# SG-4 — Hypercharge Table + Anomaly Witness (GUT Gate 4 / §6.3 + §6.5): Per-Gate Closure-Attack Dossier

> **What this is.** The closure-attack packet for **SG-4 (the hypercharge table + the hand-checkable anomaly
> witness)** on the **frozen 13D K₆ branch ONLY**. It recaps how our geometry fixes the charge embedding and
> inherits anomaly cancellation (concise; the full derivation lives on the published site), verifies where the
> status actually stands, names every open residual, and lays out the attack plan to push the gate further. It
> is **not** a rival comparison (that is `BATTLE_GATES/`), **not** the one-line status ledger, **not** a
> reprint of the manuscript.
>
> **Binding discipline (carry verbatim).** No status was ever upgraded. Frozen branch `dcc66f1b2685` / `a5b1e6f9d951`
> READ-ONLY. Honest throughout: anomaly cancellation is a **FILTER on the chiral content E, not a determiner**
> of it (there are infinitely many anomaly-free chiral solutions; our geometry passes the filter by *producing
> one SM generation per family*, it does not *derive* that this is the only solution); the perturbative
> ledgers and the by-hand witnesses are **genuinely hand-checkable and they vanish**, but they are evaluated
> **given-E** (the Gate-4 chiral spectrum supplied by SG-2/SG-3); the **BV-BRST descent (UQF-4)** stays
> **AUDIT / NO-KNOWN-ROUTE**; the **R4 mixed 't Hooft anomaly `ξ_R4`** stays **OPEN** — it is a `NOT-A-WALL`
> element whose *value is genuinely unknown* (the operative differential is the 3-primary `d₅ = Q₁`, not a
> 2-primary `d₃`; the untwisted default is the nonzero `ℤ₃` survivor; the `τ_K₆ = (2,2)` twist-correction is
> the sole remaining lever). Closure paths must be **non-target-loaded** (the κ³/π kill-test: a proposed axiom
> counts only if it would be written WITHOUT knowing the answer). **given-E ≠ derivation of E.** AXIOM-CLOSED ≠
> proven; selection ≠ derivation; dissolved ≠ solved.

---

## 0. Header & Verdict

| Field | Value |
|---|---|
| **Gate id** | SG-4 — Hypercharge table + hand-checkable anomaly witness |
| **GUT manuscript gates spanned** | **Gate 3** (hypercharge & electric charge; §6.3, narrative §5.2, companion CR3, certificate `certificates/G03_charge_z6/`) **+ Gate 5** (anomaly cancellation; §6.5, narrative §5.4 → §3 worked example, certificate `certificates/G05_anomaly_cancellation/`, Appendix E′). The dossier-spec name "SG-4 (Gate 4)" bundles the charge-recovery and anomaly-closure faces that the manuscript files as Gates 3 and 5; the chirality/family-count face (manuscript Gate 4 / SG-3) is a separate dossier (SG-3, `DERIVED-GIVEN-E`). |
| **Status label (binding)** | **PARTIAL** |
| **Manuscript card status** | Both faces are *Claimed certificate pass* in the manuscript (§6.3, §6.5); PARTIAL is the honest scoped-GUT roll-up: the **perturbative** ledgers pass as filters, but the **filter-not-determiner** caveat, the **BV-BRST (UQF-4)** descent and the **R4 mixed 't Hooft (`ξ_R4`)** anomaly stay AUDIT/OPEN. |
| **Frozen hashes it rides** | Branch `dcc66f1b2685` / manifest meta `a5b1e6f9d951`. Charge face: parity table `ac4d2df3e708` (R1.3, co-read by Gate 4), the alternate Z₆-closure freeze `a68ee92a75be` (R1.3, CR3.8 / D.3.1), R1.4 hypercharge lattice. Anomaly face: surviving chiral spectrum A2.3 (Gate-4 output); convention block ($A(\bar R) = -A(R)$, $T(\text{fund}) = 1/2$, left-handed Weyl basis); E′ trace tables. R4 face (open): home `Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`; twist `τ_K₆ = τ(c̄₁(L_{K₆}))`, `c̄₁ = 2ρ mod 3 = (2,2) ≠ 0`; carrier `x̄₁ ∈ H³(BPU(3);ℤ/3)`; operative differential `d₅ = Q₁ = βP¹`. **`ξ_R4`: no value, no hash — OPEN.** |

### 0.1 Abstract — established / open / what would close it

**Target anchor(s):** terminates on the **observed spectrum E (spin-ℂ index −3)** — anomaly cancellation is a
FILTER on E, not a determiner (charge table + perturbative ledgers DERIVED-GIVEN-E); **no new measured anchor
needed**, while the **BV-BRST descent (UQF-4/7)** and **`ξ_R4`** are **COMPUTATION-DEBTS, value UNKNOWN / OPEN**
(see §A "Target anchor(s) for this gate").

**Established (given-E, given the selected & frozen geometry).** Two distinct, genuinely strong things are
established, both **given** the Gate-4 chiral spectrum E:

1. **The charge table is a geometric output, not a fit.** The weak sphere $S^2$ supplies $T_3$, the folded
   hypercharge circle $S_Y^{\,1}/\mathbb{Z}_2$ supplies the $U(1)_Y$ direction, and a **global $\mathbb{Z}_6$
   centre-locking rule** ties the colour-$\mathbb{Z}_3$, weak-$\mathbb{Z}_2$ and a sixth-root of the
   hypercharge phase into the closure $\omega_3^{k_3}\omega_2^{k_2}\omega_6^{6Y}=1$, which **forces hypercharge
   onto the $\tfrac16\mathbb{Z}$ lattice** and yields the full one-generation $Q = T_3 + Y$ table
   ($Q_L:+\tfrac16$, $u_R:+\tfrac23$, $d_R:-\tfrac13$, $L_L:-\tfrac12$, $e_R:-1$, $H:+\tfrac12$) with **no
   per-multiplet adjustment**. This is hand-checkable in five minutes (CR3.9–3.10).
2. **The six perturbative anomaly ledgers vanish exactly, by hand.** On that frozen one-generation spectrum,
   the two pure-hypercharge witnesses ($\sum Y = 0$, $\sum Y^3 = 0$), the two mixed witnesses
   ($[SU(3)]^2U(1)_Y$, $[SU(2)]^2U(1)_Y$ — the latter's by-hand form $3\cdot\tfrac16 - \tfrac12 = 0$), the
   $[SU(3)]^3$ trace, the $[\mathrm{grav}]^2 U(1)_Y$ trace, and the **mod-2 Witten** doublet-parity count
   ($3+1=4$, even) all close in exact rational arithmetic (§3.3, §3.6.2, Appendix E′). Anomaly cancellation is
   **inherited** because the geometry's index output is exactly one SM generation per family.

**Open (the honest deductions).** (1) Anomaly cancellation is a **FILTER, not a determiner** — there are
infinitely many anomaly-free chiral assignments; passing the filter certifies *this geometry produces a
consistent spectrum*, it does **not** derive that the SM content is the unique solution. (2) Only **~10 of the
16** independent quantum-consistency classes (the perturbative + discrete-global set our geometry is checked
against) are demonstrably closed at certificate grade; the remainder are AUDIT/OPEN. (3) The **BV-BRST descent
(UQF-4)** — the *non-perturbative*, coset-twist / descended-anomaly statement that the BRST differential is
nilpotent and the descent measure exists — stays **AUDIT / NO-KNOWN-ROUTE**. (4) The **R4 mixed 't Hooft
anomaly `ξ_R4`** — the geometry's *own* candidate anomaly object (the $\mathbb{Z}_3$ colour-centre one-form
symmetry of the pure-glue projection, twisted by the K₆ data) — is **OPEN**: well-posed as a finite twisted
Spin-c bordism class, but its value is uncomputed (`NOT-A-WALL`, value `UNKNOWN`).

**What would close it (the ladder).** DERIVED-CLOSED on the *determiner* gap is **unreachable** — anomaly
cancellation is a filter by its mathematical nature; the honest ceiling is to **state the filter-not-determiner
boundary precisely** (AXIOM-/DISCLOSED-CLOSED). For the perturbative face, the realistic win is **VERIFIED**:
mount and re-run the `certificates/G03_charge_z6/` and `certificates/G05_anomaly_cancellation/` exact-rational
checks target-blind. For UQF-4, the ceiling is **AXIOM-CLOSED** (name the BV-BRST descent as an
inherited-standard-QFT consistency import) or a specialist construction (DERIVED, hard). For `ξ_R4`, the
ceiling is a **specialist twisted-bordism computation** (the `τ_K₆`-twist-correction to `d₅`) — DERIVED-CLOSED
if the value is computed, sharper-OPEN if it remains a finite-but-unsettled `ℤ₃`-linear-algebra question.

### 0.2 Website source-of-truth (link, don't duplicate)

The full construction, the worked charge tables, the six-ledger trace table, and the freeze records are the
**published manuscript**, Paper I:

- **GUT.html §6.3** (Gate-3 card, hypercharge/charge) and **§6.5** (Gate-5 card, anomaly) —
  <https://physics.magflowmeters.com/articles/GUT.html>
- **§5.2 / §5.4** narrative modules; **§3** (the worked anomaly example, end-to-end — §3.3 the six ledgers, §3.6.2
  the one-generation trace table, §3.7 where the claim is actually closed); **Appendix CR3** (hypercharge reader
  companion — CR3.5 $Q=T_3+Y$, CR3.7 the $S_Y^1/\mathbb{Z}_2$ source, **CR3.8 the $\mathbb{Z}_6$ closure**,
  CR3.10 the full table, CR3.13 remove-one-term tests); **Appendix CR5** (anomaly reader companion); **Appendix
  D** (representation-level recovery, charge tables D.2/D.3.1); **Appendix E** (chirality half) + **Appendix E′**
  (anomaly closure — full six-ledger trace table, conventions); **Appendix R0/R1** (freeze records).
- The **R4 / UQF-4 / `ξ_R4`** open objects are *not* in GUT.html (they live in Paper III/IV's quantum-completion
  audit and the Gap-02 program); see the corpus map in §5.

This dossier recaps only what is needed to attack; the site controls all common material.

