# SG-5 — Electroweak Embedding / EWSB (GUT Gate 5): CLOSURE RESULT

> **Status — binding header.**
> **Gate-level roll-up: OPEN** (any open piece ⇒ OPEN, per the grading rubric).
> **Charge/embedding leg: AXIOM-CLOSED + DERIVED-GIVEN-E.** Q=T₃+Y and SU(2)_L×U(1)_Y→U(1)_em are
> DERIVED-GIVEN-E; the minimal-actor selector + n_H=1 + Z₆ table are AXIOM-CLOSED; v_EW is accepted as a
> measured-but-irreducible second dimensionful anchor (alongside M_Pl), realized — not derived — by the Wilson-line VEV.
> **Custodial ρ_tree=1: OPEN — BLOCKED / computation-debt** (the headline correction). The tree-level ρ=1 is
> only the textbook single-doublet rep-algebra tautology; it is **not** a geometric readout, because the
> electroweak gauge-boson mass matrix M²_WZ is verifiably **absent** from the corpus.
> **Hierarchy / lightness of v: OPEN / RELOCATION** (SCALE firewall uncrossed — not a hierarchy derivation).
> **Open audits:** R5 η_BK provenance SUSPECT; R6 v/m_h reproducer AUDIT/BLOCKED; R8 N_b=N_f BLOCKED.
> **No status was ever upgraded.** Frozen branch `dcc66f1b2685` / `a5b1e6f9d951` READ-ONLY. Honest ceiling: serious candidate, NOT validated.
> **Provenance:** specialist result → refute-before-upgrade verification (`wh3rfp8ow`) → **Route B** blind
> mass-matrix computation (`wi00pp5t0`; physics + disposition both SOUND/HONEST). Verified 2026-06-25.

---

## §1. What closes — the charge/embedding leg

| Leg | Disposition | Basis |
|---|---|---|
| **Charge Q=T₃+Y** | **DERIVED-GIVEN-E** | Componentwise on the SM multiplets, given the frozen Z₆ parity table; the gate's strongest leg. |
| **EWSB SU(2)_L×U(1)_Y → U(1)_em** | **DERIVED-GIVEN-E** | The single Wilson-line/Hosotani doublet H∼(1,2,+½) with a neutral SU(2)_L-direction VEV (Q-preserving, T₃=−Y). |
| **Minimal EW-breaking actor (uniqueness)** | **AXIOM-CLOSED** (minimality selector) | Elimination inventory correct (singlet no-break; real triplet ρ=2; complex triplet ρ=½; higher fail/non-minimal). Uniqueness is **genuinely conditional on minimality** — exotic multiplets *can* give ρ=1, so minimality travels as the named axiom. **"Custodial" must be stripped from the axiom name** — bundling it RELABEL_FAILs (custodial ≈ "gives ρ=1"). `AXIOM-CUSTODIAL-FROM-DOUBLET` stays *named-but-not-banked*. |
| **n_H=1 winding · Z₆ table** | **AXIOM-CLOSED** | Minimal admissible winding; selected center-locking table (=Z₆ finest-ness OPEN). |
| **v_EW** | **measured-but-irreducible** | The second dimensionful anchor; realized by `v_EW = θ_H*/(2πR_γ)`, **not** derived. |

## §2. The headline correction — custodial ρ=1 is BLOCKED, not DERIVED

**Route B (target-blind) result: ρ=1 = OPEN / BLOCKED-computation-debt.** The tree algebra
`M_W²=g²v²/4, M_Z²=(g²+g′²)v²/4, M_γ=0 ⇒ ρ_tree=1` is correct **only as SM rep-algebra on the doublet label**.
It is **not** a readout from the frozen geometry, because:

- **The EW gauge-boson mass matrix M²_WZ is never built on the frozen branch.** The corpus has only
  coupling-level normalization (`1/g_em² = 1/g₁² + 1/g₂²`; `e = gg′/√(g²+g′²)`) and `M_W ← v` via Fermi — it
  **never** computes M_Z from `|D_μH|²`, never the (W³,B) block, never ρ. The SM θ_W mixing is **written by hand**, not derived. (The dossier's W11 records the custodial object as "DOES NOT EXIST.")
- **Three un-discharged premises** block DERIVED-GIVEN-E: (1) no certificate the protected zero-mode spectrum
  carries **no second VEV source** (triplet/extra doublet); (2) the **S²-vs-S¹_Y kinetic co-normalization** of
  the off-diagonal `−gg′v²/4` term is never formed (the two gauge factors live on geometrically distinct
  sub-manifolds); (3) no **EW KK gauge-boson mixing** (the gauge-Higgs-unification T-parameter) computation.

