1951 — The a₁ Classification Is Topology-General: All 6,552 Full-Rank One-Cycle Nine-Link Textures, and the Exact CP-Triviality Rule (WP23a)
Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated, on the full-rank one-cycle nine-link support space; derivation from first principles (WP23b) in progress Supersedes: nothing. Extends 1950 (WP22) from the 755-texture viable ensemble to the entire admissible support space of 1054.
1. Question
1950 classified a₁ on the 755 viable textures into four classes with graph-combinatorial selection rules. Left open: is the classification an artifact of the viable ensemble, or does it hold on every admissible support? And what happens on supports where the viable ensemble never lands?
2. The enumeration
All connected nine-link support pairs (mask_u, mask_d), splits (3,6)/(4,5)/(5,4)/(6,3) = 2106/8019/8019/2106 = 20,250 connected topologies (anchor reproduced exactly). Imposing the 1054 hypothesis — both sectors full-rank (perfect matching each) — leaves 6,552 topologies (468/2808/2808/468). Phase placed canonically on the first sorted cycle edge (one representative per rephasing orbit of placements).
For each topology the full WP22 analysis runs symbolically with algebraically independent edge magnitudes.
3. Result: the four classes cover everything, plus one new class
| class | count |
|---|---|
| 4cycle_trivial (B = ±χ_b(s)) | 3456 |
| 6cycle_row_diff (B = ±χ_b(s)·(H_ii − H_jj)) | 1296 |
| diagonal (B = ±g₀₁g₀₂g₁₂) | 720 |
| 4cycle_leafcol_diff (B = ±χ_b(s)·(col Gram diff)) | 432 |
| detC_zero (detC ≡ 0 identically) | 648 |
- All 5,904 topologies with detC ≢ 0 verify B = ±F with the WP22 selection rules — zero anomalies, zero new classes. The WP22 classification is topology-general, not a viable-ensemble artifact.
- The 648 detC_zero topologies are CP-trivial by support alone: J ≡ 0 for every choice of magnitudes and phase. No viable texture can live on them (consistent with 1950: all 755 viable verify).
4. The exact CP-triviality rule
Certified as an if-and-only-if on all 6,552 (cross-tab: 648/648 true-positive, 5,904/5,904 true-negative):
detC ≡ 0 ⟺ (decomposed sector diagonal AND connected sector has ≥ 1 disjoint row pair) ∨ (decomposed sector block-2+1 AND connected sector has ≥ 2 disjoint row pairs)
where "disjoint row pair" means two Q-rows whose column supports in the connected sector are disjoint (the corresponding Gram off-diagonal vanishes identically). Mechanism: for a diagonal decomposed sector, detC is proportional to x₁₂x₂₃x̄₁₃ − x₁₃x̄₁₂x̄₂₃, and a single zero off-diagonal kills both 3-cycle products; the block-2+1 reduction needs two. All 648 cases are 4-cycles; no 6-cycle topology is CP-trivial.
Boundary: the rule is certified on this support space, phase on the canonical cycle edge. Phase-placement independence within a topology is certified by 1054 for the support theorem but not re-audited here for the class labels.
5. Meaning for the program
- Roughly 9.9% of admissible one-cycle topologies are CP-trivial by support. The paper's viable textures avoid them; the overlap-graph predicate says exactly which.
- The classification theorem is now stated on the natural domain (full-rank one-cycle supports) with no viability input anywhere — it is a pure support theorem, the right substrate for the WP23b first-principles derivation.
- Nothing changes the typing: all factors remain chart data.
6. Durable verification
- Checker:
research/flavor/checkers/wp23_topology_enumeration.py(enumeration + classification + detC_zero rule cross-tab) - Certificate:
research/flavor/results/wp23_topology_enumeration.json(n_topologies 6552, n_verified 5904, split anchor ok, detc_zero_rule_ok true) - WP22 checker patch (detC_zero distinguished from support anomaly): verified not to alter the committed WP22 census (755/755 still sign)
- Commit: bab761b2 (pushed)
- Sequence claim: seqclaim-731fe404d8a1f68a11538825 (value 1951)
- Epistemic event: ev-000000002443 (claim + test + reports to Nima/Benincasa; admitted via tools/mcp_stdio_client.py, standing loader disclosure)