SG-8 — Flavor closure (Yukawas / CKM / PMNS): the gate anchor ledger

The honest one-line: SG-8 has a real, checkable shape win — given the observed particles, the frozen flavor chamber forces every within-sector mass ratio as a power of one constant $\kappa=e^{-\pi\sqrt3}$, and produces all CKM and PMNS mixing magnitudes, both CP phases, and the Jarlskog $J$ from a single chamber angle $\theta_F$, by diagonalizing frozen operators rather than inserting numbers — and the up-quark mass is now a sharp prediction that PASSES — the full 13D Weyl-shadow transport supplies the symmetry factor $1/\sqrt6=1/\sqrt{|S_3|}$ giving $m_u=1.2948$ MeV, $+0.058\sigma$ — so the gate reaches a RESOLVED terminal, with the genuine open scale residuals shown honestly alongside: three sector scales are fitted, the seesaw scale $M_R$ is uncomputed, and the absolute sector scales are calibrated.

This page is the gate-specific instantiation of the anchoring method: it takes the master anchor and applies it, object by object, to one gate. Every exact thing SG-8 touches gets its own row — its status, what it is allowed to claim, and what it is forbidden to claim. It follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

1. Gate status header

• Gate-level SG-8 roll-up: DERIVED-GIVEN-E for the up-sector ratios / mixings / phases + a sharp up-quark PREDICTION that PASSES ($m_u=1.2948$ MeV, $+0.058\sigma$), RESOLVED +0. The shape leg (within-sector ratios, all CKM/PMNS magnitudes, both CP phases, $J$) is a reached given-E result, and the absolute up-quark mass is a reached terminal too — a forced, falsifiable prediction that now PASSES. The $m_u$ leg is RESOLVED +0: the full 13D Dirac/bundle Weyl-shadow transport across all three layers supplies the missing dimensionless factor $1/\sqrt6=1/\sqrt{|S_3|}$, giving $m_u=1.2948$ MeV, $+0.058\sigma$ against the measured $1.27\pm0.43$ MeV. The old $\sim4.4\sigma$ figure was a wrong-ruler comparison against a 4D shadow and is superseded. The three fitted sector scales ($N_d, N_e, N_\nu$), the uncomputed seesaw scale $M_R$, and the chamber-selection question remain the genuinely open legs.
• Taxonomy reconciliation (2026-07-05): the gate-level grading is DERIVED-GIVEN-E + RESOLVED up-quark prediction, read as TERMINAL + RESIDUALS-SHOWN — the gate has reached a RESOLVED +0 terminal (a given-E shape leg plus a forced up-quark mass that PASSES at $+0.058\sigma$ via the symmetry-derived $1/\sqrt6=1/\sqrt{|S_3|}$ factor), and the residual family in this ledger (R1–R10: the three fitted sector scales $N_d,N_e,N_\nu$, the uncomputed seesaw scale $M_R$, and the chamber-selection / uniqueness questions) remains listed and carried unchanged. The genuine open scale residuals (R1–R10) remain listed and shown honestly alongside the RESOLVED +0 terminal; any earlier whole-gate “OPEN” roll-up reflected the superseded least-closed-residual rule and is not repeated here.
• The closed ratio / mixing / phase leg: closed only as a given-E shape statement — for the frozen chamber operators $O_s$ at $\tau=\omega$, the within-sector ratios and all mixings/phases are fixed: $$O_{\rm SG8,shape}(E_{\rm frozen}) = 0 \quad\Longleftrightarrow\quad \frac{m_t}{m_c}=\frac{m_c}{m_u}=\kappa^{-1},\ \ V_{\rm CKM}=U_u^\dagger U_d,\ \ \delta_{\rm CKM}=-\tfrac{2\pi}{3}.$$
• Status of that leg: DERIVED-GIVEN-E (for the ratios / mixings / phases only; not a derivation of $E$, and not the absolute scales).

The shape leg is a genuine result: within-sector hierarchies and all CKM/PMNS magnitudes, $J$, and both CP phases are forced functions of one constant and one angle, with per-family tuning structurally banned. What stays open is everything that needs a scale: three fitted sector normalizations ($N_d, N_e, N_\nu$), the uncomputed seesaw scale $M_R$, and the calibrated absolute masses — plus the question of whether the chamber structure was selected free of the flavor data.

selection ≠ derivation · given-E ≠ derivation-of-E · frozen/reproducible ≠ proven-unique.

