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Entry 1936 · 2026-08-23 · marici.Figueiredo · research

1936 — The π/8 Accounting: Branch Cut, Exact Fiber in the Valley Set, and a Recorded Residual (WP19)

Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated; one unresolved observation recorded in §5 Supersedes: nothing. Completes the flavor brief's WP4 at the level of the paper's ensemble claim. Consistent with 1911 (WP14b inheritance), 1929/1934 (WP16–WP18 fiber structure).

1. Question

Account for the fitted ensemble's loop-phase clustering near multiples of π/8 (the brief's fourth work package), keeping the three claims separate:

  1. phase values empirically cluster near π/8 multiples;
  2. a Yukawa-triangle angle equals a CKM angle at leading order;
  3. a UV symmetry dynamically selects the simple phase.

All exact-fit solves below are physical10-fiber solves over the fit point p*. Per the fiber-claim policy they certify finite ambiguity, never uniqueness; every converged root's sheet membership is certified by its physical16 image against the two quotient sheets over p*.

2. Branch cut (exact)

The two cos δ sheets over p* (WP18) are not both viable against the paper's 17-observable fit:

The ensemble of viable textures lives entirely on the cos δ > 0 sheet. The two-sheetedness of the measured ten-coordinate projection (1934) is therefore resolved by data with overwhelming margin: the second sheet never entered the paper's fit, and no ensemble property may be attributed to it.

3. The exact fiber is contained in the scan's valley set (certified)

A dense multi-start completion of the viable-sheet exact fiber at p* (class-phase anchors + 100 uniform phase anchors × 40 restarts per orbit; residual < 1e-7; physical16 branch gate) was run for every orbit whose WP15b dominant classes (χ²_min < 4) were not already matched to the sampled atlas at 1°. Result: in every orbit, each exact viable pinned phase has a dominant scan class within 1° — usually within 0.01°:

orbit exact pinned phases (deg) matching scan classes (deg)
0 42.831, 69.723 42.83, 69.72
4 46.861, 89.717 46.86, 89.73
5 21.671, 22.821 21.67, 22.82
7 89.717 89.69, 89.73
10 22.866, 23.890 22.87, 23.89
11 89.733 89.69, 89.73
13 89.717 89.69, 89.73
14 22.821 22.82
15 89.717 89.69, 89.73
16 68.922, 89.513 68.92, 89.51
17 89.733 89.69, 89.73

(orbits 2, 6, 9 needed no completion: all their dominant classes matched the sampled atlas phases in the first pass). The exact fit the paper implicitly solves is therefore a subset of the scan's viable minima, and the ensemble's phase content includes the exact fiber's phases.

4. The scan's extra valleys, and a falsified intermediate hypothesis

The intermediate hypothesis "every dominant class phase is a small perturbation of its own orbit's exact pinned phase" is falsified: 34 class records sit 20–66° from their orbit's viable pinned set. These are not defects and not bookkeeping mismatches (orbit indexing verified aligned between WP15b and WP18; identical χ² floor 3.36 at both valley types). They are additional, distinct viable χ² minima in the same charts — other physical quotient points inside the 3σ data ellipsoid.

Two structural facts about them:

5. The accounting, and its boundary

Established at the level of the paper's claim:

Recorded residual (unresolved observation). The extra valleys of §4 also hug the window centers within ~2°, and inheritance at p* does not explain their tightness: each extra valley is a different physical point on a different sheet of the chart map, with its own ρ, and arcsin(J/ρ) on those sheets has no established reason to land near a π/8 multiple. Candidate mechanisms: per-valley inheritance with sheet-specific ρ; χ²-floor selection of φ through next-to-leading observable dependence. Deciding between them requires per-valley physical16 data that WP15b did not store. This is recorded, not smoothed over: the "accounting" of the clustering is complete for the exact fiber and empirical for the extra valleys.

6. Consequence for the sector question

WP4 is hereby closed at the level of the paper's evidence. The π/8 clustering is the measured near-π/8-ness of the CKM angles transported into chart phase coordinates by inheritance, plus binning — not an independent quantization phenomenon. This removes the last candidate "conspicuous coincidence" that could have been promoted to a UV law, and it leaves the sector assessment where WP11–WP18 put it: the flavor carrier is the physical quotient with its invariant readout; sparse textures are charts; the loop phase is chart data.

7. Artifacts

Reproduce:

cd research/flavor
./.venv/Scripts/python checkers/wp19_pi8_accounting.py   # slow: dense LM solves
./.venv/Scripts/python checkers/wp19b_valley_coverage.py # fast: stored artifacts only

View source on GitHub