---

## 1. How OUR geometry closes SG-4 — the closure claim (concise)

> Full derivation: **GUT.html §6.3 + §6.5 + §3 + Appendices CR3/CR5/D/E′**. This section is the attack-grade
> recap, not the derivation.

SG-4 has **two faces** that must be attacked separately because they fail (or hold) for different reasons.

### 1.1 Face A — the hypercharge table (geometric, hand-checkable, strong)

The charge embedding is generated by three load-bearing factors, named so a reader can attack each:

1. **The weak sphere $S^2$ ($\times$-layer)** supplies the isospin generator $T_3$ on every doublet
   (CR3.6). (Attack handle: $T_3$ is *given-E* — it rides SG-2's gauge recovery, which is itself a cross-framework
   TIE, not a Fable-discriminating determiner.)
2. **The folded hypercharge circle $S_Y^{\,1}/\mathbb{Z}_2$ ($\times/\oplus$-layer)** supplies the $U(1)_Y$
   direction and, via the orbifold boundary, the no-mirror projection (CR3.7). The hypercharge **line bundle
   $L_Y$** ($\otimes$-layer) carries the actual $Y$ labels.
3. **The global $\mathbb{Z}_6$ centre-locking rule ($\oplus$-layer)** — the decisive structural object. The SM
   group is the quotient $G_{\rm SM} = [SU(3)_c \times SU(2)_L \times U(1)_Y]/\mathbb{Z}_6$, where the
   $\mathbb{Z}_6$ acts by $k \mapsto (\zeta_3^k, (-1)^k, e^{2\pi i k/6})$. The consistency condition for a
   multiplet $(k_3,k_2,k_6)$ is the **closure**
   $$ \omega_3^{k_3}\,\omega_2^{k_2}\,\omega_6^{6Y} = 1 \quad \text{in } \mathbb{Z}_6, \qquad \omega_n=e^{2\pi i/n}, $$
   which is **exactly what forces hypercharge onto the $\tfrac16\mathbb{Z}$ lattice** — finer than the integer
   lattice a bare circle would give, by precisely the factor six the PDG charges require (CR3.8; D.3.1; freeze
   `a68ee92a75be`).

**The output is the full one-generation table** via $Q = T_3 + Y$ applied five times after the $S^2$ and
$S_Y^1/\mathbb{Z}_2$ labels are fixed — no per-multiplet fit (CR3.9–3.10):

| Multiplet | $SU(2)_L$ | $T_3$ | $Y$ | $Q = T_3 + Y$ |
|---|---|---|---:|---|
| $Q_L=(u_L,d_L)$ | doublet | $\pm\tfrac12$ | $+\tfrac16$ | $+\tfrac23,\,-\tfrac13$ |
| $u_R$ | singlet | $0$ | $+\tfrac23$ | $+\tfrac23$ |
| $d_R$ | singlet | $0$ | $-\tfrac13$ | $-\tfrac13$ |
| $L_L=(\nu_L,e_L)$ | doublet | $\pm\tfrac12$ | $-\tfrac12$ | $0,\,-1$ |
| $e_R$ | singlet | $0$ | $-1$ | $-1$ |
| $H$ | doublet | $\pm\tfrac12$ | $+\tfrac12$ | $+1,\,0$ |

The neutrino comes out **exactly neutral** and the down quark at **exactly $-\tfrac13$** without either value
being chosen — this is the genuine win of Face A.

### 1.2 Face B — the anomaly witnesses (inherited filter, hand-checkable)

On the **frozen Gate-4 chiral spectrum** (one SM generation per family, A2.3), six quantum-consistency ledgers
are required to vanish. Each is a finite sum of fixed representation weights over the particle list — *nothing
to tune* (§3.3). With the convention block ($A(\bar R) = -A(R)$, $T(\text{fund})=\tfrac12$, left-handed Weyl
basis; right-handed singlets entered through their left-handed conjugate descriptions), the six are:

| # | Ledger | Form | Witness |
|---|---|---|---|
| L1 | $[U(1)_Y]^3$ | $\sum Y^3 = 0$ | $(1 - 32 + 4 - 9 + 36)/36 = 0$ (the §3.6.2 table) |
| L2 | $[\mathrm{grav}]^2 U(1)_Y$ | $\sum Y = 0$ | $+1-2+1-1+1 = 0$ |
| L3 | $[SU(2)]^2 U(1)_Y$ | $\sum_{\rm doublets} Y = 0$ | $3\cdot\tfrac16 - \tfrac12 = 0$ (the by-hand mixed witness) |
| L4 | $[SU(3)]^2 U(1)_Y$ | $\sum_{\rm triplets}Y = 0$ | $2\cdot\tfrac16 - \tfrac23 + \tfrac13 = 0$ |
| L5 | $[SU(3)]^3$ | $\sum_{\rm triplets} A(R) = 0$ | $Q_L$ ($\mathbf 3$) + $u_R^c$ ($\bar{\mathbf 3}$) + $d_R^c$ ($\bar{\mathbf 3}$) $\Rightarrow 1 - 1 = 0$ by vector-likeness in colour |
| L6 | $[SU(2)]^3$ (Witten mod-2) | # weak doublets even | $3 + 1 = 4$, even |

**The clean diagnostic that the cancellation is not cheap.** The *square* of the hypercharge does **not**
vanish: $\sum Y^2 = \tfrac{10}{3} \neq 0$ (§3.3). Exactly the six consistency-critical combinations vanish and
nothing innocent does — five unpaired fractions conspiring to zero. That specificity is the content of the
filter. **But** the cancellation is **inherited**: the geometry earns its anomaly pass by *producing the right
list* (one SM generation per family), and the SM's own cancellation comes along. The certificate verifies the
pass on the list actually produced, **with no content added afterward** (§3.5 closing note).

### 1.3 What "closed" means here, and its conditionality

"Closed" for SG-4 is **a geometric charge embedding + an inherited consistency filter passing on the produced
spectrum** — strictly **not**:

- not "anomaly cancellation derives the SM content" (it is a filter; see R1);
- not "the hypercharge assignments are unique across all geometries" (Face A is computed *given* the SM gauge
  group and the produced spectrum);
- not "all quantum consistency is established" (the non-perturbative BV-BRST descent and the R4 mixed anomaly
  are AUDIT/OPEN; see R3, R4).

It is conditional on:

- **given-E** — the SM chiral content (and the family index $-3$) supplied by SG-2/SG-3; SG-4 inherits SG-3's
  conditionality and does not derive E.
- **given-the-selected-geometry** — the frozen 13D K₆ branch, the folded hypercharge circle, and the
  $\mathbb{Z}_6$ quotient as *declared* global structure.

**The genuine, defensible content** (the part that survives the honest accounting):

| Genuine output | Mechanism | Grade |
|---|---|---|
| Full one-generation $Y$ / $Q$ table on the $\tfrac16\mathbb{Z}$ lattice | $S^2$ ($T_3$) + $S_Y^1/\mathbb{Z}_2$ ($Y$) + $\mathbb{Z}_6$ closure | hand-checkable |
| Six perturbative ledgers vanish; $\sum Y^2 \neq 0$ diagnostic | exact rational arithmetic on the frozen spectrum | hand-checkable |
| The geometry produces exactly one SM generation per family | spin-ℂ index $-3$ (SG-3) feeding the anomaly filter | inherited |

**The conditional / non-genuine content:**

| Claim | Honest reality |
|---|---|
| "Anomaly cancellation selects the SM" | **FALSE as a determiner** — it is a filter; infinitely many anomaly-free chiral solutions exist (R1) |
| "All 16 quantum classes close" | **~10/16** demonstrably close at certificate grade; the rest AUDIT/OPEN (R2) |
| "The theory is quantum-consistent non-perturbatively" | BV-BRST descent (UQF-4) is **AUDIT / NO-KNOWN-ROUTE** (R3) |
| "Every 't Hooft anomaly is accounted for" | the R4 mixed `ξ_R4` is **OPEN / value UNKNOWN** (R4) |

So the precise statement of closure: **the geometry genuinely produces the SM charge table from the $\mathbb{Z}_6$
closure, and the six perturbative anomaly ledgers vanish by hand on the produced spectrum — but anomaly
cancellation is a consistency FILTER passed by inheritance, not a derivation of the content, and the
non-perturbative (BV-BRST) and geometry-specific (R4) anomaly statements remain open.**

---

## 2. Verify the status — is PARTIAL real?

This is the verification a skeptic would run. Each witness gets a grade — **hand-checkable** / **symbolic** /
**machine-lane** — and an honest **reproduces?** flag.

### 2.1 The witness ledger

| # | Witness | What it asserts | Grade | Reproduces? |
|---|---|---|---|---|
| W1 | **$\mathbb{Z}_6$ closure** $\omega_3^{k_3}\omega_2^{k_2}\omega_6^{6Y}=1$ | hypercharge forced onto $\tfrac16\mathbb{Z}$; full $Y$ table | hand-checkable | **Yes** — each row's closure recomputes by hand (CR3.8–3.10); $6Y \in \mathbb{Z}$ checks for all six multiplets |
| W2 | **$Q = T_3 + Y$ table** | every $Q$ matches PDG | hand-checkable | **Yes** — five applications of one formula; $\nu_L$ neutral, $d_R$ at $-\tfrac13$ recover exactly (CR3.9) |
| W3 | **$\sum Y = 0$, $\sum Y^3 = 0$** | the two pure-$Y$ ledgers vanish | hand-checkable | **Yes** — the §3.6.2 table sums to 0/36 in the consistent left-handed bookkeeping (note: $\sum Y^3$ returns $-4/9$ in mixed conventions — a real trap, disclosed) |
| W4 | **Mixed by-hand witness** $3\cdot\tfrac16 - \tfrac12 = 0$ | $[SU(2)]^2U(1)_Y$ closes | hand-checkable | **Yes** — the headline five-minute check |
| W5 | **$\sum Y^2 = \tfrac{10}{3} \neq 0$** | cancellation is specific, not generic | hand-checkable | **Yes** — the diagnostic that distinguishes a filter from a triviality |
| W6 | **Witten mod-2** ($3+1=4$ even) | $[SU(2)]^3$ global anomaly absent | hand-checkable | **Yes** — doublet count per family even |
| W7 | **$[SU(3)]^3$ + $[SU(3)]^2U(1)_Y$ traces** | the colour ledgers close | symbolic | **Yes** — colour vector-likeness; tabulated in E′ |
| W8 | **`certificates/G03_charge_z6/`** (exact-fraction check) | the charge table regenerates from frozen labels | machine-lane | **AUDIT** — referenced (reuses the G05 spectrum file); not independently re-run here |
| W9 | **`certificates/G05_anomaly_cancellation/run.sh`** (exact-rational ledger) | all six ledgers vanish from the frozen spectrum | machine-lane | **AUDIT** — referenced; the exact-rational script is short and error-bar-free but not re-executed in this audit |
| W10 | **R4 well-posing** (`ξ_R4 ∈ Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`) | the geometry's own anomaly is a finite, finitely-checkable bordism question | symbolic | **Partial** — the *home* is verified; the *value* is OPEN (next computation = the `τ_K₆`-twist-correction to `d₅`) |
| W11 | **UQF-4 BV-BRST descent** | nilpotent BRST + descent measure exists | — | **No route** — NO-KNOWN-ROUTE (OPEN_WALLS_REGISTER): "closes when the coset-twist / descended anomaly classes are constructed by a specialist" |

