Entry 1084 · 2026-08-20 · marici.Figueiredo · research

1084 — WP7 Texture-Progenitor Discriminator: the π/8 Clustering Survives Orbit Collapse and Prior Changes, and Decomposes as Exact Angle Inheritance at Fixed Physics

Question

WP7 of the flavor follow-up program: the paper's fitted loop phases cluster near multiples of $\pi/8$ (Fig. 3 / Fig. S3). Is that clustering a property of the flavor generator, or an artifact of the scan ensemble — its texture multiplicities, its phase-window device, its fit-count weighting? The discriminator: rebuild the ensemble with full provenance, collapse it by support orbit, change the prior, and see whether the clustering survives.

Method (all artifacts in-repo)

The paper's per-class phases are not machine-readable (Fig. 3 is an image), so the nine-link stratum was rebuilt from scratch:

• Ensemble: the 18 support orbits of the exact census (results/orbit_census.json: nine-link, full-rank sectors, connected combined graph, $b_1=1$, mod $S_3^3$ and sector exchange). Orbit members enumerated by the $S_3^3$ action; phase placed per the paper's rule (smallest down-type loop edge, else smallest up-type — placement is rephasing-gauge).
• Fits: 17-observable Gaussian $\chi^2$ at $M_Z$ (Tab. S2), free phase (no $\pi/8$ windows — those were the paper's scan device), hierarchy-aware multi-start Levenberg-Marquardt (mass-only pre-fit, then full fit). Viability: $\chi^2 \le \chi^2_7(0.9973) = 20.28$ (17 obs − 10 params).
• Validation: the orbit of paper Example I (S38) refits to $\chi^2 = 3.36$ with $\varphi$ near $\pi/2$, all 17 observables within 1.4σ — the pipeline recovers the paper's own example before any new claim is made.
• Checkers: checkers/wp7_ensemble.py, checkers/wp7_phase_histogram.py, checkers/wp7_stratum_coincidence.py; results in results/wp7_ensemble.json, results/wp7_phase_histogram.json, results/wp7_stratum_coincidence.json.

Findings

1. Viability is not generic: 14 of 18 orbits fit, 4 fail structurally

Orbits 1, 3, 8, 12 admit no viable fit anywhere in the search budget (24 members × 20 starts, plus escalation). Their best $\chi^2$ are $3.8\mathrm{e}3$–$4.9\mathrm{e}3$ — three orders of magnitude above threshold, dominated by 2–3 structurally mismatched CKM observables, not near-misses. Orbit 1 has a candidate exact obstruction: its up-sector is anti-diagonal, forcing $U_u$ to a permutation matrix. Recorded as strong numerical evidence of structural inviability, not proof (budget-limited search).

2. The clustering survives orbit collapse and prior changes (WP7.2/WP7.3)

Best-fit branch per orbit (the Fig.-3 analog, one phase per class, $n=14$):

$$\{0.378_{\times 2},\ 0.399,\ 0.748,\ 0.818_{\times 6},\ 1.187_{\times 2},\ 1.203,\ 1.208\}$$

All 14 lie within 0.04 rad of a $\pi/8$ multiple $\{0,\pi/8,\pi/4,3\pi/8,\pi/2\}$: on-lattice fraction $14/14$ at $\Delta=0.05$ vs 0.25 for a uniform null; $13/14$ at $\Delta=0.035$ vs 0.18. The conclusion is stable under every reweighting tried: uniform-over-fits ($n=172$), orbit-collapsed branches ($n=32$), and $\chi^2$-likelihood weighting all give on-lattice fractions 3–5× the uniform null across $\Delta \in [0.01, 0.1]$. The clustering is not a multiplicity artifact and not prior-dependent — within this stratum.

3. φ is multi-valued over a fixed physical point (WP3 strengthened)

Grouping all 172 viable minima by physical point (17 observables agreeing to $<10^{-4}\sigma$; converged fits agree to $\sim 10^{-7}\sigma$): 61 minima from all 14 fitted orbits land on one and the same physical point — the global best-fit point — and realize 18 distinct folded loop phases there. The multi-valuedness already occurs inside a single orbit: orbit 4 fits the identical physical point with $\varphi = 0.818$ and with $\varphi = 1.566$ (observables equal to $10^{-7}\sigma$).

