SG-4 — Hypercharge & anomaly: the gate anchor ledger

The honest one-line: SG-4 has a real, checkable local win — the Standard Model's anomaly books balance exactly, by rational arithmetic, for the frozen spectrum — and on the live board the gate stands RESOLVED +0 (DERIVED-GIVEN-anchor, ratified 2026-07-08), with the quantum/global-completion residual family below shown openly and carried unchanged; the local win does not select the Standard Model or close the quantum/global completion.

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 SG-4 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It is also the template for the per-gate anchor ledgers; the other gates follow the same shape.

1. Gate status header

• Gate-level SG-4 roll-up: DERIVED-GIVEN-E — RESOLVED +0 on the live board (ratified 2026-07-08). Frozen mid-audit reading: OPEN (under the superseded least-closed-residual rule); the residual family below is carried unchanged.
• Taxonomy reconciliation (2026-07-05): under the ratified closure taxonomy (board 2026-07-08) the gate-level grading is a reached terminal — RESOLVED +0 — read as TERMINAL + RESIDUALS-SHOWN — the local anomaly/charge leg is banked (DERIVED-GIVEN-E) and the quantum/global-completion residual family listed in this ledger (ℝ-family: ℤ₆-finestness, R4 mixed anomaly, BV–BRST descent, 16/16 coverage, G03/G05 machine certs) remains listed and carried unchanged. Facts are unchanged: the earlier OPEN roll-up reflects the superseded least-closed-residual rule, not different facts. No individual residual below is closed, re-graded, or removed.
• Local gauge-representation leg: closed only as local gauge-representation admissibility — the statement $$O_{\rm SG4,local}(E_{\rm frozen}) = 0 .$$
• Status, split so it cannot be misread:
  • Local arithmetic leg: DERIVED-GIVEN-E — the anomaly/charge cancellation is exact arithmetic, given $E$. The local cancellation is not itself axiom-open.
  • Gate-level SG-4: RESOLVED +0 (frozen mid-audit roll-up: OPEN, superseded).
  • Declared / global-form ingredients (the $\mathbb Z_6$ quotient, the descent data): AXIOM-OPEN / declared.

Given $E_{\rm frozen}$, the six local anomaly ledgers vanish by exact rational arithmetic, with no per-ledger retuning. What stays open is everything beyond the local arithmetic — the global form, the quantum/global completion, and whether the spectrum is selected rather than merely admitted.

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

2. Frozen inputs (what SG-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 $E_{\rm frozen}$ is given / charged / inherited — it enters SG-4 as input. SG-4 does not derive $E$. Every "passes" below is a statement about this $E$, not a derivation of it.

3. The spectrum anchor (given-E)

One generation of left-handed Weyl fermions, $(SU(3),SU(2))_Y$: $$ Q_L=(3,2)_{1/6},\quad u^c_L=(\bar 3,1)_{-2/3},\quad d^c_L=(\bar 3,1)_{1/3}, $$ $$ L_L=(1,2)_{-1/2},\quad e^c_L=(1,1)_{1},\quad \nu^c_L=(1,1)_{0}. $$ $\nu^c_L$ is optional and anomaly-neutral. Status: GIVEN-E / upstream-inherited — not SG-4-derived.

4. Root and master-anchor traceability

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

Scale and causal order are not primary load-bearing anchors for the SG-4 local arithmetic.

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the declared $\mathbb Z_6$ table · open-residual discipline.

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

6. The arithmetic — the local win, in full

For one generation, every local anomaly coefficient is a finite rational sum over the spectrum. All six vanish for $E_{\rm frozen}$:

(1) Cubic hypercharge $[U(1)_Y]^3=\sum (\text{mult})\,Y^3$. Scaling by $36$ to clear denominators, the per-multiplet contributions are $$\{\,Q_L,\;u^c_L,\;d^c_L,\;L_L,\;e^c_L\,\}\;\to\;\{+1,\,-32,\,+4,\,-9,\,+36\},\qquad \sum = 0 .$$

(2) Mixed gravitational $[\mathrm{grav}]^2 U(1)_Y=\sum(\text{mult})\,Y$: $$\{+1,\,-2,\,+1,\,-1,\,+1\}\;\to\;\sum = 0 .$$

(3) Mixed weak $[SU(2)]^2U(1)_Y$ (weak doublets only): $\;3\cdot\tfrac16-\tfrac12 = 0.$

(4) Mixed color $[SU(3)]^2U(1)_Y$ (color triplets only): $\;2\cdot\tfrac16-\tfrac23+\tfrac13 = 0.$

(5) Cubic color $[SU(3)]^3$: $\;2-1-1 = 0$ — color is vector-like across $(Q_L,u^c_L,d^c_L)$.

(6) Witten $SU(2)$ mod-2: number of $SU(2)$ doublets $=3+1=4$, which is even, so there is no Witten anomaly.

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

The obstruction map. Collect the local pieces: $$O_{\rm SG4,local}(E)=\big(O_{\rm charge}(E),\,O_{\rm anomaly}(E),\,O_{\rm Witten}(E)\big),\qquad O_{\rm SG4,local}(E_{\rm frozen})=0 .$$ The full gate obstruction additionally carries the descent/global piece: $$O_{\rm SG4}(E)=\big(O_{\rm descent}(E),\,O_{\rm charge}(E),\,O_{\rm anomaly}(E),\,O_{\rm Witten}(E)\big).$$ We do not assert $O_{\rm SG4}(E_{\rm frozen})=0$: the descent/global residual is not closed.

