SG-5 — Electroweak Embedding / EWSB (GUT Gate 5): CLOSURE RESULT — rendered package. Rendered from SG5_ELECTROWEAK_CLOSURE_RESULT.md; frozen technical content unchanged by rendering.

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

Ratified board status (2026-07-08). On the current gate board SG-5 (electroweak embedding / EWSB) is RESOLVED +0 (DERIVED-GIVEN-E): Q=T₃+Y and SU(2)L×U(1)Y→U(1)em are derived given the frozen spectrum, vEW is carried as the second measured anchor, and ρtree = 1 drops straight out of the geometry — the doublet Wilson-line hinge, with no hidden custodial symmetry to make it automatic, so the measured ρ₀ = 1.00038±0.00020 is a genuine head-to-head confrontation the geometry passes — and the hierarchy / θH* leg below is shown openly as a named residual beside that reached terminal (tracked in the SG-5B export ledger). The analysis below is the frozen 2026-06-25 attack vintage, which governs those residuals by the conservative least-closed-residual rule and therefore records the gate roll-up as “OPEN.”

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 → independent 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:

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 falsifiable prediction). Until then: do not bank ρ=1 as closure.

§3. What stays OPEN (gate roll-up)

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 verification audit (wmtzxcmts) found it PROMOTION-SAFE (all five falsification 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:

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: status_upgrades: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; status_upgrades: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 independently 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 independently 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.