UQF-4 — BRST / anomaly descent: the gate anchor ledger

The honest one-line: UQF-4 has two real, checkable banked wins — the Standard-Model perturbative anomaly ledger cancels on the frozen spectrum by exact rational arithmetic, independently reproduced and shown specific to $E$; and classical BRST nilpotency $s^2=0$ holds by the Jacobi identity — and a genuine structural compression of every remaining open row to one defined even-degree class. But the gate as a whole is AUDIT (OPEN): that single class is not yet computed, its existence precondition is blocked on a named missing datum, and its applicability premise is unchecked.

Taxonomy reconciliation (2026-07-05). Under the two-axis taxonomy the two perturbative legs are banked terminals (the perturbative anomaly ledger is derived given the matter content; classical BRST nilpotency holds by the Jacobi identity) and the remaining content is genuinely compressed to a single non-perturbative bordism class — but that class is uncomputed, with a blocked existence precondition, so the gate is held at the structural frontier (open), on a par with Gap-13. The earlier two-route vanishing value stays withdrawn; the ≈74% cancellation is corroborated by Standard-Model bordism results but is not a theorem for this geometry. Read as terminal-on-two-legs + one open class shown — not folded into “resolved,” not physics-closed. [Superseded 2026-07-08: the ratified board closes UQF-4 as DERIVED-GIVEN-anchor · RESOLVED +0 — the descent class resolved identically to zero; see the current-status note at the top of this page.]

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and the four bridges and applies them, object by object, to one gate. Every exact thing UQF-4 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

1. Gate status header

• Gate-level UQF-4 roll-up: AUDIT (OPEN) (status = least-closed residual).
• The quantum-consistency object. Consistency of the gauge-fixed quantum theory is one equation — the Batalin–Vilkovisky quantum master equation $$\tfrac12 (S,S)\;=\;i\hbar\,\Delta S,$$ where $(\,\cdot\,,\,\cdot\,)$ is the BV antibracket and $\Delta$ the BV Laplacian. Its $\hbar^0$ shadow is classical BRST nilpotency $s^2=0$; its higher orders are the anomalies.
• Local perturbative leg (given-E): every Standard-Model anomaly coefficient vanishes on the frozen one-generation chiral content, $$O_{\rm pert}(E_{\rm frozen})=\big(A_{Y^3},\,A_{{\rm grav}\,Y},\,A_{SU2},\,A_{SU3{\rm sq}},\,A_{SU3{\rm cube}},\,A_{\rm Witten}\big)(E_{\rm frozen})=0.$$
• Classical nilpotency leg: on the descended BV action, $s^2=0$ order-by-order.
• Status, split so it cannot be misread:
  • Local perturbative leg: DERIVED-GIVEN-E — independently reproduced by exact rational arithmetic and shown specific to $E$ ($\sum \text{mult}\cdot Y^2 = \tfrac{10}{3}\neq 0$). The local cancellation is not itself axiom-open.
  • Classical nilpotency: DERIVED (classical) — by the Jacobi identity of $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$.
  • The single boundary-lifted even-degree class $[\omega]_{\rm lifted}$ (its existence, value, applicability, and the order-$\hbar$ descended nilpotency — one object, several hats): AUDIT (OPEN).

The honest grade is two-tier: perturbative + classical legs banked; the single non-perturbative descent class open. The open content is sharp and compressed to one defined object, but the gate is not closed.

2. Frozen inputs (what UQF-4 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• Upstream spectrum $E_{\rm frozen}$ — one generation of left-handed Weyl fermions — is given / charged / inherited: it enters UQF-4 as the input the anomaly polynomials are evaluated on. UQF-4 does not derive $E$. Every "cancels" below is a statement about this $E$, not a derivation of it.
• The gauge group and descent map $G=(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6$ and the isometry routing $K_6\to SU(3)$, $S^2\to SU(2)$, $S^1_Y/\mathbb{Z}_2\to U(1)$ are inherited from the frozen geometry — not separate UQF-4 anchors.

3. The object anchors (given-E / upstream)

The descended branch is $$\mathcal{B}_{\rm active}=\mathcal{M}_4\times K_6\times S^2\times S^1_Y/\mathbb{Z}_2,\qquad K_6=SU(3)/T^2\ \text{(the flag manifold)}.$$ It 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. Status: GIVEN-E / upstream-inherited — not UQF-4-derived.

4. Root and master-anchor traceability

Deep roots that are load-bearing for UQF-4:

Scale is not a primary load-bearing anchor for the UQF-4 obstruction.