Each is a real geometric channel that could break custodial SU(2). So **neither ρ=1 (case a) nor ρ≠1 (case b)
is earned** — custodial breaking is *possible-but-undemonstrated*. Naming ρ=1 a closure would be tuning to the known answer.

### To unblock (the concrete computation-debt)
Freeze (i) the SU(2)_L reduction integral over S² and the U(1)_Y reduction over S¹_Y/ℤ₂ in a **common
normalization**; (ii) `|D_μ⟨H⟩|²` assembled into `{W±; W³,B}` with the common `v=θ_H*/(2πR_γ)`; (iii) a
zero-mode-spectrum certificate (no 2nd VEV source); (iv) the EW KK mixing correction to M_Z/M_W. Then ρ reads
off **blind** — and either earns ρ=1 = DERIVED-GIVEN-E or finds it breaks vs the measured ρ₀=1.00038±0.00020
(a live falsifier). Until then: **do not bank ρ=1 as closure.**

## §3. What stays OPEN (gate roll-up)

- **R1/R2 — hierarchy / lightness of v:** OPEN / RELOCATION. θ_H* is read from the one-loop minimum (~85% of
  the hierarchy undelivered); the SCALE firewall returned RELOCATION (μ_cell has no v-independent readout) and
  that verdict is carried, not overturned. v_EW is a genuine second anchor; its **lightness is undelivered**.
- **R3 — custodial ρ=1:** OPEN (BLOCKED/computation-debt, §2).
- **R5 — η_BK provenance:** SUSPECT. **R6 — v/m_h reproducer:** AUDIT/BLOCKED. **R8 — N_b=N_f:** BLOCKED_INPUTS.

The re-scoping (close SG-5 as the EW embedding/actor gate, v_EW a transparent measured anchor, hierarchy out)
is **HONEST** — it buries no 246 GeV and keeps the hierarchy loudly OPEN. But the **gate is governed by its
least-closed residual**, so it stays **OPEN**. The charge/embedding leg is the closest-to-closed; ρ=1 is the
weakest (BLOCKED).

## §4. Honest one-line

> SG-5's **charge/embedding leg is AXIOM-CLOSED + DERIVED-GIVEN-E** (Q=T₃+Y, EWSB to U(1)_em, minimal Wilson-line
> doublet, v_EW a measured anchor). **Custodial ρ=1 is OPEN/BLOCKED** — the geometry never builds the W/Z mass
> matrix, so tree ρ=1 is a rep-algebra tautology, not a derivation. The **hierarchy is OPEN/RELOCATION**, and
> the **gate is OPEN**. NOT a hierarchy derivation; NOT a proof that the geometry yields ρ=1. Serious candidate, NOT validated.

---

## §5. The completion endpoint — SG-5A / SG-5B split (formalized 2026-06-25)

A specialist "completion-stack" pass froze SG-5's **strongest honest endpoint**, and an independent
refute-before-upgrade audit (`wmtzxcmts`) found it **PROMOTION-SAFE** (all five kill-tests pass; it closes
nothing new — it *confirms* the disposition above). The endpoint formalizes SG-5 as two sub-objects, with the
**gate itself OPEN**:

- **SG-5A — Electroweak embedding / actor / mass matrix.** Status: **OPEN**, with one open computation-debt leg.
  *Closed legs:* Q=T₃+Y, EWSB→U(1)_em (DERIVED-GIVEN-E); minimal actor / n_H=1 / Z₆ (AXIOM-CLOSED); v_EW
  (measured-but-irreducible anchor). *Open leg (R3):* the geometric W/Z mass matrix + no-second-VEV + KK Schur
  correction — i.e. ρ=1 must still be **earned**, not asserted. (Avoid "near-terminal" — it is OPEN with one
  named computation-debt leg.)
- **SG-5B — Electroweak hierarchy / θ_H\* / v-lightness.** Status: **OPEN / RELOCATION**, **exported** to the
  tracked **`SG5B_EW_HIERARCHY_EXPORT_LEDGER.md`** (SCALE / SG-6). v_EW is *not* derived inside SG-5A; the
  lightness of v is a separate transmutation problem.

