multiple times is one lineage).UQF4_XI_EXISTENCE/ → BLOCKED_MISSING_CHARACTERISTIC_DATA. The OPEN-1 existence check (§3.5): O3 and O5/B2 open, missing datum = the target-blind $w_2 + f^*\zeta$ evaluation plus the relative/Pin$^c$ boundary object.The candidate "$[\omega]_{\text{lifted}} = 0$" rests on three contested moves (10_ROW17_CONTESTED_DISPOSITION_2026-06-25.md): (a) a degree relocation that contradicts Phase 1's own placement of the class — the $d\to d+1$ inflow shift must not double as the vanishing move; (b) a contaminated citation provenance — DGL establishes $\Omega_5 = 0$ for pure $G_{\text{SM}}/\mathbb Z_6$ in Spin bordism and in the spin$^c$ context where B–L is gauged, a provenance not re-audited on this load-bearing torsion claim; (c) a probable object-substitution vs the canonical refined-PSU(3) R4 target. Governance flagged this as a CLAIM_BOUNDARY_VIOLATION — a contested degree-convention judgment dressed as a target-blind theorem. It is not recorded as a vanishing, and this dossier does not revive it. What is safe to record is Phase-1 OMEGA_EXISTS and the contested status of the value.
R3_PARTIAL_AUDIT)The five boundary/Dai–Freed/inflow rows split into LOCAL (free-part) and GLOBAL (torsion-part) rows (12_R3_FIVE_ROW_TARGET_BLIND_AUDIT_2026-06-26.md):
| # | Row | Facet | Disposition |
|---|---|---|---|
| 1 | $S^1_Y/\mathbb Z_2$ wall fermion anomaly (global) | torsion $\mathrm{Ext}(\Omega_5)$ | UNDEFINED/CONTESTED — contested degree; conditional on OPEN-1 |
| 2 | Hořava–Witten bulk-CS inflow (local) | free $\mathrm{Hom}(\Omega_6)$ | DERIVED-GIVEN-E (solid; R1 arithmetic) |
| 3 | Dai–Freed global on boundary | torsion $\mathrm{Ext}(\Omega_5)$ | UNDEFINED/CONTESTED — same $\Omega_5$ fact, contested degree |
| 4 | Dai–Freed global on $K_6\times S^2$ ($\mathbb Z_3$/$Q_1=\beta P^1$) | torsion $\mathrm{Ext}(\Omega_5)$ | CONTESTED/UNCOMPUTED — this IS row 17; not recordable as zero |
| 5 | bulk–boundary inflow balance (local) | free $\mathrm{Hom}(\Omega_6)$ | DERIVED-GIVEN-E (solid; R1) |
So two local rows are solid given-E; three global rows are undefined/contested, conditional on the unresolved OPEN-1. No row is established nonzero; no global row is a recordable zero.
An external finding (project_dai_freed_global_anomaly_2026-06-28, propagated into 01_DOSSIER.md "Cross-gate propagation note 2026-06-29") chips the R3 gauge $\Omega_5$ Dai–Freed leg ONLY, with no bearing on R4 (the coset-twist class is a distinct object). The orbifold mod-8 Dai–Freed object on the active branch is the same Pin/spin$^c$ lift bit BG-10 posits as $\sigma_\nu$; its default value, fixed by the global index $\chi = -3$, is $\sigma = 5 \bmod 8 = e^{-3i\pi/4}$. The separate gauge Dai–Freed class of that object computes to ~74% vanishing — but CONDITIONAL: the AHSS differentials $d_3/d_5$ were not line-run for the active branch. So R3's grade is unchanged (AUDIT-with-roadmap) but now carries a partial, conditional computation rather than "nothing computed." "~74% conditional" ≠ "computed-to-cancel"; the firewall still applies. (BG-10's own status is honestly disfavored on this finding — its posited $\sigma_\nu = +1$ needs a $-4\bmod 8$ Pin$^-$ flip the frozen record does not force — but that is BG-10's disposition, not UQF-4's.)
SG4_ANOMALY_CLOSURE_RESULT.md §3.OMEGA_REAL/03 (ξ), 04 ($A$), 06 ($[\omega]_{\text{bulk}}$, the Cartan-model construction and the FOS $d\le3$/$d=4$ split), 07 (the lift $L_\partial$). The value is not in any certificate as a recordable result — it is owed.The gate has one mathematical object open, wearing several hats that are not all the same computation, plus the existence precondition. Each hole below is a work-package. The strict dependency order is OPEN-1 (existence) → identity/degree repairs → value → the two-route reproduction: do not compute a value before the object exists and its identity is pinned.
