# SG-5 — Electroweak Embedding ($Q=T_3+Y$ / EWSB to $U(1)_{\rm em}$): Per-Gate Closure-Attack Dossier

> **⚠ STATUS SUPERSEDED (2026-06-25) — this is the *attack* dossier; it predates the closure pass.** Its
> "PARTIAL" labels are **retired**: by the gate-grading rubric (*any open piece ⇒ OPEN*) the current SG-5 status
> is **OPEN**, with the charge/embedding leg **DERIVED-GIVEN-E + AXIOM-CLOSED** and custodial ρ=1
> **BLOCKED/computation-debt**. The authoritative current disposition is **`SG5_ELECTROWEAK_CLOSURE_RESULT.md`**
> (with the SG-5A/SG-5B completion endpoint + the verified `SG5_complete_endpoint_stack/`). Read this dossier as
> the *attack plan*, not the status.

> **What this is.** The closure-attack packet for **SG-5 (Electroweak embedding $Q=T_3+Y$ / electroweak
> symmetry breaking to $U(1)_{\rm em}$)** on the **frozen 13D K₆ branch ONLY**. It recaps concisely how our
> geometry embeds electric charge and breaks $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ via a single
> Wilson-line/Hosotani doublet (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: $Q=T_3+Y$ on every multiplet and the $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$
> breaking are **GIVEN-E** (computed with the SM chiral content + the selected geometry as input); $v=246.02$,
> $m_h=123.82$ are **near-hits, frozen outputs given the anchors**, NOT a hierarchy derivation; the
> **gauge-hierarchy lightness of $v$ is UNDELIVERED** — it rests on $\theta_H^\star$ **read from a one-loop
> minimum** (carrying ~85% of the hierarchy), and the **SCALE firewall verdict is RELOCATION** ($\mu_{\rm cell}$
> has no $v$-independent readout; the only available anchor $\partial_\sigma V=0$ *is* the hierarchy);
> **custodial $\rho=1$ is UNDELIVERED** (no custodial-symmetry object exists anywhere in the manuscript).
> 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 target). **given-E ≠ derivation of E.** AXIOM-CLOSED ≠ proven; selection ≠
> derivation; dissolved ≠ solved.

---

## 0. Header & Verdict

| Field | Value |
|---|---|
| **Gate id** | SG-5 — Electroweak embedding $Q=T_3+Y$ / EWSB to $U(1)_{\rm em}$ |
| **GUT manuscript gates spanned** | The SG-5 scope is **distributed** across three manuscript gate cards (the manuscript never bundles "EW embedding" into one card): **Gate 3** (charge recovery $Q=T_3+Y$ + the $\mathbb{Z}_6$ rule; §6.3/§5.2; cert `certificates/G03_charge_z6/`); **Gate 5** (anomaly closure on the surviving EW-embedded chiral content; §6.5/§5.4/§3; cert `certificates/G05_anomaly_cancellation/`); **Gate 8** (Higgs protection = the EWSB mechanism + the $v$/$m_h$ outputs; §6.8/§5.7; cert `certificates/G08_higgs_protection/`). **Note the naming hazard:** the manuscript's *card* labelled "Gate 5" is **Anomaly Cancellation**, not the EW embedding; the SG-5 *dossier* (per the SCOPED ledger) is the EW-embedding + EWSB roll-up that draws on Gates 3/5/8. |
| **Status label (binding)** | **PARTIAL** |
| **Target anchor(s)** | **$M_{\rm Pl}$ + $v_{EW}$** (the two dimensionful anchors; no measured invariant beyond these). $Q=T_3+Y$ is structural / DERIVED-GIVEN-E (no anchor of its own). The hierarchy-lightness of $v_{EW}$ is the **open leg** — it rests on $\theta_H^\star$ read-from-min, so $v_{EW}$ is a genuine second dimensionful anchor, **OPEN**; closing it needs the $\theta_H^\star$ derivation, not a new anchor. (See §A end.) |
| **Manuscript card statuses** | Gate 3: *Claimed certificate pass*. Gate 5 (anomaly): *Claimed certificate pass*. Gate 8 (Higgs): *Claimed certificate pass at one-loop; Diagnostic only at higher loops* (H.9.6/H.10.1). PARTIAL is the honest scoped-GUT roll-up of the three. |
| **Frozen hashes it rides** | Branch `dcc66f1b2685` / manifest meta `a5b1e6f9d951`. Charge/EW primitives: parity table (realizes $[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6$) `ac4d2df3e708`; Higgs Wilson-line bundle `2a0462b8aab9`; Wilson-line cycle $\gamma$ `640e1d7f7773`; integer winding $n_H=1$ `f65094fd8fd1`; finite determinant $\eta_{BK}=0.009721281516312024$ `84e94518d3f5`; $y_t$ anchor `548d7099ef18`; $\lvert V_{us}\rvert$ anchor `a1bc510bc7cd`; comparison scale $M_Z$ `a6852c7a6b00`. UV reference $M_*=7.467\times10^{16}$ GeV (A1.10). **$\theta_H^\star$: read from the chamber one-loop $V_{\rm Hos}$ minimum — no first-principles value (H.9.7).** **Custodial $\rho=1$: no object, no hash — UNDELIVERED.** |

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

**Established (given-E, given the selected & frozen geometry + anchors).**

1. **Electric-charge embedding $Q=T_3+Y$ is exact and componentwise.** Hypercharge $U(1)_Y$ is the rotation of
   $S^1_Y/\mathbb{Z}_2$; electromagnetism is the unbroken combination $Q=T_3+Y$. Two exact *architectural
   predicates* hold per multiplet — $Q=T_3+Y$ componentwise, and the $\mathbb{Z}_6$ rule
   $\tfrac{t}{3}+\tfrac{d}{2}+Y\in\mathbb{Z}$ — making non-conforming hypercharges *inconsistent on the
   geometry*, not merely unobserved. The down quark lands at exactly $-\tfrac13$, charge neutrality of the atom
   to $1$ part in $10^{21}$ is reproduced structurally (§6.3, §5.2, CR3; Appendix D charge tables).
2. **The breaking $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ is delivered by ONE doublet.** A single
   Wilson-line/Hosotani Higgs $H\in(\mathbf 1,\mathbf 2,+\tfrac12)$ — a holonomy mode of the gauge connection
   on the $SU(2)_L$-direction cycle $\gamma\subset K_{\rm gauge}$, with **integer winding $n_H=1$
   (topologically protected)** — acquires a VEV at the one-loop $V_{\rm Hos}$ minimum and breaks the EW group
   to electromagnetism. One doublet, not a scalar inserted by hand (H.1, H.2.1).
3. **The numbers near-hit.** $v=246.02\pm3.5$ GeV ($0.06\sigma_{\rm th}$ vs PDG) and $m_h=123.82\pm1.8$ GeV
   ($0.48\sigma_{\rm th}$ vs PDG $125.10\pm0.14$), both as **post-freeze outputs** of the same finite
   determinant $\eta_{BK}$ that fixes $\lvert y_t/y_b\rvert$ — *one structural source, two SM outputs* (H.3,
   H.4). This one-input-two-outputs compression is the genuine, non-trivial content.
4. **The Higgs-mass quadratic destabilization is structurally avoided.** The $\delta m_H^2\sim M_*^2$ tree
   counterterm is topologically forbidden (it would require a non-integer change in $n_H$); $V_{\rm Hos}$ is
   finite, periodic, cutoff-independent (H.2.1, H.9, H.10). This is a *proven, banked* sub-result.

**Open (the honest deductions).**

- **(a) The gauge-hierarchy lightness of $v$ is UNDELIVERED.** The protection ratio $\sqrt{\eta_{BK}}/(2\pi)\sim
  10^{-2}$ is **twelve orders too large** to be $v/M_{\rm Pl}\sim10^{-16}$ (H.2 correction, 2026-06-23). The
  full descent requires $\theta_H^\star\approx2.46\times10^{-14}$ via $v_{EW}=\theta_H^\star/(2\pi R_\gamma)$,
  and **$\theta_H^\star$ is read from the chamber one-loop minimum, not derived** (H.9.7). That read modulus
  carries **~85%** of the electroweak hierarchy (SG-6 line; HIERARCHY_RG). So $v=246.02$ is a *near-hit of a
  read quantity*, not a derived lightness.
- **(b) Custodial $\rho=1$ is UNDELIVERED.** $\rho=M_W^2/(M_Z^2\cos^2\theta_W)=1$ at tree level is the
  hallmark constraint of EWSB by a doublet, protected by a custodial $SU(2)$. **No custodial-symmetry object,
  no $\rho$ computation, no $T$-parameter bound exists anywhere in the corpus** (verified: zero matches for
  custodial / $\rho=1$ / $\rho_0$ / $T$-parameter in `GUT.md`). The single-doublet content makes $\rho=1$
  *plausible by structure*, but it is **asserted-by-omission**, not delivered.
- **(c) The SCALE firewall verdict is RELOCATION (not derivation).** The question "is the SCALE cell
  $\mu_{\rm cell}$ anchorable independently of the EW hierarchy?" was put to the no tuning to the known answer firewall and
  **failed**: $\mu_{\rm cell}$ has no $v$-independent UV readout; the map $\mu_{\rm cell}\to v$ is
  invertible-by-construction; the only available anchor $\partial_\sigma V=0$ *is* where $v$ is generated
  (FIREWALL_MUCELL_INDEPENDENCE; CLOSURE_CAMPAIGN_RESULT §3; SCALE_HIERARCHY_FINAL_VERDICT). As a *derivation*
  the SCALE route is **confirmed circular**.
- **(d) The whole gate is GIVEN-E and given-the-selected-geometry.** Charge, breaking, and the numbers are all
  computed with the SM chiral content E and the frozen branch supplied as input — SG-5 does not derive E, nor
  prove this geometry unique.

**What would close it (the ladder).** DERIVED-CLOSED would require a *target-blind* derivation of
$\theta_H^\star$ (hence the smallness of $v$) from the frozen geometry, OR a target-blind custodial-symmetry
object that forces $\rho=1$. The realistic ceiling per residual is **AXIOM-CLOSED**: name one explicit,
target-blind posit (e.g. "the EW VEV is the UV floor value $\Delta_0\sim\hbar$ at the frozen reference,
adopted a priori"; "the doublet's $SU(2)_L$ holonomy carries a custodial $SU(2)_{\rm cust}$ from the $S^2$
isometry") that pays the debt in plain sight. On the hierarchy-lightness residual the *most likely honest
outcome is sharper-OPEN / RELOCATES* — the firewall has already ruled that the only escape is a named axiom
that relocates the entire hierarchy into the log-sensitivity of $v$ to $\log(\mu_{\rm cell}/M_*)$.

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

The full construction, the charge tables, the Wilson-line/Hosotani derivation, and the freeze records are the
**published manuscript**, Paper I:

- **GUT.html §6.3** (Gate-3 charge card) + **§6.5** (Gate-5 anomaly card) + **§6.8** (Gate-8 Higgs card) —
  <https://physics.magflowmeters.com/articles/GUT.html>
- **§5.2 / §5.4 / §5.7** narrative modules; **§3** the worked anomaly example; **Appendix D** (charge
  tables / SM recovery); **Appendix E′** (chirality + anomaly closure); **Appendix H** (Higgs protection:
  H.1 origin, H.2.1 explicit Wilson-line formula + $V_{\rm Hos}$, H.4 outputs, H.9 protection checklist with
  the H.9.7 non-claims box, H.10 correction-class ledger, H.11 authority summary); **Appendix CR** modules
  **CR2 / CR3 / CR5 / CR8** (reader companions); **Appendices A1/A2/R0** (full-precision constants + freeze
  records).
- Downstream cross-references: **TOE.html** (Paper IV) carries the SCALE/hierarchy firewall conclusions —
  <https://physics.magflowmeters.com/articles/TOE.html>.

