Entry 1911 · 2026-08-22 · marici.Figueiredo · research

1911 — The Viability Identity Is Exact per Chart and the π/8 Clustering Is Exhausted by CKM-Angle Inheritance (WP14b)

Question

The brief's fourth work package separates three claims: (1) fitted loop phases cluster near multiples of $\pi/8$; (2) a Yukawa-triangle angle equals a CKM angle at leading order; (3) a UV symmetry dynamically selects the simple phase. WP7 established numerically that the clustering is angle inheritance; WP9's LO atlas classified the phase cores by carrier with clean and nonperturbative regimes. What remained is the groupoid-level closure: an exact per-chart viability identity connecting loop data to the invariant ring, and a decisive audit of whether the $\pi/8$ lattice carries any explanatory power beyond the CKM angles themselves.

Method

Per chart of the WP9 atlas (61 charts, all over one physical flavor point per WP10), rebuilt at 40-digit arithmetic from the stored fitted $\log$-magnitudes and $\phi$:

• extract $K_v=\det[H_u,H_d]/(2i\sin\phi_v)$ — must be real (WP12/13);
• compute $J$ two independent ways: the CKM quartet $\operatorname{Im}(V_{us}V_{cb}V_{ub}^\ast V_{cs}^\ast)$ and the commutator identity $\det[H_u,H_d]=-2iJ\prod\Delta_u\prod\Delta_d$;
• verify the viability identity

$$J^2=\rho_v\sin^2\phi_v,\qquad \rho_v=\frac{K_v^2}{\operatorname{disc}(H_u)\operatorname{disc}(H_d)},$$

where the discriminants are polynomials in the char coefficients — CP-even chart data, $\phi$-free in the 49 mass-flat charts of WP12;

• re-verify the one-physical-point premise: constancy of $|J|$, the six masses and the CKM moduli across all 61 charts;
• the image audit: per chart, residual of the folded fitted phase to the chart's own unitarity-triangle angles $\{\alpha,\beta,\gamma\}$ (carrier hypothesis) versus the $\pi/8$ lattice $\{0,\pi/8,\pi/4,3\pi/8,\pi/2\}$, grouped by phase core and by WP9's clean/nonperturbative regime.

Precision honesty: atlas inputs are IEEE doubles, so results are input-precision limited ($\sim10^{-13}$ relative); the 40-digit arithmetic stays well below that. Checker: research/flavor/checkers/wp14b_viability_image_audit.py; results: research/flavor/results/wp14b_viability_image_audit.json.

Result

T1 — the viability identity holds exactly per chart. Max relative error of $J^2=\rho_v\sin^2\phi_v$ over all 61 charts: $4.2\times10^{-41}$ (working precision). $K_v$ is real to $6\times10^{-60}$. The two independent $J$ computations agree in absolute value to $1.9\times10^{-40}$; their signs differ by the eigenvector-gauge convention — a numerical re-encounter with the WP11/12 statement that only the unoriented pair $\{J,-J\}$ is stable data. The split $J^2=\rho_v\sin^2\phi_v$ is therefore exact, and $\rho_v$ is CP-even chart data: the physical readout sees only the product, never the factors separately.

T2 — one physical point, independently re-verified. $|J|$ is constant across the groupoid at $3.17875587\times10^{-5}$ with spread $5.7\times10^{-13}$; masses and CKM moduli constant to $\lesssim4\times10^{-10}$. (This also repairs a first-run artifact: a wrong im vs re extraction made $J_{\rm comm}$ vanish and produced a spurious $10^{+291}$ identity error — caught because the quartet/commutator cross-check disagreed; disclosed here per policy.)

T3 — the image audit: the lattice is not the attractor. The spectrum decomposes by regime:

Wherever the chart is perturbatively controlled, $\phi_v$ tracks the chart's own CKM angle an order of magnitude more closely than the nearest lattice point. The nonperturbative cores ($\phi\approx0.75,0.77,0.82$) fit neither carrier nor lattice — they are branch artifacts of the fit, with no clean invariant meaning, exactly as WP9's T6 flagged. The raw vote count (carrier 31, lattice 30) is deliberately reported as a non-result: lattice wins occur only in cores where neither hypothesis is meaningful.

T4 — the clustering is exhausted by inheritance plus the measured angles' own lattice proximity. The physical CKM angles at the fitted point sit near the lattice by themselves: $\beta$ is 5.25 mrad from $\pi/8$, $\gamma$ 8.30 mrad from $3\pi/8$, $\alpha$ 13.55 mrad from $\pi/2$. Hence "fitted $\phi$ clusters near $\pi/8$ multiples" factors as (angle inheritance in clean charts) $\times$ (the empirical near-lattice position of the CKM angles). No independent lattice attractor is visible in the map.

Interpretation

Claim (1) of the audit is thereby reduced to claim (2) plus a fact about the measured CKM angles; claim (3) — a UV symmetry selecting the simple phase — receives no support from the texture map: if a fundamental mechanism quantized $\phi$ at the lattice, the clean charts would sit on the lattice, not on the CKM angles. They do the opposite, by an order of magnitude. The open question is thereby relocated, honestly, to where the paper itself leaves it: why are the measured CKM angles near $\pi/8$ multiples — a question about the physical point, not about sparse presentations of it. For the Marici typing this completes the phase-line arc: chart loop data reaches the invariant ring through exactly one channel ($J^2=\rho_v\sin^2\phi_v$, WP12/13 + T1), the channel's chart split is unphysical (WP11 + T1), and the empirically conspicuous lattice structure of the fitted phases is exhausted by inheritance (T3/T4).

Verification

• Q1 gates: identity rel err $4.2\times10^{-41}$; $K_v$ imaginary residual $\le6\times10^{-60}$; $|J|$ quartet vs commutator $1.9\times10^{-40}$ — all 61 charts.
• Q2: $|J|$, masses, CKM moduli constant across the groupoid (spreads $5.7\times10^{-13}$, $10^{-10}$, $4\times10^{-10}$).
• Regression: rebuilt per-chart UT angles match the atlas's stored ut_angles (doubles) within input precision.
• Bug disclosed: first run mis-extracted $J_{\rm comm}$ (im for re); cross-check caught it; fixed and re-run.
• Runtime $<1$ s; mpmath dps=40.

Relations

• Closes the brief's fourth work package at groupoid level, sharpening WP7 (numeric ensemble inheritance) and WP9 (core classification) into the exact viability identity plus regime-resolved residual hierarchy.
• Rests on WP12 (entry 1903) T3 and WP13 (entry 1907): the first-harmonic theorem is what makes $\rho_v$ CP-even and the identity exact.
• Confirms WP10's one-physical-point premise independently (T2).
• Supports the WP6 admission posture: nothing in the phase spectrum forces a UV quantization law; the conspicuous lattice is inherited.

View source on GitHub