(a) Precise statement. Decide whether the descended $M_4 \times K_6 \times S^2$-over-$S^1_Y/\mathbb Z_2$ branch admits the tangential structure ξ ($\mathrm{Spin}^c\text{-}(SU(3)\times SU(2)\times U(1))/\mathbb Z_6$ + the five twists). Concretely: evaluate $w_2(X) + f^*\zeta = 0$ in $H^2(X;\mathbb Z_2)$ on the branch's actual bundle data, and build the B2 relative/Pin$^c$ boundary object at the $S^1_Y/\mathbb Z_2$ walls. This is a characteristic-class / orientability / integrality question — not an anomaly-vanishing question. (UQF4_XI_EXISTENCE/00_RESULT.md.)
(b) Why it's hard / prior-attempt lessons. The required inputs are not yet assembled: the parities of (a) $SU(2)_L$ flux on $S^2$, (b) $U(1)_Y$ flux on $S^2$, (c) Wilson-line/flux content over the two 2-cycles of $K_6$. Traps the verifier already caught — do not repeat them: (i) Do not discharge O3 with DGL arXiv:1910.11277 — DGL compute $\Omega^{\mathrm{Spin}}_*(B(G_{\text{SM}}/\mathbb Z_6))$, the bordism of the bare gauge bundle, which presupposes the structure (cohomology-of-$BG$ $\neq$ the branch's $w_2$/flux) — this is the forbidden EXISTENCE→IDENTITY→VALUE inversion. (ii) Do not assume "yes" to pass the gate — a "yes" assumed is target-fitting.
(c) Exactly what closes it. Compute $w_2 + f^*\zeta$ target-blind on the actual bundle data. $= 0$ ⇒ ξ exists ⇒ the class is defined (proceed to Hole B). $\neq 0$ with no compensating twist ⇒ XI_NOT_ADMITTED ⇒ the gate is REFUTED in the strongest way (the invariant is nonexistent, not zero — a genuine, valid close). O1 is already DISSOLVED (relative Dai–Freed reframe); O2, O4 are BANKED; the live obstructions are O3 and O5/B2.
(d) Machinery & inputs. Obstruction theory for lifting $\tau_X: X\to BO$ through $\xi: B\xi\to BO$; $\mathrm{Spin}^c$/$\mathrm{Spin}$-$G$ characteristic classes; relative/Pin$^c$ Dai–Freed boundary structure. Start from UQF4_XI_EXISTENCE/00_RESULT.md, OMEGA_REAL/03_TANGENTIAL_STRUCTURE.md §§2–5, and the frozen flux data on the active branch.
(e) Leverage. OPEN-1 is the single residual that gates everything: Holes B, C, D are all downstream of it (the class is not an element of any group until ξ exists). O5/B2 is the shared blocker that also moves UQF-3 (the physical-Hilbert gate). Closing OPEN-1 is the highest-leverage single move in the gate.
(a) Precise statement. Evaluate $[\omega]_{\text{lifted}} = L_\partial([\omega]_{\text{bulk}}) \in A = (I\Omega^\xi)^5(\mathrm{pt})$ on the active branch, target-blind: the relevant bordism group plus the η-invariant phase on the generators, with the $d_5 = Q_1 = \beta P^1$ differential at $p=3$ evaluated on the operative class. The $d\le 3$ part is provably zero (FOS Cor 7.5). The $d=4$ part is unprotected (FOS Cor 7.6 fails — $H^2(SU(3)/T^2) = \mathbb Z^2 \neq 0$ and $SU(3)$ has a cubic Casimir). (06_COSET_WZ_CLASS.md; OMEGA_REAL/04.)
(b) Why it's hard / prior-attempt lessons. No published computation of the descended/boundary-lifted $SU(3)/T^2$ gauged-WZ obstruction exists — this is a genuine frontier. The exact trap the campaign fell into (and that was withdrawn): the first-pass plug declared OMEGA_VANISHES via $\mathrm{Ext}(\Omega^{\mathrm{Spin}^c}_5(B(G_{\text{SM}}/\mathbb Z_6))) = \mathrm{Ext}(0) = 0$, but this was a CLAIM_BOUNDARY_VIOLATION for three reasons that must be repaired first (the R-deg/R-id/R-2rt repairs of UQF4_ROW17_REPAIR): the degree was relocated ($(I\Omega^\xi)^5$ vs $(I\Omega^\xi)^6$; $\Omega_4$ vs $\Omega_5$) so the $d\to d+1$ inflow shift double-counted as the vanishing move; the object identity was probably substituted (bare $B(G_{\text{SM}}/\mathbb Z_6)$ vs the canonical refined-PSU(3)/one-form-center target; spin$^c$-with-B–L vs pure-gauge); and DGL read several times is one lineage, not two independent routes. Do not relocate the degree; do not substitute the object; do not count one citation as two routes.