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

---

## 1. How OUR geometry closes the EW embedding — the closure claim (concise)

> Full derivation: **GUT.html §6.3 + §6.5 + §6.8 + Appendices D / E′ / H + CR2/CR3/CR5/CR8**. This section is
> the attack-grade recap, not the derivation.

### 1.1 The mechanism end-to-end

SG-5 is one forward chain on the frozen SM-routing backbone
$K_{\rm gauge}=K_6\times S^2\times S^1_Y/\mathbb{Z}_2$ (plus $\mathcal M_4$):

```
 isometries(K_gauge) --> SU(3)_c × SU(2)_L × U(1)_Y     [SG-2, given]
   --S^1_Y/Z2 + Z6 quotient--> Q = T_3 + Y on every multiplet   [Gate 3]
   --one Wilson-line/Hosotani doublet H (n_H=1)-->  V_Hos(θ_H) minimum θ_H*
   --SU(2)_L×U(1)_Y → U(1)_em-->  v = θ_H*/(2πR_γ),  m_h^2 = V''_Hos(θ_H*)/(2πR_γ)^2
   --same η_BK (finite determinant)--> { v=246.02, m_h=123.82, |y_t/y_b| }
```

The load-bearing factors, named so a reader can attack each:

1. **Charge carrier $S^1_Y/\mathbb{Z}_2$ + the $\mathbb{Z}_6$ global quotient.** Hypercharge is the circle
   rotation; the $\mathbb{Z}_2$ fold + center actions realize $[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6$.
   The charge claim is **two exact predicates**: $Q=T_3+Y$ componentwise, and
   $\tfrac{t}{3}+\tfrac{d}{2}+Y\in\mathbb{Z}$ ($t=+1,-1,0$ for $\mathbf3,\bar{\mathbf3},\mathbf1$; $d=1,0$ for
   doublet/singlet). Frozen parity table `ac4d2df3e708` (R1.3). (Attack handle: the parity/center assignments
   on the fold were *selected and frozen*, not searched — §5.2 says "searched no manifolds; selected the
   parity/center assignments; froze the parity table." Charge is forced *given* that frozen table.)
2. **The single Wilson-line/Hosotani doublet $H=(\mathbf1,\mathbf2,+\tfrac12)$.** Identified with the holonomy
   mode $W_\gamma=\mathcal P\exp(i\oint_\gamma A)$ on the $SU(2)_L$-direction cycle $\gamma$ (cycle
   `640e1d7f7773`, bundle `2a0462b8aab9`). $E_{\rm Higgs}=L_\gamma\otimes V_{SU(2),{\rm doub}}\otimes
   L_{Y=+1/2}$ (A2.5). Exactly **one** Higgs ($n_H=1$); lower winding $n_H=0$ gives $v=0$ (no breaking),
   $n_H\ge2$ gives $v\approx492$ GeV and breaks the one-input-two-outputs alignment (failure ledger
   L6368/L7650). (Attack handle: $n_H=1$ is *minimal admissible integer*, a selection, not a derivation that
   nature picks 1.)
3. **The Hosotani phase and the breaking.** $A_\gamma=\tfrac{\theta_H}{2\pi R_\gamma}T_H$, $\theta_H\in[0,2\pi)$;
   the VEV is $\langle H\rangle=\theta_H/(2\pi R_\gamma)$, so $v_{EW}=\theta_H^\star/(2\pi R_\gamma)$ with
   $\theta_H^\star$ the minimum of the one-loop Coleman–Weinberg/Hosotani potential
   $$ V_{\rm Hos}(\theta_H)=-\frac{3}{64\pi^6R_\gamma^4}\sum_{n=1}^\infty\frac{1}{n^5}\big[N_b\cos(n\theta_H)-N_f\cos(n\theta_H)\big], $$
   which is finite (the $n^{-5}$ sum converges), periodic (so $n_H$ is a topological invariant), and
   cutoff-independent. The non-zero $\theta_H^\star$ **is** the EWSB — the unbroken direction is $U(1)_{\rm em}$
   with $Q=T_3+Y$. (Attack handle: **$\theta_H^\star$ is read from this minimum** — the single soft spot;
   §1.2 (a).)
4. **The finite determinant $\eta_{BK}=0.009721281516312024$ (`84e94518d3f5`).** Sets the Higgs-mass scale
   and the between-sector ratio simultaneously via the structural identity
   $$ \frac{1}{\eta_{BK}}=e^{2\beta_1^\star}=32\pi\exp\!\Big(\frac{\sqrt3}{24\pi}\Big)\approx102.87=\lvert y_t/y_b\rvert \quad(\text{raw determinant; certified }M_Z\text{ value}\approx58,\text{ App J}). $$
   One geometric quantity, two SM outputs ($v$, $m_h$) plus the Yukawa ratio — the structural over-determination
   test. (Attack handle: the $\eta_{BK}$ provenance is the same kind of constant flagged for SG-8 Route A — is
   it written from the geometry or reverse-engineered to hit $\lvert y_t/y_b\rvert\approx58$? See §4.3.)
5. **The mass relations.** $m_h^2=V''_{\rm Hos}(\theta_H^\star)/(2\pi R_\gamma)^2\sim g^2/(2\pi R_\gamma)^2\cdot
   \mathcal O(1)$; the post-RG outputs are $v=246.02\pm3.5$, $m_h=123.82\pm1.8$ (H.4, A1.10). Note the
   $N_b-N_f$ structure: if $N_b=N_f$ the $\theta_H$-dependence vanishes (no breaking) — EWSB requires the boson/
   fermion KK content to differ, an attackable structural fact (§3 R5).

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

"Closed" for SG-5 is **three things at once, each at a different grade**:

- **Charge ($Q=T_3+Y$): the strongest leg** — an exact componentwise architectural identity + the $\mathbb{Z}_6$
  consistency rule, *given the frozen parity/center table*. This is genuine and hand-checkable.
- **The breaking ($\to U(1)_{\rm em}$ by one doublet): genuine mechanism, given-E** — a single Wilson-line
  doublet with $n_H=1$ does break the group correctly and produces a VEV. The *mechanism* is real; the
  *smallness of the VEV* is not derived.
- **The numbers ($v$, $m_h$): near-hits, frozen-output, given-the-anchors** — post-freeze outputs of the same
  $\eta_{BK}$, agreeing with PDG, *but* with $\theta_H^\star$ read from a minimum (so $v$'s magnitude is a read
  modulus's near-hit) and $m_h$'s scale set by a determinant whose provenance must pass the kill-test.

It is conditional on:

- **given-E** — the SM chiral content and the EW group $SU(2)_L\times U(1)_Y$ supplied from SG-2/SG-3/SG-4.
- **given-the-selected-geometry** — the frozen branch, the selected parity table, the selected cycle $\gamma$,
  $n_H=1$ as the minimal admissible integer.
- **given-the-anchors** — $y_t$ and $\lvert V_{us}\rvert$ (the flavor anchors that fix $\eta_{BK}$'s
  calibration interface), plus the **read** $\theta_H^\star$ and the **uncomputed** hierarchy smallness.

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

| Genuine output | Mechanism | Grade |
|---|---|---|
| $Q=T_3+Y$ componentwise on every multiplet | $S^1_Y/\mathbb{Z}_2$ + $\mathbb{Z}_6$ quotient, given parity table | hand-checkable |
| $\mathbb{Z}_6$ rule $\tfrac t3+\tfrac d2+Y\in\mathbb{Z}$ | global center consistency | hand-checkable |
| $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ by one doublet | Wilson-line holonomy, $n_H=1$ | symbolic |
| Higgs-mass quadratic destabilization **avoided** | topological $n_H$ + finite $V_{\rm Hos}$ | symbolic (1-loop) |
| $v$, $m_h$ from **one** determinant ($\eta_{BK}$) | one-input-two-outputs compression | machine-lane (AUDIT) |

**The conditional / non-genuine content** (the honest "near-hit, not derived"):

| Quantity | Honest reality |
|---|---|
| $v=246.02$ magnitude (the lightness) | **read modulus** — $\theta_H^\star\approx2.46\times10^{-14}$ read from $V_{\rm Hos}$ min; ~85% of hierarchy undelivered |
| $m_h=123.82$ scale | set by $\eta_{BK}$ whose provenance must pass the κ³/π kill-test |
| $\rho=1$ (custodial) | **not delivered** — no custodial object, asserted-by-omission |
| hierarchy $v/M_{\rm Pl}\sim10^{-16}$ | **RELOCATION** per SCALE firewall — circular as a derivation |

So the precise statement of closure: **the geometry genuinely embeds electric charge ($Q=T_3+Y$) and breaks the
EW group with a single protected doublet, and near-hits $v$/$m_h$ from one determinant; but the lightness of $v$
is a read modulus (not derived), custodial $\rho=1$ is undelivered, and the SCALE route to the hierarchy is a
confirmed relocation. Genuine embedding + breaking; undelivered lightness + custodial.**

---

## 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 | **$Q=T_3+Y$ componentwise** (Gate 3; D charge tables) | every SM multiplet gets the observed $Q$ | hand-checkable | **Yes** — e.g. $Q_L$: $T_3=\pm\tfrac12$, $Y=+\tfrac16\Rightarrow Q=\{+\tfrac23,-\tfrac13\}$; recomputes by hand for all multiplets given the frozen $Y$ table |
| W2 | **$\mathbb{Z}_6$ rule** $\tfrac t3+\tfrac d2+Y\in\mathbb{Z}$ | non-conforming $Y$ inconsistent on geometry | hand-checkable | **Yes** — e.g. $Y(Q_L)=\tfrac15$ gives $\tfrac{31}{30}\notin\mathbb{Z}$ (killed); the SM triples satisfy it field-by-field (corroborated by A4-Z6 disposition: Tong congruence $q=3z_2-2z_3\bmod6$) |
| W3 | **Single doublet breaks $\to U(1)_{\rm em}$** (H.1, H.2.1) | one Wilson-line $H$ at $n_H=1$ does EWSB | symbolic | **Yes (structurally)** — $\langle H\rangle=\theta_H^\star/(2\pi R_\gamma)$ in the $SU(2)_L$ direction leaves $Q=T_3+Y$ unbroken; standard Hosotani algebra |
| W4 | **$V_{\rm Hos}$ finite + cutoff-independent** (H.2.1) | no $\delta m_H^2\sim\Lambda^2$; $n^{-5}$ sum converges | symbolic | **Yes** — the CW sum converges absolutely; the $\theta_H$-independent divergences cancel $N_b$ vs $N_f$; periodicity protects $n_H$ |
| W5 | **$v=246.02\pm3.5$, $m_h=123.82\pm1.8$** (H.4) | post-freeze outputs, $0.06\sigma/0.48\sigma$ vs PDG | machine-lane | **AUDIT** — values present (H.4, A1.10), claimed regenerated by the R0 reproducer; **not independently re-run here** (same class as SG-7/SG-8 absent-harness flag) |
| W6 | **$1/\eta_{BK}=32\pi\exp(\sqrt3/24\pi)\approx102.87$** (A1.13, L7612) | one determinant → $v$, $m_h$, $\lvert y_t/y_b\rvert$ | hand-checkable (algebra) | **Yes for the identity**; $32\pi\,e^{\sqrt3/24\pi}=102.87$ recomputes — **but provenance of the *form* is the open question (κ³/π)** |
| W7 | **$n_H=1$ winding + lower-winding exclusion** (H.8a) | $n_H=0\Rightarrow v=0$; $n_H\ge2\Rightarrow$ PDG miss | hand-checkable | **Yes** — the exclusion is a clean structural statement; $n_H=2\Rightarrow v\approx492$ GeV breaks alignment |
| W8 | **Higgs-protection checklist** (H.9, 8 rows) | structural protection, scope-honest | symbolic/audit | **Conditional** — an auditable *checklist*; H.9.6 explicitly reports higher-loop protection as **Diagnostic only**, not proven |
| W9 | **2026-06-23 correction box** (H.2) | $\sqrt{\eta_{BK}}/(2\pi)\sim10^{-2}$ is **12 orders too large** for the hierarchy | hand-checkable | **Yes** — $10^{-2}$ vs $10^{-16}$; this is the authoritative honest figure; the hierarchy is **not** $\eta_{BK}$ |
| W10 | **$\theta_H^\star\approx2.46\times10^{-14}$ read from the min** (H.9.7; HIERARCHY_RG) | the smallness is a read modulus | symbolic/audit | **Reproduces as a read value, NOT as a derivation** — $\theta_H^\star=2\pi R_\gamma v_{EW}$ inverts the observed $v$; deriving it blind is OPEN |
| W11 | **Custodial $\rho=1$** | — | — | **DOES NOT EXIST** — zero corpus matches; undelivered |