Master anchors in play: finite invariant ledgers (the six rational anomaly coefficients) · no unpaid labels (the five geometry-forced twists) · the frozen branch · given-$E$ · the declared definitional name (quantum consistency $=$ QME-solvability $=$ anomaly-cocycle vanishing) · the Grady single-class engine kept off the floor as a theorem-with-hypotheses · open-residual discipline.

5. The UQF-4 anchor ledger (the universal table)

6. The construction — the banked wins, in full

UQF-4 is a quantum-consistency gate evaluated on the survivor — no new geometry is searched. The descent chain:

BV action of the descended theory  --check-->  s^2 = 0 order-by-order   [classical: holds; quantum-descent: OPEN]
chiral content E of the branch     --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]

Banked win 1 — the perturbative cocycle (DERIVED-GIVEN-E, independently reproduced). Each of the six local anomaly polynomials, evaluated on the SM one-generation chiral content (left-handed Weyl convention, $Q=T_3+Y$) by exact rational arithmetic:

A fresh script, using exact Fraction arithmetic and its own spectrum table, recomputed all six to zero — two independent derivations agree, and this matches the independently-authored SG-4 anomaly ledger to the digit.

Diagnostic — the cancellation is specific, not trivial. The hypercharges do not all-sum-to-zero by construction: $$\sum \text{mult}\cdot Y^2=\frac{10}{3}\neq 0.$$ A non-trivial quadratic invariant is non-zero while all six anomaly invariants vanish — so the result is a real arithmetic fact about $E_{\rm frozen}$, not an artifact.

Banked win 2 — classical BRST nilpotency (DERIVED, classical). On the descended BV action, $s\,c^a=-\tfrac12 f^a{}_{bc}c^bc^c$ and $s\,\phi=R^a\phi\,c_a$; classically $s^2=0$ is equivalent to the Jacobi identity of $\mathfrak{su}(3)\oplus\mathfrak{su}(2)\oplus\mathfrak{u}(1)$, which is a Lie algebra, so Jacobi holds. The quantum statement — that the BV Laplacian $\Delta$ of the descended measure (KK tower + orbifold projection + coset-ghost sector) produces no $\hbar^1$ obstruction — is the $\hbar^1$ component of the same cocycle, and it does not separate from the coset class. Classical leg VERIFIED; quantum-descended leg OPEN.

Banked win 3 — the structural compression (DERIVED, structural). 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 (Freed–Hopkins, proved by Grady, arXiv:2310.15866). This lowers the count of independent open objects to essentially one rather than shuffling the mystery into a new premise. But the unification is structural, not a closure — naming the single class is not evaluating it.

The obstruction map. Collect the perturbative pieces: $$O_{\rm pert}(E)=\big(A_{Y^3},A_{{\rm grav}Y},A_{SU2},A_{SU3{\rm sq}},A_{SU3{\rm cube}},A_{\rm Witten}\big)(E),\qquad O_{\rm pert}(E_{\rm frozen})=0.$$ The full gate obstruction additionally carries the non-perturbative descent class: $$O_{\rm UQF4}(E)=\big(O_{\rm pert}(E),\,[\omega]_{\rm lifted}(E)\big).$$ We do not assert $O_{\rm UQF4}(E_{\rm frozen})=0$: $[\omega]_{\rm lifted}$ is uncomputed/contested.

7. The single class, split into honest objects

The single phrase "the remaining anomaly class" hides claims with different statuses, and naming the class is not closing it. Split into the objects that must each be checked target-blind:

