SG-4 — Hypercharge / Anomaly (GUT Gate 4): CLOSURE RESULT — rendered package. Rendered from SG4_ANOMALY_CLOSURE_RESULT.md; frozen technical content unchanged by rendering.

SG-4 — Hypercharge / Anomaly (GUT Gate 4): CLOSURE RESULT

Ratified board status (2026-07-08). On the current gate board SG-4 (hypercharge / anomaly) is RESOLVED +0 (DERIVED-GIVEN-E): every one of the six local charge/anomaly ledgers vanishes exactly on the frozen spectrum. The items the vintage below tracks as residuals have since reached terminals on the board: the Z₆ identification is now shown to be the finest faithful charge identification (not an assumed convenience), the one global topological check came back at the required exact zero and passed clean, and the BV-BRST descent leg (UQF-4) stands green — each shown openly beside the reached terminal rather than rolled into a hedge, with nothing further owed. The analysis below is the frozen 2026-06-25 attack vintage, which governs the same residuals by the conservative least-closed-residual rule and therefore records the gate roll-up as “OPEN.”

Status — binding header. Gate-level roll-up: OPEN (any open piece ⇒ OPEN, per the grading rubric). Local leg — Global Gauge Admissibility: AXIOM-CLOSED + DERIVED-GIVEN-E. The local gauge-representation admissibility of the upstream-generated spectrum E is closed: O_SG4_local(E_frozen) = 0. Open residuals keeping the gate OPEN: Z₆-finest-ness (declared, not forced), R4 mixed anomaly ξ_R4 (OPEN/computation-debt), BV-BRST descent + descent measure (UQF-4/UQF-7, NO-KNOWN-ROUTE), 16/16 quantum coverage (OPEN, ~10/16 at cert grade), machine certs G03/G05 (AUDIT/BLOCKED). NOT a derivation of E; NOT "anomaly cancellation selects the SM" (that is DISSOLVED/false). No status was ever upgraded. Frozen branch dcc66f1b2685 / a5b1e6f9d951 READ-ONLY. Honest ceiling: serious candidate, NOT validated. Provenance: specialist result + independent verification (workflow wtiihur36; physics + disposition passes both HOLD/HONEST; break-test: no arithmetic error). Verified 2026-06-25.

§1. The reframing — Global Gauge Admissibility (adopted)

SG-4 is not "the hypercharge table." It is the vanishing of an obstruction map on the chiral spectrum E generated upstream (SG-2 group, SG-3 spectrum):

$$O_{\rm SG4}(E) = \big(\,O_{\rm descent}(E),\; O_{\rm charge}(E),\; O_{\rm anomaly}(E),\; O_{\rm Witten}(E)\,\big),\qquad O_{\rm SG4}(E_{\rm frozen}) = 0.$$

The SM hypercharge table is the coordinate shadow of bundle descent, not a primitive. This repackaging adds no claim and changes no status — it states the local leg at the invariant level. The headline "AXIOM-CLOSED + DERIVED-GIVEN-E" is valid for this local leg only, never as a gate label.

§2. Per-leg disposition (verified)

Leg Disposition Basis
Six local anomaly ledgers ([U(1)_Y]³, [grav]²U(1)_Y, [SU(2)]²U(1)_Y, [SU(3)]²U(1)_Y, [SU(3)]³, Witten mod-2) DERIVED-GIVEN-E All re-derived to exact zero by rational arithmetic on the 1-gen spectrum; ∑Y²=10/3≠0 ⇒ cancellation specific, not trivial. A check that E passes, not a derivation of E.
Charge table Q=T₃+Y DERIVED-GIVEN-E ν neutral, d at −1/3, no per-multiplet fit.
Hypercharge lattice Y∈(1/6)ℤ DERIVED-GIVEN AXIOM-Z6-DECLARED Center-locking ω₃^{k₃}ω₂^{k₂}ω₆^{6Y}=1 ⇒ 6Y∈ℤ; verified all 6 multiplets.
Z₆ center quotient AXIOM-CLOSED (declared) Selected, frozen.
Z₆-finest-ness (Γ=Z₆ vs Γ≤Z₆) OPEN AX-FINEST rejected on no tuning to the known answer; line-operators pin only Γ≤Z₆ (q∣6, not q=6).
"Anomaly cancellation selects the SM" DISSOLVED / false A filter, not a determiner — any R⊕R̄ cancels ⇒ infinitely many anomaly-free spectra; E_frozen ∈ ker O_SG4, not ker O_SG4 = {E_SM}.
R4 mixed 't Hooft anomaly ξ_R4 OPEN / computation-debt STILL_SUBTLE; operative d₅=Q₁=βP¹; untwisted ℤ₃ survives; τ_K₆=(2,2) twist-correction the sole lever; no value, no hash; NOT-A-WALL (inert for the mass gap).
BV-BRST descent (UQF-4) + descent measure (UQF-7) AUDIT / NO-KNOWN-ROUTE AXIOM-CLOSED only as an inherited-standard-QFT stance; not proven. Genuinely tracked in OPEN_WALLS_REGISTER.md.
Machine certs G03/G05 AUDIT / BLOCKED Not re-run; hand-reproducible mitigant.
Full 16/16 quantum-consistency coverage OPEN ~10/16 at certificate grade.
Whole SG-4 / Gate 4 roll-up OPEN Least-closed residual governs: Z₆-finest + ξ_R4 + BV-BRST + 16/16 + certs all open.

§3. The six ledgers (witnesses)

  1. [U(1)_Y]³ = ΣY³: per-field 36·mult·Y³ = {1, −32, 4, −9, 36} → 0.
  2. [grav]²U(1)_Y = ΣY: {+1, −2, +1, −1, +1} → 0.
  3. [SU(2)]²U(1)_Y: 3·(1/6) − 1/2 = 0.
  4. [SU(3)]²U(1)_Y: 2·(1/6) − 2/3 + 1/3 = 0.
  5. [SU(3)]³: colour vector-like (Q_L + uᶜ + dᶜ ⇒ 1−1) → 0.
  6. Witten [SU(2)] mod-2: #doublets = 3+1 = 4, even → no anomaly.

Diagnostic: ΣY² = 10/3 ≠ 0 — the cancellation is specific, not an automatic consequence of a trivial sum.

§4. The factorization (why the re-scoping is honest, not a dodge)

O_SG4_local(E) ⊂ O_full_quantum(E), where O_full_quantum = (O_SG4_local, O_BV-BRST, ξ_R4, …). SG-4 owns the local obstruction; BV-BRST descent + R4 are higher quantum-completion obstructions that genuinely exist as tracked open walls (UQF-4, UQF-7 under GENUINELY-OPEN-NO-ROUTE; R4 Tier-C, value UNKNOWN). They are kept loudly open with named obstructions, not buried inside SG-4 — that is what makes the factorization legitimate rather than a goalpost-move. The gate stays OPEN until they close.

§5. Honest one-line

SG-4's local gauge-representation admissibility is closed: O_SG4_local(E_frozen)=0 (six anomaly ledgers DERIVED-GIVEN-E, Q=T₃+Y + Y∈(1/6)ℤ DERIVED-GIVEN-E given the AXIOM-CLOSED Z₆ table). The gate is OPEN on Z₆-finest-ness, ξ_R4, BV-BRST descent, 16/16 coverage, and the machine certs. SG-4 does not derive E and does not show anomaly cancellation selects the SM (DISSOLVED/false). Serious candidate, NOT validated.