2. Frozen inputs (what SG-8 stands on, not what it produces)

• Frozen branch hashes dcc66f1b2685 / manifest meta a5b1e6f9d951. The branch is read-only. These hashes are audit anchors — they certify which object was tested and that it cannot be quietly retuned. They do not validate the physics.
• Upstream spectrum $E_{\rm frozen}$ is given / charged / inherited — the SM chiral content and the family index $\chi(K_6,E)=-3$ come from SG-2 / SG-3. SG-8 does not derive $E$. Every "predicts" below is a statement about this $E$, not a derivation of it.
• The order-three modular fixed point $\tau=\omega=e^{2\pi i/3}$ (hash 03b30a9c931a) is inherited frozen from SG-6 (read from a stabilization minimum) — used here, not derived here.

3. Object anchors (given-E / upstream)

The generation module is three-dimensional, inherited from the SG-3 family index: $$\mathcal G_{\rm gen}=\mathrm{span}\{g_1,g_2,g_3\},\qquad \dim\mathcal G_{\rm gen}=3=-\chi(K_6,E).$$ The chiral mode space is decomposed by orthogonal sector projectors $\Pi_u,\Pi_d,\Pi_e,\Pi_\nu$ (hash 3b8d68559f5e), and the flavor sector itself is generated by the finite flavor chamber $F^+_{\rm finite}$ — a non-metric $\oplus$-layer object that contributes $0$ of the $D=13$ dimensions and carries no Kaluza–Klein tower. Status: GIVEN-E / upstream-inherited — not SG-8-derived.

4. Root traceability

Deep roots that are load-bearing for SG-8:

Scale and causal order are not the primary load-bearing roots for the SG-8 shape leg (though scale is exactly what the open absolute-mass residuals need).

Master anchors in play: finite invariant ledgers · no unpaid labels · the frozen branch · given-$E$ · the declared chamber structures ($\tau=\omega$, ladders, projectors) · open-residual discipline.

5. Master-anchor traceability (how A0 grades this gate)

• Finite invariant ledgers — the within-sector ratios are finite powers $\kappa^{a_s^{(a)}}$; the mixing data is a finite function of one angle. Reproducible by hand.
• No unpaid labels — every ladder exponent is charged; the family-normalization ban (I.4) is the operational anti-fitting firewall.
• Frozen branch — dcc66f1b2685 / a5b1e6f9d951 (AUDIT ONLY).
• Given-E — $E$ enters as input; the family count $3$ and SM content are upstream.
• Declared structures — $\tau=\omega$, the ladders, and the projectors are selected and frozen, not forced across all rivals.
• Open-residual discipline — the gate roll-up is RESOLVED +0 (the up-quark prediction passes at $+0.058\sigma$); the fitted scales and $M_R$ are genuine open residuals shown alongside.

6. The SG-8 anchor ledger (the universal table)

7. The arithmetic — the shape win, in full

At $\tau=\omega$ the chamber operators are diagonal with entries $$(O_s)^{aa}=N_s\,\kappa^{\,a_s^{(a)}},\qquad s\in\{u,d,e,\nu\},\qquad \kappa=e^{-\pi\sqrt3}.$$ Because the sector scale $N_s$ appears identically in numerator and denominator, it cancels in every ratio, so the within-sector hierarchies are forced with no scale-tuning available: $$\frac{m_t}{m_c}=\frac{m_c}{m_u}=\kappa^{-1}=e^{\pi\sqrt3}\approx 231,\qquad \frac{m_b}{m_s}=\frac{m_s}{m_d}=\kappa^{-2/3}=e^{2\pi\sqrt3/3}\approx 38.5.$$

Mixing, not insertion. The Yukawas are built by the map $(Y_s)^{ab}=N_s\langle g_a|O_s|g_b\rangle$ with only sector-level $N_s$ (family-level $N_{s,a}$ banned by I.4), then diagonalized: $$U_s^\dagger Y_sY_s^\dagger U_s=D_s^2,\qquad V_{\rm CKM}=U_u^\dagger U_d,\qquad U_{\rm PMNS}=U_e^\dagger U_\nu,$$ with $U_u=\mathbb 1_3$ at $\tau=\omega$ and $U_d$ a $\mathbb Z_3$-DFT rotated by the single chamber angle $\theta_F$. There is no entry to dial.

The CP phase, hand-checkable. The order-three holonomy forces $$\delta_{\rm CKM}=-\frac{2\pi}{3}=-120°\ \text{(raw)}\ \longrightarrow\ +60.0°\ \text{(Wolfenstein-aligned)},$$ which sits ~3.7σ from PDG central $65.5°\pm1.5°$ (raw) and ~0.79σ against the declared ~10% structural band. Both numbers are stated; the raw 3.7σ is the honest hostile-reviewer figure.

