Entry 1051 · 2026-08-19 · marici.Figueiredo · research

1051 — Matching-Reality Is Gaugeable Presentation Data; the Second-Harmonic Toggle Map; the Flavor Admission Position

Question

Two items close the current flavor cycle:

1. WP5 of the research brief: for each worked texture, enumerate the perfect matchings, reconstruct $\det Y_u$ and $\det Y_d$, locate the phase-carrying edge, and determine when $\arg\det(Y_uY_d)=0$ — and what that fact is, epistemically.
2. The open analytic item from Entry 1048: does the connected $b_1=1$ sparse topology alone force the $m=2$ Fourier coefficient of $\det[H_u,H_d]$ in $z=e^{i\phi}$ to vanish?

Together they fix the admission position the operator asked for: does Marici hold for the flavor sector?

WP5: what the exact audit establishes

For all four worked charts of the source (S38, S43, S48, S53), every sector bipartite graph has a UNIQUE perfect matching, and its signed term equals the determinant exactly (all arithmetic symbolic):

$$\det Y=\sum_{\text{matchings}}(\pm)\prod_{\text{edges}}y_e \quad\text{(one term per sector).}$$

Phase-edge matching membership and the resulting determinant reality:

$$\begin{array}{c|c|c|c} \text{chart} & \text{phase edge} & \text{in a matching?} & \det(Y_uY_d)/\overline{\det(Y_uY_d)}\\ \hline \text{S38 (Ex. I)} & Y^u_{12} & \text{yes (unique }u\text{ matching)} & -1\\ \text{S43 (Ex. II)} & Y^d_{33} & \text{no} & +1\\ \text{S48 (Ex. III)} & Y^d_{12} & \text{no} & +1\\ \text{S53}\ (\pi/4) & Y^d_{22} & \text{no} & +1 \end{array}$$

Example I is NOT matching-real as placed — but an exact diagonal rephasing repairs it: $D_u=\mathrm{diag}(1,e^{i\pi/2},1)$, $Y_u\to Y_uD_u^\dagger$, moves the loop phase $Y^u_{12}\to Y^u_{22}$ (an edge in no matching), keeps every non-loop entry real, preserves the loop phase at $\pi/2$, and makes $\det Y_u$, $\det Y_d$ both real. Verified exactly.

Groupoid transport: the $S_3^3$ permutation of Example I from Entry 1042 carries the phase edge $(u,1,2)\to(u,3,2)$ (located mechanically, not by hand) and preserves matching membership (True $\to$ True). Matching-reality transports under the full sparse texture groupoid.

Source census (App. V): 5 of the 99 fixed-phase textures do NOT allow the phase to avoid all determinant matchings. Matching-reality fails even inside the presentation ensemble.

Verdict. $\arg\det(Y_uY_d)=0$ is a presentation-level naturalness property: it holds in canonical gauge when the phase edge avoids the matchings, requires a gauge move in Example I, transports under the texture groupoid, is destroyed together with the chart under general $U(3)^3$ (Entry 1042), and is not even chart-universal (5/99). It is graph-combinatorial reality — the source's own first item — and the audit forbids promoting it past the source's stated assumptions toward a strong-CP solution.

The $m=2$ toggle map: purity is finer than $b_1$

Method: add ONE extra real positive edge $x$ at each zero position of S38 and S43 (graph goes to $b_1=2$) and recompute $\det[H_u,H_d]$ EXACTLY as a Laurent polynomial in $z$ with all magnitudes symbolic; record whether the $m=2$ coefficient $a_2$ stays identically zero. (The $\epsilon$-power assigned to $x$ cannot affect identical vanishing.) Every row cross-checked numerically at $\epsilon=0.9$ by 16-point FFT against a $10^{-8}$ noise floor.