### 2.2 What reproduces, plainly

- **Face A reproduces fully by hand.** The $\mathbb{Z}_6$ closure forces the $\tfrac16\mathbb{Z}$ lattice and the
  $Q = T_3 + Y$ table falls out with no fit. This is the strongest leg of SG-4 and is genuine geometry-to-charge
  recovery (given the SM group and the produced spectrum).
- **Face B's six perturbative ledgers reproduce by hand.** The mixed witness $3\cdot\tfrac16 - \tfrac12 = 0$,
  the cubic $\sum Y^3 = 0$, the linear $\sum Y = 0$, the colour traces, and the Witten parity all close in exact
  rational arithmetic. The $\sum Y^2 = \tfrac{10}{3}$ diagnostic confirms the cancellation is *specific*, the
  feature that makes it a genuine filter and not a triviality.
- **The exactness is real.** Because every ledger is a finite sum of rationals, the certificate is a short
  exact-rational script with **no error bars** — a clean pass/fail. This is the manuscript's cleanest gate by
  that measure.

### 2.3 What does NOT yet reproduce (honest gaps in the status)

- **The machine certificates (W8/W9) are AUDIT.** The exact-fraction charge check and the exact-rational anomaly
  ledger are *referenced*; the load-bearing executable (the run.sh regenerating the ledger byte-equal from the
  frozen spectrum file) is **not independently re-run** in this audit — the same class as SG-7's absent
  δ-harness and SG-8's J.6/K.5 mount. Until re-run target-blind, "the ledgers regenerate from the frozen hashes"
  is **declared-not-independently-verified here.** (Mitigant: unlike SG-7, the underlying arithmetic is
  *hand-reproducible*, so the AUDIT is lower-stakes — a skeptic can verify the math directly.)
- **The "anomaly cancellation selects E" reading does NOT reproduce — it is mathematically false.** Anomaly
  cancellation is a *filter*: there are infinitely many anomaly-free chiral $U(1)$ and non-abelian assignments
  (the constraint is a finite set of Diophantine/polynomial conditions with an infinite solution variety). The
  geometry passes the filter by *producing one SM generation per family*; it does **not** derive that this is the
  only solution. The manuscript is already honest here (§3.5 closing note, §3.6.3 "what it does not show"), but
  any roll-up that reads SG-4 as "the SM is selected by anomaly cancellation" overclaims.
- **`ξ_R4` does not reproduce — it does not yet have a value.** The R4 mixed 't Hooft anomaly is well-posed
  (a finite twisted Spin-c bordism group) but **uncomputed**; the untwisted default is the nonzero `ℤ₃`
  survivor, and whether the frozen `τ_K₆ = (2,2)` twist deforms it is an unsettled finite `ℤ₃`-linear-algebra
  question. No value, no hash.
- **UQF-4 (BV-BRST descent) does not reproduce — there is no route.** The quantized BV-BRST descent measure and
  the coset-twist / descended anomaly classes are **not constructed**; flagged NO-KNOWN-ROUTE.

### 2.4 Why PARTIAL (not higher, not lower)

- **Not DERIVED-GIVEN-E** (the SG-2/SG-3 tier): SG-4 is not a single rigid integer/representation recovery. Its
  charge face is strong and given-E, but its anomaly face is a *filter* (not a determiner), only ~10/16 quantum
  classes close, and two anomaly statements (BV-BRST, R4) are AUDIT/OPEN — too many open residuals for the
  DERIVED tier.
- **Not OPEN/DECLARED-FROZEN:** the genuine wins are real and hand-checkable — the $\mathbb{Z}_6$-forced charge
  table and the six vanishing ledgers are *not* reparameterizations; the $\sum Y^2 \neq 0$ diagnostic proves the
  cancellation is specific.
- **PARTIAL is exactly right:** a strong, hand-checkable geometric charge embedding + inherited perturbative
  anomaly closure, with a precise filter-not-determiner boundary, ~10/16 classes closed, and the BV-BRST and R4
  residuals named AUDIT/OPEN. This matches the SCOPED_GUT ledger SG-4 line verbatim.

---

## 3. The attack surface — what to attack (ranked by leverage)

Every open residual as a named object with its precise obstruction. **Leverage** = how much closing it moves
the gate (and how much it sharpens the claim boundary).

### 3.1 The residual register

| ID | Residual (named object) | Precise obstruction | Status | Leverage |
|---|---|---|---|---|
| **R1** | **Anomaly = FILTER, not determiner** | Anomaly cancellation is a finite set of Diophantine/polynomial conditions with an **infinite** solution variety; passing certifies *this geometry produces a consistent spectrum*, NOT that the SM content is forced. Any "anomaly selects the SM" reading overclaims. | DISCLOSED (boundary stated in §3.5/§3.6.3; must stay stated) | **HIGHEST** — it is the claim-boundary that conditions the whole anomaly face |
| **R2** | **Only ~10/16 quantum classes close** | The full quantum-consistency set our geometry should be checked against is ~16 independent classes (the 6 perturbative ledgers + discrete-global/center/parity/descent classes). Only ~10 are demonstrably closed at certificate grade; the rest (global-discrete, descent, the R4 mixed) are AUDIT/OPEN. | OPEN (partial coverage) | **HIGH** — defines exactly how much of "quantum consistency" is actually delivered |
| **R3** | **BV-BRST descent (UQF-4) AUDIT** | The *non-perturbative* statement — nilpotent BRST differential + a constructed quantized descent measure + the named coset-twist / descended anomaly classes — is **NO-KNOWN-ROUTE**. Perturbative ledger vanishing does not establish it. | AUDIT / NO-KNOWN-ROUTE | **HIGH** — the only path to "quantum-consistent" beyond one-loop perturbation theory |
| **R4** | **R4 mixed 't Hooft anomaly `ξ_R4` OPEN** | The geometry's OWN candidate anomaly: the $\mathbb{Z}_3$ colour-centre one-form symmetry of the pure-glue projection, twisted by `τ_K₆`. Well-posed as `Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`; **value UNKNOWN**. Operative differential is the 3-primary `d₅ = Q₁ = βP¹` (NOT a 2-primary `d₃`, which is identically zero on 3-torsion); untwisted default = nonzero `ℤ₃` survivor; `τ_K₆ = (2,2)` twist-correction the sole lever. `NOT-A-WALL`. | OPEN (value), well-posed (home) | **MEDIUM-HIGH** — geometry-specific; the most computationally tractable open object, but `NOT-A-WALL` (value is inert for the mass gap) |
| **R5** | **Machine certificates AUDIT** (`G03_charge_z6/`, `G05_anomaly_cancellation/run.sh`) | The exact-fraction charge check and exact-rational anomaly ledger are referenced, not independently re-run here. | AUDIT | **MEDIUM** — cheapest to close (owner artifact); makes "certificate pass" machine-real; lower-stakes because hand-reproducible |
| **R6** | **Charge table is given-E + given-group** | Face A computes $Y$/$Q$ *given* the SM gauge group (SG-2, a cross-framework TIE) and the produced spectrum (SG-3, given-E). It does not derive the SM group or content. | DISCLOSED (inherited conditionality) | **LOW-MEDIUM** — honest framing; no new physics |
| **R7** | **$\mathbb{Z}_6$ quotient is declared global structure** | The $\mathbb{Z}_6$ centre-locking is a *declared* admissible quotient (one of $\{1,\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_6\}$); "the finest quotient is $\mathbb{Z}_6$" is pinnable only by already wanting the $\tfrac16\mathbb{Z}$ lattice (the A4-Z6 sub-claim B finding). | OPEN (selection ≠ forcedness) | **LOW-MEDIUM** — selection-inside-a-category; sharpens what $\mathbb{Z}_6$ does and does not derive |
| **R8** | **The $\bar3$-vs-$3$ orientation bit** | `ξ_R4 ≠ 0` (if the class survives) additionally needs the $\bar c_1(L_{K_6})$ pushforward to land on it **plus** one owner orientation bit (colour rep $3$ vs $\bar3$). The bare object $\xi_{R4}=0$ is forced; the nonzero value needs the bit. | OPEN (owner bit) | **LOW** — only matters conditional on R4 surviving; a single discrete choice |

### 3.2 Leverage ranking (attack order)

1. **R1** (filter-not-determiner) — the single most important claim-boundary; state it precisely and bind it.
2. **R2** (~10/16 classes) — the honest measure of how much quantum consistency is delivered.
3. **R3** (BV-BRST descent) — the only path past perturbation theory; NO-KNOWN-ROUTE, so the realistic move is
   to name it as an inherited-standard-QFT import or hand it to a specialist.
4. **R4** (`ξ_R4`) — geometry-specific and most computationally tractable; the specialist twisted-bordism target.
5. **R5** (certificate AUDIT) — cheapest; owner artifact; makes the pass machine-real.
6. **R6 / R7 / R8** — inherited conditionality / declared-quotient selection / owner orientation bit.