**The R3 upgrade path (frozen recipe to earn ρ=1 = DERIVED-GIVEN-E).** Build, target-blind: (i) the SU(2)_L (S²)
+ U(1)_Y (S¹_Y/ℤ₂) reduction integrals in **one common normalization**; (ii) the assembled `∫|D_μ⟨H⟩|²` →
`{W±; W³,B}` mass matrix with the common v; (iii) a **no-second-VEV** zero-mode certificate; (iv) the EW KK
**Schur-complement** correction `M_eff² = M_00² − M_0n²(M_nn²)⁻¹ M_n0²` (or a bound). Then ρ reads off blind:
ρ=1 → DERIVED-GIVEN-E · ρ≠1 → FINDING (vs ρ₀=1.00038±0.00020) · data absent → still BLOCKED.

**Audit flags — resolved 2026-06-25:** (a) the SG-5B export now has a **tracked destination ledger**,
`SG5B_EW_HIERARCHY_EXPORT_LEDGER.md` (status OPEN/RELOCATION) — tracked, not merely asserted; (b) the
completion-stack artifacts are now **archived + re-run-verified** (`SG5_complete_endpoint_stack/`, 23 files,
manifest **22/22 intact**): `validate_endpoint.py` → **PASS (exit 0, fail-closed)**; `compute_textbook_rho.py`
→ ρ=1 (rep-algebra only, correctly labeled); `reproduce_eta_BK.py` → η_BK = 0.009721281516312024 (exact, 0
error; provenance still BLOCKED_AUDIT). The validator's PASS is now a **re-run certificate**, not self-reported —
and it passes *because nothing is promoted* (`verdict.json`: promotions:0, target_blind:true,
original_SG5_rollup:OPEN, all R3/R5/R6/R8 evidence flags false).

**Net:** SG-5 stays **OPEN**. The endpoint is the cleanest frozen disposition **plus** the precise recipe to
close ρ=1 — it does **not** close it. No status was ever upgraded.

---

## §6. Completion routes + axiom reduction (all-paths stack — re-verified 2026-06-25)

A follow-on specialist pass produced **`SG5_all_paths_to_completion_stack/`** (archived in the corpus;
**re-run-verified** by direct examination: `validate_all_paths.py` → **PASS** (fail-closed; promotions:0; status
OPEN); `reproduce_eta_BK.py` → η_BK = 0.009721281516312023; missing certificates honestly marked `.MISSING.md`).
It lays out **all four completion routes** + the **axiom floor** and — like the endpoint — **closes nothing new**
(R3 BLOCKED, gate OPEN).

**The four routes** (pick per residual, by the evidence):
- **Path A — DERIVED-GIVEN-E:** supply the R3 certificates (common-normalization integrals · no-second-VEV ·
  KK-Schur bound) → read ρ blind → DERIVED-GIVEN-E iff the geometry yields C_W=C_3=C_Y=C_3Y=1, Δ_KK=0/bounded.
- **Path B — Reduced to Axiom:** **only for walls, NOT for missing computations** (explicit guard against
  relabeling computation-debt).
- **Path C — FINDING:** record ρ≠1 / custodial breaking as a falsifiable finding.
- **Path D — BLOCKED:** name the exact missing owner artifacts (the current state of R3/R6/R8).

