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

> **Status — binding header.**
> **Gate-level roll-up: OPEN** (any open piece ⇒ OPEN, per the grading rubric).
> **Local leg — Global Gauge Admissibility: AXIOM-OPEN + 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 refute-before-upgrade verification (workflow `wtiihur36`; physics + disposition passes both HOLD/HONEST; break-test: no arithmetic error). Verified 2026-06-25.
> **⏩ Atomicity correction (2026-06-25):** all `AXIOM-CLOSED` labels herein are **demoted to `AXIOM-OPEN`** per the atomicity sweep — no axiom (Z₆ quotient, BV-BRST stance) passed the atomic test (proven-irreducible or measured); the Z₆ quotient is selected-not-forced (Γ≤Z₆) and the hypercharge lattice rests on E + the lcm spacing, not on a closed Z₆ axiom. Gate disposition **unchanged (OPEN)**; No status was ever upgraded.

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## §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-OPEN + 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-E** | Center-locking `ω₃^{k₃}ω₂^{k₂}ω₆^{6Y}=1` ⇒ 6Y∈ℤ (spacing 1/6 = lcm(3,2) from the center denominators; does **not** rest on a closed Z₆ axiom); verified all 6 multiplets. |
| **Z₆ center quotient** | **AXIOM-OPEN (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-OPEN 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-OPEN 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.