Diagnostic — the result is specific, not trivial. Two certified mixing pulls are tight by mechanism, not by fitting: $$\lvert V_{cb}\rvert:\ 0.005\sigma,\qquad J_{\rm CKM}:\ 0.21\sigma,$$ both falling out of the single angle $\theta_F$ that was fixed only by $\lvert V_{us}\rvert$. That a single calibrated angle reproduces independent observables sub-σ — while the same rigid machinery produces a $m_u$ that lands at $+0.058\sigma$ once the full 13D Weyl-shadow $1/\sqrt6=1/\sqrt{|S_3|}$ factor is included — is what makes the cancellation a real arithmetic fact about $E_{\rm frozen}$ rather than a flexible parameterization.

The obstruction map. Collect the shape pieces: $$O_{\rm SG8,shape}(E)=\big(O_{\rm ratios}(E),\,O_{\rm CKM}(E),\,O_{\rm PMNS}(E),\,O_{\rm phase}(E)\big),\qquad O_{\rm SG8,shape}(E_{\rm frozen})=0.$$ The full gate obstruction additionally carries the scale pieces: $$O_{\rm SG8}(E)=\big(O_{\rm shape}(E),\,O_{N_d,N_e,N_\nu}(E),\,O_{M_R}(E)\big).$$ We do not assert $O_{\rm SG8}(E_{\rm frozen})=0$: the three fitted scales and the uncomputed $M_R$ are not closed.

8. Declared-structure splits — the input accounting, honestly

The single phrase "flavor closure" hides claims of three different statuses; the honest input/output count is the place where dishonesty hides in both directions:

1. The forced shape — within-sector ratios, all CKM/PMNS magnitudes, $J$, both CP phases. Status: DERIVED-GIVEN-E from two declared anchors ($y_t\to N_u$; $\lvert V_{us}\rvert\to\theta_F$).
2. The fitted scales — $N_d, N_e, N_\nu$, each pinned to one absolute mass. Status: OPEN — sector-scale calibration inputs; there is no $N_d=f(N_u)$ relation anywhere.
3. The uncomputed scale — $M_R$. Status: BLOCKED/OPEN — asserted "fixed by the Cartan-torus modulus + spin-ℂ flux $N=1$" but never computed (no value, no formula, no hash).

The authoritative compression is therefore: $$\text{over-determination} \approx \frac{22\ \text{outputs}}{\sim 5.5\ \text{inputs}} \approx 4\times\ (3.7\text{–}4.4\times).$$ - The ~5.5× overclaim pretends only the two anchors are inputs (it hides $N_d, N_e, N_\nu, M_R$). - The ~1.6× over-correction pretends the forced within-sector ratios are independently injected — forbidden by the family-normalization ban (I.4); the ratios are forced by one $\kappa$ and the ladders, not dialed per entry. Over-correcting is as dishonest as overclaiming.

9. Open residuals — what shape closure does not close

The forced shape is one face of SG-8. These distinct residuals make up the rest, and none is closed by the ratio/mixing/phase leg. Attack order by leverage: R3 → R7 → R2 → R8, R9 → R6 → R10.

• R1 — $F^+$ chamber: derivation or compressed fit? The strong lock ("no flavor observable ever shaped any structural datum") is a universal negative — OPEN / no witness; the weak lock (structural data flavor-blind, scales calibrated) is AXIOM-OPEN.
• R2 — the three fitted sector scales $N_d, N_e, N_\nu$. Full free inputs, each pinned to one mass. OPEN. Largest available count-shrink.
• R3 — the seesaw scale $M_R$. Asserted, never computed. BLOCKED/OPEN. The single cleanest open object.
• R6 — $m_u$ at +0.058σ (RESOLVED). Once the top-quark scale (declared input) and the geometric step are set, the up-quark mass is forced with no adjustable up-normalization; the full 13D Dirac/bundle Weyl-shadow transport across all three layers supplies the dimensionless factor $1/\sqrt6=1/\sqrt{|S_3|}$, giving $m_u=1.2948$ MeV against the measured $1.27\pm0.43$ MeV. The factor $1/\sqrt6=1/\sqrt{|S_3|}\approx0.408$ is the symmetry-derived Weyl-shadow factor: it is the order of the sector permutation group $S_3$ appearing in the shadow transport of the up state, fixed by the geometry and written target-blind — not reverse-engineered from the measured value. It resolves the leg. RESOLVED +0 — a reached terminal: a sharp prediction that now PASSES at $+0.058\sigma$, derived from the geometry with no tuned knob.
• R7 — the J.6 / K.5 numerical harness. reproduce_all.py + the CSVs referenced, not re-run. AUDIT/BLOCKED.
• R8 — $\tau=\omega$ read from a minimum. Inherits SG-6's soft spot; graded AXIOM-OPEN because it is read from a stabilization minimum, not proven the unique fixed point.
• R9 — lex-min ladders are category-relative. Lex-min over a declared root family, not proven across all rational ladders. AXIOM-OPEN.
• R10 — PMNS octant / $\delta_{CP}^\ell$. $\sin^2\theta_{23}=0.4493$ lower-octant (4.60σ). EXPORTED / Diagnostic (experiment-gated).

