Entry 1908 · 2026-08-22 · marici.Figueiredo · research

1908 — Determinant Phase-Avoidance Is Chart Data: the Strong-CP Route Does Not Transport Across the Flavor Carrier Groupoid (WP14)

Question

Residue of the brief's fifth work package, in the form of its high-value falsifier #4: do perfect-matching determinant properties survive allowed chart changes? The paper's App. V route to real $\det(Y_uY_d)$ rests on textures where the unique loop phase can avoid every perfect matching of the Yukawa bipartite graphs, making the determinant a real monomial (up to sign). WP5 established this for the four worked textures. The decisive groupoid form of the question is new: the 61 fitted charts of the WP9 atlas all lie over one physical flavor point (WP10, entry-level certification via the WP10 permutation gate). Is phase-avoidance uniform across them — a property of the physical point — or mixed, hence chart data?

Method

Pure combinatorics; no floating point, no fitting. For each nine-link support $(\mathrm{mask}_u,\mathrm{mask}_d)$ with the loop phase on a chosen cycle edge, enumerate the perfect matchings of the two $3\times3$ bipartite sector graphs (rows $Q$, columns $u^c$ / $d^c$). Each matching contributes $\operatorname{sgn}(\pi)\prod m_e$ to $\det Y$; the phase edge multiplies its monomial by $z=e^{i\phi}$ iff it belongs to that matching. Three exhaustive classes:

• real_monomial — the phase edge is in no matching of its sector: $\det(Y_uY_d)$ real monomial, $\arg\det\in\{0,\pi\}$ structurally;
• z_times_monomial — the phase edge is in every matching (unique matching in all occurring cases): $\det(Y_uY_d)=z\cdot(\text{real})$, so $\arg\det(Y_uY_d)=\pm\phi\ \mathrm{mod}\ \pi$ exactly;
• A_plus_zB — mixed membership in a two-matching sector: $\arg\det=\arg(A+Be^{i\phi})$, magnitude-dependent.

Census levels: (a) all 6552 viable connected one-cycle full-rank supports (the WP13 class), orbit-reduced under $S_3^3\times$swap to 18 orbits, phase placed on each cycle edge — 80 orbit cases; (b) the full support-level ensemble, 28944 phase placements; (c) the 61 fitted WP9 carrier-groupoid vertices. Regression gate: the four WP5 worked textures must reproduce results/wp5_matching_reality.json. Checker: research/flavor/checkers/wp14_matching_determinant_census.py; results: research/flavor/results/wp14_matching_determinant_census.json.

Typing caution, stated once and meant throughout: $\arg\det(Y_uY_d)$ is not invariant under the $U(1)^3$ part of the weak-basis group; the physical combination is $\bar\theta=\theta+\arg\det(Y_uY_d)$. This census measures a chart-combinatorial property. It says nothing by itself about spontaneous CP, radiative stability, or a solution of the strong CP problem.

Result

T1 — exact trichotomy over all one-cycle nine-link topologies. The 80 orbit cases split $31/27/22$ into real_monomial / z_times_monomial / A_plus_zB. Every multi-matching sector in the census has matching counts $(1,2)$ (one matching up-type, two down-type) with mixed phase membership — the A_plus_zB mechanism is a single Cauchy–Binet interference, mirroring WP12's $\cos\phi$ mass-term mechanism. At full support level, $13392/28944\approx46.3\%$ of phase placements are phase-avoiding.

T2 — falsifier #4 decided: phase-avoidance does not transport. Over the 61 fitted charts — all above the same physical point — the classes are mixed:

$$24\ \text{real}\quad/\quad 31\ \arg\det=\pm\phi\ \text{exactly}\quad/\quad 6\ \arg\det=\arg(A+Be^{i\phi})\ (\text{all orbit 2}).$$

The same physical flavor point is presented by charts whose $\det(Y_uY_d)$ is structurally real, charts whose determinant carries the entire loop phase, and charts whose determinant phase is a magnitude-dependent interpolation. Determinant phase-avoidance is therefore chart data, not physical data: no statement about $\arg\det$ per chart descends through the carrier groupoid. (55 of 61 fitted charts have matching counts $(1,1)$ — both determinants monomials — so the classification there is purely the combinatorics of which edge the unique matching uses.)

T3 — the existential refinement survives. What can descend is a fiber-level existential: the fiber over the fitted physical point contains phase-avoiding charts (24 of them, spanning orbits 0, 6, 10, 11, 13, 14, 15, 16, 17). Whether every physical point in the viable region admits a phase-avoiding chart is open — the support-general 46% avoidance rate does not by itself survive the viability selection, which is known to correlate with chart structure (WP7/WP9).

T4 — disjoint from the WP13 zero dichotomy. All 12 phase cases in WP13's CP-trivial orbits $(84,119),(85,118),(85,220)$ — where $\det[H_u,H_d]\equiv0$ identically — are phase-in-det cases. No real_monomial orbit case is CP-trivial. Structural determinant reality and structural CP-triviality are disjoint chart properties in this census.

Ensemble reconciliation with the paper: App. V's "5 of 99 fixed-phase textures do not allow the phase to avoid all determinant matchings" concerns the paper's fitted fixed-phase ensemble (including ten-link classes); our 24/61 counts the nine-link LO atlas over one physical point. Different ensembles, no contradiction; the qualitative paper statement ("most viable textures allow avoidance") survives in the existential form T3.

Interpretation

This is the WP6/WP11 pattern recurring at the determinant layer: a conspicuous chart quantity that organizes presentations without descending to the physical quotient. The strong-CP-friendliness of the framework types precisely as: if a UV completion sets $\theta=0$ (spontaneous CP or otherwise, with radiative stability addressed separately) and the Yukawa sector is realized in a phase-avoiding presentation, then $\bar\theta=0$ at tree level — and T3 says such a presentation exists over the fitted point. But T2 says nature selecting "a nine-link texture" does not select that presentation: 37 of 61 fitted charts over the same physics fail it, 31 of them maximally ($\arg\det=\pm\phi$). For the H2LR typing: determinant phase-membership belongs to the chart atlas $\mathfrak F_9^{\mathrm{sparse}}$, not to $\mathcal O_{\mathrm{flavor}}$; the only candidate physical statement is the existential fiber property, which is exactly the kind of "chart-existence" clause WP6 already isolated for the flavor admission verdict.

Verification

• WP5 regression: all four worked textures (S38/S43/S48/S53) reproduce the prior phase-membership booleans and per-sector matching counts — pass.
• Exhaustiveness: all 6552 viable supports enumerated (matches WP13's independent count); 18 orbits, 80 orbit phase cases; 61/61 fitted charts classified, class counts sum to 61.
• The A_plus_zB label was audited for the theoretical subcase "phase in all of $\ge2$ matchings" (which would be $z\cdot$real polynomial, arg $=\pm\phi$): zero occurrences in the census.
• Runtime $\sim2$ s; stdlib only.

Relations

• Closes the brief's fifth work package residue and decides falsifier #4.
• Extends WP5 (wp5_matching_reality.json, four worked textures) to all topologies and to the full fitted groupoid.
• Cross-references WP13 (entry 1907): T4 is the disjointness statement against its zero dichotomy.
• Depends on WP10's certification that the 61 charts lie over one physical point (the premise that makes T2 decisive).
• Repeats the WP6 (ev-758) admission-verdict pattern at the determinant layer: presentation property, existential fiber residue.

View source on GitHub