(c) Exactly what closes it. Compute the value in manifest bordism/η form (so scheme-independence is automatic), at the degree Phase 1 fixed, on the canonical refined target. Vanishes ⇒ the binding obstruction is gone (proceed toward DERIVED-GIVEN-E). Nonzero ⇒ the theory is REFUTED on the active branch (a valid close). Regulator doing the cancellation work ⇒ REFUTES the row (hidden knob).
(d) Machinery & inputs. AHSS / James spectral sequence for $MT\xi$ (inputs: $H^*(BG)$, $H^*(SU(3)/T^2) = \mathbb Z[t_1,t_2,t_3]/(\sigma_1,\sigma_2,\sigma_3)$, the $\mathbb Z_6$/$\mathbb Z_2$ twists) + Anderson duality; the $\mathbb Z_3$ untwisted-survival + $\tau_{K_6} = (2,2)$ lever (OMEGA_REAL/04, 06; OMEGA_COMPUTE/01). The Cartan-model construction of $[\omega]_{\text{bulk}}$ is in 06 §2.
(e) Leverage. This is the same class as Holes C and the quantum leg of the nilpotency check (§3.9) — settling it settles all three at once. It is the gate's binding blocker.
(a) Precise statement. Verify the quantum master equation $\tfrac12(S,S) = i\hbar\,\Delta S$ order-by-order on the descended BV action — i.e. that the BV Laplacian $\Delta$ of the descended (orbifold + KK + coset-ghost) measure produces no $\hbar^1$ obstruction. Classical $s^2 = 0$ already holds (§3.3). (01_DOSSIER.md §F.3.)
(b) Why it's hard / prior-attempt lessons. The $\hbar^1$ piece is not independent of the coset sector: $\Delta$ on the flag-manifold ghost system is precisely where $[\omega]$ enters. So this does not separate from Hole B at one loop — it is a facet of the same class. The trap is treating it as a separable symbolic check that could "pass" while Hole B is open.
(c) Exactly what closes it. Either an order-by-order symbolic/machine-lane verification through $\hbar^1$ (which, by §3.9, is the $[\omega]_{\text{lifted}}$ evaluation) → VERIFIED; or, if it rests on a structural assumption, name it value-free (AXIOM-BV-NILPOTENCY-DESCENT) and mark AXIOM-CLOSED-pending = OPEN; failure ⇒ the gauge-fixing is inconsistent through the descent ⇒ REFUTED.
(d) Machinery & inputs. Descended BV action with the descended ghost sector / gauge-fixing fermion; the BV Laplacian of the orbifold+KK+coset-ghost measure (OMEGA_REAL/05).
(e) Leverage. Closes simultaneously with Hole B (same class). Independent value only as a consistency cross-check on the bordism computation.
(a) Precise statement. Confirm that the descended coset-WZ anomaly is captured by the invertible Freed–Hopkins/Grady classification at all (vs requiring relative / non-invertible anomaly machinery). Grady's theorem has three load-bearing hypotheses — reflection-positivity, invertibility, fixed ξ — all of which must be checked for the descended branch. (01_DOSSIER.md §D.2; OMEGA_REAL/03.)
(b) Why it's hard / prior-attempt lessons. The Freed–Hopkins framework is invertible-theory machinery, while the gauged flag-manifold coset-WZ sector is a genuinely interacting structure. Whether its descent anomaly is organized by the invertible classification is an unverified premise that was silently smuggled inside "single class α" in early framings. The trap is assuming applicability because it is convenient for the collapse.
(c) Exactly what closes it. Check reflection-positivity + invertibility + the fixed ξ-structure for the descended branch (a bordism/orientability computation that overlaps with Hole A). If applicable ⇒ the single-class collapse is licensed and the anomaly is fully in $(I\Omega^\xi)^{d+1}$. If not ⇒ the rows do not collapse to one invertible class and the open content is larger than stated — sharper-OPEN (a valid, honest negative).
(d) Machinery & inputs. Grady arXiv:2310.15866 hypotheses; the structure of gauged-WZ anomaly theories as invertible/bordism-classified. The candidate resolution noted in OMEGA_COMPUTE/00_RESULT.md §2 ("a gauged-WZ anomaly theory is invertible") is plausible but not independently established on the canonical target — treat it as a check, not a result.
(e) Leverage. A facet of the same class (§3.9); resolves with Holes B/C. A negative here is the most informative outcome because it would re-expand the gate's open content.
(a) Precise statement. Show every computed row is regulator-independent — or that the regulator is fixed by the geometry, not chosen to cancel. (01_DOSSIER.md §F.6.)
