# SG-8 — Flavor Closure (Yukawas / CKM / PMNS, GUT Gate 8): CLOSURE RESULT

> **Honest ceiling.** Serious candidate / partial unification **signal** — **NOT validated**. Gate status:
> **OPEN** (by the least-closed-residual rule). The genuine win is real and scoped; the gate does **not** close.
> **No status was ever upgraded.** Frozen branch `dcc66f1b2685` / `a5b1e6f9d951` **READ-ONLY**. Independently verified
> (adversarial pass + cross-check against the dossier). Updated 2026-06-25.

## §1. The genuine win — the ratio/mixing/phase leg is DERIVED-GIVEN-E

Given the frozen branch, the **selected** F⁺ flavor chamber, and the anchor set {E, α_i(M_Z), y_t, |V_us|}, the
following are **DERIVED-GIVEN-E**:

- **Within-sector mass RATIOS** as forced powers of κ = e^(−π√3): m_t/m_c = m_c/m_u = e^(π√3) ≈ 231,
  m_b/m_s = m_s/m_d = e^(2π√3/3) ≈ 38.5 (the sector scale N_s **cancels** in the ratio — not smuggled in).
- **δ_CKM = −2π/3** from the order-three holonomy (read, not fit).
- **All CKM & PMNS mixings + the Jarlskog J** from a **single chamber angle θ_F** (calibrated by the |V_us|
  anchor): |V_cb| 0.005σ, J 0.21σ; PMNS angles + δ_CP^ℓ from the second-cycle Berry phase.
- **Diagonalization-not-insertion** (the anti-fitting firewall): V_CKM = U_u†U_d is the misalignment of two
  frozen diagonalizations, **not** an inserted unitary; family-level normalizations N_{s,a} are **structurally
  forbidden**. This is what makes the chamber a *mechanism*, not a list of fitted numbers.

## §2. The five mandatory caveats (these ride with the DERIVED-GIVEN-E label, always)

1. **Scoped to RATIOS + MIXINGS + PHASES + J only — NOT absolute masses.** The geometry supplies the
   *structure*; the absolute sector scales (m_b, m_τ, Δm²) are **measured-but-irreducible** and N_d/N_e/N_ν are
   **fitted** calibration inputs — there is **no** N_d = f(N_u) relation.
2. **δ_CKM agreement is BAND-RELATIVE ~0.79σ** against a declared ~10% structural band — **not** vs raw PDG
   ±1.5° (the +60.0° vs 65.5° pull is **~3.7σ** against the raw error). Both numbers must be stated.
3. **m_u carries a forced ~4.4σ prediction-vs-data tension** (m_u/m_t = κ² once y_t pins N_u; **no axiom can
   rescue it** — it is forced, not fit). This is the leg's **sharpest falsifier** and is disclosed as such.
4. **"GIVEN-E" and "given the SELECTED chamber" are part of the label.** given-E ≠ derivation of E; selection ≠
   derivation; the chamber is selected-inside-a-declared-category (R1/R9 uniqueness not proven). The leg is
   DERIVED-GIVEN-E **relative to the three named axioms** (AXIOM-CLOSED ≠ proven).
5. **M_R is UNKNOWN / OPEN** — the absolute neutrino-scale leg is not closed.

## §3. The axiom floor (value-free; verified not relabels) + what was refused

**Banked (Reduced to Axiom, value-free, AXIOM-CLOSED ≠ proven):**
- `AXIOM-MODULAR-FIXED-POINT` — τ = ω, the order-three modular fixed point of the F⁺ Cartan torus (a symmetry
  statement; axiom-closed, **not** a uniqueness theorem).
- `AXIOM-LEXMIN-LADDER` — the action ladders from root system + lex order (axiom-closed, **not** DERIVED; full
  rational A₂/Ã₂ uniqueness unproven).
- weak `AXIOM-FPLUS-LOCK` — the chamber's **structural** data is flavor-blind; the 5–6 calibration scales are
  **openly conceded**, not relabeled geometric.