### 2.2 What reproduces, plainly

- **The charge embedding reproduces by hand.** $Q=T_3+Y$ on every multiplet and the $\mathbb{Z}_6$ rule are
  exact arithmetic given the frozen $Y$ table. This is the strongest, most defensible leg of SG-5 — comparable
  in rigidity to SG-2/SG-3's "given-E" recoveries.
- **The breaking + protection reproduce symbolically.** A single $n_H=1$ Wilson-line doublet breaks
  $SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ leaving $Q=T_3+Y$ unbroken; $V_{\rm Hos}$ is finite and
  cutoff-independent; the quadratic destabilization is genuinely avoided. This is the *proven banked
  sub-result* (SCALE_HIERARCHY_FINAL_VERDICT §"PROVEN sub-result": mass protected to $\sim10^{14}$ GeV).
- **The $\eta_{BK}$ identity reproduces algebraically.** $32\pi\,e^{\sqrt3/24\pi}\approx102.87$ is a clean
  recompute. The *value* reproduces; the *provenance of the formula* is the live question (§4.3).

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

- **The lightness of $v$ does NOT reproduce as a derivation.** $v=246.02$ matches PDG only because
  $\theta_H^\star$ is read from the chamber minimum; the required $\theta_H^\star\approx2.46\times10^{-14}$ is
  not produced from first principles (H.9.7). The "Gate-8 exponent PASS" comparing
  $I_{\rm Hos}=\ln(M_{\rm Pl}/246.02)$ to $I_{\rm obs}=\ln(M_{\rm Pl}/246.22)$ is **circular** — the identity
  $I_{\rm Hos}-I_{\rm obs}=\ln(v_{\rm obs}/v_{\rm corpus})$ makes it bit-identical to comparing $v$ vs $v$
  (HANDOFF_T_HIERARCHY_RG_EXPONENT). So "v emerges" is a re-log of the standing input.
- **The SCALE hierarchy does NOT reproduce — it is RELOCATION.** $\mu_{\rm cell}$ (the would-be bridge) has no
  $v$-independent readout; $\mu_{\rm cell}\to v$ is invertible-by-construction (exponential); the only
  available anchor $\partial_\sigma V=0$ *is* where $v$ is set (FIREWALL_MUCELL_INDEPENDENCE; the firewall is
  *posed correctly and currently UNCROSSED*). Round-1 campaign **demoted** the SCALE-firewall closure from
  AXIOM_CLOSED to **OPEN** under the OVERCLAIM test.
- **Custodial $\rho=1$ does NOT reproduce — it does not exist.** No custodial $SU(2)$, no $\rho$/$T$ object,
  no value, no hash. The single-doublet content makes $\rho=1$ *natural* in the SM sense, but the corpus
  **never states or computes it**. This is an undelivered required EW observable.
- **The $v/m_h$ harness is AUDIT.** The R0 reproducer is *referenced*; the executable byte-equal regeneration
  of $v=246.02$/$m_h=123.82$ from the frozen $\eta_{BK}$ + geometry is not independently re-run in this audit.

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

- **Not DERIVED-GIVEN-E** (the SG-2/SG-3 tier): although the *charge* leg is rigid-given-E, SG-5 as a whole
  carries an undelivered hierarchy lightness (read $\theta_H^\star$), an undelivered custodial $\rho=1$, a
  RELOCATION verdict on the SCALE route, and an AUDIT harness — too many open residuals for the DERIVED tier.
- **Not OPEN/DECLARED-FROZEN:** the wins are real and non-trivial — $Q=T_3+Y$ is exact, the breaking is
  delivered by one protected doublet, the quadratic destabilization is genuinely avoided, and $v$/$m_h$
  near-hit from a single determinant. This is well clear of "not claimed."
- **PARTIAL is exactly right:** genuine charge + breaking + protection, near-hit numbers, with named open
  residuals (lightness, custodial, SCALE-firewall) and downgrade rules with teeth. Matches the SCOPED_GUT
  ledger SG-5 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.

### 3.1 The residual register

| ID | Residual (named object) | Precise obstruction | Status | Leverage |
|---|---|---|---|---|
| **R1** | **The gauge-hierarchy lightness of $v$ ($\theta_H^\star$ read from a minimum)** | $v_{EW}=\theta_H^\star/(2\pi R_\gamma)$ requires $\theta_H^\star\approx2.46\times10^{-14}$; this value is **read from the chamber one-loop $V_{\rm Hos}$ minimum, not derived** (H.9.7). It carries **~85%** of the EW hierarchy. The protection ratio $\sqrt{\eta_{BK}}/(2\pi)\sim10^{-2}$ is 12 orders too large to be it. | OPEN (shared with SG-6) | **HIGHEST** — it is the single biggest "near-hit vs derivation" gap; closing it would convert $v$ from read to derived |
| **R2** | **The SCALE firewall: $\mu_{\rm cell}$ vs the hierarchy = RELOCATION** | $\mu_{\rm cell}$ has no $v$-independent UV readout; $\mu_{\rm cell}\to v$ invertible-by-construction; the only anchor $\partial_\sigma V=0$ *is* the hierarchy. Confirmed circular as a derivation (FIREWALL; campaign demotion to OPEN). | OPEN / RELOCATION (firewall uncrossed) | **HIGHEST (tied)** — it is the *reason* R1 is hard; the firewall already proved the only escape relocates |
| **R3** | **Custodial $\rho=1$ undelivered** | $\rho=M_W^2/(M_Z^2\cos^2\theta_W)=1$ is a required tree-level EW observable protected by custodial $SU(2)$. **No custodial object, no $\rho$/$T$ computation exists in the corpus.** Asserted-by-omission. | OPEN (absent) | **HIGH** — a named EW observable the gate must deliver to be "EWSB-complete"; currently a silent gap |
| **R4** | **$\theta_H^\star$ uniqueness / read-not-derived (the SG-6 inheritance)** | $\theta_H^\star$ is the minimum of $V_{\rm Hos}$; SG-6 reads it from a one-loop $V$ minimum (the acute SG-6 soft spot). The CP/breaking depend on it being the *right* minimum. | OPEN (inherited from SG-6) | **HIGH** — R1's mechanical root; shared with SG-6 |
| **R5** | **$\eta_{BK}$ provenance (target-blind or reverse-engineered?)** | $1/\eta_{BK}=32\pi\,e^{\sqrt3/24\pi}\approx102.87=\lvert y_t/y_b\rvert$ (raw). Is the *form* written from the Wilson-line determinant, or reverse-engineered to hit $\approx58$ at $M_Z$? Same class as SG-8 Route A; the κ³/π cautionary pattern. | SUSPECT (provenance unaudited) | **MEDIUM-HIGH** — if reverse-engineered, the "one-input-two-outputs" headline is true-by-construction |
| **R6** | **$v$/$m_h$ reproduction harness AUDIT** | The R0 reproducer regenerating $v=246.02$/$m_h=123.82$ byte-equal from frozen $\eta_{BK}$ + geometry is referenced, not independently re-run. Same class as SG-7's absent δ-harness. | AUDIT | **MEDIUM** — required for "claimed certificate pass" to be machine-real |
| **R7** | **Higher-loop Higgs protection Diagnostic-only** | H.9.6/H.10.1: one-loop protection survives (gauge invariance preserves $n_H$); all-orders loop-resummation is **not proven** — Diagnostic only. | DISCLOSED (scope limit) | **LOW-MEDIUM** — a disclosed limitation, not a hidden gap; a hard theorem to close |
| **R8** | **$N_b\ne N_f$ requirement for a non-trivial minimum** | $V_{\rm Hos}\propto(N_b-N_f)\cos(n\theta_H)$; if $N_b=N_f$ the $\theta_H$-dependence vanishes (no breaking). The breaking *requires* a boson/fermion KK imbalance in the protected projection. | OPEN (structural prerequisite) | **LOW-MEDIUM** — is the imbalance forced by the geometry, or assumed? |
| **R9** | **Parity/center table selected-not-searched** | $Q=T_3+Y$ + $\mathbb{Z}_6$ are forced *given* the frozen parity table `ac4d2df3e708`, which §5.2 says was *selected* ("searched no manifolds; selected the parity/center assignments; froze the table"). | OPEN (selection ≠ derivation) | **LOW** — the charge leg is strong given the table; the table's selection is the residue |
| **R10** | **$n_H=1$ minimality (selection)** | $n_H=1$ is the *minimal admissible integer*; the geometry does not *derive* that nature picks 1 (it excludes 0 and shows $\ge2$ misses PDG, which is a post-hoc data check). | DISCLOSED (selection) | **LOW** — clean structural exclusion, but selection not forcedness |

### 3.2 Leverage ranking (attack order)

1. **R2** (SCALE firewall = RELOCATION) — it is the *reason* the hierarchy is hard; the firewall already
   adjudicated the only escape relocates. The single highest-value (and most adversarially-tested) target.
2. **R1** (lightness of $v$ / read $\theta_H^\star$) — the biggest near-hit-vs-derivation gap; mechanically
   rooted in R4, scale-rooted in R2.
3. **R3** (custodial $\rho=1$) — a named EW observable the gate currently does not deliver; cheapest *physics*
   gap to at least *axiomatize* honestly.
4. **R4** ($\theta_H^\star$ read-from-min) — shared with SG-6; the mechanical root of R1.
5. **R5** ($\eta_{BK}$ provenance) — decides whether the one-input-two-outputs headline is genuine.
6. **R6** (harness AUDIT) — cheapest to close (owner artifact); makes the certificate machine-real.
7. **R7 / R8 / R9 / R10** — disclosed limit / structural prerequisite / selections.

> **The cardinal honest point.** R1 and R2 are where the **hierarchy** lives, and the SCALE firewall has
> *already* run the no tuning to the known answer blade on them and returned **RELOCATION**. Any closure path that
> "derives" $\theta_H^\star$ or $v$ by introducing a UV-floor axiom **relocates the entire hierarchy into the
> log-sensitivity of $v$ to $\log(\mu_{\rm cell}/M_*)$** — the κ³/π failure mode. A path here counts only if it
> survives the firewall's six gates *blind*. R3 (custodial) is a genuine *missing* deliverable, not a
> relocation — it is the cleanest place to make honest progress (name the custodial axiom or compute $\rho$).
> R6 is the *non-physics* residual (reproducibility); closing it improves machine-reality but moves no input.

---

## 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 target value — the κ³/π kill-test), the **specialist target** / **owner artifact**
needed, the **math to attempt**, and the **success ladder** (DERIVED-CLOSED rare → AXIOM-CLOSED likely →
sharper-OPEN → REFUTED). Latest campaign dispositions folded in
(`CLOSURE_CAMPAIGN_RESULT_2026-06-24.md` §3 + round 2).

---

### 4.1 R2 — The SCALE firewall: $\mu_{\rm cell}$ vs the hierarchy (the load-bearing relocation)

**Why first.** R2 is the *reason* R1/R4 are hard. The SCALE-frontier convergence reduces the whole
hierarchy-derivation question onto one object, $\mu_{\rm cell}$ (the uniform-cell $\Delta_0$ spectral value),
and the firewall asks whether $\mu_{\rm cell}$ can be anchored independently of $v$. This has been attacked
already; the dossier's job is to carry the verdict precisely and state what (if anything) could move it.