(b) Why it's hard / prior-attempt lessons. Scheme-independence is asserted for the uncomputed rows. A regulator quietly doing the cancellation work is the single most dangerous hidden knob in the gate.
(c) Exactly what closes it. Deliver each row in manifest bordism/η form, where scheme-independence is automatic (a corollary, off the floor). A regulator doing the cancellation REFUTES the row.
(d) Machinery & inputs. Bordism/η representatives from Holes B–D; the no-target-fitting falsification test (OMEGA_REAL/09).
(e) Leverage. Not closeable independently; it is the discipline that makes Holes B–D count.
The gate resolves the moment OPEN-1 is discharged and the single class is evaluated target-blind on the canonical object at the correct degree, with two independent routes and a manifest-bordism firewall:
Every outcome is a valid close; none may be assumed.
What is claimed. Given the spectrum E, the perturbative Standard-Model anomalies cancel (independently reproduced by exact arithmetic; specific to E by the $\sum Y^2 = 10/3 \neq 0$ check); classical BRST nilpotency $s^2 = 0$ holds; and the scattered open rows are a single Anderson-dual / Dai–Freed even-degree class whose $d\le 3$ part provably vanishes (FOS Cor 7.5) and whose object is now canonically defined (Phase-1 OMEGA_EXISTS).
What is NOT claimed. The gate is not closed. The single class's value is uncomputed/contested (the candidate vanishing was withdrawn as a claim-boundary violation), its existence precondition is blocked on a named missing datum (OPEN-1), and its applicability premise is unchecked. No anomaly is asserted to vanish that was not computed.
The distinctions that bound the claim. - dissolved ≠ solved. "Anomaly selects the spectrum" is a dissolved category error; the perturbative win is a filter on E, not a selector of E (infinitely many anomaly-free spectra exist). - given-E ≠ derivation of E. The perturbative cancellation is DERIVED-GIVEN-E. Deriving E itself is an upstream gate (SG-2/SG-3) — anchor-transfer, not elimination. - selection ≠ derivation; "machinery exists" ≠ "calculation done"; a captured log ≠ an independent reproduction; ATOMIC floor ≠ closed gate.
The anchors paid. UQF-4 rests on an ATOMIC floor of one measured anchor + one definition: ATOM-E (the chiral content E on which the obstruction is evaluated — the irreducible floor-of-one; the group and descent map are inherited from the frozen geometry via the isometries, not separate anchors) and ATOM-Q (the definitional name: quantum consistency = QME-solvability = anomaly-cocycle vanishing). The Grady single-class engine is correctly kept off the floor as a theorem-with-hypotheses. A floor of one measured invariant is the minimum honest bookkeeping permits — the correct outcome, not a deficiency.
The dissolved unicorns, framed as shared ceilings (never as our weakness, never as proven). That no accepted theory has a complete non-perturbative anomaly-descent certificate for any realistic compactified model is a community-wide open problem — demanding this program alone solve in full what nobody has solved is not a defect specific to us. And the earlier "no construction route in any prior work for the coset-twist descent" framing was itself rejected on internal audit as an over-broad universal-negative: the honest residual is a bounded even-degree bordism/η computation, not a no-go, and it must not be revived as a hole.
Scope ceiling. Even a fully clean UQF-4 is an anomaly-consistency leg only — no Osterwalder–Schrader/Wightman existence, no nontriviality, no spectral gap — so it cannot touch Gap-02 (the mass gap stays Precisely-OPEN) (10_ROW17_CONTESTED_DISPOSITION_2026-06-25.md §"Scope reminder"). Downstream, UQF-4's AUDIT status is contagious: it caps the unification paper at AUDIT and downgrades the gates that consume it (UQF-7, UQF-6 one tier each; UQF-3 loses its certificate) — exported to the downstream-gate dependency ledger to update automatically if UQF-4 moves.
Bottom line. UQF-4 is a serious candidate / partial unification, NOT validated — an honestly-held AUDIT (OPEN) gate that has been sharpened and corrected, not closed. The genuine gains are real: a verified, target-blind perturbative cancellation specific to E; a verified classical nilpotency; a structural compression of the open content to one defined class; and two logged self-corrections (the odd-degree no-go and the over-stated $d\le4$ protection). None is a promotion. The path forward is the bounded, target-blind program of §6 — existence first, then one bordism/η evaluation of the even-degree class — where vanishing ⇒ DERIVED-GIVEN-E, nonzero or no-ξ ⇒ REFUTED, computed, never assumed.
Status: AUDIT (OPEN). No status promoted. Frozen branch read-only. Given-E ≠ derivation of E. Anchored ≠ derived. Assuming the open class vanishes: REJECTED.