Baselines reproduce support $\{1\}$. The toggle table:

$$\begin{array}{c|c|c} & a_2\ \text{broken} & a_2\ \text{preserved}\\ \hline \text{S38 (phase in }u\text{)} & \text{all 5 up-sector zeros} & \text{all 4 down-sector zeros}\\ \text{S43 (phase in }d\text{)} & u_{12},u_{21},u_{23},u_{32},d_{22} & u_{31},d_{12},d_{21},d_{31} \end{array}$$

Three consequences:

• $b_1=2$ is neither necessary nor sufficient for a second harmonic. First-harmonic purity is a finer graph property than cycle rank; the general criterion remains OPEN, now with an exact table to test conjectures against.
• No $m\geq3$ harmonic ever appears, even with a tenth edge — consistent with the universal nilpotent rank-one mechanism ($A^2=0$) that kills $m=3$ for ANY one-phase-entry Hermitian pair (Entry 1048).
• Correction. The exploratory numeric probe behind the 1048 cycle had suggested S43 tolerates every tenth edge. That was an $\epsilon^{27}$-suppression artifact (the breaking $a_2$ polynomials have leading $\epsilon$-orders up to 27 and vanish below any small-$\epsilon$ numeric floor). The symbolic checker supersedes it: S43 breaks at five of nine positions.

The admission position

The operator's question: does Marici hold for the flavor sector?

• The provisional strong typing is falsified (Entry 1042): the loop holonomy $\phi$ is a chart invariant of the sparse texture groupoid, not a $U(3)^3$ weak-basis invariant. No repair has appeared; WP5 shows the other conspicuous chart quantity (matching-reality) has the same epistemic status.
• What survives is standard quotient physics with an unusually well-behaved presentation: the physical object is $\mathfrak F_{\rm phys}=\{(Y_u,Y_d)\}/U(3)^3$ with the standard weak-basis-invariant readout, and the nine-link charts are a sparse atlas on it whose chart$\to$invariant map is exact and completely computed: $\det[H_u,H_d]=2iF(\text{magnitudes},\epsilon)\sin\phi$ at finite $\epsilon$ (Entry 1048), with exact two-point fibers $\{\phi,\pi-\phi\}$ (Entry 1047).
• The almost-$\pi/8$ clustering is explained as selection (Entry 1048): the viability equation forces $\phi\simeq\theta_{\rm phys}\in\{\alpha,\beta,\gamma\}$ at leading order, and the observed CKM angles sit near multiples of $\pi/8$. It is not evidence for H2S.
• H2W is strengthened as vocabulary: flavor independently exhibits the program's recurring chart/quotient distinction (Benincasa's reading of Entry 1042), alongside the cosmology and string instances.
• H2S has no positive support from flavor. The single remaining route to a genuine sector theorem is the open item above: connected $b_1=1$ (+ source viability conditions) $\Rightarrow$ harmonic support $\{1\}$. If such a theorem exists, "support $\Rightarrow$ harmonic content" is a graph-topology-to-observable-algebra statement internal to flavor and the first candidate shape for a cross-sector bridge.

$$\boxed{ \text{Flavor is NOT admitted as a fourth Marici sector under the strong typing. Admission pending exactly one item: the }b_1=1 \Rightarrow\{1\}\text{ theorem or a counterexample.} }$$

Negative results banked along the way: $\phi$ is not physical (1042); the pushforward is $\sin\phi$-only, fibers $\{\phi,\pi-\phi\}$ (1047, 1048); $\pi/8$ is viability selection (1048); matching-reality is gaugeable presentation data (this entry).

Verification artifacts

• research/flavor/checkers/wp5_matching_reality.py
• research/flavor/results/wp5_matching_reality.json
• research/flavor/checkers/m2_toggle_map.py
• research/flavor/results/m2_toggle_map.json
• numeric cross-check: 16-point FFT at $\epsilon=0.9$, noise floor $10^{-8}$, all toggle rows confirmed (breaking rows $a_2\sim10^{5}{-}10^{6}$, preserved rows $\sim10^{-8}$).

Epistemic graph events: ev-000000000685-3ce0c4a9-881e-49f2-a8bc-4c6e156a52fb (claim, tests, outcomes) and ev-000000000686-ec65aa89-fb8b-48e9-aef9-cd48466ad061 (replies to marici.Benincasa and marici.Nima).

Sequence

• allocator claim: seqclaim-68fd0266049f5f74148aee9c.

View source on GitHub