**Technique: T5 (no tuning to the known answer firewall).** The firewall criterion is an exact 4-part conjunction
(FIREWALL_MUCELL_INDEPENDENCE): (1) UV-reference readout not IR-fixing; (2) $v$-independence of the anchor;
(3) freeze-before-compare; (4) non-invertibility of $\mu_{\rm cell}\to v$. **Current standing: gates (2) and
(4) FAIL, gate (3)'s $\eta$-window is OPEN.**

**The firewall, gate by gate (carried verbatim from the campaign):**

| Gate | What it demands | Verdict |
|---|---|---|
| (1) UV readout not IR-fixing | $\mu_{\rm cell}$ pinned at a UV reference ($M_*$/$M_U$) by $v$-independent data | the form is right (renorm-scale residue) — **form verified, provenance not** |
| (2) $v$-independence of anchor | the pinning datum contains no EW input; $\partial_\sigma V=0$ disqualifies | **FAILS** — the only specified anchor $\partial_\sigma V=0$ *is* where $v$ is set |
| (3) freeze-before-compare | $\mu_{\rm cell}$ frozen before any $v$/$c_{\rm loop}$ computation; $\eta$-window closed | **OPEN** — uniform $\Delta_0$ unproven (live scheme knob W3) |
| (4) non-invertibility | $\mu_{\rm cell}\to v$ not algebraically invertible | **FAILS** — $\mu_{\rm cell}=I^{-1}(\ln(M_{\rm Pl}/v_{\rm obs}))$ closed-form invertible (κ³/π trap) |

**Named axiom it could reduce to (κ³/π kill-test applied).**

> **AXIOM-UV-CELL-FLOOR** (target-blind form): *"the SCALE cell $\mu_{\rm cell}$ (the uniform-cell $\Delta_0$
> spectral value) is fixed at the frozen UV reference $M_*$, value $\sim\hbar$-class, a priori, BEFORE solving
> $\partial_\sigma V=0$."*

This is writable with no EW number in sight (it references only $\hbar$, $M_*$, the spectrum). **But the
firewall already ruled it does not establish independence**, for two reasons it states in plain sight: (i) it
is a *named axiom, not a derivation* — it posits the value rather than computing it from the spectrum; and (ii)
it **relocates the entire hierarchy into the log-sensitivity of $s_{\rm min}$ to $\log(\mu_{\rm cell}/M_*)$** —
$v/M_{\rm Pl}\sim2\times10^{-17}$ comes out only through $e^{-6s_{\rm min}}$, and $s_{\rm min}$ is
logarithmically sensitive to $\mu_{\rm cell}$, so a tiny shift in $\mu_{\rm cell}$ moves $v$ *exponentially* —
the hallmark of a knob. **Unless the floor axiom fixes $\mu_{\rm cell}$ to within the log-tolerance by a
principle that has nothing to do with 246 GeV, "v emerges" is true-by-construction.** The campaign also found
the closure banks a "unifies two of three roots onto one cell" consequence that
`SCALE_HIERARCHY_FINAL_VERDICT` line 22 directly contradicts ("the cell and the hierarchy stay TWO SEPARATE
floor-values … the link CAN'T be made"). **Internal corpus conflict ⇒ must not be banked.**

**Specialist target (to hand off).** Either (a) exhibit a $v$-independent UV readout for $\mu_{\rm cell}$ —
$\Delta_0$'s spectral value at the $K_6/M_U$ scale — and show it pins $\mu_{\rm cell}$ with **no EW datum**,
*and* prove uniform $\Delta_0$ (closing the $\eta$-window, gate 3); OR (b) prove the dimensional near-no-go
that *forbids* it (Path A: a second mass cannot be built from $\{M_{\rm Pl},\hbar,\text{dimensionless
geometry}\}$ without a $\sigma$-carrying length — Buckingham-π). Path A is the *already-reached* honest endpoint.

**Math to attempt.** Run the firewall's six establishment steps **blind**: strike $\partial_\sigma V=0$ from the
anchor list; freeze+hash $\mu_{\rm cell}$ before any $c_{\rm loop}$/$v$ computation; run the $v$-independence
and non-invertibility audits; show log-tolerance robustness (does $\sigma_{\rm min}$ reproduce $v\approx246$
with $\mu_{\rm cell}$ held at the UV value, AND is that UV value forced by $\Delta_0\approx\hbar+M_*$ to within
the log-tolerance, *not* chosen so 246 comes out?). The corpus numerics already report a **~34× threshold-weight
swing across the $\sigma$-band** (v-independence audit FAILS) and an exponential $\mu_{\rm cell}\to v$ map.

**Success ladder.**
- DERIVED-CLOSED: a $v$-independent UV readout pins $\mu_{\rm cell}$ + uniform $\Delta_0$ proven → hierarchy
  becomes a prediction. **Judged unreachable in this geometry** (Path A dimensional near-no-go).
- AXIOM-CLOSED: AXIOM-UV-CELL-FLOOR named — **but the campaign DEMOTED this to OPEN**, because gates (2)/(4)
  fail and it banks an internally-refuted root unification. So even AXIOM-CLOSED is *not* honestly available
  here as a *derivation* claim.
- **sharper-OPEN (the honest standing):** $v_{EW}$ is a **genuine second dimensionful anchor**, irreducible in
  this structure, with a dimensional (Path A) + mechanistic (Path B) near-no-go *explaining why*. This is
  stronger than "we failed" — it is "here is the structural reason it can't be reduced here."
- REFUTED: someone exhibits a blind UV readout that survives all six gates → the firewall is crossed.

**Honest disposition: OPEN / RELOCATION — the firewall is correctly posed and UNCROSSED.** As a derivation the
SCALE route is confirmed circular; the real banked win is the *near-no-go itself* (genuine, non-promoting). Do
**not** bank AXIOM-UV-CELL-FLOOR as a closure — the campaign already demoted it.

---

### 4.2 R1 — The lightness of $v$ ($\theta_H^\star$ read, not derived)

**The target without loading.** $v=246.02$ matches PDG only because $\theta_H^\star\approx2.46\times10^{-14}$
is *read* from the $V_{\rm Hos}$ minimum. Closing R1 means **deriving $\theta_H^\star$ (hence $v$'s smallness)
blind from the frozen geometry, without a new tuning.** The number it must reproduce **without a new knob**:
$\theta_H^\star=2\pi R_\gamma v_{EW}$ with $1/(2\pi R_\gamma)=1.0\times10^{16}$ GeV $\Rightarrow$
$\theta_H^\star\approx2.46\times10^{-14}$ (i.e. $I_{EW}=\ln(M_{\rm Pl}/v_{EW})\approx36.83$ derived blind).

**Technique: T1 (axiom-floor) via dimensional transmutation; gated by R2's firewall.** Dimensional
transmutation is the *right mechanism shape* — $v_{EW}=M_{\rm Pl}\,e^{-I_{EW}}$ removes the second anchor **iff**
$I_{EW}\approx36.83$ is derived blind. The honest state (HANDOFF_T_HIERARCHY_RG_EXPONENT;
REVIEW_HIERARCHY_TRANSMUTATION):

- **It legitimately reduces the residual to ONE dimensionless number** $I_{EW}$ — a real simplification.
- **It does NOT yet discharge the burden.** As produced, $36.83$ is just $-\ln(v_{\rm corpus}/M_{\rm Pl})$,
  computed *from* the assumed $v$. The "exponent PASS" is the circular re-log (W2 auto-FAIL).
- **The mechanism is the wrong kind for a periodic minimum.** $\theta_H^\star$ is a stationary point of a
  *periodic* potential, **NOT a running coupling**, so there is no transmutation exponent
  $t_\star=2\pi/(b\alpha)$ to bound (Path B, SCALE_HIERARCHY_FINAL_VERDICT). The field-count bound governs only
  $\ln(M_{\rm Pl}/M_U)$ (the frozen ~15%), never the ~85% hierarchy carried by $\theta_H^\star$.

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

> **AXIOM-HIERARCHY-IS-UV-EXPONENT** (target-blind form): *"$v_{EW}=M_{\rm Pl}\,e^{-I_{EW}}$, with $I_{EW}$ the
> dimensionless RG/geometric exponent computed from the frozen UV boundary vector $\Gamma_{\rm Pl}$
> (dimensionless data at $M_{\rm Pl}$, NO electroweak input)."*

Writable with no "246" in sight — it references only $M_{\rm Pl}$ and the frozen dimensionless UV data. **The
kill-test passes on the *statement*** (no target inside); the test is then whether $I_{EW}$ *computes to*
$\approx36.83$ blind. **Current status: the only available production of $36.83$ is circular** (it re-logs
$v$), so the axiom is *writable* but **not yet yielding a blind value** — and Path B says the periodic-minimum
mechanism cannot supply the exponent at all.

**Specialist target / owner artifact.** (1) Freeze and hash the **UV dimensionless boundary vector
$\Gamma_{\rm Pl}=\mathcal G(\mathcal B_{\rm active})$** — gauge/Yukawa/scalar couplings + the geometry's
dimensionless inputs, with **no EW dimensionful input** — *before any comparison*. (2) Compute $t_\star$ blind
and test $t_\star\stackrel?=I_{EW}=36.83$ with $\lvert z\rvert\le2$. (3) **Automatic FAIL** if the "derivation"
reduces to $-\ln(v_{\rm corpus}/M_{\rm Pl})$ or if $\theta_H^\star$ is supplied rather than derived
(HANDOFF_T_HIERARCHY_RG_EXPONENT G1–G6). The owner artifact is the frozen $\Gamma_{\rm Pl}$ + a blind
$I_{EW}$-computation harness — *neither exists*.

**Math to attempt.** Recompute $V_{\rm Hos}(\theta_H)$ from the frozen KK spectrum (the $N_b$, $N_f$ content)
and find $\theta_H^\star$ **without inserting the observed $v$** — i.e. derive the minimum's location from the
geometry's $N_b-N_f$ imbalance alone. If the natural minimum sits at $\mathcal O(1)$ rather than
$2.46\times10^{-14}$, the read-off is *not* reproduced blind and R1 stays open (the expected outcome, since a
generic CW minimum is $\mathcal O(1)$, not $10^{-14}$).

**Success ladder.**
- DERIVED-CLOSED: $\Gamma_{\rm Pl}$ frozen + $I_{EW}$ computes to $36.83$ blind → $v$'s smallness derived;
  SCALE collapses $2\to1$ anchor. **Judged unreachable** (Path B: periodic minimum, no exponent; Path A:
  dimensional near-no-go).
- AXIOM-CLOSED: AXIOM-HIERARCHY-IS-UV-EXPONENT named — but it **inherits R2's firewall fate**: adopting it
  relocates the hierarchy into log-sensitivity, so it is OPEN, not bankable as a closure.
- **sharper-OPEN (honest standing):** $\theta_H^\star$ (hence $v$'s lightness) is **read from a minimum**, and
  the transmutation route is rigorously ruled out for a periodic minimum. The ~85% is undelivered, *with a
  reason*.
- REFUTED: a blind $V_{\rm Hos}$ recompute yields $\theta_H^\star$ near $10^{-14}$ from geometry alone →
  would be a genuine derivation (not expected).

**Honest disposition: OPEN, with the mechanism rigorously diagnosed.** The genuine banked win is the *proven
mass-protection* (no $\delta m_H^2\sim M_*^2$), honestly bounded at **12 orders short** of the hierarchy. The
lightness is a posit — and now we know *why it must be* in this geometry.