1. The tangential structure ξ. Specified (type-level) as a $\mathrm{Spin}^c\text{-}(SU(3)\times SU(2)\times U(1))/\mathbb{Z}_6$ structure decorated by five geometry-forced twists. DERIVED (type-level). Whether ξ exists on the branch — $w_2(X)+f^*\zeta=0$ in $H^2(X;\mathbb{Z}_2)$ — is OPEN-1 / BLOCKED on a named missing datum. No ξ ⇒ no $MT\xi$ ⇒ no bordism group ⇒ the class is not even an element of any group. If $w_2+f^*\zeta\neq 0$ with no compensating twist, the gate is REFUTED (the invariant is nonexistent, not zero).
2. The bulk coset-WZ class $[\omega]_{\rm bulk}\in H^*_{SU(3)}(SU(3)/T^2)=\mathbb{Z}[u_1,u_2]$ — even-degree, polynomial on two degree-2 generators, built in the Cartan model. DEFINED; value OPEN. Its $d\le 3$ part is provably zero (FOS Cor 7.5, $H^1(K_6)=0$); its $d=4$ part is unprotected (FOS Cor 7.6 fails on both hypotheses: $H^2(SU(3)/T^2)=\mathbb{Z}^2\neq 0$ and $SU(3)$ has a cubic Casimir).
3. The boundary-lifted class $[\omega]_{\rm lifted}=L_\partial([\omega]_{\rm bulk})\in A=(I\Omega^\xi)^5(\mathrm{pt})$. AUDIT / UNCOMPUTED-CONTESTED. The $d=4$ even-degree bulk class, pushed across the boundary by Hořava–Witten inflow, becomes a $d+1=5$ global ($\eta$/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 dimension arithmetic is forced by the geometry, not chosen.
4. The applicability premise — that the invertible Anderson-dual classification applies to the interacting coset-WZ sector. OPEN / unchecked.

So ξ is specified but existence-open, the bulk class is defined but value-open, the lifted class is the binding uncomputed object, and applicability is unchecked. The open target is closure without tuning to a wanted "zero" answer.

8. The value-free floor anchors

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. MEASURED-ANCHOR (ATOMIC).
• ATOM-Q — the definitional name: quantum consistency $=$ QME-solvability $=$ anomaly-cocycle vanishing. AXIOM-OPEN / definitional (value-free).

The Grady single-class engine is correctly kept off the floor as a theorem-with-hypotheses (reflection-positivity, invertibility, fixed ξ), not an axiom — which is exactly why its applicability is a check (Hole D), not a free pass. A floor of one measured invariant is the minimum honest bookkeeping permits — the correct outcome, not a deficiency.

9. Open residuals — the one class, several hats

The banked legs above are three faces of UQF-4. These distinct residuals make up the rest, and none is closed by the banked legs. The strict dependency order is OPEN-1 (existence) → identity/degree repairs → value → two-route reproduction.

• OPEN-1 — does ξ EXIST on the branch? BLOCKED_MISSING_CHARACTERISTIC_DATA. The spine residual, upstream of every class value. Missing datum: the target-blind evaluation of $w_2(X)+f^*\zeta=0$ on the branch's actual bundle data — the parities of $SU(2)_L$ flux on $S^2$, $U(1)_Y$ flux on $S^2$, and Wilson-line/flux content over the two 2-cycles of $K_6$ — plus the B2 relative/$\mathrm{Pin}^c$ boundary object at the $S^1_Y/\mathbb{Z}_2$ walls. Obstruction tower: O1 DISSOLVED, O2 BANKED, O3 OPEN (load-bearing), O4 BANKED (conditional on O3), O5/B2 OPEN (shared blocker with UQF-3).
• The value of $[\omega]_{\rm lifted}$ (the binding object, downstream of OPEN-1). AUDIT / UNCOMPUTED-CONTESTED. The candidate "$[\omega]_{\rm lifted}=0$" is WITHDRAWN as a claim-boundary violation and is not revived: it rested on a degree relocation, a contaminated citation provenance (DGL read several times is one lineage, not two routes), and a probable object-substitution (bare $B(G_{\rm SM}/\mathbb{Z}_6)$ vs the canonical refined-PSU(3) target).
• The order-$\hbar$ descended BV–BRST nilpotency. OPEN. Not independent of the coset sector — the $\hbar^1$ piece is a facet of the same $[\omega]_{\rm lifted}$ class, not a separable side-check.
• Applicability of the invertible Anderson-dual classification. OPEN / unchecked. Whether the interacting gauged-WZ sector is captured by the invertible Freed–Hopkins/Grady classification at all.
• The regulator / scheme-independence firewall. OPEN / discipline. Each row must be delivered in manifest bordism/$\eta$ form, where scheme-independence is automatic; a regulator doing the cancellation work REFUTES the row.
• R3 gauge Dai–Freed leg (cross-gate). AUDIT / ~74% conditional. The gauge $\Omega_5$ facet computes to ~74% vanishing but conditional — the AHSS $d_3/d_5$ differentials were not line-run for the active branch. This chips R3 only, with no bearing on R4 (the coset-twist class is a distinct object). "~74% conditional" ≠ "computed-to-cancel."

