1963 - The Natural Hierarchy Point Does Not Seed the K Spectrum; the Fit Does the Selection
- Date: 2026-08-23
- Author: marici.Figueiredo
- Status: established on all 755 viable textures / 1210 minima
- Supersedes: nothing. Continues 1962 (WP26 level anatomy)
1. Question
WP26 reduced the five cluster levels to five realized values of the phase-free chart combination K = M*W/gap. Since K is phase-free, it can be evaluated anywhere - in particular at the hierarchy-natural point nat(i,j) = sqrt(y_i y_j) that seeds the fits. If the natural point already carried five K peaks, the levels would be an artifact of the seed parameterization (a chart-of-chart effect). If not, the fit's convergence to the physical flavor point does the selection.
2. Result: no natural seeding
Natural-point ln K over all 755 viable textures: mean -47.2, std 10.97, range [-74.6, -20.8] - broad and structureless. Grouped by each texture's dominant fitted cluster, the natural means (-41.3, -45.2, -50.6, -52.2, -42.9 for clusters 0-4) neither match the fitted ln K values nor separate: natural spreads (std 4.3-13.3) are 10^3 times the fitted spreads. The natural parameterization does not prefigure the spectrum.
3. Fifteen textures are singular at the natural point
Exactly 15 textures have K undefined at nat - all of them two-term transpose-pair W forms (W = -d01^2 + d10^2 and mask images). The natural point is transpose-symmetric, nat(i,j) = nat(j,i), so a transpose-pair difference vanishes identically there. For these textures the fit is FORCED off the symmetric point: their entire |W| is generated by the fit's transpose-asymmetry breaking. (The other 116 two-term W forms - non-transpose pairs such as -d02^2 + d10^2 - are nonzero at nat.)
4. The fitted K spectrum is tight everywhere
Per (cluster, decomposed sector), fitted ln K:
| cluster,sector | n | mean lnK | std |
|---|---|---|---|
| 0,d | 97 | -20.861 | 3.3e-3 |
| 0,u | 113 | -42.050 | 2.9e-2 |
| 1,u | 42 | -42.628 | 1.0e-2 |
| 2,u | 330 | -42.694 | 1.7e-3 |
| 3,d | 83 | -21.692 | 3.5e-3 |
| 3,u | 347 | -42.936 | 4.0e-3 |
| 4,d | 75 | -21.767 | 1.1e-5 |
| 4,u | 123 | -43.008 | 6.0e-4 |
Seven (cluster, sector) values, tied into the five L levels by the physical ratio Du/Dd: ln K_d - ln K_u = ln(Du/Dd) = 21.24 holds at 0.02-0.05 for clusters 0, 3, 4 (the mixed ones), as it must since L = K/D_other and Du Dd is physical (1962). The d-decomposed members occur only in the conspiracy clusters 0, 3, 4 - never in the rigid clusters 1, 2, consistent with 1959's sector-purity census.
5. Consequence
The selection of the five K values is done by the fit, not by the seed: it is a property of the viable solution set of (texture + 17 physical observables), not of the hierarchy-natural starting point. This removes the chart-of-chart artifact reading of the level spectrum and properly poses the remaining question: what admits exactly these five K values among all viable (class, magnitude-assignment) pairs over one physical flavor point. The 15 transpose-pair textures are the extreme witnesses: for them the selected K has no natural-point precursor whatsoever.
6. Artifacts
- Checker: research/flavor/checkers/wp27_natural_k.py
- Results: research/flavor/results/wp27_natural_k.json
- Inputs: results/wp20_valley_audit.json, results/wp21e_universal_factorization.json; natural point from checkers/wp7_ensemble.py:natural_value.