> **The cardinal honest point.** R1 is **not a problem to be solved** — it is a true mathematical fact about
> what anomaly cancellation *is*. The attack on R1 is to **state the boundary**, not to manufacture a
> determination; any "we derived the SM from anomaly cancellation" closure would be a category error, not a
> reduction. R3 (BV-BRST) and R4 (`ξ_R4`) are the genuine-physics open objects; both carry an explicit κ³/π
> kill-test so a tuned or target-loaded closure cannot be banked. R5 is the non-physics residual
> (reproducibility); closing it improves machine-reality but moves no claim.

---

## 4. THE ATTACK PLAN — closure paths (the core)

For each residual: the **technique** (closure playbook T1 axiom-floor / T4 no-go / T5 firewall / T6
all-operator-conditional / T7 eliminative / T10 selector), the **named axiom it could reduce to** (stated so it
would be written WITHOUT the answer — the κ³/π kill-test), the **specialist target** (theorem to hand off) or
the **owner artifact** (certificate/computation) needed, the **math to attempt**, and the **success ladder**
(DERIVED-CLOSED rare → AXIOM-CLOSED likely → sharper-OPEN → REFUTED).

---

### 4.1 R1 — Anomaly cancellation is a FILTER, not a determiner (the claim-boundary)

**Why first.** The entire anomaly face's *force* depends on reading it correctly. Read as a determiner, SG-4
overclaims and invites the fatal objection "you assumed the SM content." Read as a filter, SG-4's anomaly face
is exactly as strong as it honestly is: a consistency check the produced spectrum passes.

**Technique: T5 (no tuning to the known answer firewall) + T2-guarded restatement.** This is **not** a problem to solve
with math — it is a **wrong premise to drop**. The premise "anomaly cancellation determines the chiral content"
is false; the audit's job is to drop it and state what replaces it.

**Named axiom / principle it reduces to (κ³/π-clean):**

> **PRINCIPLE-ANOMALY-IS-A-FILTER** (target-blind): *"Anomaly cancellation is a pass/fail consistency
> constraint on a GIVEN chiral spectrum, with an infinite solution variety; it eliminates inconsistent
> spectra, it does not single out the SM content. The geometry's contribution is to PRODUCE one SM generation
> per family (via the spin-ℂ index $-3$); the cancellation is then INHERITED."*

This is writable without any charge number — it is a statement about *what kind of object* anomaly cancellation
is. It carries no target value, so it passes the kill-test trivially. It is **already substantially the
manuscript's position** (§3.5 closing note: "the anomaly cancellation … is not an independent miracle this
geometry performs — it is the Standard Model's own cancellation, *inherited*"; §3.6.3 "what it does not show").

**Specialist target (optional sharpening).** A bounded **selector-strength theorem**: "Within the declared
search category (admissible compact factors + the $\mathbb{Z}_6$ quotient), is the SM chiral content the
*unique* anomaly-free survivor, or do other geometries in the category also pass?" If a specialist shows the
category is narrow enough that the filter *plus the index constraint* leaves only the SM content, that would
upgrade the filter toward a category-relative determiner — but **expect this to fail**: the filter is infinite
and the category is broad. The honest expected result is that anomaly cancellation **never** becomes a
determiner.

**Math to attempt.** Exhibit two distinct anomaly-free chiral assignments (e.g. the SM plus any vector-like
addition, or a known anomaly-free non-SM $U(1)$ charge solution) to make the infinite-variety point concrete
and un-disputable in the dossier. This is a five-line example, not a research computation.

**Success ladder.**
- DERIVED-CLOSED: **unreachable** — a determiner claim is mathematically false.
- **DISCLOSED-CORRECTED (the ceiling, essentially already standing):** the filter-not-determiner boundary stated
  precisely and bound to the roll-up; "anomaly selects the SM" explicitly retired.
- sharper-OPEN: the selector-strength theorem is attempted and the category is shown broad (filter ≠ determiner
  confirmed).
- REFUTED: someone reads SG-4 as a determiner anywhere in the public package → that line must be patched.

**Honest disposition: DISCLOSED-CORRECTED.** The closure is *stating the boundary*, not raising the claim. This
is the cleanest, lowest-risk win in the gate and requires only that the roll-up never read "anomaly determines E."

---

### 4.2 R2 — Only ~10/16 quantum-consistency classes close (the coverage measure)

**The obstruction.** "Quantum consistency" is not a single check. The honest enumeration of the classes our
geometry should be checked against is ~16 independent obligations: the **6 perturbative ledgers** (L1–L6 above,
demonstrably closed, hand-checkable) **plus** discrete-global / center / parity / descent classes (e.g. the
Witten-type global checks beyond $SU(2)$, the discrete-$\theta$ / one-form anomalies, the BV-BRST descent, the
R4 mixed). Only ~10 of the 16 are at certificate grade; the remainder are R3/R4 and the discrete-global tail.

**Technique: T6 (all-operator-conditional) + T7 (eliminative).** The move is to **enumerate the 16 classes
explicitly** and grade each — converting "~10/16" from a vague fraction into a named, auditable table.

**Named axiom it could reduce to (κ³/π-clean):**

> **AXIOM-QUANTUM-CLASS-COVERAGE** (target-blind): *"SG-4's anomaly closure is asserted at certificate grade
> ONLY for the enumerated perturbative + closed-discrete classes; the remaining classes (descent, R4 mixed,
> any un-enumerated global-discrete) are AUDIT/OPEN and explicitly not claimed."*

This is a *coverage declaration*, not a value — it passes the kill-test by construction.

**Specialist target / owner artifact.** (1) Owner: write the explicit 16-row class table (which 6 are
perturbative-closed, which are the discrete-global tail, which are R3/R4). (2) Specialist: for each
discrete-global class, determine whether it is *inherited-standard-QFT* (automatically satisfied by any
consistent theory with this content — then DISCHARGED-AS-STANCE, not proven) or a *genuine geometry-specific
obligation* (then AUDIT pending construction).

**Math to attempt.** Tabulate the discrete-global anomalies for $G_{\rm SM} = [SU(3)\times SU(2)\times
U(1)]/\mathbb{Z}_6$ on the produced spectrum (the new-anomaly literature for the SM with the $\mathbb{Z}_6$
quotient is mostly settled — most discrete-global anomalies of the SM are known to vanish or be inherited). Map
each onto our 16-class enumeration.

**Success ladder.**
- DERIVED-CLOSED: all 16 classes shown closed (the 6 perturbative + the discrete-global tail inherited + R3/R4
  resolved). **Gated on R3/R4** — so not reachable until those close.
- AXIOM-CLOSED: the coverage table written, the discrete-global tail discharged as inherited-standard-QFT
  stance, R3/R4 left named-OPEN → "~10/16 closed, 6 named-open with dispositions."
- sharper-OPEN: a discrete-global class turns out geometry-specific and uncomputed → it joins R3/R4 as a named
  open obligation.
- REFUTED: a discrete-global anomaly is shown *non-vanishing* on the produced spectrum → SG-4 anomaly face
  downgrades to *Open / not claimed* (the §6.12 Gate-5 downgrade rule).

**Honest disposition: AXIOM-CLOSED on coverage (write the 16-row table; discharge the inherited tail as stance;
leave R3/R4 named-OPEN).** This converts a vague "~10/16" into an auditable boundary. The number 16 itself is a
claimed enumeration that the owner should pin precisely — the dossier reports it as the corpus figure, not as
an independently re-derived count.

---

### 4.3 R3 — BV-BRST descent + descended anomaly classes (UQF-4 / UQF-7) AUDIT / NO-KNOWN-ROUTE

> **Wall-ID note (per OPEN_WALLS_REGISTER, the granular authority).** There are **two** distinct
> NO-KNOWN-ROUTE objects here: **UQF-4 = the named coset-twist / descended anomaly classes** (their
> vanishing) and **UQF-7 = the quantized BV-BRST descent measure** (its existence). The Quantum-paper
> rosetta *bundles* these under "UQF-4 — BRST/BV nilpotency and anomaly closure," which is why this dossier
> elsewhere tags the descent as "UQF-4"; the precise split is UQF-4 (anomaly classes) + UQF-7 (descent
> measure), and **both** are owed for Gate-4. The closure plan below attacks both.

**The obstruction.** The perturbative ledgers establish *one-loop* gauge-anomaly cancellation. They do **not**
establish the non-perturbative statement: that the **BV-BRST differential is nilpotent** ($s^2 = 0$) on the
full interacting theory, that the **quantized descent measure exists** (UQF-7), and that the **named coset-twist /
descended anomaly classes** vanish (UQF-4). Both are flagged **NO-KNOWN-ROUTE** in OPEN_WALLS_REGISTER ("closes
when the coset-twist / descended anomaly classes are constructed by a specialist; no short path exists today").

**Technique: T1 (axiom-floor) for the honest import; specialist construction for the genuine path.**

**Route A (the honest import — most likely outcome).** The BV-BRST descent / nilpotency is a property **any
consistent gauge QFT must satisfy** — it is not a new wall this framework invents. The OPEN_WALLS_REGISTER
already records the perturbative-anomaly/descent KIND as **INHERITED-STANDARD-QFT** ("consistency any consistent
theory must satisfy; not a new wall of this framework").

> **AXIOM-BRST-DESCENT-INHERITED** (target-blind): *"The BV-BRST differential is nilpotent and the descent
> machinery is well-defined for the framework's gauge sector at the same standard as for the SM itself; this is
> imported as a standard-QFT consistency property, not derived here."*

Writable with no framework-specific value; passes the kill-test. **But** importing it is a **STANCE, not a
proof** — and the κ³/π-discipline caveat applies: an EFT "not-owed" adjudication is a stance, nothing built. The
honest disposition stays AUDIT, with the import named.

**Route B (the genuine construction — hard, specialist).** Construct the quantized BV-BRST descent measure for
the *specific* coset/orbifold structure of the frozen branch (the $K_6 = SU(3)/T^2$ coset, the
$S_Y^1/\mathbb{Z}_2$ fold, the $\mathbb{Z}_6$ quotient) and exhibit the descended anomaly classes vanishing.
This is a research-grade construction with **no short path today**.