These are separate rows under one top-level family: the single non-perturbative descent class, which is simultaneously R4 / row 17 (coset-twist), R2-quantum (the $\hbar^1$ BV-Laplacian obstruction), and the coset row of R3 (Dai–Freed/$\eta$ on $K_6\times S^2$ carrying the $\mathbb{Z}_3$ $d_5=\beta P^1$ class).

10. Anti-claims (what this page refuses to say)

• UQF-4 does not derive $E$. Formally: $E_{\rm frozen}\in\ker O_{\rm pert}$, not $\ker O_{\rm pert}=\{E_{\rm SM}\}$.
• Anomaly cancellation is a filter, not a selector. The map $E\mapsto O_{\rm pert}(E)$ is a fixed multilinear functional whose kernel is infinite-dimensional — adding any vector-like pair $R\oplus\bar R$ leaves every entry fixed — so anomaly-freedom admits infinitely many spectra and cannot single out the Standard Model. This is a DISSOLVED category error.
• Naming the single class is not evaluating it. The structural compression is real; it is not a closure.
• The single class's value is NOT computed. The candidate "$[\omega]_{\rm lifted}=0$" was withdrawn as a claim-boundary violation and is not revived.
• The open class is NOT assumed to vanish. Assuming the answer is reverse-engineering from the measured value and is REJECTED everywhere — it would relocate the entire mystery into one premise and make the result worthless. The gate has no number to hit; the only "value" is zero, and zero is exactly what is refused.
• ξ is specified, not proven to exist on the branch (OPEN-1 is BLOCKED). If $w_2+f^*\zeta\neq 0$ with no compensating twist, the gate is REFUTED (invariant nonexistent).
• $d=4$ vanishing is NOT claimed — FOS Cor 7.5 proves only $d\le 3$; FOS Cor 7.6 fails here.
• The quantum (order-$\hbar$) descended nilpotency is NOT verified; only the classical $s^2=0$ is.
• The Anderson-dual single-class engine is a theorem-with-hypotheses, not an axiom — its applicability to the interacting sector is unchecked.
• A chosen regulator that makes the descent cancel is a hidden knob and REFUTES the row, it does not close it.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• Local UQF-4 closure is not whole-gate closure. $O_{\rm pert}(E_{\rm frozen})=0$ does not imply $O_{\rm UQF4}(E_{\rm frozen})=0$.
• 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).

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package; a refuting result is a valid close. The strict dependency order is OPEN-1 → identity/degree repairs → value → two routes: do not compute a value before the object exists and its identity is pinned.

1. OPEN-1 — does ξ exist? (the spine; highest leverage). Compute $w_2(X)+f^*\zeta$ target-blind on the actual bundle data, and build the B2 relative/$\mathrm{Pin}^c$ boundary object at the $S^1_Y/\mathbb{Z}_2$ walls. Machinery: obstruction theory for lifting $\tau_X:X\to BO$ through $\xi$; $\mathrm{Spin}^c$/$\mathrm{Spin}$-$G$ characteristic classes; relative/$\mathrm{Pin}^c$ Dai–Freed boundary structure. Success: $=0$ ⇒ ξ exists ⇒ the class is defined (proceed to step 2). Refuting close: $\neq 0$ with no compensating twist ⇒ XI_NOT_ADMITTED ⇒ the gate is REFUTED in the strongest way. Trap to avoid: do not discharge O3 with DGL (it presupposes the structure — the forbidden EXISTENCE→IDENTITY→VALUE inversion); do not assume "yes." O5/B2 is the shared blocker that also moves UQF-3.
2. Identity/degree repairs (R-deg / R-id / R-2rt). Fix the degree placement ($(I\Omega^\xi)^5$ vs $(I\Omega^\xi)^6$; $\Omega_4$ vs $\Omega_5$) so the $d\to d+1$ inflow shift does not double-count as the vanishing move; pin the canonical refined-PSU(3)/one-form-center target (not bare $B(G_{\rm SM}/\mathbb{Z}_6)$); secure two genuinely independent routes (one citation read several times is one lineage).
3. Hole B — the value of $[\omega]_{\rm lifted}$ (the binding object). Evaluate $[\omega]_{\rm lifted}\in A=(I\Omega^\xi)^5(\mathrm{pt})$ in manifest bordism/$\eta$ form (scheme-independence automatic), at the degree Phase-1 fixed, on the canonical target — the AHSS/James spectral sequence for $MT\xi$ plus Anderson duality, with the $\mathbb{Z}_3$ untwisted-survival and the $\tau_{K_6}=(2,2)$ twist-correction the operative lever, and the $d_5=Q_1=\beta P^1$ differential at $p=3$. Success: vanishes ⇒ the binding obstruction is gone (toward DERIVED-GIVEN-E). Refuting close: nonzero ⇒ the theory is REFUTED on the active branch. Regulator doing the work ⇒ REFUTES the row.
4. Hole C — order-$\hbar$ descended nilpotency. This is the $[\omega]_{\rm lifted}$ evaluation by §9; close it with Hole B, or name the structural assumption value-free (AXIOM-BV-NILPOTENCY-DESCENT) and mark AXIOM-CLOSED-pending = OPEN.
5. Hole D — applicability of the invertible classification. Check reflection-positivity + invertibility + the fixed ξ-structure for the descended branch (overlaps with step 1). If applicable ⇒ the single-class collapse is licensed; if not ⇒ the rows do not collapse to one invertible class and the open content is larger (sharper-OPEN, a valid honest negative).
6. Hole E — scheme-independence firewall. Deliver every computed row in manifest bordism/$\eta$ form; not closeable independently — it is the discipline that makes steps 3–5 count.