---

### 4.3 R3 — Custodial $\rho=1$ undelivered (the cleanest missing deliverable)

**The obstruction.** $\rho=M_W^2/(M_Z^2\cos^2\theta_W)=1$ at tree level is *the* signature of EWSB by an
$SU(2)$ doublet, protected by a custodial $SU(2)_{\rm cust}\subset SU(2)_L\times SU(2)_R$. The corpus
**contains no custodial object**: no $\rho$, no $T$-parameter, no $SU(2)_{\rm cust}$ — verified by exhaustive
search of `GUT.md` (zero matches). The gate breaks the EW group with one doublet but **never states or computes
the constraint that single-doublet breaking is supposed to deliver.** This is a genuine *missing deliverable*,
not a relocation.

**Technique: T1 (axiom-floor) to name the custodial source; or T7 (eliminative) to compute $\rho$ directly.**

**Route A (custodial from the doublet structure — the standard SM fact, target-blind).** In the SM, $\rho=1$
at tree level follows automatically from the Higgs being a single $SU(2)_L$ **doublet** (hypercharge
$+\tfrac12$): the would-be Goldstone structure has an accidental global $SU(2)_L\times SU(2)_R\to SU(2)_{\rm
cust}$ symmetry under which $(W^1,W^2,W^3)$ form a triplet, forcing $M_W=M_Z\cos\theta_W$. **Our Higgs is
exactly one such doublet** ($E_{\rm Higgs}=L_\gamma\otimes V_{SU(2),{\rm doub}}\otimes L_{Y=+1/2}$, A2.5).

- **Named axiom (κ³/π-clean):** **AXIOM-CUSTODIAL-FROM-DOUBLET** — *"the Wilson-line Higgs is a single
  $SU(2)_L$ doublet of $Y=+\tfrac12$ ($n_H=1$), and the EW-breaking sector inherits the accidental custodial
  $SU(2)_{\rm cust}$ of a single-doublet potential; hence $\rho=1$ at tree level."* Writable with no $\rho$
  value in sight (it references only the doublet rep + the standard accidental symmetry). **PASSES the
  kill-test by construction.** The test is then whether the *geometric* $V_{\rm Hos}$ respects the custodial
  symmetry — i.e. whether the $S^2$/$S^1_Y$ routing breaks it.
- **κ³/π / RELOCATION check:** PASS *as a statement*, **but with a real risk**: the custodial $SU(2)$ is an
  *accidental* SM symmetry that is **broken by $U(1)_Y$ gauging and by Yukawa splitting** ($y_t\ne y_b$). Our
  $V_{\rm Hos}\propto(N_b-N_f)$ and the hypercharge routing on $S^1_Y/\mathbb{Z}_2$ could *break* the custodial
  symmetry geometrically — so the axiom must be *checked*, not assumed. The genuine question: **does the
  frozen geometry preserve or break $SU(2)_{\rm cust}$ in the breaking sector?**

**Route B (compute $\rho$ from the geometric mass matrix — the falsifiable path).** $M_W$ and $M_Z$ are set by
the same $\langle H\rangle=\theta_H^\star/(2\pi R_\gamma)$ in the $SU(2)_L$ direction; with the frozen gauge
couplings $g$, $g'$ from SG-2/SG-7, compute $M_W^2=\tfrac14 g^2 v^2$, $M_Z^2=\tfrac14(g^2+g'^2)v^2$ and form
$\rho=M_W^2/(M_Z^2\cos^2\theta_W)$ **target-blind**. If a single doublet sources both, $\rho=1$ falls out; if
the $S^2$/$S^1_Y$ routing introduces a triplet contamination or a custodial-breaking term, $\rho\ne1$ and the
single-doublet claim is *refuted*.

- **Specialist target:** "Given the frozen Higgs bundle $E_{\rm Higgs}$ and the SG-2 gauge embedding, compute
  the $W$/$Z$ mass matrix from $\langle H\rangle$ and report $\rho$ at tree level, target-blind." Either it
  yields $\rho=1$ (discharges R3 — and this would be a genuine DERIVED-GIVEN-E result, since $\rho=1$ is a
  *structural* consequence not a fit), or it reveals a custodial-breaking geometric term (R3 becomes a
  prediction-vs-data tension at the per-mille level — the live falsifier is the measured $\rho_0=1.00038\pm
  0.00020$).

**Math to attempt.** (1) Write the EW gauge-boson mass matrix from the single-doublet VEV in the geometric
basis. (2) Check whether the $S^2$ (weak) + $S^1_Y$ (hyper) routing respects the custodial $SU(2)_{\rm cust}$.
(3) Compute $\rho$ blind. This is a **bounded, clean computation** — arguably the most tractable *physics*
target in SG-5, because $\rho=1$ for a single doublet is a near-automatic SM result and the only question is
whether the geometry spoils it.

**Success ladder.**
- DERIVED-CLOSED: $\rho=1$ falls out of the single-doublet geometric mass matrix target-blind → R3 closes as a
  genuine DERIVED-GIVEN-E output (the *missing* EW deliverable supplied). **Plausible and cheap** — this is the
  best realistic outcome in the gate, because $\rho=1$ is structural, not tuned.
- AXIOM-CLOSED: AXIOM-CUSTODIAL-FROM-DOUBLET named + the geometry shown not to break it at the level needed →
  $\rho=1$ reduced to one standard, target-blind posit.
- sharper-OPEN: the custodial check is not run → R3 stays a disclosed missing deliverable.
- REFUTED: the geometry breaks $SU(2)_{\rm cust}$ and yields $\rho\ne1$ beyond $\rho_0=1.00038\pm0.00020$ →
  a genuine falsifier of single-doublet EWSB on this branch.

**Honest disposition: OPEN but the most tractable HONEST-PROGRESS target in SG-5.** Run Route B (compute
$\rho$ blind) first — it can reach DERIVED-CLOSED because $\rho=1$ is a structural single-doublet consequence,
and it surfaces any geometric custodial-breaking term as a clean falsifier. **The current silent omission of
$\rho$ is the gate's most fixable gap.**

---

### 4.4 R4 — $\theta_H^\star$ read from a minimum (the SG-6 inheritance)

**The obstruction.** $\theta_H^\star$ — on which the breaking *location* and $v$'s magnitude both depend — is
the minimum of the one-loop $V_{\rm Hos}$, inherited from SG-6, where it is the chamber-center witness *read
from a one-loop $V$ minimum*, not independently derived. SG-6's acute soft spot propagates directly into SG-5
(this is R4 = SG-8's R8 analogue for the EW sector).

**Technique: T1 (axiom-floor), shared with SG-6.** Two honest reductions, in increasing strength:

> **AXIOM-THETA-FROM-MIN (weak, current standing):** *"$\theta_H^\star$ is the location of the global minimum
> of the frozen one-loop $V_{\rm Hos}(\theta_H)$."* This is *true by construction* and carries no target value
> — but it is the "read from a minimum" framing SG-6 already concedes is the soft spot.

> **AXIOM-THETA-SYMMETRY (strong, the goal):** *"$\theta_H^\star$ is fixed by a symmetry/topology of the
> Wilson-line cycle (e.g. a modular or reflection symmetry of $S^1_Y/\mathbb{Z}_2$), not by minimizing a
> potential."* If provable, this upgrades $\theta_H^\star$ from "read" to "symmetry-fixed" — the same upgrade
> SG-8 seeks for $\tau=\omega$.

**The decisive obstruction (carried from Path B).** Unlike SG-8's $\tau=\omega$ (an order-three modular
*fixed point*, symmetry-protected), $\theta_H^\star\approx2.46\times10^{-14}$ is **not at a symmetric point** —
it is a tiny generic displacement of a periodic potential. A symmetry of the cycle would force
$\theta_H^\star\in\{0,\pi\}$ (the symmetric points), giving $v=0$ or $v=$ maximal — **neither is
$10^{-14}$.** So the strong axiom, if it applied, would give the *wrong* value. This is *why* $\theta_H^\star$
must be read: the hierarchy is precisely the smallness of a *non-symmetric* minimum.

**Specialist target.** "Is there any symmetry of the frozen Wilson-line cycle $\gamma$ / the $S^1_Y/\mathbb{Z}_2$
fold that fixes $\theta_H^\star$ at a non-symmetric point $\sim10^{-14}$?" Expected answer: **no** — symmetries
fix symmetric points, and $10^{-14}$ is not one. This is a clean group-theory falsification.

**Success ladder.** AXIOM-THETA-FROM-MIN (weak) is *already standing* (AXIOM-CLOSED at the honest floor);
AXIOM-THETA-SYMMETRY would be DERIVED-CLOSED but is **expected to FAIL** (the minimum is not symmetric);
sharper-OPEN is the realistic endpoint. **Honest disposition: AXIOM-CLOSED at the weak axiom; the strong
symmetry upgrade is REFUTED-in-advance because $10^{-14}$ is not a symmetric point — which is exactly the
content of R1's near-no-go.**

---

### 4.5 R5 — $\eta_{BK}$ provenance (the one-input-two-outputs headline)

**The obstruction.** The headline "one geometric quantity, two SM outputs" rests on
$1/\eta_{BK}=32\pi\exp(\sqrt3/24\pi)\approx102.87=\lvert y_t/y_b\rvert$ (raw determinant). If the *form*
$32\pi\,e^{\sqrt3/24\pi}$ was written from the Wilson-line finite determinant *target-blind*, the headline is
genuine. If it was reverse-engineered to land at $\lvert y_t/y_b\rvert\approx58$ (at $M_Z$) / $\approx102.87$
(raw), it is **true-by-construction** — the κ³/π cautionary pattern (the $a:=4\kappa/\sqrt3$ keystone; the
$\eta_B=6.05\times10^{-10}$ "manifest"). This is the same class as **SG-8 Route A** ($\eta_{BK}$,
$K_{tb}^{\rm crit}$ provenance).

**Technique: T5 (no tuning to the known answer firewall) on the constant's provenance.**

- **κ³/π kill-test:** PASS only if $\eta_{BK}$ is demonstrably target-blind — a record that
  $32\pi\,e^{\sqrt3/24\pi}$ was computed from the Gate-8 Wilson-line/Berezin–Kontsevich determinant *before*
  and *independently of* the $\lvert y_t/y_b\rvert$ value. **Current status: SUSPECT.** $32\pi\approx100.53$ is
  suspiciously close to $\approx102.87$ with the $e^{\sqrt3/24\pi}\approx1.0233$ correction doing the
  fine-adjust; the test demands the determinant was written from geometry, not from $\approx58/102.87$.
- **Math to attempt:** recompute the Berezin–Kontsevich finite determinant on the active branch from scratch,
  target-blind, and check whether $1/\eta_{BK}$ falls out as $32\pi\,e^{\sqrt3/24\pi}$ *without*
  $\lvert y_t/y_b\rvert$ in its inputs. This is a bounded computation handed to the Gate-8 determinant
  specialist (shared with SG-8 Route A — close them together).

**Named axiom?** None — this is a *provenance audit*, not an axiom. The honest fallback if SUSPECT cannot be
cleared:

> **DISCLOSED-CAUTION:** state that the one-input-two-outputs compression is genuine **iff** $\eta_{BK}$ is
> target-blind, and flag $\eta_{BK}$ provenance as unaudited pending the determinant recompute — exactly as
> SG-8 flags Route A.

**Success ladder.** VERIFIED (determinant recompute yields $32\pi\,e^{\sqrt3/24\pi}$ blind → headline genuine);
RELOCATES (reverse-engineered → headline true-by-construction, the over-determination claim folds);
sharper-OPEN (provenance record absent → SUSPECT, disclosed). **Honest disposition: SUSPECT — do not bank the
one-input-two-outputs headline as a forced compression until the determinant recompute passes the kill-test.**
Couple this audit to SG-8 Route A (same constant).

---

### 4.6 R6 — $v$/$m_h$ reproduction harness (AUDIT)

**The obstruction.** "$v=246.02$ and $m_h=123.82$ regenerate from the frozen $\eta_{BK}$ + geometry via the R0
reproducer" is the load-bearing reproducibility claim, but the executable regeneration is *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 harness.

