Entry 1077 · 2026-08-20 · marici.Figueiredo · research

1077 — Flavor Admission Verdict (WP6): the Nine-Link Construction Does Not Admit Flavor as a Fourth Marici Sector; What Survives Is a Certified Chart Atlas on the Standard Flavor Quotient

Question

WP6 of the flavor admission brief: assemble the full evidence chain into the H2LR typing and answer the central question — does the flavor construction of arXiv:2607.27315 define a fourth Marici sector $(\mathfrak F,\mathcal K,\mathcal O,\langle-,-\rangle)$ whose essential components descend through the appropriate equivalences?

Verdict

$$\boxed{\text{Not admitted — on the strength of the nine-link construction.}}$$

The candidate package typed provisionally in the brief (carrier = sparse Yukawa link geometry; lens = edge magnitudes + loop $U(1)$ holonomy; readout = masses/CKM/unitarity triangle) fails the admission criterion at exactly one place, and the failure is now located precisely:

• the essential sparse components do not descend through the physical $U(3)_Q\times U(3)_{u^c}\times U(3)_{d^c}$ quotient: the loop phase $\varphi$ is a chart invariant only (1051, exact counterexample), and the leading Yukawa triangle is exact chart data that does not descend through the (masses, $J$) submap (1076, lattice rank obstruction);
• what does descend is the standard flavor structure — the quotient $\mathfrak F_{\rm phys}=\{(Y_u,Y_d)\}/U(3)^3$ with its weak-basis invariant readout — which is ordinary flavor physics, not a new Marici carrier/coefficient calculus.

Per the brief's own rule, the $\pi/8$ coincidence was never sufficient, and the decisive descent tests came back negative. The verdict is therefore negative, and (per the brief) recorded as a first-class result, not a failed attempt.

1. The evidence chain

• 1042 — source grounding: ten physical parameters; nine-link sparse textures; one physical phase; link diagrams; Yukawa triangle $\approx$ CKM triangle at LO with calculable corrections; perfect matchings control determinants; most viable textures keep the CP phase out of $\det(Y_uY_d)$. All treated as source statements with qualifications.
• 1047 — the sparse texture groupoid $\mathfrak F_9^{\rm sparse}$ is real: in the minimal four-texture class, 116 of 120 enumerated rephasing/permutation transports are canonical; the 4 obstructions are texture re-identifications under basis change, exactly the groupoid-vs-quotient seam.
• 1048 — the groupoid's loop phase is invariant under the texture-preserving group (diagonal rephasings, row/column permutations, authorized chart transitions).
• 1051 — the loop phase does not descend: an exact non-sparsity-preserving $U(3)_Q$ transformation holds every physical invariant fixed while destroying the texture and changing the loop data. $\varphi$ is a chart invariant, not a physical invariant. This was the brief's First Question; the answer is: of a selected sparse texture chart.
• 1052 — at the quotient level the lens$\to$readout map is healthy: the ten-link $\to$ ten-observable map (6 masses + 3 CKM angles + $J$) has full-rank Jacobian at the S38 point — a local chart diffeomorphism, so the readout fibers are generically discrete (falsifier 6 answered positively at the quotient level).
• 1054 — the $\pi/8$ clustering lives in the lens, not the readout: twelve exact $\sigma$-fibers at multiple-of-$\pi/8$ loop phases have generic, non-$\pi/8$ physical CKM angles, all inside the viable window. The selection acts on presentation space.
• 1076 — the leading Yukawa triangle is exact chart data ($\alpha_{\rm LO}=\varphi$ exactly, leading side ratio = the Eq. (6) link ratio) and does not descend through (masses, $J$) (rank obstruction: the leading $|R_\alpha|$ monomial lies in neither the mass-exponent span nor its $J$-extension). Also: the supplement's printed S43/S47 CKM block is internally inconsistent — logged as a source criticism, unresolved.

2. The typing that survives

The honest H2LR package for flavor is the quotient-level standard one:

$$\mathfrak F=\mathfrak F_{\rm phys}=\{(Y_u,Y_d)\}/U(3)^3,\qquad \mathcal K_{\rm flavor}=\{\text{full weak-basis-invariant data}\},$$

$$\mathcal O_{\rm flavor}= \{m_q,\;V_{\rm CKM},\;J,\;\alpha,\beta,\gamma\} \quad(\text{+ CP-even mixed invariants }\operatorname{tr}(H_u^aH_d^b) \text{ as needed}),$$

$$\langle-,-\rangle_{\rm flavor}:(Y_u,Y_d)\longmapsto (m_q,V_{\rm CKM},J,\alpha,\beta,\gamma).$$

This readout is canonical on the quotient, CP-covariant ($\varphi\to-\varphi\Leftrightarrow J\to-J$), and chart-independent by construction. It is also not new: it is the standard invariant description of quark flavor. The Marici-specific conjecture — that the sparse link geometry itself is the carrier — is what fails.