- **Specialist target:** "Construct the BV-BRST complex for the frozen-branch gauge sector with its coset-twist
  data and show (i) $s^2 = 0$ on the interacting fields, (ii) the descent equations close, (iii) the
  coset-twist / descended anomaly classes are exact (vanish in cohomology)."
- **κ³/π kill-test:** PASSES trivially (no target value is involved — it is a structural construction).

**Math to attempt.** None bounded enough for this dossier — Route B is a multi-month specialist construction.
The actionable step is **Route A**: name the inherited-standard-QFT import explicitly in the claim boundary.

**Success ladder.**
- DERIVED-CLOSED: Route B completed (descent measure constructed, classes shown exact). **Unlikely soon
  (NO-KNOWN-ROUTE).**
- **AXIOM-CLOSED (likely):** AXIOM-BRST-DESCENT-INHERITED named as a standard-QFT import; AUDIT retained with the
  import recorded.
- sharper-OPEN: the import is judged *not* automatically inherited for this coset structure → genuine
  geometry-specific obligation, AUDIT stays.
- REFUTED: a descended anomaly class is shown obstructed → the framework's gauge sector is inconsistent
  (would be a major failure; no indication of this).

**Honest disposition: AUDIT, reducible to AXIOM-BRST-DESCENT-INHERITED (stance, not proof); Route B is the only
DERIVED path and has no short route.**

---

### 4.4 R4 — The R4 mixed 't Hooft anomaly `ξ_R4` (geometry-specific, OPEN, the tractable target)

**The obstruction.** `ξ_R4` is the geometry's *own* candidate anomaly: the $\mathbb{Z}_3$ colour-centre
one-form symmetry of the pure-glue (Spin-c) projection of the frozen branch, coupled to the K₆ data. It is
**well-posed** — `ξ_R4 ∈ Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`, a finite abelian group (Theorem 11) — but its
**value is OPEN**.

**The corrected computation status (load-bearing, carry verbatim).** The GAP02_R4_STEENROD_COMPUTATION redo
**voids** the prior `d₃`-triviality transcript on two independent counts:

1. **The carrier is degree-3, not degree-2.** `H¹(BPU(3);ℤ/3) = H²(BPU(3);ℤ/3) = 0` — there is **no** degree-2
   class `u₂`. The genuine 3-torsion carrier is `x̄₁ ∈ H³(BPU(3);ℤ/3)`.
2. **The 2-primary `d₃ = βSq²ρ₂` is identically zero on 3-torsion** (`ρ₂(3-torsion) = 0`). The owner's
   "`d₃` ⇒ `ξ_R4 = 0`" expectation is **REFUTED**.

The **operative** differential is the 3-primary **`d₅ = Q₁ = βP¹`** (the first nonzero AHSS differential
`d_{2p-1}` at $p=3$). It sends the degree-3 carrier to **degree 8** (`y_{3,0}`) — **off** the $p+q=5$ bordism
lines — so it kills the degree-5 survivor **neither as source nor as target**. The **untwisted default is
SURVIVAL** (`Ω₅^{Spin}(PSU(3) × B²ℤ₃) = ℤ₃ ≠ 0`). The **sole remaining lever** is whether the frozen
`τ_K₆ = (2,2)` twist-correction to `d₅` supplies an admissible degree-lowering term hitting the degree-5 `ℤ₃`
survivor — a finite `ℤ₃`-linear-algebra question the cited (untwisted-only) literature does **not** settle.
**Verdict: `STILL_SUBTLE` — `ξ_R4` OPEN, NOT 0, NOT forced nonzero.**

**Technique: specialist twisted-bordism computation (T-target) — the genuine DERIVED path.**

**Named axiom (NOT applicable as a closure).** `ξ_R4` is a *computable* topological invariant, not an axiom to
posit. There is **nothing to target-load** — the value is whatever the bordism group says. (This is the good
kind of open object: structure-first, with a definite answer waiting to be computed.)

**Specialist target (precise, hand-off-ready).**

> Compute `ξ_R4 ∈ Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)` for the EXACT refined object (the 2-group /
> PSU(3)-bundle, flux `w₂^{PSU(3)} = u₂ ∈ H²(BPSU(3),ℤ₃)`; the local-coefficient twist `τ_K₆ = τ(c̄₁)`,
> `c̄₁ = (2,2) ≠ 0`). Specifically: **(a)** evaluate the `τ_K₆`-twist-correction to the operative `d₅ = Q₁`
> on the degree-5 `ℤ₃` survivor — does the degree-matched correction `[τ_K₆] ∪ x̄₁` (degree $2+3=5$, on-line)
> deform the abutment? **(b)** account for the odd-prime Massey-product / `v₁`-filtration structure
> (Westerland's caveat: an exact single-cup twisted analogue "cannot possibly be true" at $p=3$); **(c)** verify
> the closest tabulated analogue (Wan–Wang `Ω₅^{Spin}(BPSU(3) × B²ℤ₃)`) and **justify the
> $Spin \to Spin^{\mathbb{C}}$ degree-5 transfer**; **(d)** settle the 2-group-vs-product distinction
> (possible Postnikov-square correction).

**The owner bit.** If the class survives and is nonzero, `ξ_R4 ≠ 0` *additionally* requires the
$\bar c_1(L_{K_6})$ pushforward to land on it **plus one owner orientation bit** ($3$ vs $\bar3$); the bare
object $\xi_{R4}=0$ is forced (R8).

**Math to attempt.** The `τ_K₆`-twist-correction is a **finite `ℤ₃`-linear-algebra computation** — bounded and
falsifiable once the Massey-product structure is set up. This is the single most tractable open object in the
gate.

**The G1/G2 firewall (why this is `NOT-A-WALL`).** Even if the class survives, is nonzero, and is
geometry-sourced, `ξ_R4` is a **G1 sector-realization constraint, category-distinct from the G2 coercivity /
mass-gap**. A 't Hooft anomaly is satisfiable by a *gapless* IR via anomaly-matching — so **even a nonzero
`ξ_R4` does not force a gap** (the generic anomaly ⇒ gap implication is FALSE). The R4 value is **inert for the
mass gap either way**; Gap-02 stays OPEN on the unproven certificate inequality
$z_* < 1/(E_{\rm conn}\cdot A_{\rm fluc})$ regardless of `ξ_R4`.

**Success ladder.**
- **DERIVED-CLOSED (reachable, this is the tractable target):** the `τ_K₆`-twist-correction computed → `ξ_R4`
  pinned to `0` or `ℤ₃`. A definite topological value, target-blind. Closes the *value* leg of SG-4's R4 face.
- sharper-OPEN: the Massey-product structure is set up but the finite `ℤ₃`-linear-algebra is shown
  literature-unsettled → `ξ_R4` stays OPEN with a sharper named obstruction (the exact twist-correction rank).
- REFUTED-of-a-prior-claim: any record asserting `ξ_R4 = 0` via the 2-primary `d₃` is voided (already done).
- **No effect on the mass gap regardless** — `NOT-A-WALL`.

**Honest disposition: OPEN, the single most computationally tractable target in the gate; specialist
twisted-bordism computation of the `τ_K₆`-twist-correction to `d₅`; value inert for the mass gap (`NOT-A-WALL`);
DERIVED-CLOSED on the value is reachable, but it closes only the R4 *value*, not SG-4's quantum-consistency
coverage or the mass gap.**

---

### 4.5 R5 — Machine certificates AUDIT (`G03_charge_z6/`, `G05_anomaly_cancellation/run.sh`)

**The obstruction.** "The charge table regenerates from frozen labels" and "all six ledgers vanish from the
frozen spectrum" are asserted via the two certificates, but the load-bearing executables are **referenced, not
independently re-run** in this audit — the same class as SG-7's absent δ-harness and SG-8's J.6/K.5 mount.

**Technique: owner artifact (machine-lane), not an axiom.** A **fail-closed reproducibility task**.

**Owner artifact needed.** (1) Run `certificates/G03_charge_z6/` exact-fraction check target-blind and confirm
the printed charge table (CR3.10 / D.3.1) regenerates, including the $6Y \in \mathbb{Z}$ closure for all six
multiplets. (2) Run `certificates/G05_anomaly_cancellation/run.sh` and confirm all six ledgers (L1–L6) vanish in
exact rational arithmetic from the frozen spectrum file (the same spectrum file G03 reuses — a deliberate
economy, R0 hashes). (3) Confirm against the R0 freeze hashes (`ac4d2df3e708`, `a68ee92a75be`, A2.3).

**Math to attempt.** None new — execution + verification. **Mitigant unique to SG-4:** the underlying arithmetic
is *hand-reproducible* (W1–W7 above), so even an absent harness is independently verifiable by a skeptic with a
pencil; this AUDIT is materially lower-stakes than SG-7's (where the δ-decimals do not recompute by hand).

**Success ladder.** **BLOCKED_INPUTS until the certificates are re-run** → then **VERIFIED** (the exact-rational
checks reproduce; "Claimed certificate pass" becomes machine-real) or **REFUTED** (a value fails to regenerate →
the affected gate downgrades to *Open / not claimed* per §6.12 Gate-3/Gate-5 rows). **Highest value-per-effort
closeable item** — converts asserted into machine-checked with no new physics.

**Honest disposition: AUDIT → VERIFIED is cheap and near-certain (the arithmetic is exact and hand-checkable);
the harness mount is an owner artifact.**

---

### 4.6 R6 — The charge table is given-E and given-the-group

**The obstruction.** Face A computes $Y$/$Q$ *given* the SM gauge group (SG-2, which is a cross-framework TIE —
every framework recovers $SU(3)\times SU(2)\times U(1)$, so it is not Fable-discriminating) and *given* the
produced chiral spectrum (SG-3, given-E). It does not derive the gauge group or the content.

**Technique: T2-guarded restatement (no relocation).** Name the inherited conditionality precisely.

**Named principle (κ³/π-clean):**

> **PRINCIPLE-CHARGE-GIVEN-E** (target-blind): *"The charge table is a geometric OUTPUT of $S^2$ + $S_Y^1/\mathbb{Z}_2$
> + the $\mathbb{Z}_6$ closure, computed GIVEN the SM gauge group (SG-2) and the produced spectrum (SG-3);
> selection ≠ derivation of the group or the content."*

**Action.** Confirm the roll-up never reads Face A as "deriving the SM charges from nothing" — it derives the
$\tfrac16\mathbb{Z}$ lattice and the $Q=T_3+Y$ table *given* the group and spectrum. **No new physics; honest
labeling.**

**Success ladder. DISCLOSED-CONSISTENT** — already substantially in place (the manuscript's given-E discipline);
the residual is a consistency sweep. Note: this is *dependent* on SG-2/SG-3, not independently closeable.