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

**Owner artifact needed.** (1) Run the R0 reproducer against the frozen inputs ($\eta_{BK}$ `84e94518d3f5`,
$n_H=1$ `f65094fd8fd1`, cycle $\gamma$, $M_Z$ `a6852c7a6b00`), **target-blind**. (2) Confirm it regenerates
$v=246.02\pm3.5$ and $m_h=123.82\pm1.8$ (the H.4/A1.10 values) to the declared precision. (3) Re-hash the R1.5/
R1.6 rows and confirm the meta-hash recomputes to `a5b1e6f9d951`.

**Math to attempt.** None new — execution + verification. Fail-closed: if $v$ or $m_h$ cannot be regenerated,
Gate 8 downgrades per the §6.12 / H.10.3 rule.

**Success ladder.** **BLOCKED_INPUTS until the reproducer is mounted/re-run** → then **VERIFIED** ($v$/$m_h$
regenerate; "claimed certificate pass" becomes machine-real) or **REFUTED** (a value fails → downgrade).
**Highest value-per-effort closeable item** — converts an asserted status into a machine-checked one with no
new physics.

---

### 4.7 R7 — Higher-loop Higgs protection (Diagnostic only)

**The obstruction.** One-loop protection survives because gauge invariance preserves the integer winding $n_H$;
all-orders loop-resummation protection is **not proven** — H.9.6/H.10.1 report it as **Diagnostic only**, a
deliberate scope limitation.

**Technique: T1 (axiom-floor) on the topological-protection statement; or specialist no-go/resummation.**

> **AXIOM-WINDING-PROTECTION-ALL-LOOPS** (κ³/π-clean): *"the integer winding $n_H$ is preserved to all loop
> orders because an integer cannot deform continuously under any perturbative gauge-invariant correction."*
> Writable with no mass value in sight; it is a topological statement.

**The honest gap.** The axiom is *plausible* (integers don't deform) but **not proven** — cross-sector and
higher-loop corrections from outside the protected manifold are bounded, not provably zero (H.9.6,
H.10.1 cross-sector row). The specialist target is a **loop-resummation theorem**: "perturbative corrections
within the protected sector preserve $n_H$ at all orders," handed to a Hosotani/gauge-Higgs-unification
specialist.

**Success ladder.** AXIOM-CLOSED (name the all-loop topological-protection axiom) is the realistic floor;
DERIVED-CLOSED requires the resummation theorem (hard); the gate already DISCLOSES this honestly as Diagnostic.
**Honest disposition: DISCLOSED → AXIOM-CLOSED at the topological-protection axiom; the all-loop theorem is a
genuine open specialist target, correctly scoped as Diagnostic, not hidden.**

---

### 4.8 R8 — $N_b\ne N_f$ requirement for a non-trivial minimum

**The obstruction.** $V_{\rm Hos}\propto(N_b-N_f)\cos(n\theta_H)$. If $N_b=N_f$ in the relevant projection the
$\theta_H$-dependence vanishes — *no breaking*. EWSB on this branch **requires** a boson/fermion KK imbalance.
Is that imbalance *forced* by the frozen spectrum, or assumed?

**Technique: T7 (eliminative) — compute the imbalance from the frozen KK content.**

**Math to attempt.** Tabulate $N_b$, $N_f$ for the protected projection from the frozen $K_{\rm gauge}$ KK
spectrum (the same spectrum used in the G heat-kernel ledger and the EXTRACT_Q shape-doublet sum). Check the
sign and magnitude of $N_b-N_f$ — and whether it produces a minimum at non-zero $\theta_H$ (necessary for
breaking) at all. **Caution (carried from EXTRACT_Q_SHAPE_DOUBLET):** the relevant multiplicity table
$d_{(p,q)}$ on $SU(3)/T^2$ is **present-but-BLOCKED on disk** — the $T^2$-invariant zero-weight multiplicity
rule $m_0(p,q)$ is not supplied, so the graded sum (and hence a clean $N_b-N_f$) is **fail-closed**. The frozen
heat-kernel ledger is ~3:1 boson-dominant (suggestive of a non-trivial minimum) but this is a dimensional hint,
**not** a computed imbalance.

**Named axiom?** None needed if the imbalance computes; if it cannot (blocked), the honest statement is that
the *existence* of a non-trivial breaking minimum rests on the blocked $m_0(p,q)$.

**Success ladder.** VERIFIED ($N_b-N_f$ computes and forces a non-trivial minimum → breaking is structural);
BLOCKED_INPUTS (the $m_0(p,q)$ rule is absent → the breaking's existence is fail-closed on a missing
group-theory datum); REFUTED ($N_b=N_f$ → no breaking, the doublet would not condense). **Honest disposition:
BLOCKED_INPUTS — the breaking *mechanism* is sound, but the *existence* of a non-trivial $\theta_H^\star$ rests
on the same blocked $m_0(p,q)$ table that fail-closes the shape-doublet $q$. Supply $m_0(p,q)$ to unblock.**

---

### 4.9 R9 — Parity/center table selected-not-searched

**The obstruction.** $Q=T_3+Y$ + the $\mathbb{Z}_6$ rule are *exact given* the frozen parity table
`ac4d2df3e708`, which §5.2 states was **selected** ("searched no manifolds; selected the parity/center
assignments on the fold; froze the parity table"). So the charge leg is forcedness *conditional on a selection*.

**Technique: T10 (selector) + T4 (no-go on alternatives).**

> **AXIOM-PARITY-Z6** (κ³/π-clean): *"the $\mathbb{Z}_2$ fold + center actions realize
> $[SU(3)\times SU(2)\times U(1)_Y]/\mathbb{Z}_6$ — the finest quotient consistent with the surviving spectrum
> and the $SU(3)\times SU(2)$ centers."* Writable with no charge value in sight.

**The honest standing (from the A4-Z6 disposition).** The campaign found $\Gamma=\mathbb{Z}_6$ is **OPEN**: the
line-operator/maximality posit pins only $\Gamma\le\mathbb{Z}_6$ ($q\mid6$, not $q=6$), and "finest" is
pinnable only by already wanting $\mathbb{Z}_6$ — the symmetric minimal cover $\Gamma=1$ is a priori equally
motivated (no tuning to the known answer rejects AX-FINEST). **But** the SM hypercharge triples were verified to satisfy
the Tong congruence $q=3z_2-2z_3\bmod6$ **field-by-field**, and the $\chi=-3$ trap is honored — so the *charge
assignments* are consistent, even if the *quotient's finest-ness* is not forced.

**Specialist target.** "Is $\mathbb{Z}_6$ forced (not merely $\le\mathbb{Z}_6$) by the surviving spectrum +
the requirement that $Q=T_3+Y$ be globally well-defined?" Expected: $\le\mathbb{Z}_6$ forced, $=\mathbb{Z}_6$
selected.

**Success ladder.** AXIOM-CLOSED (AXIOM-PARITY-Z6 named; $Q=T_3+Y$ exact given it) is *already standing* and is
the gate's strongest leg; DERIVED-CLOSED requires forcing $=\mathbb{Z}_6$ (OPEN per A4-Z6). **Honest
disposition: AXIOM-CLOSED — the charge embedding is rigid given the frozen parity table; the table's selection
(and the $\mathbb{Z}_6$ finest-ness) is the residue, OPEN per the A4-Z6 campaign.**

---

### 4.10 R10 — $n_H=1$ minimality (selection)

**The obstruction.** $n_H=1$ is the *minimal admissible integer* (H.8a): $n_H=0$ gives $v=0$ (no breaking),
$n_H\ge2$ gives $v\approx492$ GeV and breaks the one-input-two-outputs alignment. The geometry **excludes**
$n_H=0$ structurally and $n_H\ge2$ by a *post-hoc PDG check*; it does not *derive* that nature picks 1.

**Technique: T10 (selector).**

> **AXIOM-MINIMAL-WINDING** (κ³/π-clean): *"the Wilson-line Higgs carries the minimal non-zero admissible
> integer winding $n_H=1$."* Writable with no $v$ value in sight (it references only minimality + admissibility).

**The honest split.** $n_H=0$ exclusion is **structural** (no VEV — a clean a-priori kill). $n_H\ge2$ exclusion
is **data-dependent** ($v\approx492$ misses PDG) — that is a selection by the target, not a forcedness. So the
clean part (exclude 0) is target-blind; the "pick 1 over 2" part leans on $v$.

**Success ladder.** AXIOM-CLOSED (AXIOM-MINIMAL-WINDING) is the floor; DERIVED-CLOSED would require an
*a-priori* reason (not $v\approx492$ misses PDG) to prefer 1 over 2 — none is given. **Honest disposition:
AXIOM-CLOSED at minimal-winding; the $n_H=0$ exclusion is structural and clean, the $n_H\ge2$ exclusion is a
data check (selection), correctly disclosed.**

---

### 4.11 Attack-plan roll-up

| Residual | Technique | Named axiom (κ³/π-clean) | Specialist target / owner artifact | Realistic endpoint |
|---|---|---|---|---|
| R2 SCALE firewall | T5 | AXIOM-UV-CELL-FLOOR | blind 6-gate firewall pass; or Path A dimensional near-no-go | **OPEN / RELOCATION** (firewall uncrossed; campaign-demoted) |
| R1 lightness of $v$ | T1 | AXIOM-HIERARCHY-IS-UV-EXPONENT | freeze $\Gamma_{\rm Pl}$ + blind $I_{EW}$ harness | **sharper-OPEN** (periodic-min → no exponent; 12 orders short) |
| R3 custodial $\rho=1$ | T1 / T7 | AXIOM-CUSTODIAL-FROM-DOUBLET | **compute $\rho$ blind from the single-doublet mass matrix** | **OPEN → DERIVED-CLOSED plausible** (most tractable physics target) |
| R4 $\theta_H^\star$ read | T1 | AXIOM-THETA-FROM-MIN (weak) | symmetry-fixing check (expected to fail at $10^{-14}$) | AXIOM-CLOSED (weak); strong upgrade REFUTED-in-advance |
| R5 $\eta_{BK}$ provenance | T5 | (none — provenance audit) | recompute Berezin–Kontsevich determinant blind (with SG-8 Route A) | **SUSPECT** → VERIFIED or RELOCATES |
| R6 $v$/$m_h$ harness | machine-lane | (none) | run R0 reproducer + re-hash | BLOCKED_INPUTS → VERIFIED (best value/effort) |
| R7 higher-loop protection | T1 | AXIOM-WINDING-PROTECTION-ALL-LOOPS | loop-resummation theorem | DISCLOSED → AXIOM-CLOSED |
| R8 $N_b\ne N_f$ | T7 | (none) | compute $N_b-N_f$ (needs $m_0(p,q)$) | **BLOCKED_INPUTS** ($m_0(p,q)$ absent) |
| R9 parity/$\mathbb{Z}_6$ table | T10 + T4 | AXIOM-PARITY-Z6 | force $=\mathbb{Z}_6$ (OPEN per A4-Z6) | AXIOM-CLOSED (charge rigid given table) |
| R10 $n_H=1$ minimality | T10 | AXIOM-MINIMAL-WINDING | a-priori 1-over-2 reason (none given) | AXIOM-CLOSED |

**REDUCE-vs-RELOCATE verdict on the plan.** The plan does **not** turn one hard problem into three harder
ones — but it must be *brutally honest that two of its highest-leverage residuals RELOCATE*. R1 and R2 (the
hierarchy lightness) are **confirmed relocations as derivations**: the SCALE firewall already proved
$\mu_{\rm cell}$ has no $v$-independent readout, and the periodic-minimum mechanism rigorously cannot supply a
transmutation exponent — so any "derivation" of $v$'s smallness relocates the hierarchy into the
log-sensitivity of $v$ to $\log(\mu_{\rm cell}/M_*)$ (the κ³/π trap). The honest endpoint there is **a genuine
second dimensionful anchor with a near-no-go explaining why it is irreducible** — a real, non-promoting result,
not a closure. The *one place real physics progress is available* is **R3 (custodial $\rho=1$)**: it is a
genuine *missing* deliverable (not a relocation), it is a bounded computation, and $\rho=1$ for a single
doublet is a *structural* SM consequence — so R3 can plausibly reach **DERIVED-CLOSED** and supply the EW
observable the gate currently omits. The cheap machine-lane win is **R6** (run the $v$/$m_h$ reproducer).
**R5** ($\eta_{BK}$ provenance) is a SUSPECT that must be cleared or the one-input-two-outputs headline folds.
**R8** is fail-closed on the same missing $m_0(p,q)$ group-theory datum that blocks the shape-doublet.