**Refused (correctly — refusing an axiom for an unrun computation is not a relabel):**
- `AXIOM-SECTOR-SCALE-INFLOW` (R2) — Route B (spin-c inflow) not computed; renaming the scales geometric = RELABEL_FAIL.
- `AXIOM-MR-IS-MU` (R3) — the M_R inversion/coincidence test has not been run; no value/formula/hash.

## §4. Per-residual ledger

| Residual | Disposition | How |
|---|---|---|
| R1 — F⁺ chamber lock | **CLOSED (weak)** | weak `AXIOM-FPLUS-LOCK` (structural data flavor-blind; scales conceded). Strong "whole chamber derived" stays **OPEN** (no witness). |
| R2 — sector scales N_d/N_e/N_ν | **OPEN** | fitted calibration inputs pinned to m_b/m_τ/Δm²; no N_d=f(N_u). Axiom **refused**. |
| R3 — seesaw M_R | **BLOCKED/OPEN** | no lab witness; pinning via η_B = tuning to the known answer (excluded). Axiom **refused**. |
| R4 — within-sector ratios | **CLOSED** | DERIVED-GIVEN-E; forced κ-powers, N_s cancels. |
| R5 — absolute masses m_b, m_τ, Δm² | **EXPORTED** | measured-but-irreducible anchor-consistency diagnostics. |
| R6 — m_u ~4.4σ | **OPEN (falsifier-grade)** | forced m_u/m_t = κ²; no axiom rescues it. The sharpest falsifier. |
| R7 — J.6/K.5 numeric harness | **BLOCKED** | CSVs + reproduce_all.py not mounted; pulls unverifiable by re-execution. |
| R8 — τ=ω modular fixed point | **CLOSED** | `AXIOM-MODULAR-FIXED-POINT` (value-free; axiom-closed not a theorem). |
| R9 — lex-min ladder | **CLOSED** | `AXIOM-LEXMIN-LADDER` (root system + lex order; axiom-closed not DERIVED). |
| R10 — PMNS octant / δ_CP^ℓ | **EXPORTED (Diagnostic)** | 4.60σ vs upper-octant; DUNE/JUNO-gated falsifier. |
| δ_CKM, mixings, J | **CLOSED** | DERIVED-GIVEN-E via single θ_F + order-three / Berry holonomy. |

## §5. The live blockers + the fail-closed forcing harness (continuation path)

Four rows decide whether SG-8 can ever move past OPEN: **R2, R3, R6, R7.** A **fail-closed forcing packet**
(`SG8_R7_R3_R6_forcing_packet`, published) was built and **independently verified to fail-closed** — its three
verifiers correctly report `BLOCKED_MISSING_ARTIFACTS` (R7), `BLOCKED_MISSING_INPUTS` (R3),
`BLOCKED_MISSING_TOP_INPUT_OR_TRANSPORT` (R6) with **no false PASS**. Dropping the real files into
`certificates/G09_flavor/` forces each row to VERIFIED / REFUTED / sharper-OPEN with no narrative wiggle room.
The continuation artifacts needed: the R7 CSVs + `reproduce_all.py`; the R3 M_R-inversion inputs (N_ν, O_ν
eigenvalues, Δm² convention, M_U, R₀⁻¹); the R6 frozen R1.7 transport recipe (hash `f531205a9159`).

## §6. Honest gate roll-up

**SG-8: OPEN** (least-closed residual). The ratio/mixing/phase **leg** is DERIVED-GIVEN-E (with the §2 caveats);
the gate is held OPEN by R2/R6 (OPEN) and R3/R7 (BLOCKED). Axiom floor = {AXIOM-MODULAR-FIXED-POINT,
AXIOM-LEXMIN-LADDER, weak AXIOM-FPLUS-LOCK}. No fitted number (N_d/N_e/N_ν, M_R, m_u) was renamed geometric; no
new axiom banked for an unrun computation. **No status was ever upgraded.** Serious candidate / partial-unification signal —
**NOT validated.**