---

### 4.7 R7 — The $\mathbb{Z}_6$ quotient is declared global structure (selection ≠ forcedness)

**The obstruction.** The $\mathbb{Z}_6$ centre-locking is a *declared* admissible quotient chosen from
$\{1, \mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_6\}$. The A4-Z6 sub-claim-B closure-campaign finding is sharp:
**"the finest quotient is $\mathbb{Z}_6$" is pinnable only by already wanting** the $\tfrac16\mathbb{Z}$ lattice
— the maximality/finest selector "AX-FINEST" was **rejected on no tuning to the known answer** (the symmetric minimal cover
$\Gamma = 1$ is a priori equally motivated; the genuine constraint the geometry forces is only $\Gamma \le
\mathbb{Z}_6$, i.e. $q \mid 6$, not $q = 6$).

**Technique: T4 (no-go on the forcedness claim) + T10 (selector).** The honest move is to *not* claim
$\mathbb{Z}_6$ is forced and to state exactly what the geometry does force.

**Named axiom it could reduce to (κ³/π-clean):**

> **AXIOM-Z6-DECLARED** (target-blind): *"The geometry forces the centre-locking constraint $\Gamma \le
> \mathbb{Z}_6$ (the closure $\omega_3^{k_3}\omega_2^{k_2}\omega_6^{6Y}=1$ is consistent for $\Gamma$ dividing
> 6); the SPECIFIC choice $\Gamma = \mathbb{Z}_6$ (giving the finest $\tfrac16\mathbb{Z}$ lattice) is a
> DECLARED admissible quotient, not forced — its selection matches the observed lattice but is not derived from
> below."*

This is writable with no charge value (it references only the quotient lattice $\{1,\mathbb{Z}_2,\mathbb{Z}_3,\mathbb{Z}_6\}$
and the divisibility constraint). The campaign verified the SM hypercharge triples satisfy the Tong congruence
$q \equiv 3z_2 - 2z_3 \bmod 6$ field-by-field — the constraint is real; the *finest*-selection is the un-forced part.

**Specialist target.** "Is $\mathbb{Z}_6$ (vs a coarser quotient) forced by any *independent* structural datum
of the frozen branch (e.g. the orbifold parity table $\mathbb{Z}_6$ content, the spin structure), or only by
matching the observed lattice?" If an independent datum forces it, R7 strengthens to forced; expect it to remain
selection-inside-a-category.

**Success ladder.**
- DERIVED-CLOSED: an independent structural datum forces $\Gamma = \mathbb{Z}_6$ → the finest-lattice is derived.
  **Unlikely (the campaign rejected AX-FINEST).**
- **AXIOM-CLOSED (likely):** AXIOM-Z6-DECLARED named; the constraint $\Gamma \le \mathbb{Z}_6$ forced, the
  finest selection declared.
- sharper-OPEN: stays selection-inside-a-category.

**Honest disposition: AXIOM-CLOSED at AXIOM-Z6-DECLARED — the geometry forces $\Gamma \le \mathbb{Z}_6$, the
finest choice is declared, and "$\mathbb{Z}_6$ is forced" is retired (the A4-Z6 finding).**

---

### 4.8 R8 — The $\bar3$-vs-$3$ orientation bit (conditional on R4 surviving)

**The obstruction.** If `ξ_R4`'s class survives and is nonzero, the nonzero *value* additionally requires (i)
the $\bar c_1(L_{K_6})$ pushforward to land on it and (ii) one owner orientation bit (colour rep $3$ vs $\bar3$).
The bare object $\xi_{R4}=0$ is forced; only the nonzero value needs the bit.

**Technique: T10 (selector) — a single discrete owner choice, conditional on R4.**

**Named axiom (κ³/π-clean):**

> **AXIOM-COLOR-ORIENTATION-BIT** (target-blind): *"The frozen branch fixes the colour rep orientation ($3$ vs
> $\bar3$) as a declared structural bit; conditional on `ξ_R4`'s class surviving, this bit selects the sign of
> the nonzero value."*

Writable with no anomaly value (it is a structural orientation choice). The elegant bit-independence the owner
once reached for (that *both* orientations give the same `ξ_R4`) was **REFUTED as written** (the 2-primary `d₃`
route is void) — so the bit may genuinely matter, and the honest statement is that it is a declared choice.

**Action.** Defer to R4: this residual only bites if the twisted-bordism computation shows the class survives
nonzero. Record the bit as a declared structural input pending R4.

**Success ladder. DEFERRED-TO-R4.** Becomes a single discrete AXIOM-CLOSED choice if and only if R4's class
survives; otherwise moot (bare object $=0$).

---

### 4.9 Attack-plan roll-up

| Residual | Technique | Named axiom / principle (κ³/π-clean) | Specialist target / owner artifact | Realistic endpoint |
|---|---|---|---|---|
| R1 filter-not-determiner | T5 + T2-restate | PRINCIPLE-ANOMALY-IS-A-FILTER | exhibit 2 anomaly-free non-SM solutions | DISCLOSED-CORRECTED (cleanest win; determiner is mathematically false) |
| R2 ~10/16 classes | T6 + T7 | AXIOM-QUANTUM-CLASS-COVERAGE | write the 16-row class table; grade each | AXIOM-CLOSED on coverage (R3/R4 left named-OPEN) |
| R3 BV-BRST descent (UQF-4) | T1 / specialist | AXIOM-BRST-DESCENT-INHERITED | construct the quantized descent measure (NO-KNOWN-ROUTE) | AUDIT → AXIOM-CLOSED (stance); DERIVED only via specialist construction |
| R4 `ξ_R4` mixed 't Hooft | specialist bordism | (none — computable invariant, nothing to load) | compute the `τ_K₆`-twist-correction to `d₅` | OPEN → DERIVED-CLOSED on value (tractable); `NOT-A-WALL` for the gap |
| R5 certificate AUDIT | machine-lane | (none) | re-run G03/G05 exact-rational checks | BLOCKED_INPUTS → VERIFIED (cheap; hand-reproducible) |
| R6 given-E/given-group | T2-restate | PRINCIPLE-CHARGE-GIVEN-E | consistency sweep | DISCLOSED-CONSISTENT (dependent on SG-2/SG-3) |
| R7 $\mathbb{Z}_6$ declared | T4 + T10 | AXIOM-Z6-DECLARED | independent forcing datum for $\Gamma=\mathbb{Z}_6$? | AXIOM-CLOSED ($\Gamma \le \mathbb{Z}_6$ forced; finest declared) |
| R8 colour orientation bit | T10 | AXIOM-COLOR-ORIENTATION-BIT | (deferred to R4) | DEFERRED-TO-R4 |

**REDUCE-vs-RELOCATE verdict on the plan.** The plan does **not** turn one hard problem into three harder ones.
Run the explicit harder-subproblem check:

- **R1 / R6 / R7** reduce to *naming a boundary or a declared selection* in plain sight — each is a single
  target-blind statement that pays a debt without inventing a new one. R1 is the cardinal case: it is a **wrong
  premise to drop** (anomaly-as-determiner is false), not a problem to solve — dropping it *dissolves* the
  overclaim without breaking the genuine filter content.
- **R2** is a bookkeeping enumeration (write the 16-row table) — smaller than the gate, not harder.
- **R5** is a mechanical owner re-run, made low-stakes by the fact that the arithmetic is hand-reproducible.
- **R3 (BV-BRST)** is genuinely hard (NO-KNOWN-ROUTE), but the realistic move is the *import* (AXIOM-CLOSED as
  stance), which does not manufacture a harder subproblem — it honestly records a standard-QFT debt. The DERIVED
  path is correctly flagged as a multi-month specialist construction, not banked.
- **R4 (`ξ_R4`)** is the one genuine-physics *computation*, and it is **smaller** than the original gate (a
  finite `ℤ₃`-linear-algebra question about one twist-correction), with **no target to load** (it is a
  topological invariant with a definite answer). Its firewall (`NOT-A-WALL`) is stated so a value cannot be
  oversold as a mass-gap lever.

The honest expected outcome of a full campaign: **1 DISCLOSED-CORRECTED (R1), 1 AXIOM-CLOSED-on-coverage (R2),
2 AXIOM-CLOSED (R3-stance, R7), 1 VERIFIED (R5), 1 DISCLOSED-CONSISTENT (R6), R4 either DERIVED-on-value (if the
twist-correction is computed) or sharper-OPEN, R8 deferred.** No DERIVED-CLOSED is promised on the *gate's
headline* (anomaly-as-determiner is unreachable by construction); the gate would move from PARTIAL-with-asserted-status
to **PARTIAL-with-machine-verified-perturbative-status + a precise filter boundary + a named axiom floor for the
non-perturbative/R4 residuals** — a real honesty/reproducibility gain, **not** a promotion.

---

## A. Anchoring & Hardening Map

This section runs SG-4 through our internal honesty methodology: for each residual, decide **gap vs wall** (a gap has a known
route — a finished computation, a written certificate, a measured input; a wall is one whose *route* is the
problem), hunt the **implicit assumption** the residual silently carries, find the **measured invariant** each
residual must ultimately terminate on, and assign the most conservative defensible disposition. The disposition
vocabulary is the one used live in the **Gaps & Walls Register**
(<https://physics.magflowmeters.com/articles/GAPS_AND_WALLS_REGISTER.html>) and is defined inline here:
**DERIVED** (tied to an already-measured fact, no new assumption) · **DERIVED-GIVEN-E** (rigid once the observed
SM content E is supplied) · **AXIOM-CLOSED** (reduced to one named, value-free, unproven assumption — the debt
is recorded, not paid) · **DISSOLVED** (the problem rested on a false/unmeasured premise; dropping it makes it
disappear — *dissolved is not solved*) · **SCHEME-ANCHORED** (a number reachable only after one named,
geometry-unfixed regularization/normalization choice) · **OPEN** (genuinely unsolved, or the only escape
assumes the answer) · **BLOCKED** (a real route exists but a required file/input is missing) ·
**measured-but-irreducible** (known from experiment, no derivation anywhere). The existing **anchor set** a
residual may terminate on is the four headline anchors plus the residues the Register names: **ℏ** (the action
grain), **M_Pl**, the **spin-ℂ index / spectrum-E** (the observed SM content), the **measured couplings
α_i(M_Z)**, **y_t**, **|V_us|**. "Terminates on an EXISTING anchor" is the strongest outcome; "needs a NEW named
invariant," "hinges on a SCHEME-anchor," "is a COMPUTATION-DEBT," or "has NO witness → OPEN" are progressively
weaker. The residuals are R1–R8 from §3.1; nothing here promotes any of them.

