1947 — The Universal a₁ Factorization: the Reduced Form a₁/(D_uD_d) = M·W/(gap·D_other), and the Two Cluster Archetypes (WP21)
Date: 2026-08-23 Author: marici.Figueiredo Status: established as stated, on the certified viable ensemble; the π/8 question is located, not solved Supersedes: nothing. Continues 1937 (WP20), which migrated the π/8 question to the magnitude sector; 1924 (WP13 support theorem); 1911 (WP14b inheritance). One tooling bug (WP21c v1) is disclosed and repaired in §3.
1. Question
1937 showed: at all 1210 viable minima the inheritance identity J² = ρ_v sin²φ holds with ρ_v = (a₁/(D_uD_d))², the folded phases form discrete clusters, and the cluster values are window-tight. The question that remained: what is a₁ structurally, and which factor of a₁/(D_uD_d) carries the discreteness?
2. The factorization theorem (certified on all 755 viable textures)
For every one of the 755 unique viable nine-link textures (mask_u, mask_d, phase edge; WP15b viable minima, χ² < 4):
-
detC(z) = a₁(z − z⁻¹), support {±1} (1924, rechecked per texture).
-
a₁ = M·B with M the monomial gcd of a₁'s support.
-
Exactly one sector is row-sharing decomposable (the other is row-connected). Two cases exhaust the ensemble:
- (2+1) block (682 textures): H of the decomposed sector is a 2×2 block ⊕ singleton. Its Vandermonde factors EXACTLY: D_sec² = disc_b · χ_b(s)², where χ_b is the char poly of the 2×2 block, s the singleton diagonal entry (= its eigenvalue), disc_b the block discriminant (the squared eigenvalue gap of the block). The identity is verified symbolically for all 682 textures (0 failures). B is ALWAYS divisible by χ_b(s): B = ±χ_b(s)·W, with W a ±1-coefficient alternating quadratic form in the OTHER sector's squared magnitudes: 471 textures W = ±1, 131 textures W two-term (e.g. −d01²+d10²), 153 textures W three-term (e.g. −d01²−d02²+d10²). No other coefficient ever occurs.
- diagonal (73 textures): H of the decomposed sector is diagonal, D_sec = |g₀₁g₀₂g₁₂| with g_ab = H_aa − H_bb, and B = ±(g₀₁g₀₂g₁₂)·W with W = ±1 in every case.
-
Therefore, universally on the viable ensemble:
a₁/(D_uD_d) = M·W / (gap · D_other)with gap = √disc_b (2+1 case) or the product of remaining gaps (diagonal case; there, none remain, so gap = 1). Numerically verified at one exact-rational point per texture: max relative error 1.6e-13 over all 755 textures.
Certificate: research/flavor/checkers/wp21e_universal_factorization.py,
research/flavor/results/wp21e_universal_factorization.json
(755/755 factored, zero anomalies).
The 2+1 Vandermonde identity is elementary (eigenvalues {s, λ±}, (λ+−s)(λ−−s) = χ_b(s)); the content of the theorem is that a₁'s alternating bracket ALWAYS contains exactly this χ_b(s), once and only once, leaving a ±1-coefficient residual W of at most three terms.
3. Disclosure: WP21c v1 bug
The first census (wp21c v1) reported 0/755 absorbed. That was a
mechanical artifact, not a scientific result: v1 tested absorption by
sp.div(D², B², *squared_generators), which raises PolynomialError
("contains an element of the set of generators") for every texture.
v2 replaces the test by sp.cancel(D²/B²) plus a polynomiality check
and reproduces the WP21a verified instance (85/234) before running.
The v1 file was overwritten by the v2 output; the bug and fix are
recorded in the v2 checker docstring. Lesson logged: a census that
reports universal failure through an exception path is a bug until
the exception path is excluded.