The honest expected outcome of a full campaign: **R3 plausibly DERIVED-CLOSED (compute $\rho$); R6 VERIFIED
(run reproducer); R4/R7/R9/R10 AXIOM-CLOSED at honest floors; R1/R2 sharper-OPEN/RELOCATION (with the
near-no-go banked); R5 SUSPECT-pending-audit; R8 BLOCKED_INPUTS.** No DERIVED-CLOSED is promised on the
hierarchy. The gate would move from PARTIAL-with-an-undelivered-custodial-and-asserted-status to
**PARTIAL-with-$\rho=1$-delivered + a machine-verified $v$/$m_h$ + a named axiom floor + a banked
hierarchy near-no-go** — a real honesty/reproducibility/physics gain on R3/R6, **not** a promotion on the
hierarchy.

---

## A. Anchoring & Hardening Map

This gate, run through **our internal honesty methodology**: for each open residual we ask **gap-or-wall?** (does a route
exist, or is the route itself the problem), **hunt the implicit assumption** the residual silently rests on,
then ask **which measured invariant truth must it terminate on** — does it reduce to an EXISTING anchor
($\hbar$ / $M_{\rm Pl}$ / spectrum-$E$ / $\alpha_i(M_Z)$ / $y_t$ / $\lvert V_{us}\rvert$), need a NEW named
invariant, hinge on a SCHEME-anchor (one geometry-unfixed convention), sit as a COMPUTATION-DEBT (the route
exists but the finished number/harness is absent), or have NO witness at all → OPEN. The framework and the
disposition vocabulary used below are the live **Gaps & Walls Register**
(<https://physics.magflowmeters.com/articles/GAPS_AND_WALLS_REGISTER.html>); the labels mean exactly what they
mean there — **DERIVED** (tied to an already-measured fact, no new assumption) / **DERIVED-GIVEN-E** (rigid once
the observed content $E$ is supplied) / **AXIOM-CLOSED** (reduced to one named value-free posit; the gap stays
open) / **DISSOLVED** (a false/unmeasured premise dropped) / **SCHEME-ANCHORED** (reachable only after one named
geometry-unfixed convention) / **OPEN** (genuinely unsolved, or the only escape assumes the answer) /
**BLOCKED** (a real route exists but a required input is missing/contradictory) / **measured-but-irreducible**
(known from experiment, no derivation here). Ceiling, unchanged: **serious candidate, NOT validated.**

### A.1 Per-residual anchoring & hardening table

| Residual (from §3) | GAP or WALL (+kind) | Measured-invariant truth it must terminate on | Honest disposition | What would HARDEN it (concrete next step) |
|---|---|---|---|---|
| **R1** — lightness of $v$ ($\theta_H^\star$ read from a minimum) | **WALL** (no-known-route / only escape assumes the answer) | Needs a **NEW named invariant**: a target-blind exponent $I_{EW}=\ln(M_{\rm Pl}/v_{EW})\approx36.83$ anchored on $M_{\rm Pl}$ alone (no EW input). It does **not** reduce to an existing anchor — the only available "derivation" re-logs the observed $v$ (circular), and a periodic CW minimum supplies no transmutation exponent. | **OPEN** (mechanism rigorously diagnosed; ~85% of the hierarchy undelivered, protection ratio 12 orders short) | Freeze + hash the UV dimensionless boundary vector $\Gamma_{\rm Pl}$ (no EW dimensionful input), then recompute $V_{\rm Hos}$ from the frozen KK content and test whether $\theta_H^\star\sim10^{-14}$ falls out **blind** — expected to fail (generic minimum is $\mathcal O(1)$), which sharpens the OPEN with a reason. |
| **R2** — SCALE firewall: $\mu_{\rm cell}$ vs the hierarchy | **WALL** (the route itself is the problem — RELOCATION) | The implicit assumption is that $\mu_{\rm cell}$ can be anchored independently of $v$. The firewall shows it cannot: its only anchor $\partial_\sigma V=0$ **is** where $v$ is set, and $\mu_{\rm cell}\to v$ is invertible-by-construction. Terminates on **no $v$-independent invariant** — $v_{EW}$ is itself a genuine second dimensionful anchor. | **OPEN / RELOCATION** (firewall correctly posed, UNCROSSED; AXIOM-UV-CELL-FLOOR campaign-DEMOTED to OPEN) | Exhibit a $v$-independent UV readout for $\Delta_0$ at the $K_6/M_U$ scale **and** prove uniform $\Delta_0$ (close the $\eta$-window); OR bank the Path-A dimensional near-no-go (a second mass cannot be built from $\{M_{\rm Pl},\hbar,$ dimensionless geometry$\}$) as the honest endpoint — do **not** bank the floor axiom. |
| **R3** — custodial $\rho=1$ undelivered | **GAP** (a route exists; a deliverable is simply missing) | Terminates on the **measured $\rho_0=1.00038\pm0.00020$** via the existing anchors: $\rho=M_W^2/(M_Z^2\cos^2\theta_W)$ is fixed by the single-doublet VEV + the frozen gauge couplings (the $\alpha_i(M_Z)$ line). $\rho=1$ is a **structural** single-doublet consequence, not a fit. | **OPEN** — but the gate's one plausibly **DERIVED-GIVEN-E** target (the most tractable physics gap) | Run Route B: write the $W$/$Z$ mass matrix from the single-doublet $\langle H\rangle$ in the geometric basis, check whether the $S^2$/$S^1_Y$ routing preserves $SU(2)_{\rm cust}$, and compute $\rho$ **blind** — $\rho=1$ discharges R3; any custodial-breaking term surfaces as a clean falsifier vs $\rho_0$. |
| **R4** — $\theta_H^\star$ read from a minimum (SG-6 inheritance) | **GAP→WALL boundary** (weak axiom stands; strong upgrade has no route) | The weak form is **true by construction** (no invariant needed). The strong upgrade would terminate on a **symmetry/topology** invariant of the cycle $\gamma$ — but $10^{-14}$ is **not a symmetric point** (symmetries force $\theta_H^\star\in\{0,\pi\}$), so no symmetry invariant can supply it. | **AXIOM-CLOSED** at AXIOM-THETA-FROM-MIN (weak); strong symmetry upgrade **REFUTED-in-advance** | The clean falsification is already available: confirm no symmetry of $\gamma$ / the $S^1_Y/\mathbb{Z}_2$ fold fixes a non-symmetric $\theta_H^\star\sim10^{-14}$ — recording it as a group-theory no-go hardens R1's near-no-go. |
| **R5** — $\eta_{BK}$ provenance (one-input-two-outputs) | **GAP** (provenance audit — a route exists) | Hinges on whether $1/\eta_{BK}=32\pi\,e^{\sqrt3/24\pi}$ was written **target-blind** from the Wilson-line determinant, or reverse-engineered to land on $\lvert y_t/y_b\rvert$ (the **$y_t$ anchor**). If blind, it reduces to geometry; if not, the headline is true-by-construction (the κ³/π pattern). | **SUSPECT** (provenance unaudited) — do not bank the over-determination headline until cleared | Recompute the Berezin–Kontsevich finite determinant on the active branch **target-blind** (with $\lvert y_t/y_b\rvert$ absent from inputs); couple to SG-8 Route A (same constant). Pass → VERIFIED; reverse-engineered → RELOCATES. |
| **R6** — $v$/$m_h$ reproduction harness | **GAP** (COMPUTATION-DEBT — route exists, harness not re-run) | Terminates on the **frozen spectrum / $\eta_{BK}$ + the $M_Z$ comparison scale**: the R0 reproducer regenerating $v=246.02$, $m_h=123.82$ from frozen inputs. No new invariant; pure machine-reality. | **AUDIT → BLOCKED_INPUTS** until the reproducer is mounted/re-run | Run the R0 reproducer **target-blind** against the frozen inputs ($\eta_{BK}$ `84e94518d3f5`, $n_H=1$, cycle $\gamma$, $M_Z$), confirm $v$/$m_h$ to declared precision, and re-hash to `a5b1e6f9d951`. Highest value-per-effort closeable item; fail-closed downgrade if a value misses. |
| **R7** — higher-loop Higgs protection | **GAP** (disclosed scope limit; route to a theorem exists) | Terminates on a **topological invariant** ($\hbar$-class quantization): the integer winding $n_H$ cannot deform continuously under perturbative gauge-invariant corrections. One-loop is proven; all-orders is the open theorem. | **DISCLOSED → AXIOM-CLOSED** at AXIOM-WINDING-PROTECTION-ALL-LOOPS (honestly scoped Diagnostic, not hidden) | Hand a loop-resummation theorem to a gauge-Higgs-unification specialist: "perturbative corrections within the protected sector preserve $n_H$ at all orders," bounding the cross-sector contributions H.9.6/H.10.1 flag as not-provably-zero. |
| **R8** — $N_b\ne N_f$ for a non-trivial minimum | **GAP** (BLOCKED — required group-theory datum missing) | Terminates on the **frozen spectrum-$E$** (the KK content of the protected projection): the imbalance $N_b-N_f$ must compute from the $SU(3)/T^2$ multiplicity table. Blocked by the absent $T^2$-invariant zero-weight rule $m_0(p,q)$. | **BLOCKED_INPUTS** — breaking *mechanism* sound; *existence* of $\theta_H^\star\ne0$ fail-closed on $m_0(p,q)$ | Supply the $m_0(p,q)$ zero-weight multiplicity rule (the same datum that blocks the shape-doublet $q$), tabulate $N_b$, $N_f$, and confirm $N_b\ne N_f$ forces a minimum at non-zero $\theta_H$. |
| **R9** — parity/center table selected-not-searched | **GAP** (selection $\neq$ derivation; charge rigid given the table) | The charge leg $Q=T_3+Y$ + the $\mathbb{Z}_6$ rule terminate on the **frozen spectrum-$E$** (the SM hypercharge triples satisfy the Tong congruence $q=3z_2-2z_3\bmod6$ field-by-field, $\chi=-3$ honored), **given** the frozen parity table. The table's finest-ness ($=\mathbb{Z}_6$ vs $\le\mathbb{Z}_6$) needs no new invariant — it is a selection. | **AXIOM-CLOSED** at AXIOM-PARITY-Z6 (the gate's strongest leg, hand-checkable); $=\mathbb{Z}_6$ finest-ness OPEN per A4-Z6 | Attempt to force $=\mathbb{Z}_6$ (not merely $\le\mathbb{Z}_6$) from the surviving spectrum + global well-definedness of $Q=T_3+Y$; expected outcome $\le\mathbb{Z}_6$ forced, $=\mathbb{Z}_6$ selected — recording the residue cleanly. |
| **R10** — $n_H=1$ minimality (selection) | **GAP** (selector; clean structural part + a data-leaning part) | The $n_H=0$ exclusion is **structural** ($v=0$, target-blind). The "pick 1 over 2" leans on the **measured $v$** ($n_H\ge2\Rightarrow v\approx492$ misses PDG) — a selection by the target, not a new invariant. | **AXIOM-CLOSED** at AXIOM-MINIMAL-WINDING ($n_H=0$ kill clean; $n_H\ge2$ kill is a disclosed data check) | Find an *a-priori* reason (independent of $v\approx492$ missing PDG) to prefer $n_H=1$ over $n_H=2$ — none is currently given; absent that, the honest floor is the named minimality axiom. |