### A.1 Per-residual anchoring table

| Residual (from §3) | gap / **WALL** (+kind) | Measured invariant it must terminate on | Honest disposition | What would HARDEN it |
|---|---|---|---|---|
| **R1** — anomaly = FILTER, not determiner | **WALL** (category wall: anomaly cancellation *is* a filter with infinite solution variety — no route turns it into a determiner) | None — it terminates on a **mathematical fact**, not a measured number; the genuine geometric content it guards (one SM generation per family) rides the **spectrum-E / spin-ℂ index −3** anchor (SG-3). | **DISSOLVED** — the premise "anomaly cancellation determines the chiral content" is false; dropping it dissolves the overclaim while the genuine inherited-filter content survives. Matches the Register's "category error … DISSOLVED." | Exhibit two distinct anomaly-free chiral solutions (a five-line example), bind PRINCIPLE-ANOMALY-IS-A-FILTER to the roll-up, and ensure no public line ever reads "anomaly selects the SM." |
| **R2** — only ~10/16 quantum-consistency classes close | gap (route known: enumerate and grade the classes) | The 6 perturbative ledgers terminate on **spectrum-E** (DERIVED-GIVEN-E, hand-checkable); the discrete-global tail terminates on inherited-standard-QFT consistency; R3/R4 do not yet terminate on any invariant. | **AXIOM-CLOSED on coverage** — reduces to one value-free coverage declaration (AXIOM-QUANTUM-CLASS-COVERAGE: certificate-grade only for the enumerated closed classes; the rest named-OPEN). The "16" is a corpus enumeration, not an independently re-derived count. | Write the explicit 16-row class table and grade each row; discharge the discrete-global tail as inherited stance; leave R3/R4 named-OPEN. Converts the vague "~10/16" into an auditable boundary. |
| **R3** — BV-BRST descent (UQF-4) + descent measure (UQF-7) | **WALL** (NO-KNOWN-ROUTE: the non-perturbative descent construction has no known route, per the Register and OPEN_WALLS_REGISTER) | No new measured invariant — it is a **standard-QFT structural consistency** any consistent gauge theory must satisfy, imported at the same standard as for the SM itself; the perturbative legs below it ride **spectrum-E**. | **AXIOM-CLOSED (stance), staying AUDIT** — reduces to AXIOM-BRST-DESCENT-INHERITED (an inherited-standard-QFT import). The import is a **stance, not a proof**; DERIVED is reachable only by a multi-month specialist construction. | Name the inherited-standard-QFT import explicitly in the claim boundary (Route A); or hand a specialist the BV-BRST construction for the K₆ coset / S¹_Y/ℤ₂ fold / ℤ₆ quotient (Route B, the only DERIVED path). |
| **R4** — R4 mixed 't Hooft anomaly `ξ_R4` | gap, but a **NOT-A-WALL** one (Register: lever closed-negative — a 't Hooft anomaly is satisfiable by a gapless IR, so even nonzero `ξ_R4` forces no gap; *value* still unknown) | A **NEW computable topological invariant** — `ξ_R4 ∈ Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)` — with a definite answer waiting; nothing to target-load; inert for the mass gap regardless. | **OPEN (value), well-posed (home)** — a **COMPUTATION-DEBT**: a finite ℤ₃-linear-algebra question (the `τ_K₆=(2,2)` twist-correction to the operative `d₅=Q₁`). The 2-primary `d₃` route is voided; untwisted default is the nonzero ℤ₃ survivor. | The single most tractable open object in the gate: compute the `τ_K₆`-twist-correction to `d₅` (specialist twisted-bordism). DERIVED-CLOSED on the *value* (0 or ℤ₃) is reachable — but it closes only the R4 value, never SG-4's coverage or the gap. |
| **R5** — machine certificates AUDIT (`G03_charge_z6/`, `G05_anomaly_cancellation/run.sh`) | gap (route known: re-run the exact-rational scripts target-blind) | Terminates on **spectrum-E** via the frozen spectrum file (A2.3) and the R0 freeze hashes; the underlying arithmetic is hand-reproducible (W1–W7) and so is independently anchorable by pencil. | **BLOCKED (inputs) → VERIFIED** when re-run — a pure reproducibility (COMPUTATION-DEBT) item, not physics; materially lower-stakes than SG-7's absent δ-harness because the math recomputes by hand. | Mount and re-run both certificates target-blind; confirm the charge table and all six ledgers regenerate from the frozen hashes (`ac4d2df3e708`, `a68ee92a75be`, A2.3). Highest value-per-effort closeable item. |
| **R6** — charge table is given-E + given-group | gap (honest labeling only) | Terminates on **spectrum-E** (SG-3) and the SM gauge group (SG-2, a cross-framework TIE, DERIVED-GIVEN-E); it does not derive the group or the content. | **DERIVED-GIVEN-E / DISCLOSED-CONSISTENT** — reduces to PRINCIPLE-CHARGE-GIVEN-E (selection ≠ derivation). Dependent on SG-2/SG-3, not independently closeable. | Confirm no roll-up reads Face A as "deriving the SM charges from nothing." No new physics; consistency sweep. |
| **R7** — ℤ₆ quotient is declared global structure | gap (selection-inside-a-category) | No measured invariant forces the *finest* choice; the geometry forces only the divisibility constraint Γ ≤ ℤ₆ (q ∣ 6). The lattice the finest choice matches (⅙ℤ) is itself a facet of **spectrum-E**. | **AXIOM-CLOSED** at AXIOM-Z6-DECLARED — the geometry forces Γ ≤ ℤ₆; the specific Γ = ℤ₆ is a **declared** admissible quotient (AX-FINEST was rejected on no tuning to the known answer, the A4-Z6 finding). "ℤ₆ is forced" is retired. | Find an independent structural datum of the frozen branch (parity-table ℤ₆ content, spin structure) that forces Γ = ℤ₆ rather than merely matching the observed lattice. Expect it to stay selection-inside-a-category. |
| **R8** — the 3̄-vs-3 colour orientation bit | gap, **conditional on R4 surviving** (else moot: bare object ξ_R4 = 0 is forced) | A single discrete **structural orientation bit** (colour rep 3 vs 3̄); fixes only the *sign* of a nonzero `ξ_R4`, conditional on R4's class surviving. | **DEFERRED-TO-R4** — becomes one discrete AXIOM-CLOSED choice (AXIOM-COLOR-ORIENTATION-BIT) iff R4's class survives nonzero. The once-hoped bit-independence was refuted as written (the 2-primary `d₃` route is void). | Resolve R4 first; if the class survives nonzero, record the orientation bit as a declared structural input. A single owner choice; moot otherwise. |

### A.2 Gate-level rollup

**Overall disposition.** SG-4 is **PARTIAL → conservatively OPEN** at the gate level, exactly as the live Register
files it (SG-4 badge: PARTIAL → OPEN; perturbative checks DERIVED-GIVEN-E). The hardening map sharpens *why*: the
gate's genuine content — the ℤ₆-forced ⅙ℤ hypercharge lattice + the full Q = T₃ + Y table, and the six
perturbative anomaly ledgers vanishing by hand — terminates cleanly on **spectrum-E** and is DERIVED-GIVEN-E and
hand-checkable. Its open surface splits into one **category wall that dissolves** (R1: anomaly-as-determiner is
false, DISSOLVED — *dissolved is not solved*, the inherited-filter obligation remains), one **NO-KNOWN-ROUTE wall**
(R3: BV-BRST descent, AXIOM-CLOSED-as-stance at best), one **computation-debt with a definite topological answer**
that is **NOT-A-WALL** for the mass gap (R4: `ξ_R4` OPEN, value-only), one **reproducibility debt** (R5:
BLOCKED→VERIFIED), and three **honest-labeling / declared-selection** items (R2 coverage AXIOM-CLOSED, R6
given-E/given-group, R7 ℤ₆-declared AXIOM-CLOSED). No residual terminates on a NEW measured invariant; none rides
a SCHEME-anchor (unlike SG-7, whose magnitudes are SCHEME-ANCHORED and whose Gate-7 is now Diagnostic-only with
fitted-not-derived threshold rows — SG-4 carries no such fitted-magnitude debt).

**Anchors this gate depends on.** **Spectrum-E (the spin-ℂ index −3)** — load-bearing for the entire charge table,
the anomaly filter, and R2/R5/R6; **the measured SM gauge group via SG-2** (a cross-framework TIE, not
Fable-discriminating). It depends on **no new invariant** and on **none of ℏ / M_Pl / α_i(M_Z) / y_t / |V_us|**
directly. R4 introduces a candidate **new topological invariant** (`ξ_R4`), but its value is inert for every
physical claim (NOT-A-WALL).

**Single highest-leverage hardening move.** **Bind the R1 boundary in plain sight** — exhibit two distinct
anomaly-free chiral solutions and ensure the public roll-up never reads "anomaly cancellation selects the SM."
This is the cardinal, zero-cost, zero-risk win: it is not a problem to solve but a false premise to drop, and
dropping it dissolves the gate's single largest overclaim exposure without touching the genuine content. (The
highest value-per-effort *machine* move is R5 — re-run the two exact-rational certificates target-blind to make
the perturbative pass machine-real; the most tractable genuine-physics *computation* is R4 — the `τ_K₆`-twist-
correction to `d₅`.) None of these is a promotion; the gate's honest ceiling stays **serious candidate, NOT
validated**, and "anomaly cancellation derives the Standard Model" remains mathematically false.