4. Factor-discreteness audit (all 1210 viable minima)
wp21d2 evaluates the log-identity
ln a₁/(D_uD_d) = ln M + ln|W| − ln gap − ln D_other
at every WP20 viable minimum (reduced form re-verified pointwise to 4.8e-12) and measures the four log-factors. Results:
- Ensemble: the individual factors fluctuate enormously (std 5.0–8.7 in log units) while their combination has std 0.32. The magnitude sector is vast; the physical combination lives in a thin shell.
- Two cluster archetypes:
- Rigid (43.17°, n=42, 42 textures; 46.91°, n=330, 190 textures): EVERY factor is frozen within the cluster (std ≤ 0.01), across hundreds of distinct texture charts. The minima are isolated (10 chart coordinates vs 10 observables) and all describe essentially the same physical point; M and W, though chart coordinates, are pinned by the fit.
- Conspiracy (23.15°, n=210; 68.39°, n=430; 89.57°, n=198): factors vary internally with std 4–10 while the combination is frozen to ≤ 0.033. A continuous valley of magnitude configurations shares one level set of a₁/(D_uD_d).
- No factor correlates with φ across the ensemble (|corr| ≤ 0.28). The discreteness of the phase spectrum is NOT carried by any single factor of the reduced form.
5. What this establishes
- Chart/physical separation is now exact. In the reduced form, gap and D_other are physical invariants (mass-squared differences), while M and W are chart coordinates. The inheritance identity a₁/(D_uD_d) = |J|/sinφ factorizes as (chart M · chart W)/(physical gap · physical D_other).
- The cluster structure is a level-set structure. Clusters are level sets of the physical combination |J|/sinφ; φ is pinned at every viable minimum (1937: zero trenches), and the cross-texture coincidence of φ within rigid clusters follows from the minima being the same physical point — consistent with the leading-order triangle identification (1903, 1921) making φ effectively a function of CKM data at viable minima.
- The π/8 question is located, not solved. What remains empirical: WHICH level sets are realizable (which φ values admit χ²-viable fits at all), and why those cluster near π/8 multiples. Any future derivation must explain the realizable set of M·W/(gap·D_other) values, not the phase sector directly.
- W is the next exact object. Its coefficients are always ±1 and it has ≤ 3 terms; it looks like a Plücker/cofactor form of the connected sector. An invariant identification of W (and thereby a closed form for a₁ in weak-basis invariants times chart data) is the natural WP22 question.
6. Boundaries
- The factorization is certified for the 755 textures present in the WP15b viable ensemble. A topology-general proof over ALL one-cycle nine-link graphs (in the sense of 1924's support theorem) is open; the census found zero counterexamples and the mechanism (block char poly dividing the bracket) is graph-theoretic, so a general proof is plausible but not claimed.
- W's evaluation at a minimum is chart data. Nothing here promotes M or W to weak-basis invariants; the separation in §5.1 is the point.
- Cluster archetypes (rigid/conspiracy) are a measured taxonomy of THIS ensemble, not a theorem about all ensembles.
7. Durable verification
research/flavor/checkers/wp21a_a1_symbolic.py→results/wp21a_a1_symbolic.json(85/234 symbolic base case; B = −χ_b(s) at the singleton, numeric residual 0).research/flavor/checkers/wp21c_a1_cancellation_census.py(v2) →results/wp21c_a1_cancellation_census.json(471/755 absorbed; v1 bug disclosed in §3).research/flavor/checkers/wp21e_universal_factorization.py→results/wp21e_universal_factorization.json(755/755: 682 block21- 73 diagonal; Vandermonde identity exact 755/755; numeric max rel err 1.6e-13; W ∈ {1,2,3}-term ±1-forms).
research/flavor/checkers/wp21d2_factor_discreteness_full.py→results/wp21d2_factor_discreteness_full.json(1210/1210 records, reduced form max rel err 4.8e-12; archetype table).- Ledger sequence claim: seqclaim-7094050b1769f3f76db5a396 (value 1947 from the shared marici-ledger-entry allocator).