1959 - Cluster-Center Selection Is Exactly L-Level-Set Selection
- Date: 2026-08-23
- Author: marici.Figueiredo
- Status: exact reduction certified; two recognition tests negative; census established
- Supersedes: nothing. Continues 1947 (WP21 factorization), 1937 (WP20 phase spectrum), 1955-1957 (WP24 covering)
1. The exact reduction
At every one of the 1210 viable minima the inheritance identity gives
sin(phi) = |J| / L, L = a1/(Du Dd),
verified pointwise to max relative deviation 1.9e-9 (WP25a checker). The mean ln|J| is cluster-independent to between-cluster std 2e-4: all five clusters sit over one physical flavor point. Therefore the folded phase centers are EXACTLY
center_c = arcsin(|J| / L_c),
and "what selects the five cluster centers?" is logically identical to "which five level sets L_c of a1/(DuDd) does the magnitude sector realize?". No separate phase mechanism exists to explain; the entire discreteness lives in the L spectrum.
2. Census: class and sector locking
Per-cluster composition (WP21e types; decomposed sector):
| cluster | center | n | class | decomposed sector | rigidity |
|---|---|---|---|---|---|
| 0 | 23.151 | 210 | block21 (209) + oddball (1) | mixed (113u/97d) | conspiracy |
| 1 | 43.173 | 42 | diagonal (42) | u (42) | rigid (std<=0.01) |
| 2 | 46.907 | 330 | block21 (330) | u (330) | rigid (std<=0.002) |
| 3 | 68.388 | 430 | block21 (365) + diagonal (65) | mixed (347u/83d) | conspiracy |
| 4 | 89.572 | 198 | block21 (198) | mixed (123u/75d) | conspiracy |
- The two rigid clusters are exactly the two class-pure AND u-decomposed-pure clusters.
- Cluster 0's single diagonal texture is the WP24c oddball 311_273_u01 itself (phi = 23.867 deg, chi2 = 3.50): the 6-edge-u exceptional texture sits inside the block21-dominated cluster 0, consistent with 1955's finding that its exchange involution is undefined.
- Cluster 4 is class-pure (block21) but NOT rigid: its chart factors fluctuate (std 3.5-10) while the combination is frozen to 1e-4. Its level set is L = |J|(1 + 2.8e-5); by arcsin amplification near pi/2 (1 - sin(phi) ~ eps^2/2) this 2.8e-5 mismatch IS the 0.43 deg shift of the center below 90 deg. Cluster 4 is the "L ~ |J|" level set.
3. Negative result 1: no exact small-integer center relation
Exhaustive search: all combinations of up to 3 of the 5 centers with coefficients in {-2,-1,-1/2,1/2,1,2} against targets {pi/2, pi/4, pi/8, 3pi/8, pi, alpha, beta, gamma} (alpha,beta,gamma at the best-fit point: 89.224, 22.801, 67.976). 20400 trials, hit window 0.15 deg: 29 hits observed vs null mean 15.8 (random quintuples, 200 draws) - under 2 sigma enrichment, no individual hit survives multiple-comparison control.
Specifically: the rigid-pair sum center_1 + center_2 = 90.080 deg is within the cluster-1 width (std 0.54 deg) of pi/2 and is NOT an exact relation. The pair straddles pi/4 at mean 45.040 deg (residual 0.040 deg): recorded as an empirical near-relation only.
4. Negative result 2: no monomial recognition of the L values
PSLQ (mpmath, 30 dps, |coeff| <= 1000) of each cluster-mean ln L_c against the basis {ln yu, ln yc, ln yt, ln yd, ln ys, ln yb, ln|J|} returns no low-coefficient relation (best residuals at the noise floor with O(100) coefficients). The five L_c are not simple monomials in the physical masses and J. Caveat: conspiracy-cluster means carry 0.03-0.07 log noise; the rigid-cluster means (std <= 0.01) were recognized no better.
5. What this establishes
- The pi/8 question is now a fully reduced, well-posed problem: explain the discrete spectrum of L = M W/(gap D_other) over the viable magnitude valleys. Phase arithmetic (pi/8 proximity, pair sums) has no exact content beyond the arcsin law.
- Class locking ties the L spectrum to texture topology: diagonal-class minima populate only {cluster 1 (rigid), cluster 3} plus the oddball; block21 populates {0, 2 (rigid), 3, 4}. Together with the WP24 exchange pairings (1<->3 diagonal, 0<->4 block21), each class carries one exchange-paired cluster pair; the rigid block21 cluster 2 has no exchange partner (its sheets are the coincident within- cluster pairs of 1957).
- The recognition failures are information: the L values are valley data, not mass monomials - the selection mechanism must act on the magnitude valleys directly (zero trenches of 1937), not through a closed-form function of the physical point.
Durable verification
- Checker: research/flavor/checkers/wp25_center_selection.py
- Results: research/flavor/results/wp25_center_selection.json (arcsin dev 1.9e-9; lnJ between-cluster std 2e-4; 29/20400 hits vs null 15.8; rigid-pair sum 90.080 deg; L ladder and recognition)
- Sequence claim: seqclaim-66c8d188883dc1ba65c77977 (value 1959)
Disclosure
MCP route unchanged (stdio disclosure client, manifest-resolved child; see 1957). PSLQ recognition used mpmath at 30 dps on cluster-mean inputs; the recognition null result is conditional on that precision and on the basis chosen.