1950 — The Identification of W: a Gram-Norm Difference Across the Graph Cut, and the Complete a₁ Classification (WP22)
Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated, on the certified viable ensemble; the topology-general proof over all 20,250 one-cycle nine-link textures remains open Supersedes: nothing. Completes 1947 (WP21), which factored a₁ = M·χ_b(s)·W universally but left W unidentified.
1. Question
1947 established, on all 755 unique viable nine-link textures:
detC(z) = a₁(z − z⁻¹), a₁ = M·B, B = χ_b(s)·W,
with M the monomial gcd of a₁'s support, χ_b(s) the 2×2 block's characteristic polynomial evaluated at the gap endpoint s, and W a residual taking ±1-valued one-, two-, or three-term forms (471/131/153 textures). §5.4 of 1947 left the structural identity of W open. WP22 answers: what is W?
2. The classification theorem (certified 755/755)
On the certified viable ensemble, B = ±F exactly, where F is one of four forms, selected by the cycle structure of the decomposed sector:
| class | count | cycle | F |
|---|---|---|---|
| diagonal | 73 | 4 | g₀₁·g₀₂·g₁₂ (= ±D_sec, the full gap product) |
| 4cycle_trivial | 398 | 4, singleton off cycle | χ_b(s) |
| 4cycle_leafcol_diff | 50 | 4, singleton off cycle | χ_b(s)·[(Y†Y){l₁l₁} − (Y†Y){l₂l₂}] |
| 6cycle_row_diff | 234 | 6, singleton on cycle | χ_b(s)·[H_ii − H_jj] |
So W = ±1 in 471 cases (73 + 398) and W is an exact difference of two Gram norms in all 284 multi-term cases (234 row-Gram + 50 column-Gram). The two-/three-term presentations of W recorded in 1947 collapse, in every case, to a single two-norm difference.
3. The selection rules are graph-combinatorial, not numerical
- 6-cycle (234): {i, j} — the row indices of the Gram difference — are exactly the two Q-rows touched by the decomposed sector's cycle edges. Certified as an if-and-only-if on all 234: the predicted pair equals the fitted pair in every case.
- 4-cycle with two leaf columns in the connected sector (145 = 95 + 50): if a leaf column attaches to the singleton row, W = ±1 (95 textures, absorbed into 4cycle_trivial); if no leaf column attaches to the singleton row, W is the difference of the two leaf columns' Gram norms (50 textures). Clean iff, no exceptions.
- The overall sign B = ±F is phase-convention dependent (placement of the loop edge within the texture) and is recorded per texture but carries no invariant meaning.
4. What W is
W measures a Gram-norm asymmetry across the graph cut defined by the cycle/tree split of the decomposed sector:
- when the cycle is balanced against the tree (diagonal sector, or 4-cycle with a leaf anchoring the singleton), the asymmetry vanishes and W reduces to ±1;
- when two rows (6-cycle) or two columns (4-cycle, unanchored singleton) sit on opposite sides of the cut with no balancing attachment, W is exactly the difference of their Gram norms.
This completes the structural reading of the WP21 reduced form:
a₁/(D_uD_d) = M·W/(gap·D_other),
every factor of which is now identified: M (monomial support), gap (block gap), D_other (connected-sector Vandermonde), W (Gram-norm asymmetry across the cut).
Typing caution, unchanged and now sharper: every factor on the right-hand side is chart data — defined relative to a sparse texture presentation — not a weak-basis invariant. The Gram norms themselves are basis-dependent. The identification of W does not promote a₁ to a physical invariant; it makes the chart-level mechanism fully explicit, which is what the realizable-level-set question (1947 §6) requires as input.
5. Boundary
- Certified on the 755 unique viable nine-link textures (WP15b viable minima, χ² < 4), each verified exactly.
- The full one-cycle nine-link texture space has 20,250 connected textures (splits (3,6)/(4,5)/(5,4)/(6,3) = 2106/8019/8019/2106). A topology-general proof — or a full enumeration — is open (candidate WP23).
6. Durable verification
- Checker:
research/flavor/checkers/wp22_w_identification.py - Certificate:
research/flavor/results/wp22_w_identification.json(n_textures = 755, n_verified = 755, anomalies = 0; class_counts diagonal 73 / 4cycle_trivial 398 / 4cycle_leafcol_diff 50 / 6cycle_row_diff 234; per-texture class, sign, and selection-rule witnesses) - Commit: a86485ed (checker + certificate, pushed)
- Sequence claim: seqclaim-b6073256bc72b489f8962244 (value 1950)
- Epistemic event: ev-000000002435 (claim + test entities, tests and
depends_on relations to the WP21 claim, reports to marici.Nima and
marici.Benincasa). Disclosed exception: admitted via
research/flavor/tools/mcp_stdio_client.py(agent-side loader emission fault, standing disclosure).