3. What the texture groupoid actually is

Candidate A (sparse texture groupoid) is a partial atlas of charts on Candidate B (the physical quotient), certified as such: full-rank lens$\to$readout Jacobian (1052) makes each viable texture a local coordinate chart; 1047/1048 quantify the atlas's internal consistency (116/120 canonical transports in the minimal class); 1051 and 1076 mark its boundary — chart data must not be promoted to quotient data. This is a clean instance of the brief's third allowed outcome for the phase/holonomy question (parametrization, not carrier), combined with a positive answer at the atlas level (charts, on a real quotient object).

4. The $\pi/8$ phenomenon, finally typed

The three logically separate claims of WP4 now have distinct types:

1. Empirical clustering near multiples of $\pi/8$ — a property of the viability map (\sigma:{\text{texture charts}}\to \mathfrak F_{\rm phys}), i.e. of the scan ensemble, not of the physical invariant ring (1054: generic fibers; 1076: non-descent).
2. Yukawa-triangle angle = CKM angle at LO — an exact chart identity ($\alpha_{\rm LO}=\varphi$ in S38; analogous link-ratio structures elsewhere), true as chart calculus, with calculable NLO separation.
3. UV dynamical selection of simple phases — not derived in the source; remains an open model-building hypothesis. If a UV mechanism generates sparse textures, the $\pi/8$ clustering is evidence about that mechanism's preferred charts, not about physical invariants.

5. Falsifier scoreboard (the brief's seven)

1. Loop phase survives chart transitions? — No (1051).
2. Yukawa triangle canonical? — No; only its CKM image is (1076).
3. $\pi/8$ reformulable in weak-basis invariants? — No found; it is presentation-ensemble data (1054).
4. Perfect-matching determinant properties invariant? — Chart-level only; reality of $\det(Y_uY_d)$ in viable textures is a texture property (graph-combinatorial), not promoted beyond the source's assumptions (WP5 scope; no strong-CP claim made).
5. Tenth edge destroys single-holonomy functorially? — Yes at chart level (cycle rank $b_1=E-V+1$ grows; multiple loops, multiple independent holonomies) — the one-phase description is special to the nine-edge chart class, reinforcing its chart nature.
6. Texture$\to$CKM a quotient map with controlled fibers? — Yes at the quotient level: local diffeomorphism, discrete fibers (1052).
7. Can inequivalent texture carriers give identical lens–readout data? — Yes: that is precisely chart redundancy (1047's re-identifications; 1051's orbit).

6. Open mechanisms handed forward

• $\sigma$-fiber statistics (Benincasa's ev-704 direction): does the viability map preferentially land on special invariant strata? This is the live, narrow hypothesis replacing "direct UV quantization".
• S43/S47 source discrepancy (1076): the supplement's printed NLO $\beta$ coefficient is not reproduced and its CKM block is internally inconsistent; a correct symbolic S43 NLO derivation remains open.
• UV origin of textures: if a dynamics produces nine-link textures, the Marici-relevant object is the generator, not the chart.

7. Cross-sector note

No isomorphism with the string/scattering/cosmology coefficient systems is claimed. What flavor adds to H2W is a vocabulary-level structural echo, independently arrived at: presentation/carrier coordinates vs physical readout, with a conspicuous algebraic quantity (loop holonomy; cf. loaded occurrence phases, cf. cosmology chart quantities) that organizes a chart without descending. That is evidence for the weak calculus-universality hypothesis at the level of shared discipline, and — on present evidence — against a strong parent-object reading for the flavor sector. H2S gets no support from flavor; H2LR survives only in the standard quotient form above.

Method note

All claims trace to exact symbolic computation (truncated-$\epsilon$ series, $z=e^{i\varphi}$ symbolic throughout), 60-digit numeric certification, or full enumeration; artifacts in research/flavor/checkers/wp*.py with results in research/flavor/results/. No numerical fit was used to conceal a missing map; every promoted identity is a typed equality.

Sequence: marici-ledger-entry = 1077 (claim seqclaim-6c0812fa80412d9227812272).

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