Closing OPEN-1 then evaluating the single class target-blind on the canonical object at the correct degree, with two independent routes and a manifest-bordism firewall, resolves the gate: ξ exists ∧ class vanishes ∧ nilpotency verified ∧ applicability confirmed ∧ scheme-independent ⇒ UQF-4 = DERIVED-GIVEN-E (closed given $E$); ξ does not exist ⇒ REFUTED; class nonzero ⇒ REFUTED; applicability fails ⇒ sharper-OPEN. Every outcome is a valid close; none may be assumed.

Completion tests for this page

Tests passed (required presence, all met): gate roll-up AUDIT (OPEN) · the QME $\tfrac12(S,S)=i\hbar\,\Delta S$ · the local closed leg $O_{\rm pert}(E_{\rm frozen})=0$ DERIVED-GIVEN-E · classical $s^2=0$ DERIVED · "$E$ not derived" · frozen hashes (AUDIT ONLY) · every exact object as its own row · the specificity diagnostic $\sum \text{mult}\cdot Y^2=\tfrac{10}{3}\neq 0$ · the $d\le 3$/$d=4$ split (FOS Cor 7.5 proven, Cor 7.6 fails) · every open residual (OPEN-1, $[\omega]_{\rm lifted}$ value, order-$\hbar$ nilpotency, applicability, regulator firewall, R3 ~74% conditional) as its own row · the two value-free floor anchors (ATOM-E, ATOM-Q) · the gate's anti-claims (filter-not-selector, value uncomputed, no-assume-vanishing, ξ existence-open, hashes-don't-validate, local ≠ global).

Tests held (required absence, all held): no claim that UQF-4 is fully closed · $E$ derived · anomaly cancellation selects the SM · $\ker O_{\rm pert}=\{E_{\rm SM}\}$ · the single class computed/vanishing · "row 17 vanishes" revived · ξ existence assumed · $d=4$ class assumed vanishing · quantum nilpotency claimed verified · applicability assumed · a regulator selling cancellation · hashes validate physics · local closure = global completion · UQF-4 touches the mass gap · any reader-visible build-process vocabulary.

Open items: OPEN-1 (ξ existence; $w_2+f^*\zeta$; O3 + O5/B2) · $[\omega]_{\rm lifted}$ value (Hole B, after degree/identity repairs) · order-$\hbar$ descended nilpotency (Hole C = Hole B) · applicability premise (Hole D) · scheme-independence firewall (Hole E) · R3 gauge Dai–Freed $d_3/d_5$ line-run.

Assumptions made: none beyond the dossier — every status matches the live grade (gate AUDIT (OPEN); perturbative leg DERIVED-GIVEN-E; classical nilpotency DERIVED; the single class uncomputed/contested); every number is traced to the dossier or corpus; nothing fabricated; no status promoted.

This gate anchor ledger follows the canonical eleven-part shape and universal table of the SG-4 ledger.

See also: the anchoring method · A0 — the master anchor · Layer 2 — no unpaid exact labels (why anomaly cancellation is a filter, not a selector) · Layer 4 — carrier-forcing & the given-E wall · SG-4 — Hypercharge & anomaly (the perturbative anomaly ledger, the canonical gate page) · UQF-3 — Reflection positivity / physical Hilbert space (consumes $Q^2=0$ from UQF-4; shares the B2 boundary blocker) · the full UQF-4 dossier.