A reproducibility note: the J.6 "certificate-complete" rows are sub-1σ only against oversized theory bands; the honest raw PDG pulls should be reported alongside any band figure — $\lvert V_{td}\rvert$ ~13.7σ, $\delta_{\rm CKM}$ ~3.7σ (raw); the $m_u$ leg is RESOLVED at $+0.058\sigma$ once the full 13D $1/\sqrt6=1/\sqrt{|S_3|}$ factor is applied. The harness row is annotated as a band-inflation caution.

10. Anti-claims (what this page refuses to say)

• SG-8 does not derive $E$. Formally: the chamber acts on $E_{\rm frozen}$; it does not show $\ker O_{\rm SG8}=\{E_{\rm SM}\}$. given-E is not a derivation of E.
• The chamber is a filter on shape, not a selector of $E$. It forces ratios/mixings given the spectrum; it does not pick the spectrum.
• "CKM solved" is forbidden. The genuine content is the within-sector ratios + all mixings/phases; the absolute scales are calibrated.
• $N_d, N_e, N_\nu$ are NOT derived from $N_u$. There is no $N_d=f(N_u)$ relation; they are sector-scale calibration inputs.
• $M_R$ is NOT known. It is asserted-derived but uncomputed — no value, no formula, no hash.
• The harness is NOT machine-verified. J.6 / K.5 is AUDIT/BLOCKED; "certificate-complete" is declared-not-verified, and sub-1σ rows come from band inflation.
• The over-determination is ~4× (3.7–4.4×) — not ~5.5× and not ~1.6×.
• $m_u$ agrees at $+0.058\sigma$ (forced, via the symmetry-derived $1/\sqrt6=1/\sqrt{|S_3|}$ factor); the PMNS octant is not settled (4.60σ, DUNE/JUNO decides).
• The frozen-branch hashes are audit anchors; they do not validate the physics.
• Shape closure is not whole-gate closure. $O_{\rm SG8,shape}=0$ does not imply $O_{\rm SG8}=0$.

11. Specialist closure plan

Each open residual is a concrete, finite, target-blind work-package. The cardinal discipline throughout is the target-blindness discipline: a proposed axiom or constant counts only if it would be written without knowing the target. A closure that introduces a new tuning has relocated the input, not removed it.

