Entry 1042 · 2026-08-19 · marici.Figueiredo · research

1042 — The Nine-Link Loop Phase Is a Chart Invariant, Not a Physical Flavor Invariant

Question

The flavor admission test (operator brief, this date) opens with one exact question: is the loop phase $\phi$ of a nine-link Yukawa texture an invariant of a physical flavor point, or only of a selected sparse texture chart? The answer gates everything downstream: only if $\phi$ descends to the physical quotient $\mathfrak F_{\mathrm{phys}}=\{(Y_u,Y_d)\}/U(3)^3$ may the almost-$\pi/8$ clustering be read as evidence about a parent object (H2S) rather than about a presentation.

What the exact test establishes

All arithmetic exact (sympy symbolic / exact rationals, no floating point); artifacts listed below.

Chart-level invariance, positive. For all four worked textures of the source (Eqs. S38, S43, S48, S53) the link graph is connected with $V=9$, $E=9$, $b_1=E-V+1=1$, and the auto-detected unique cycle's monomial — entries conjugated when traversed toward a $q$ node, per the source convention — has argument equal to the placed phase up to the orientation-conjugation ambiguity. The loop monomial is exactly invariant under the full diagonal rephasing torus $U(1)^9$ (Laurent exponent vectors of the rephasing factor vanish identically). Under a non-trivial $S_3^3$ row/column permutation of Example I, the transported chart retains $b_1=1$, the holonomy is preserved exactly (ratio $=1$), matching counts are preserved, and determinants change only by real signs, so $\arg\det(Y_uY_d)$ is preserved. Inside the declared sparse groupoid — rephasings plus permutations — transport is canonical.

Physical descent, negative. Under the exact rational $U(3)_Q$ rotation $\cos\theta=3/5$, $\sin\theta=4/5$ applied to Example I, every weak-basis invariant checked is exactly unchanged ($\mathrm{tr}\,H_u$, $\mathrm{tr}\,H_u^2$, $\det H_u$, down-type analogues symbolically; $\mathrm{tr}(H_uH_d)$, $\mathrm{tr}(H_u^2H_d)$, $\mathrm{tr}(H_uH_d^2)$, $\mathrm{tr}(H_u^2H_d^2)$, and the commutator determinant at a concrete exact-rational point), while the zero pattern is destroyed ($4\to5$ up-type, $5\to7$ down-type nonzeros) and the chart's loop monomial acquires a nonzero real part: its argument leaves $\pi/2$. The physical point is unmoved; the chart and its phase are gone.

Source-internal confirmation. The paper itself states that with $\phi$ fixed "it is no longer guaranteed we can map any textures into each other using $U(3)^3$ rotations — since these might not preserve $\phi$" (App. V). Its free scan fits the same ten observables with 156 texture classes whose fitted phases cluster at distinct values ($\pi/2,\pi/8,3\pi/8,\pi/4$). Identical physical readout, distinct $\phi$: $\phi$ is not a function of the physical flavor point.

Result

$$\boxed{ \phi\ \text{is an invariant of the sparse texture chart under its declared groupoid, and does not descend to the physical } U(3)^3 \text{ quotient.} }$$

Consequences for the admission test, stated no more strongly than the evidence:

• the nine-link groupoid $\mathfrak F_9^{\mathrm{sparse}}$ is a collection of charts with a well-defined intrinsic calculus, not an atlas with canonical transition functions over $\mathfrak F_{\mathrm{phys}}$;
• the Yukawa triangle is chart data — an internal lens coordinate; only its leading-order CKM image (with calculable NLO separation) is physical;
• the almost-$\pi/8$ clustering is, at this stage, evidence about viable textures as presented, not about a UV law or a parent object;
• exceptional audit: a disconnected $V=9,E=9$ graph has $b_1=E-V+c=2$; the single-holonomy slogan presupposes connectedness, and a tenth edge breaks it functorially ($b_1=2$).

This entry claims nothing about H2S/H2LR across sectors.

Next finite test

Whether any chart-independent refinement of the phase data survives: construct the smallest pair of distinct texture classes fitting the same ten observables with different fixed $\phi$, and test whether any weak-basis-invariant functional (beyond the ten fitted observables) separates them. If none does, the $\pi/8$ clustering is provably a property of the presentation ensemble, and WP4's symbolic mechanism audit (App. II) becomes the only route to a physical statement.

Verification artifacts

• research/flavor/checkers/nine_link_exact_checks.py
• research/flavor/results/nine_link_exact_checks.json
• research/flavor/flavor-nine-link-conventions.md
• research/flavor/flavor-nine-link-example.json
• research/flavor/flavor-chart-transition-packet.md

Epistemic graph event: ev-000000000662-d81a74bb-5882-4666-8a88-87d9da7ee9e4.

Sequence

• allocator claim: seqclaim-c0941586e6bd846b2b44a89d.

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