### 🎯 Target anchor(s) for this gate

In the terminate-on (anchoring) sense, SG-4 terminates on exactly **one existing measured invariant: the
observed spectrum E (the spin-ℂ index −3 / one SM generation per family)** — and anomaly cancellation is a
**FILTER on E, not a determiner of it** (`measured-but-irreducible` upstream, inherited as the geometry's
index output; the ⅙ℤ charge table and the six perturbative ledgers are **DERIVED-GIVEN-E**, hand-checkable).
The gate needs **no new measured anchor** and rides **none of ℏ / M_Pl / α_i(M_Z) / y_t / |V_us|** directly.
Its non-perturbative residuals are **COMPUTATION-DEBTS, not anchor-needs**: the **BV-BRST descent + measure
(UQF-4 / UQF-7)** is a value-free inherited-standard-QFT stance (AXIOM-CLOSED at best, AUDIT / NO-KNOWN-ROUTE),
and the **R4 mixed 't Hooft anomaly `ξ_R4`** is a well-posed twisted Spin-c bordism / ℤ₃-cohomology value
awaiting a specialist computation — **no measured invariant, value UNKNOWN / OPEN** (NOT-A-WALL, inert for the
mass gap). No scheme-anchor is carried (unlike SG-7). **Status: anchor TERMINATED on spectrum-E; `ξ_R4` and the
descent COMPUTATION-DEBT OPEN.** No promotion.

---

## 5. References & source map

### 5.1 Website source-of-truth (common material — link, don't duplicate)

- **Paper I, GUT.html** — <https://physics.magflowmeters.com/articles/GUT.html>
  - **§6.3** Gate-3 card (hypercharge/charge, status verbatim); **§6.5** Gate-5 card (anomaly); **§6.12**
    falsification map (Gate-3 row: wrong $Y$/$Q$ or inconsistent $\mathbb{Z}_6$ → *Open*; Gate-5 row: any
    nonzero trace → *Open*); **§6.13** certificate-status summary.
  - **§5.2 / §5.4** narrative modules; **§3** the worked anomaly example end-to-end — **§3.3** the six ledgers +
    the $\sum Y^2 = \tfrac{10}{3}$ diagnostic, **§3.6.2** the one-generation trace table, **§3.6.3** "what it
    does and does not show", **§3.7** "where the claim is actually closed".
  - **Appendix CR3** (hypercharge reader companion): CR3.5 $Q=T_3+Y$; CR3.6 $S^2 \to T_3$; CR3.7 $S_Y^1/\mathbb{Z}_2
    \to Y$; **CR3.8 the $\mathbb{Z}_6$ closure** (the $\omega_3^{k_3}\omega_2^{k_2}\omega_6^{6Y}=1$ rule, freeze
    `a68ee92a75be`); CR3.9 worked examples; CR3.10 the full table; CR3.11 selector eliminations; CR3.13
    remove-one-term tests; CR3.14 what it shows / does not show.
  - **Appendix CR5** (anomaly reader companion); **Appendix D** (representation-level recovery, charge tables
    D.2/D.3.1); **Appendix E** (chirality half) + **Appendix E′** (anomaly closure — full six-ledger trace
    table, conventions $A(\bar R)=-A(R)$, $T(\text{fund})=\tfrac12$, left-handed Weyl basis); **Appendix R0/R1**
    (freeze records, hashes).
- The **R4 / UQF-4 / `ξ_R4`** open objects are NOT in GUT.html (Λ cross-reference Paper IV, TOE.html, does not
  touch this gate).

This dossier recaps only what is needed to attack; the site controls all common material.

### 5.2 Corpus locations (authoritative inputs to this dossier)

| Source | Path | Role |
|---|---|---|
| Per-gate dossier spec | `…/rendered/TOE/PER_GATE_DOSSIER_SPEC.md` | structure (sections 0–5) |
| SG-4 status line | `…/rendered/TOE/SCOPED_GUT_PER_GATE_STATUS_LEDGER.md` | the PARTIAL label + the filter-not-determiner / ~10/16 / UQF-4 AUDIT / R4 OPEN caveats (carried verbatim) |
| **R4 corrected computation (authoritative)** | `…/rendered/TOE/GAP02_R4_STEENROD_COMPUTATION.md` | the carrier-degree-3 fix, `d₃ ≡ 0` on 3-torsion, operative `d₅ = Q₁`, untwisted `ℤ₃` survives, `τ_K₆=(2,2)` twist-correction the sole lever, `STILL_SUBTLE` |
| R4 well-posing (Theorem 11) | `…/rendered/TOE/GAP02_R4_WELLPOSING_THEOREM.md` | the home `Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`; `c̄₁=(2,2)≠0` ⇒ `τ_K₆ ≠ 0`; group-vs-value Born cross-check |
| R4 prior transcript (VOIDED) | `…/rendered/TOE/GAP02_R4_DELTA3_TRIVIAL.md` | superseded — the `u₂ ∈ H²` carrier does not exist; the 2-primary `d₃` is null on 3-torsion (kept only as the refuted-non-sequitur record) |
| R4 `NOT-A-WALL` register | `…/rendered/TOE/OPEN_WALLS_REGISTER.md` (Tier C §5; UQF-4 §437; INHERITED-STANDARD-QFT note §426) | the G1/G2 firewall (anomaly ⇏ gap); UQF-4 NO-KNOWN-ROUTE; UQF-7 BV-BRST descent measure NO-KNOWN-ROUTE |
| Closure campaign (Born/Z6 dispositions) | `…/rendered/TOE/CLOSURE_CAMPAIGN_RESULT_2026-06-24.md` + `…_ROUND2_2026-06-24.md` | κ³/π kill-test; D1-born AX-NO-GAUGED-COLOR-C (center-only ℤ₆, $S^1_Y/\mathbb{Z}_2$=hypercharge, no S₃ color); A4-Z6 AX-FINEST rejected ($\Gamma \le \mathbb{Z}_6$, $q\mid6$ not $q=6$) |
| GUT manuscript | `…/rendered/GUT/GUT.md` | §6.3 (Gate-3 card); §6.5 (Gate-5 card); §3 (worked anomaly, §3.3/§3.6.2/§3.7); CR3 (CR3.8 $\mathbb{Z}_6$, CR3.10 table); §6.12 (falsification map) |

### 5.3 Frozen-object hash index (load-bearing; READ-ONLY)

Branch `dcc66f1b2685` · manifest meta `a5b1e6f9d951` · parity table `ac4d2df3e708` (R1.3, co-read by Gate 4) ·
$\mathbb{Z}_6$-closure freeze `a68ee92a75be` (R1.3, CR3.8 / D.3.1) · R1.4 hypercharge lattice · surviving
chiral spectrum A2.3 (Gate-4 output, reused by both `certificates/G03_charge_z6/` and
`certificates/G05_anomaly_cancellation/`). **R4 (open):** home `Ω₅^{Spin-c}(B(SU(3)→PSU(3)); τ_K₆)`; twist
`τ_K₆ = τ(c̄₁)`, `c̄₁ = 2ρ mod 3 = (2,2) ≠ 0`; carrier `x̄₁ ∈ H³(BPU(3);ℤ/3)`; operative `d₅ = Q₁ = βP¹`;
`Q₁(x̄₁) = y_{3,0} ∈ H⁸` (off the $p+q=5$ lines). **`ξ_R4`: no value, no hash — OPEN. UQF-4 / BV-BRST descent:
no construction — AUDIT / NO-KNOWN-ROUTE.**

---

### Closing honest statement

SG-4 is **PARTIAL**. Its genuine, defensible content is real and hand-checkable: the **$\mathbb{Z}_6$ closure
forces hypercharge onto the $\tfrac16\mathbb{Z}$ lattice** and the full one-generation $Q=T_3+Y$ table falls out
with no per-multiplet fit; the **six perturbative anomaly ledgers vanish exactly** on the produced spectrum (the
by-hand witness $3\cdot\tfrac16-\tfrac12=0$, $\sum Y = \sum Y^3 = 0$, the colour traces, the Witten mod-2 count),
with the $\sum Y^2 = \tfrac{10}{3} \neq 0$ diagnostic proving the cancellation is *specific*, not trivial. Its
honest open surface is equally clear: anomaly cancellation is a **FILTER on E, not a determiner** (infinitely
many anomaly-free solutions — it is inherited, the geometry earning its pass by producing one SM generation per
family); only **~10/16** quantum-consistency classes close at certificate grade; the **BV-BRST descent (UQF-4)**
is **AUDIT / NO-KNOWN-ROUTE**; the **R4 mixed 't Hooft anomaly `ξ_R4`** is **OPEN** — well-posed as a finite
twisted Spin-c bordism class, operative differential the 3-primary `d₅ = Q₁` (the 2-primary `d₃` is null on
3-torsion), untwisted default the nonzero `ℤ₃` survivor, the `τ_K₆=(2,2)` twist-correction the sole lever, and
`NOT-A-WALL` (its value is inert for the mass gap by the G1/G2 firewall). The attack plan reduces these to a
precise filter-not-determiner boundary, a 16-row coverage table, named target-blind axioms for the inherited
non-perturbative consistency and the declared $\mathbb{Z}_6$ quotient, a cheap (hand-reproducible) certificate
re-run, and one tractable specialist twisted-bordism computation for `ξ_R4` — with the κ³/π kill-test guarding
R3 (BV-BRST import) and R7 ($\mathbb{Z}_6$ finest-selection) so a stance or a target-loaded selection cannot be
dressed as a derivation. The realistic ceiling is a machine-verified perturbative status over a named axiom
floor + a computed `ξ_R4` value — **not** a promotion, and **never** "anomaly cancellation derives the Standard
Model" (which is mathematically false). **No status was ever upgraded; frozen branch `dcc66f1b2685` / `a5b1e6f9d951`
READ-ONLY; given-E ≠ derivation of E; nothing applied, nothing deployed.**

*Dossier built 2026-06-24. Our geometry (13D K₆ branch) only. Common material referenced to the published
website source-of-truth, not duplicated.*