**Axiom-reduction ledger (banked vs candidate):**
- **BANKED:** `AXIOM-FROZEN-ADMISSIBILITY-GRAMMAR` (governing rule) · `AXIOM-MINIMAL-EW-BREAKING-ACTOR`
  (AXIOM-CLOSED, stripped of any custodial/ρ claim) · `AXIOM-EW-SCALE-ANCHOR-SEPARATION` (export discipline) ·
  `AXIOM-BLIND-STRUCTURAL-PROVENANCE` (method axiom).
- **CANDIDATE / computation-debt (NOT banked):** `AXIOM-SINGLE-EW-VEV-SOURCE`, `AXIOM-COMMON-EW-KINETIC-DESCENT`,
  `AXIOM-EW-KK-SCHUR-DECOUPLING` — each terminalizes **only** when its R3 certificate is supplied (Path A), never
  by being named. **`AXIOM-CUSTODIAL-FROM-DOUBLET` is explicitly NOT banked** (it would relabel ρ=1).
- **Three-root compression (conditional — only *if* the certificates succeed):** frozen admissibility · minimal
  single EW actor · metric-consistent descent with KK control.

The discipline holds throughout: genuine structural axioms banked; the R3 pieces kept as **computation-debt**
(candidate axioms, not relabeled closures); the hierarchy exported; ρ=1 BLOCKED until *earned* by the geometry.
**Net unchanged: SG-5 OPEN. No status was ever upgraded.**

---

## §7. Path A executed — BLOCKED (conditional-DERIVED); the co-normalization blocker dissolved *for ρ* (2026-06-25)

Path A (assemble the geometric W/Z mass matrix from the frozen geometry, read ρ blind) was executed and
adversarially verified (workflow `wex5xj2ae`; physics + disposition passes both **HONEST**; target-blind;
No status was ever upgraded). Outcome: **BLOCKED (conditional-DERIVED)** — *not* a legitimate upgrade to DERIVED-GIVEN-E — but
with a real banked advance and a sharply-narrowed remaining blocker.

**Banked advances (UPHELD — genuine, not promotions):**
1. **The co-normalization blocker is *void for ρ*.** With independent overlaps a_W (S²), a_B (S¹_Y), the assembled
   matrix gives **ρ_tree = C_W/C_3 exactly** — C_Y and C_3Y cancel, so the S²-vs-S¹_Y co-normalization **a_B
   drops out of ρ entirely** (explicit algebra, not assertion). The thing three prior passes were stuck on
   (blocker #2) is a **red herring for ρ**. (It is still owed for the *absolute* masses M_W, M_Z.)
2. **The prior pass-condition was over-strong:** ρ=1 needs only **C_W=C_3 and C_3Y²=C_3·C_Y**, not all four C's=1.
3. **Photon-masslessness (det of the neutral block = 0): DERIVED-GIVEN-E**, normalization-independent — forced
   by the Q=T₃+Y-preserving VEV (U(1)_em unbroken).

**Why it stays BLOCKED (the adversary's catch):** the headline "C_W=C_3 because one su(2) field / one S²
wavefunction" is an identity of an *assumed fundamental-doublet* covariant derivative. The frozen object is a
**non-custodial Hosotani** Higgs (A_y = θ_H/(2πR_γ)·T_H, T_H∈su(2)_L, the doublet tensored in by hand, **SO(4)
eliminated**), so the W/Z mass term descends from the **commutator** `Tr|[A_μ, A_y]|²` — and C_W=C_3 is **not**
read out of the frozen data: with no custodial symmetry to protect it, it must be earned from the VEV-weighted
commutator overlaps `Tr|[T^a, T_H]|²` (a=1,2,3 weighting equally).

**ρ_tree=1 is therefore CONDITIONAL-DERIVED on three un-discharged premises:** (i) **C_W=C_3** via the Hosotani
commutator overlaps — a **newly-named owner-artifact** `R3_hosotani_commutator_overlap` (the prior passes
mis-cited the co-normalization; *this* is the real load-bearing blocker); (ii) **no-second-VEV** (R8-gated,
needs the m₀(p,q) zero-weight table); (iii) **KK-Schur ≤ 10⁻⁴** (dimensional hint ~10⁻²⁹; the l=0→1 / n=0→1
overlap not computed).