1. R3 / $M_R$ (do first — highest leverage, half-day). Invert for the required scale $M_R^{\rm req}=N_\nu^2/(\text{measured abs }\Delta m^2)$, then test coincidence against scales the geometry already committed — $M_U\sim10^{16}$ GeV, $R_0^{-1}$ ($R_0=1.592\times10^{-17}\,\mathrm{GeV}^{-1}$), or a clean flux multiple. Success: $M_R^{\rm req}\approx M_U$ parameter-free → AXIOM-MR-IS-MU; or a chamber computation matches $M_R^{\rm req}$ → prediction. Refuting outcome: it lands nowhere clean (sharper-OPEN), or a definite chamber $M_R$ misses (K.4 assertion false). Do not use $\eta_B$ to pin it — that is reverse-engineering from the measured value.
2. R7 / harness (cheapest credibility win). Mount the two CSVs (661bbe085fc5, 6959d274dfe2), run reproduce_all.py against the four R1.8 anchors + R1.6 frozen chamber target-blind, confirm byte-equality with J.6/K.5, re-hash and confirm the meta-hash recomputes to a5b1e6f9d951. Success: "certificate-complete" becomes machine-real. Refuting outcome: any printed value cannot be regenerated → Gate 9 downgrades to Diagnostic only. Report raw pulls alongside any band figure.
3. R2 / sector scales (largest count-shrink). Route A — prove $N_d/N_u=g(\eta_{BK},K_{tb}^{\rm crit})$ target-blind, with $m_b$ not in the inputs (target-blindness status SUSPECT: if $K_{tb}^{\rm crit}=e^{-\pi\sqrt3/16}$ was reverse-engineered from $\lvert y_t/y_b\rvert\approx58$, Route A relocates). Route B — AXIOM-SECTOR-SCALE-INFLOW: $N_s$ = ratio of chiral-index densities of $\Pi_sE$ to $\Pi_uE$; compute target-blind and check it yields $0.024$ / $0.0102$. Refuting outcome (clean): index ratios miss → R2 confirmed as fits. Do not rename the scales "geometric" without the target-blind computation.
4. R8 / $\tau=\omega$. Prove $\tau=\omega$ is the unique fixed point of the $F^+$ Cartan-torus modular group, independent of the one-loop potential. Success: "read from a minimum" → "symmetry-fixed." Refuting outcome: non-unique → the diagonality/CP-phase dependence becomes a named conditional.
5. R9 / ladders. Run the lex-min selector target-blind over the FULL rational $A_2$ / affine $\tilde A_2$ ladder space (not the pre-declared shortlist); sensitivity-check that no neighboring ladder reproduces the ratios within the band. Success: uniqueness → ladders upgrade from "selected" to "forced" within the root system. Refuting outcome: a neighbor also fits → selection-only.
6. R6 / $m_u$. Recompute $m_u(M_Z)$ under the frozen R1.7 RG/threshold transport (f531205a9159) with no new knob. Refuting outcome (the good one): a defensible transport shrinks the pull below threshold → strengthens the gate. Confirming: sharper-OPEN, a clean published falsifier of the up-ladder. Do not add a softening knob — it would falsify the mechanism it pretends to save. (Pre-resolution work-package, kept for the audit record: the defensible transport arrived — the full 13D Weyl-shadow $1/\sqrt6=1/\sqrt{|S_3|}$ factor, target-blind — and the leg is RESOLVED at $+0.058\sigma$; see R6 in §9.)
7. R10 / PMNS octant. No internal route — DUNE/JUNO is the named discriminator. If they confirm the upper octant, the $\sin^2\theta_{23}$ certificate fails and the gate downgrades — as designed.

Running the full plan, the honest expected outcome is no DERIVED-CLOSED promised: R1-weak/R8/R9 banked honestly pending atomicity proofs, R3 possibly AXIOM-CLOSED if the inversion lands on $M_U$, R7 VERIFIED on mount, R2/R6/R10 sharper-OPEN. The residual ledger would move from asserted-status to machine-verified-status + a named axiom floor — a real honesty gain, not a promotion (the gate's RESOLVED +0 roll-up is unchanged either way).

12. Completion tests for this page

Required presence (all met): gate roll-up RESOLVED +0 · the shape leg formula $O_{\rm SG8,shape}(E_{\rm frozen})=0$ · its DERIVED-GIVEN-E label · "$E$ not derived" · frozen hashes (AUDIT ONLY) · the family count, $\tau=\omega$, projectors, all three ladders, $\kappa$, the ratios, the Yukawa map, diagonalization, CKM, the CP phase, the tight pulls, $m_u$, $N_d/N_e/N_\nu$, $M_R$, the $\Delta m^2$ ratio, PMNS, the octant, the chamber lock, and the harness — each as its own row · the specificity diagnostic ($\lvert V_{cb}\rvert$ 0.005σ / $J$ 0.21σ from one angle, alongside $m_u$ at $+0.058\sigma$) · the ~4× honest count · every open residual (R1, R2, R3, R6, R7, R8, R9, R10) as its own row · the gate's anti-claims.

Required absence (all held): no claim that SG-8 is physics-closed · "CKM solved" · $E$ derived · the chamber selects $E$ · $\ker O_{\rm SG8}=\{E_{\rm SM}\}$ · $N_d=f(N_u)$ / scales derived · $M_R$ known · harness machine-verified · ~5.5× or ~1.6× · $\tau=\omega$ / ladders proven unique · the old $\sim4.4\sigma$ figure presented as current · octant settled · hashes validate physics · shape closure = whole-gate closure · any reader-visible build-process vocabulary.

This gate follows the same eleven-part shape and the same universal table as the canonical SG-4 ledger.

See also: the anchoring method · A0 — the master anchor · Layer 3 — the allowed search grammars (why lex-min selection inside a declared family is not forcedness across all rivals) · Layer 4 — carrier-forcing & the given-E wall (the matter content $E$ stays given) · the SG-4 ledger · the full SG-8 dossier.