7. The $\mathbb Z_6$ structure, split into three honest objects

The single phrase "the $\mathbb Z_6$" hides three different claims with three different statuses:

1. Center-locking congruence $\;\omega_3^{k_3}\omega_2^{k_2}\omega_6^{6Y}=1 \Rightarrow 6Y\in\mathbb Z$, giving $Y\in\tfrac16\mathbb Z$ with the $\tfrac16$ spacing the $\mathrm{lcm}$ of the $SU(3)$ and $SU(2)$ center denominators, $\tfrac{1}{\mathrm{lcm}(3,2)}=\tfrac16$. Status: DERIVED-GIVEN-E (given the declared global-form/center table and $E$).
2. The declared $\mathbb Z_6$ quotient/table. Status: AXIOM-OPEN / declared — selected and frozen, not forced.
3. $\mathbb Z_6$-finestness, the claim $\Gamma=\mathbb Z_6$. Status: OPEN — the line-operator argument currently pins only $\Gamma\le\mathbb Z_6$, not equality.

So the hypercharge spacing is DERIVED-GIVEN-E, the quotient is a declared axiom, and finestness is open. The congruence is load-bearing for the lattice, but it is not a closed root theorem — the open target is to derive the quotient/finestness from line operators, global form, or bundle descent without tuning to the known Standard-Model answer.

8. Open residuals — the full quantum/global completion family

The local arithmetic above is one face of SG-4. These distinct residuals make up the rest, and none is closed by local anomaly cancellation:

• $\mathbb Z_6$-finestness — $\Gamma=\mathbb Z_6$ vs $\Gamma\le\mathbb Z_6$. OPEN.
• R4 mixed 't Hooft anomaly $\xi_{R4}$ — OPEN / computation debt. A distinct, finite computation, not the same object as BV–BRST and not a mass-gap wall. Operative description: $d_5=Q_1=\beta P^1$, the untwisted $\mathbb Z_3$ survives, and the $\tau_{K_6}=(2,2)$ twist-correction is the sole lever; no value and no hash yet.
• BV–BRST descent + descent measure — AUDIT / NO-KNOWN-ROUTE. Named separately; it is not hidden inside the local anomaly ledger and is not proven.
• Full 16/16 quantum-consistency coverage — OPEN. A coverage roll-up that references R4 and BV–BRST as part of the unresolved surface but does not replace them; partial coverage is not closure.
• Machine certificates G03 / G05 — AUDIT / BLOCKED. Hand-reproduction mitigates but does not close the machine certs.

These three — BV–BRST, R4, and the 16/16 roll-up — are separate rows under one top-level family: full quantum/global completion.

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

• SG-4 does not derive $E$. Formally: $\;E_{\rm frozen}\in\ker O_{\rm SG4,local}$, not $\;\ker O_{\rm SG4,local}=\{E_{\rm SM}\}$.
• Anomaly cancellation is a filter, not a selector. Vector-like additions $R\oplus\bar R$ cancel every anomaly, so anomaly-freedom admits infinitely many spectra; it does not uniquely pick the Standard Model.
• $\mathbb Z_6$-finestness is not proven (only $\Gamma\le\mathbb Z_6$).
• BV–BRST descent is not proven; R4 is not computed; 16/16 coverage is not closed.
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• Local SG-4 closure is not whole-gate closure. $O_{\rm SG4,local}=0$ does not imply $O_{\rm SG4}=0$.

10. Specialist closure plan

Each open residual is a concrete, finite work-package:

1. $\mathbb Z_6$-finestness — prove $\Gamma=\mathbb Z_6$ from line operators / global form / bundle descent (target-blind), or honestly keep $\Gamma\le\mathbb Z_6$.
2. R4 / $\xi_{R4}$ — compute the value via $d_5=Q_1=\beta P^1$ with the $\tau_{K_6}=(2,2)$ twist-correction; record the value and a hash, or keep it open as a named computation debt.
3. BV–BRST descent + measure — identify an exact descent route and measure, or keep it disclosed as no-known-route (inherited-standard-QFT stance).
4. 16/16 coverage — complete the quantum-consistency roll-up; do not treat partial coverage as closure.
5. Machine certs G03 / G05 — rerun or replace; pair with the hand-reproduction.

Closing all five upgrades SG-4 from "local leg closed, gate open" toward whole-gate closure — and even then, only given $E$.

11. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 (frozen mid-audit reading: OPEN) · $O_{\rm SG4,local}(E_{\rm frozen})=0$ · local leg DERIVED-GIVEN-E · $E$ not derived · frozen hashes · $Q=T_3+Y$ · $Y\in\tfrac16\mathbb Z$ · center-locking relation · $\mathbb Z_6$ quotient AXIOM-OPEN · $\mathbb Z_6$-finestness OPEN · all six anomaly ledgers · diagnostic $\sum Y^2=\tfrac{10}{3}\neq0$ · R4 residual · BV–BRST residual · 16/16 residual · G03/G05 cert residual · anomaly-is-a-filter anti-claim · hashes-don't-validate anti-claim.

Required absence (all held): no claim that SG-4 is physics-closed · $E$ is derived · anomaly cancellation selects the SM · $\ker O_{\rm SG4}=\{E_{\rm SM}\}$ · $\mathbb Z_6$-finestness proven · BV–BRST proven · R4 computed · 16/16 closed · hashes validate physics · local closure = global completion.

This is the canonical gate anchor ledger; the other gates follow the same eleven-part shape and the same universal table.

See also: the anchoring method · Layer 2 — no unpaid exact labels (why anomaly cancellation is a filter, not a selector) · Layer 4 — carrier-forcing & the given-E wall · the full SG-4 dossier.