### A.2 Gate-level rollup

- **Overall disposition.** SG-5 stays **PARTIAL**, and the hardening map confirms *why* it cannot rise higher:
  its three genuine legs split sharply by what invariant they touch. The **charge leg** (R9) is the strongest —
  **AXIOM-CLOSED / hand-checkable**, terminating on the frozen spectrum-$E$ given the parity table. The
  **breaking + protection** legs are real mechanism but carry a **BLOCKED** existence proof (R8) and an
  **AXIOM-CLOSED** all-loop scope limit (R7). The **hierarchy** legs (R1, R2) are the two **WALLs** — they
  terminate on **no $v$-independent invariant**, and the SCALE firewall has already returned **OPEN /
  RELOCATION**; $v_{EW}$ is a genuine second dimensionful anchor. **Custodial $\rho=1$** (R3) is the one true
  **GAP** with a measured invariant ($\rho_0$) and a plausible **DERIVED-GIVEN-E** route. The headline economy
  (R5) is **SUSPECT** until the determinant is recomputed blind, and the $v$/$m_h$ harness (R6) is a
  **COMPUTATION-DEBT**.
- **Anchors it depends on.** Existing anchors: **spectrum-$E$** (charge tables, the $\mathbb{Z}_6$ rule, the KK
  imbalance — R8/R9), **$y_t$** (the $\lvert y_t/y_b\rvert$ calibration that $\eta_{BK}$ rides — R5),
  **$\lvert V_{us}\rvert$** (the flavor-anchor interface), **$\alpha_i(M_Z)$** (the frozen gauge couplings
  entering $M_W$/$M_Z$ — R3), and **$M_{\rm Pl}$ / $\hbar$** (the would-be hierarchy exponent and the
  topological-winding quantization — R1/R7). It depends on **no derived hierarchy invariant** — that is exactly
  the open surface. Consistency note: SG-5's quantitative status is *milder* than **SG-7**, which the live
  Register and GUT.md now carry as **Diagnostic-only / SCHEME-ANCHORED** (threshold rows fitted-not-derived,
  magnitude scheme-anchored); SG-5's numbers are near-hits given the anchors, not fitted threshold rows, but
  share the discipline that signs/structure are firmer than magnitudes.
- **Single highest-leverage hardening move.** **Compute $\rho$ blind from the single-doublet geometric mass
  matrix (R3, Route B).** It is the only residual that is a genuine *missing deliverable* (not a relocation),
  it is a bounded computation, it terminates on a *measured* invariant ($\rho_0=1.00038\pm0.00020$), and
  $\rho=1$ is a structural single-doublet consequence — so it can plausibly reach **DERIVED-GIVEN-E** and supply
  the one required EW observable the gate currently omits, while surfacing any geometric custodial-breaking term
  as a clean falsifier. (The cheap reproducibility win is R6; the two WALLs R1/R2 admit only a banked
  near-no-go, not a closure.) **No status was ever upgraded** — none of the above is a promotion; it is the honest map of where
  each residual must terminate.

### 🎯 Target anchor(s) for this gate

In the terminate-on (anchoring) sense, SG-5 must terminate on the **two dimensionful anchors $M_{\rm Pl}$
and $v_{EW}$** — and on no measured invariant beyond those. The **charge leg ($Q=T_3+Y$ + the $\mathbb{Z}_6$
rule)** carries **no anchor of its own**: it is **structural / DERIVED-GIVEN-E** once the observed content $E$
(the SM hypercharge triples) is supplied. The breaking and the near-hit numbers ($v=246.02$, $m_h=123.82$)
ride the existing anchors ($\eta_{BK}$/spectrum-$E$, $y_t$, $\alpha_i(M_Z)$) — they are **frozen outputs given
the anchors, NOT a derived lightness**. The **single open leg is the hierarchy-lightness of $v_{EW}$**: it
rests on $\theta_H^\star$ **read from a one-loop minimum** (not derived) and terminates on **no $v$-independent
invariant** — so $v_{EW}$ stands as **a genuine second dimensionful anchor, OPEN** (the SCALE firewall returned
OPEN / RELOCATION). Closing the hierarchy needs the **$\theta_H^\star$ derivation, not a new anchor**; there is
**no measured invariant beyond $\{M_{\rm Pl},\,v_{EW}\}$** for this gate to reach for. (No scheme-anchor is
load-bearing for the target statement; $v_{EW}$'s lightness is OPEN, not SCHEME-ANCHORED.) **No status was ever upgraded.**

---

## 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 charge card; **§6.5** Gate-5 anomaly card; **§6.8** Gate-8 Higgs card; **§6.12**
    falsification map (Gate 3/5/8 rows); **§6.13** certificate summary.
  - **§5.2 / §5.4 / §5.7** narrative modules; **§3** the worked anomaly example.
  - **Appendix D** (SM recovery / charge tables); **Appendix E′** (chirality + anomaly closure).
  - **Appendix H** (Higgs protection): H.1 origin, **H.2.1** explicit Wilson-line formula + $V_{\rm Hos}$ + the
    **2026-06-23 correction box** ($\sqrt{\eta_{BK}}/(2\pi)\sim10^{-2}$ is 12 orders too large), H.4 outputs,
    **H.9** protection checklist (the **H.9.7 non-claims box**: "$m_h$ not predicted from first principles
    without chamber input"; "$\theta_H^\star$ read from the chamber minimum"), **H.10** correction-class
    ledger, **H.11** authority summary.
  - **Appendix CR** modules **CR2 / CR3 / CR5 / CR8** (reader companions).
  - **Appendices A1 / A2 / R0** (full-precision constants $\eta_{BK}$, $M_*$; $E_{\rm Higgs}$ bundle; freeze
    records — all hashes in §0 above).
- **Paper IV, TOE.html** — <https://physics.magflowmeters.com/articles/TOE.html> (SCALE / hierarchy firewall
  conclusions; the $v_{EW}$ second-anchor irreducibility).

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-8 exemplar (shape match) | `…/rendered/TOE/PER_GATE_DOSSIERS/DOSSIER_SG8_FLAVOR_CLOSURE_ATTACK.md` | format / depth template |
| SG-5 status line | `…/rendered/TOE/SCOPED_GUT_PER_GATE_STATUS_LEDGER.md` | the PARTIAL label + honest caveats (carried verbatim) |
| **SCALE firewall (authoritative R2 adjudication)** | `…/rendered/TOE/FIREWALL_MUCELL_INDEPENDENCE_2026-06-23.md` | the 4-part firewall criterion; gates (2)/(4) FAIL; RELOCATION |
| **SCALE / hierarchy final verdict** | `…/rendered/TOE/SCALE_HIERARCHY_FINAL_VERDICT_2026-06-23.md` | Path A circular + Path B wrong-mechanism; $v_{EW}$ = second anchor; "two separate floor-values" |
| **Hierarchy RG exponent handoff** | `…/rendered/TOE/HANDOFF_T_HIERARCHY_RG_EXPONENT.md` | $\theta_H^\star\approx2.46\times10^{-14}$; circular re-log auto-FAIL; $\Gamma_{\rm Pl}$ protocol |
| Closure campaign (R2 demotion) | `…/rendered/TOE/CLOSURE_CAMPAIGN_RESULT_2026-06-24.md` | A2-firewall AXIOM_CLOSED→OPEN demotion; κ³/π discipline; §3 decisive verdict |
| Closure campaign round 2 | `…/rendered/TOE/CLOSURE_CAMPAIGN_ROUND2_2026-06-24.md` | demotion-on-verify norm; RELABEL_FAIL / BLOCKED_INPUTS dispositions |
| Shape-doublet $q$ extract (R8 block) | `…/rendered/TOE/EXTRACT_Q_SHAPE_DOUBLET_2026-06-23.md` | $m_0(p,q)$ present-but-BLOCKED → $N_b-N_f$ fail-closed |
| GUT manuscript | `…/rendered/GUT/GUT.md` | §6.3 charge; §6.5 anomaly; §6.8 Higgs; §5.2/§5.4/§5.7; Appendix D; Appendix E′; **Appendix H** (H.1–H.11); A1/A2/R0 |

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

Branch `dcc66f1b2685` · manifest meta `a5b1e6f9d951` · parity table (realizes $/\mathbb{Z}_6$) `ac4d2df3e708`
· Higgs Wilson-line bundle `2a0462b8aab9` · Wilson-line cycle $\gamma$ `640e1d7f7773` · integer winding
$n_H=1$ `f65094fd8fd1` · $\eta_{BK}=0.009721281516312024$ `84e94518d3f5` · anchor $y_t$ `548d7099ef18` ·
anchor $\lvert V_{us}\rvert$ `a1bc510bc7cd` · comparison scale $M_Z$ `a6852c7a6b00` · UV reference
$M_*=7.467\times10^{16}$ GeV (A1.10). **$\theta_H^\star$: no first-principles value — read from the
$V_{\rm Hos}$ minimum (H.9.7).** **Custodial $\rho=1$: no object, no hash — UNDELIVERED.** **$m_0(p,q)$
zero-weight multiplicity (R8): present-but-BLOCKED on disk — $N_b-N_f$ fail-closed.**

---

### Closing honest statement

SG-5 is **PARTIAL**. Its genuine, defensible content is real and strong on three legs of unequal grade: the
**electric-charge embedding $Q=T_3+Y$** is exact componentwise on every multiplet (plus the $\mathbb{Z}_6$
consistency rule), *given* the frozen parity table — the strongest, hand-checkable leg; the **breaking
$SU(2)_L\times U(1)_Y\to U(1)_{\rm em}$ by a single Wilson-line/Hosotani doublet** ($n_H=1$,
integer-winding-protected) is a genuine mechanism that also **structurally avoids the Higgs-mass quadratic
destabilization** (a proven, banked sub-result); and $v=246.02$, $m_h=123.82$ **near-hit** PDG from one finite
determinant $\eta_{BK}$ (one input, two outputs). Its honest open surface is equally clear and is dominated by
the **hierarchy**: the **gauge-hierarchy lightness of $v$ is UNDELIVERED** — $\theta_H^\star\approx2.46\times
10^{-14}$ is **read from the chamber minimum** (~85% of the hierarchy), the protection ratio
$\sqrt{\eta_{BK}}/(2\pi)\sim10^{-2}$ is **12 orders too large**, and the **SCALE firewall verdict is
RELOCATION** ($\mu_{\rm cell}$ has no $v$-independent readout; $\partial_\sigma V=0$ *is* the hierarchy);
**custodial $\rho=1$ is UNDELIVERED** (no custodial object exists in the corpus); $\eta_{BK}$ provenance is
**SUSPECT** pending a target-blind determinant recompute; and the $v$/$m_h$ harness is **AUDIT**. The attack
plan reduces these to named, target-blind axioms and bounded falsifiable computations — with the κ³/π kill-test
guarding R1/R2 (the hierarchy, confirmed RELOCATION) and R5 ($\eta_{BK}$ provenance), and with the one genuine
physics-progress target being **R3 (compute $\rho$ blind from the single-doublet mass matrix → plausibly
DERIVED-CLOSED)**, the cheap machine-lane win being **R6 (run the $v$/$m_h$ reproducer)**, and **R8** fail-closed
on the same missing $m_0(p,q)$ datum that blocks the shape-doublet. The realistic ceiling is **$\rho=1$
delivered + a machine-verified $v$/$m_h$ + a named axiom floor + a banked hierarchy near-no-go** —
**not** a promotion on the hierarchy. **No status was ever upgraded; frozen branch `dcc66f1b2685` / `a5b1e6f9d951`
READ-ONLY; given-E ≠ derivation of E; selection ≠ derivation; dissolved ≠ solved; 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.*