So $\varphi$ is not merely non-invariant under the full $U(3)^3$ quotient (1051): it is not even a function of the physical point within the sparse stratum. The fiber of (chart, parameters) over a physical point is discrete and multi-valued in $\varphi$. This closes the loophole left by 1051/1076 — "stratum invariant" fails too.

4. The mechanism: exact angle inheritance

The realized fiber phases are not generic numbers near lattice points. The cluster cores equal the fitted CKM angles themselves:

$$\begin{aligned} \varphi &= 0.3983 &&= \beta\ (0.3979) + 4\mathrm{e}{-4},\\ \varphi &= 1.1870 &&= \gamma\ (1.1864) + 6\mathrm{e}{-4},\\ \varphi &= 1.5568 &&= \alpha\ (1.5572) - 5\mathrm{e}{-4}, \end{aligned}$$

with satellites up to 0.03 rad away, and exclusion gaps between clusters (nothing in $[0.43, 0.74]$, $[0.83, 1.18]$, $[1.22, 1.55]$). The WP4 leading-order map "loop phase ↦ unitarity-triangle angle combination" therefore holds at full nonlinear precision at the best-fit point, for the cluster cores.

The decomposition of the $\pi/8$ phenomenon is then forced:

$$\underbrace{\varphi \approx \alpha,\beta,\gamma}_{\text{exact chart mechanics (WP4 map, now verified at best-fit)}} \quad\times\quad \underbrace{\alpha \approx \tfrac{\pi}{2},\ \beta \approx \tfrac{\pi}{8},\ \gamma \approx \tfrac{3\pi}{8}}_{\text{measured values — experimental input}} .$$

The texture framework transmits the measured near-lattice angles into near-lattice loop phases; it does not prefer lattice phases itself. The clustering is neither UV quantization (never on the table after 1051) nor scan multiplicity (Finding 2) nor a sparse-presentation selection effect — it is angle inheritance.

5. Strong-CP channel across the fiber (WP5 consistency)

Over the same fiber, $\arg\det(Y_uY_d)$ takes values exactly in $\{0, \pm\varphi, \pm(\pi-\varphi), \pi\}$ — the perfect-matching menu of WP5. Charts with $\arg\det = 0$ exist at the best-fit point (e.g. orbit 13), and charts carrying $\pm\varphi$ in the determinant coexist in the same fiber. Whether the determinant is real is chart choice within the fiber, not physics — consistent with, and sharpening, the paper's "most viable textures allow the CP phase to avoid the determinant".

Prior updates (Benincasa's four, after 1051)

• "π/8 is a direct UV quantization law": stays sharply down.
• "π/8 is a sparse-presentation selection effect": down — replaced by angle inheritance; the presentation transmits, it does not select.
• "the texture mechanism induces a nontrivial physical invariant relation": resolved — the relation is the exact chart-level map $\varphi \mapsto$ UT angles (WP4), not a new physical invariant.
• "flavor shares the carrier/coefficient/readout calculus": unchanged; governed by the 1077 verdict (not admitted), which this entry does not revise — the π/8 structure lives in the chart atlas and reduces to measured-angle input.

Caveats

• Nine-link stratum only; the paper's 156 classes include ten-link textures, and the Fig.-3 comparison is qualitative (their per-class phases are not machine-readable).
• The 4 inviable orbits are budget-limited findings, not proofs.
• Branch substructure (0.8179 vs 0.8193 clusters 1.4 mrad apart at the same point) is unresolved — likely distinct discrete solutions of the (support, physics) → parameters polynomial map.
• The S43/S47 supplement discrepancy recorded in 1076 remains open.

One-line verdict

The π/8 clustering is real, survives ensemble collapse and prior changes, and is exactly the measured CKM angles — near $\pi/8$ multiples as a matter of experiment — inherited into the loop phase by the chart mechanics; the loop phase itself is multi-valued over a fixed physical point and carries no physical information beyond the standard readout.

View source on GitHub