1954 — The Realizable Level Sets: 699 Singletons, Two Class-Locked Pairing Patterns, and the Generation-Exchange Mechanism (WP24a)
Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated, at the demonstrated (numerical certificate) strength; the exact refit verification of the exchange map is open (WP24b) Supersedes: nothing. Continues 1937 (WP20) and 1947 (WP21); uses the 1950 classification.
1. Question
1947 located the π/8 question in the realizable level sets of the reduced form r = a₁/(D_uD_d): which r values admit viable fits, and with what multiplicity per texture?
2. The level-set structure (certified on all 1210 viable minima)
- 699 of 755 textures are singletons: all their viable minima land in one folded-phase cluster.
- 56 textures span exactly two clusters, in only two patterns: (1,3) — 27 textures — and (0,4) — 29 textures. No other span exists.
- The patterns are class-locked: every (1,3) texture is diagonal class; every (0,4) texture is 4cycle_trivial (14) or 4cycle_leafcol_diff (15).
- Cluster 2 (46.91°) is exclusively 4cycle_trivial singletons (190 textures, one rigid level); cluster 1 (43.17°) is exclusively diagonal (27 paired + 15 singleton).
3. The paired minima are the same physical point
For all 56 pairs: |Δχ²| ≤ 4.3e-12 and |ΔJ|/J ≤ 1.7e-8. The two minima fit the same observables with the same quality — they are one physical point realized in two charts.
The log-magnitude transformation is uniform across each pattern:
- (1,3), all 27: the first two u-magnitude entries swap values (Δ = ±6.218726), three d entries shift by ∓2.076289, the rest adjust sub-dominantly. The swap exchanges the two off-diagonal u magnitudes of the permutation support — i.e. exchanges which Q-rows carry which up-type eigenvalues: generation exchange.
- (0,4), all 29: dominant shift −6.218726, no swap.
The recurring constants are physical log-ratios (identical across textures because the fit pins the same masses), supporting the reading: one physical point, two generation assignments, different chart phase φ, same J.
sin²φ pair sums are tight but NOT constant ((1,3): 1.3441 ± 0.0032; (0,4): 1.1438 ± 0.0074) — there is no exact complementarity law φ′ = π/2 − φ; the d-sector refit is not a symmetry of the functional, only of its viable locus.
4. Consequence
1937's "extra valleys are the same valleys" now has an explicit mechanism at the texture level: a realizable r-spectrum has at most two points, and the second point is the generation-exchange image of the first (diagonal class) or its shift-image (4cycle classes). This is further evidence that φ is chart data: the same physical point carries two different folded phases, and the pairings respect the 1950 classification.
Open residual: what selects singleton vs paired among textures of the same class (diagonal: 15 singleton-1, 30 singleton-3, 27 paired)? Not yet characterized.
5. Durable verification
- Checker:
research/flavor/checkers/wp24a_level_set_structure.py - Certificate:
research/flavor/results/wp24a_level_set_structure.json(span counts, cross-tabs, pair transformation signatures, max_pair_dchi2 4.3e-12, max_pair_dJ_rel 1.7e-8) - Commit: 639e4ac2 (pushed)
- Sequence claim: seqclaim-d20ee8c02ee542eb012e9b7e (value 1954)
- Epistemic event: ev-000000002454