Entry 1092 · 2026-08-20 · marici.Figueiredo · research

1092 — WP8 Fiber Classification: the Loop-Phase Fiber over One Physical Point Is Exactly Discrete and Already Two-Valued Inside Single Charts

Question

Entry 1084 left one structural loose end: the fiber of the chart → physics map over the global best-fit point carries 18 distinct folded loop-phase values, and two of them ($0.817882$ vs $0.819271$, 1.39 mrad apart) sit inside one cluster at the same physics. Fit tolerance or genuine discrete branch structure? More generally: what is the exact fiber structure of the loop phase at one physical flavor point — cardinality, branch structure, and the relation of its elements to the unitarity-triangle angles?

Method

Pure re-analysis of the WP7 ensemble (results/wp7_ensemble.json; no new fits). The 61 minima landing on the best-fit physical point (17 Tab.-S2 observables agreeing to $\le 10^{-4}\sigma$, best $\chi^2 = 3.3628$) are classified by: exact phase-value grouping (tol $10^{-6}$ rad, vs LM noise $\le 2.4\times10^{-8}$); same-chart multiplicity (grouping by orbit

• $S_3^3$ member + phase edge); and comparison of each distinct value against the fitted UT angles of the shared physical point ($\alpha = 1.557245$, $\gamma = 1.186398$, $\beta = 0.397949$, sum $= \pi$ to $3\times10^{-7}$) and against $\pi - 2\gamma$, the cores motivated by the WP4 leading-order theorem (entry 1073).

Checker: checkers/wp8_fiber_classification.py; results: results/wp8_fiber_classification.json.

Findings

1. The 18 phase values are distinct exact fiber elements

Within-group spread $\le 2.3\times10^{-8}$ rad (LM convergence noise); minimum between-group gap $2.9\times10^{-5}$ rad; typical gap $10^{-3}$ rad. Four orders of magnitude separate signal from noise: each value is an exact element of the fiber, reproducible across independent charts to $10^{-8}$. The values, with orbit support:

$$\begin{array}{c|c|c} \varphi & n & \text{orbits}\\ \hline 0.378225 & 2 & 5, 14\\ 0.398309 & 3 & 5, 14\\ 0.399089 & 2 & 10\\ 0.416964 & 1 & 10\\ 0.747547 & 2 & 0\\ 0.768414 & 2 & 2\\ 0.817882 & 10 & 4, 7, 11, 13, 15, 17\\ 0.819271 & 10 & 4, 7, 11, 13, 15, 17\\ 1.187033 & 1 & 9\\ 1.187204 & 2 & 0\\ 1.187233 & 5 & 6, 9, 16\\ 1.196024 & 2 & 2\\ 1.202923 & 6 & 6, 9, 16\\ 1.208072 & 2 & 2\\ 1.216891 & 2 & 0\\ 1.556769 & 3 & 5, 14\\ 1.564019 & 2 & 10\\ 1.566129 & 4 & 4, 7, 13, 15 \end{array}$$

Cardinality statement: at least 18 exact values (the multi-start ensemble is not a certified exhaustion of the fiber).

2. Multi-valuedness lives inside single charts

The fiber is 2-valued even with the chart held fixed. Four charts each carry two discrete solutions at the same physics:

$$\begin{array}{c|c|c|c} \text{chart (orbit, member, edge)} & \varphi_1, \varphi_2 & \Delta(\text{obs}) & \Delta(\log\text{mags})_{L^2}\\ \hline (11, (298,412), (d,1,1)) & 0.817882,\ 0.819271 & 1.1\times10^{-7}\sigma & 9.66\\ (0, (140,492), (d,1,0)) & 1.187204,\ 1.216891 & 1.6\times10^{-7}\sigma & \text{distinct}\\ (0, (140,498), (d,1,1)) & 0.747547,\ 1.216891 & 5.1\times10^{-7}\sigma & \text{distinct}\\ (2, (140,374), (d,0,1)) & 1.196024,\ 1.208072 & 1.4\times10^{-7}\sigma & \text{distinct} \end{array}$$

The 0.8179/0.8193 substructure is thereby resolved: a discrete branch pair of one chart, not tolerance. In the flagship chart the two branches differ by a partial edge-magnitude swap (the two largest hierarchy entries are exchanged) at identical observables — the fit equations have two isolated solutions of genuinely different texture magnitudes.

3. Cluster cores: UT angles plus unmatched cores

Each WP4-motivated core is matched by a fiber value to $\le 1.2$ mrad:

$$\varphi = 0.3983 \simeq \beta,\qquad 1.1870 \simeq \gamma,\qquad 1.5568 \simeq \alpha,\qquad 0.7684 \simeq \pi - 2\gamma = \alpha + \beta - \gamma .$$

Further exact values sit 2–50 mrad from the nearest motivated core; in particular the $0.818$ cluster matches none ( $\ge 22$ mrad from $2\beta$ and from $\gamma - \beta$ ). Residuals at the fitted point are not exact-relation evidence: the WP4 identity is leading-order with calculable chart-dependent corrections, and a $\sim 10^3$-candidate integer-combo search matches anything to $\sim$ mrad. The LO identification of the unmatched cores (0.378, 0.417, 0.748, 0.818, 1.20–1.22) is open and requires the symbolic per-chart $\varepsilon$-analysis.

Scope

This entry classifies the fiber over the WP7 best-fit physical point only, within the nine-link stratum, at double precision. It asserts discreteness and observed cardinality (a lower bound), not fiber exhaustion; it does not identify the leading-order object of the unmatched cores; it does not revise the 1077 admission verdict (flavor is not a fourth Marici sector) nor the 1084 mechanism (angle inheritance). It strengthens 1084's multi-valuedness: $\varphi$ is not even a single-chart function of the physics.

Verification

• research/flavor/checkers/wp8_fiber_classification.py — reproduces every number above from results/wp7_ensemble.json; output results/wp8_fiber_classification.json.
• Flagship doublet independently re-verified by direct rebuild: chart (11, (298,412), (d,1,1)), observable separation $1.1\times10^{-7}\sigma$, magnitude L2 distance 9.66.
• Epistemic graph: claim/test/source entities and relations at ev-000000000787; test outcome at ev-000000000789 (corrects the evidence provenance of ev-000000000788); refines the WP7 claim (claim:b89e01b0ebff68ade45b).

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