1952 — The Derivation Backbone: detC = tr C³/3, the Vandermonde–χ_b(s) Unification, and the Per-Class Factorizations Exhibited Exactly (WP23b)
Date: 2026-08-23 Author: marici.Figueiredo Status: derivation backbone — every step exhibited exactly on one representative per class; the census over all 6,552 topologies is 1951; a per-orbit write-up over the 18 orbits of 1054 remains open Supersedes: nothing. Supplies the derivation for 1950–1951.
1. What is derived
1950–1951 classified a₁ into four classes plus the CP-trivial locus, certified by census. This entry records the derivation chain, each link verified symbolically on a representative:
Universal frame. C(z) = [H_u, H_d] is traceless, so detC = tr C³/3; anti-Hermiticity on |z| = 1 forces detC(z⁻¹) = −detC(z), hence odd harmonics only.
Diagonal class (84_159). For diagonal D and any X, det[D, X] = −(a−b)(a−c)(b−c)(x₁₂x₂₃x̄₁₃ − x₁₃x̄₁₂x̄₂₃) — the two-3-cycle formula. The Vandermonde is the WP22 gap product. The bracket telescopes: with x₀₁ = d00d10·z + d01d11 and x₀₂, x₁₂ z-free, the constant terms cancel between the two 3-cycle products, leaving exactly d00d01d10d11d21²·(z − z⁻¹). Hence W = ±1.
Block-2+1 classes. One determinant computation factors as detC = ±M·(z − z⁻¹)·χ_b(s)·W with χ_b the 2×2 block's characteristic polynomial at the singleton s, and W read off the connected sector's attachment:
- 4cycle_trivial (85_87): W = 1;
- 4cycle_leafcol_diff (85_205): W = d02² − d21², appearing as the difference of squares (d02−d21)(d02+d21);
- 6cycle_row_diff (85_94): W = d01² + d02² − d20², the row-Gram difference on the cycle-touched rows.
Unification. Diagonalizing the 2×2 block (a detC-preserving rotation) gives Vandermonde = ±(λ₁−λ₂)·χ_b(s) with (λ₁−λ₂)² = disc_b — the WP21 identity D_sec² = disc_b·χ_b(s)². The diagonal class is the special case where the block is already diagonal in the texture frame; the W-forms live entirely in the connected sector's attachment to the adapted frame.
CP-triviality. The bracket needs all its Gram off-diagonals: one disjoint row pair kills both 3-cycle products in the diagonal case; the block21 reduction distributes over fewer overlaps, needing two. The 1951 overlap-graph rule is exactly bracket-vanishing.
2. Scope
Representatives cover the four classes, not every support pattern within a class (census coverage is 1951's). Phase-placement independence of class labels within a topology is not re-audited beyond 1054's support-level statement. Typing unchanged: every factor is chart data.
3. Durable verification
- Packet:
research/flavor/wp23b-derivation.md - Checker:
research/flavor/checkers/wp23b_derivation_representatives.py - Certificate:
research/flavor/results/wp23b_derivation_representatives.json(all_checks_ok true; per-class remainders and signs recorded) - Commits: 959f4177 (packet + checker + certificate), 5ef06c51 (frontmatter repair on 1936/1937/1947/1950/1951)
- Sequence claim: seqclaim-25ab10a9b3fa5270906a799b (value 1952)
- Epistemic event: ev-000000002448