**R3: BLOCKED. GATE: SG-5 OPEN (unchanged). No status was ever upgraded.** The net is precision, not closure: blocker #2
dissolved for ρ, the det=0 leg DERIVED-GIVEN-E, and the remaining blocker sharpened from "the whole
co-normalization" to one specific commutator-overlap certificate. ρ₀=1.00038±0.00020 used as a post-hoc
falsifier only — input to nothing.

## §8. The commutator overlap COMPUTED — ρ is physical but UNDETERMINED (BLOCKED, sharper); 2026-06-25

§7's load-bearing artifact `R3_hosotani_commutator_overlap` has now been **computed** and adversarially verified
(two independent routes + an independent re-derivation; reproducer `SG5_R3_RHO_REPRODUCER.py`, all checks PASS to
machine precision). The result sharpens the blocker one level deeper and is the **resolved** R3 disposition.

**What was computed.** The VEV-weighted commutator overlaps are, exactly,
$$O_a=\big\langle \mathrm{Tr}\,|[T^a,T_H]|^2\big\rangle = \tfrac12\big(|h|^2-h_a^2\big),\qquad T_H=\textstyle\sum_b h_b T_b,$$
so the tree W/Z mass matrix gives
$$\rho_{\rm tree}=\frac{O_1+O_2}{2\,O_3}=\frac12+\frac{h_3^2}{h_1^2+h_2^2}\ \in\ \big[\tfrac12,\infty\big),
\qquad \rho=1\iff 2h_3^2=h_1^2+h_2^2\ \text{(a 54.74° cone)}.$$

**The decisive new fact — V_Hos is exactly FLAT in the Wilson-line direction.** The doublet holonomy eigenvalues
are $\pm\theta_H/2$ **independent of the direction** of $T_H$ (the KK weight-multiset is rotation-invariant), so
$V_{\rm Hos}(\theta_H,\text{direction})=V_{\rm Hos}(\theta_H)$: the one-loop potential **selects no direction.**
ρ=1 (the cone) is one ordinary, **non-preferred** level set, not a minimum.

**ρ is physical, but its value is undetermined.** The EW-neutral 3-axis (T₃) is **pinned** by the frozen Z₆
fermion-charge table (Q=T₃+Y must reproduce the observed charges), so ρ is a genuine gauge-invariant observable
(invariant under the residual Q-preserving U(1)₃) — **not** a gauge artifact. But because $V_{\rm Hos}$ is flat in
the tilt of $T_H$ relative to that pinned axis, the value of $h_3$ — hence ρ — is a **flat modulus**, undetermined
over $[\tfrac12,\infty)$. The custodial SO(4)/SO(5) coset that would force the cone was **eliminated**; there is
**no tree-level su(2)_L-anisotropic term** in the frozen data to lift the flat direction.

**Two routes, one missing datum.** Route A (numerical V_Hos) → BLOCKED (tilt physical, value flat). Route B
(structural) → leans ρ=1 by the lone-doublet accidental custodial, but only **conditional** on two embedding
facts (P1∧P2) *not present in the frozen corpus*. They agree on every computed fact and disagree only on
disposition, and the disagreement reduces to a single owner-artifact: **whether the frozen γ-cycle / Z₆ embedding
pins the EW-neutral 3-axis independently of $T_H$.** Per the reconciliation rule (material disagreement →
BLOCKED) and the core principle (**ANCHORED ≠ DERIVED**; a conditional gauge-equivalence lean is not a
derivation), the disposition is **BLOCKED**.

**Status — resolved: R3 BLOCKED (sharper). ρ=1 is neither derived nor falsified — it is UNDETERMINED.** The
frozen geometry, as given, **does not fix ρ.** Closing it requires one of: the γ-cycle/Z₆ orthogonality
certificate (does the 3-axis pin independently of $T_H$?), or the two-loop Yukawa/g′-corrected $V_{\rm Hos}$
(where $y_t\neq y_b$ and $g'\neq0$ first break the adjoint isotropy). **GATE: SG-5 OPEN. No status was ever upgraded.** We make
**no** claim the model predicts ρ=1, and introduced **no** parameter tuned to the measured value;
ρ₀=1.00038±0.00020 was an input to nothing.