1962 - Level Anatomy: Physical Vandermondes, the Exact Reduced Form, and Rigid M-Freeze vs Conspiracy Compensation
- Date: 2026-08-23
- Author: marici.Figueiredo
- Status: certified pointwise at all 1210 stored minima
- Supersedes: nothing. Continues 1947 (WP21 factorization), 1950 (WP22 W identification), 1959 (WP25a center selection), 1960 (WP25b trench landscapes)
1. The reduced form, certified pointwise
WP21e's reduced form is UNIVERSALLY
L = a1/(Du Dd) = M * W / (gap * D_other),
with gap = sqrt(disc_b) for block21 (the chi_b(s) of 1947 having cancelled against Du_sec = sqrt(disc_b) * |chi_b(s)|) and gap = the remaining-gap product for the diagonal class. A first version of this audit wrongly re-inserted ln chi_b(s) into the L identity; the residual failures (lnW up to 32.7 on W = +-1 classes) exposed exactly the cancellation and corrected the formula. Certified with the correct identity: the residual
lnW_residual = lnL - lnM + lngap + lnDo
is <= 1.9e-9 on all 712 minima of the W = +-1 classes and equals ln|W|, bounded in [6.7e-2, 16.2], on all 498 minima of the 284 multi-term Gram-norm-difference textures. n = 1210/1210.
2. The Vandermondes are physical
Du and Dd are products of squared-mass differences - weak-basis invariants. Certified: ln(Du Dd) is constant across all 1210 minima with global std 4.0e-16 (machine precision). The global spread of ln D_other (std 8.66) is exactly the u/d sector mix (D_other = Dd for u-decomposed textures, Du for d-decomposed); within sector-pure subsets it sits at fit precision (cluster 1: 2.2e-9).
Consequence: the chart content of L is entirely
K := M * W / gap = a1 / D_sec,
and the five levels are five realized values of K divided by one physical constant.
3. Rigid clusters are M-frozen
- Cluster 1 (42 diagonal, all u-decomposed): gap = 1 and W = +-1 exactly (lnW std 4.6e-13), so L = M/Dd EXACTLY (residual 4e-14). The cluster's entire residual lnL spread (std 9.96e-3) is lnM's spread: corr(lnM, lnL) = 1.0.
- Cluster 2 (330 block21, all u-decomposed): every texture has W = +-1 (lnW std 3.5e-10; no Gram-norm-difference class occurs), gap near-frozen (std 2.8e-4, mean lngap = -0.067), and again the residual spread is M's: corr(lnM, lnL) = 0.964, std(lnL) = 1.75e-3 vs std(lnM) = 1.94e-3.
So the rigid clusters are classes whose fitted magnitude assignments are near-unique: the monomial M is essentially one value per class, and L inherits it.
4. Conspiracy clusters compensate across factors
Clusters 0, 3, 4 show the opposite regime:
| cluster | std lnM | std lngap | std lnW | std lnL |
|---|---|---|---|---|
| 0 | 8.82 | 4.45 | 4.95 | 3.3e-2 |
| 3 | 4.15 | 6.83 | 5.08 | 4.0e-3 |
| 4 | 9.12 | 6.66 | 3.55 | 3.1e-5 |
Individual chart factors range over e^{+-4..9} while lnL is frozen to between 3.1e-5 and 3.3e-2. No single factor carries the freeze: |corr(factor, lnL)| <= 0.67 for every factor in every conspiracy cluster (lnW: |corr| <= 0.08). The level is a genuine multi-factor integral of motion across the viable chart classes.
5. What this establishes for the selection question
The level-selection problem now decomposes cleanly:
L_c = K_c / D_other, D_other physical (2.2e-9 .. 4e-16),
so "why five phase clusters" is exactly "why do the viable chart classes realize five values of K = M W/gap". The two rigid clusters realize K through near-unique magnitude assignments (M-frozen); the three conspiracy clusters realize K through exact compensation of wildly varying chart factors. The remaining open object is the mechanism that admits precisely these five K values - equivalently, the geometry of the viable (class, magnitude-assignment) pairs over one physical flavor point.
Typing caution unchanged: M, W, gap are chart data; Du, Dd, and D_other are physical; K is chart data whose five realized values organize the ensemble.
6. Artifacts
- Checker: research/flavor/checkers/wp26_level_anatomy.py (v2; the docstring records the v1 ln-chi error and its diagnostic value).
- Results: research/flavor/results/wp26_level_anatomy.json.
- Inputs: results/wp20_valley_audit.json (1210 stored minima), results/wp21e_universal_factorization.json (per-texture forms).