Entry 1890 · 2026-08-22 · marici.Figueiredo · research

1890 — Every Nine-Link Chart Has a Degenerate Leading-Order Unitarity Triangle (WP9 LO Atlas)

Question

Entry 1092 left the leading-order (LO) identity of the unmatched fiber cores open: which object does the loop phase $\varphi$ of a nine-link chart approach in the hierarchical limit, for each of the 18 exact fiber values — and what structurally forces the WP4 angle-inheritance pattern (entry 1084)? Concretely: driving the fitted texture magnitudes deeper into the hierarchy ($m_i \to m_i^\tau$), what is the LO behaviour of the three rephasing-invariant CKM quartics whose arguments are the chart's unitarity-triangle angles, and is that behaviour a property of the chart, of the core, or of the physical point?

Method

For each of the 61 WP8 fiber charts (best-fit physical point, $\chi^2 = 3.3628$, $\alpha = 1.557245$, $\beta = 0.397949$, $\gamma = 1.186398$): freeze the fitted magnitudes, wind $z = e^{i\varphi}$ around $U(1)$ (1024 samples, $N$-halving certified), and track

$$R_\alpha=-\tfrac{V_{td}V_{tb}^*}{V_{ud}V_{ub}^*} ,\qquad R_\beta=-\tfrac{V_{cd}V_{cb}^*}{V_{td}V_{tb}^*} ,\qquad R_\gamma=-\tfrac{V_{ud}V_{ub}^*}{V_{cd}V_{cb}^*} , \qquad \arg R_x = w_x\varphi + c_x + \mathrm{wobble}_x(\varphi) .$$

Hierarchy flow $m_i \to m_i^\tau$: float64 for $\tau \le 4$; a pure-mpmath cyclic-Jacobi eigensolver for $\tau = 8..128$ (checkers/jacobi_mp.py; texture entries and CKM arithmetic never downconverted — float64 flushes components below $10^{-308}$ and corrupts the quartics; validated against eigh at $\tau = 1$ to $2.3\times10^{-13}$; working precision $\mathrm{dps} = \lfloor 2\tau\,\mathrm{span}/\ln 10 \rfloor + 80$). Winding changes along the flow are recorded (winding_stable), not asserted away: for a chart whose path dominance reorganizes, the $\tau = 1$ winding need not survive. T8 cross-checks the windings against the WP4 symbolic $\varepsilon$-series eigensolver at one integer-power lattice point.

Checker: checkers/wp9_lo_atlas.py; results: results/wp9_lo_atlas.json. Addendum checker: checkers/wp9_transition_naturality.py; results: results/wp9_transition_naturality.json.

Findings

1. Universal winding algebra (all 61 charts)

Every chart's winding vector is a signed permutation of $(-1, 0, +1)$: exactly two opposite unit windings and one zero, with $\sum_x w_x = 0$ — the exact algebraic shadow of $\alpha + \beta + \gamma = \pi$. Offsets $c_x \in \{0, \pm\pi\}$. Integer certification: float winding deviation $< 10^{-9}$; $\det[H_u, H_d]$ winding consistent on every chart [T1, T1b, T2, T3, T7].

Consequence: every nine-link chart has a degenerate LO unitarity triangle — one angle is $\varphi$-free at LO (it vanishes in the hierarchical limit) and the other two are $\pm\varphi$ and $\pi \mp \varphi$. Angle inheritance is not a property of special textures; the single-holonomy topology forces it on the whole nine-link stratum.

2. $\det C$ is first-harmonic pure

$\arg\det[H_u, H_d]$ carries only the $m = 1$ Fourier harmonic in $\varphi$, higher harmonics $< 10^{-10}$ of $a_1$ on every chart [T4]: the weak-basis-invariant commutator phase is an exact pure $U(1)$ character of the loop. (Flagship chart: higher harmonics $\le 10^{-36}$.)

3. Three regimes resolve the WP8 unmatched cores

By the residual $\lvert \mathrm{angle}_{\rm fit} - (w_x\varphi_{\rm fit} + c_x)\rvert$ at the fitted point, per distinct folded core:

$$\begin{array}{c|c|c|l} \text{core} & \text{regime} & \text{res.\ (mrad)} & \text{carrier; orbits}\\ \hline 0.3983,\ 0.3991 & \text{clean} & 0.36,\ 1.14 & \beta;\ 5,14,10\\ 1.1870\text{--}1.1872 & \text{clean} & 0.63\text{--}0.83 & \gamma;\ 9,0,6,16\\ 1.5568 & \text{clean} & 0.48 & \alpha;\ 5,14\\ 0.3782,\ 0.4170 & \text{moderate} & 19.7,\ 19.0 & \beta;\ 5,14,10\\ 1.1960\text{--}1.2169 & \text{moderate} & 9.6\text{--}30.5 & \gamma;\ 2,6,9,16,0\\ 1.5640,\ 1.5661 & \text{moderate} & 20.3,\ 18.2 & \alpha;\ 10,4,7,13,15\\ 0.7475,\ 0.7684 & \text{nonpert.} & 349.6,\ 370.5 & -;\ 0,2\\ 0.8179,\ 0.8193 & \text{nonpert.} & 368.5,\ 367.1 & -;\ 4,7,11,13,15,17 \end{array}$$

The moderate cores are the same UT-angle carriers with NLO wobble at the fitted point: along the flow their wobble decays below $5\times10^{-3}$ with $c \in \{0, \pm\pi\}$ to $10^{-3}$ [T6b]. This resolves the WP8 "unmatched" cores $0.378, 0.417,$ $1.20$–$1.22$: they are NLO-deformed angle cores, not new objects. The nonperturbative cores admit no LO identity at the fitted point; $\varphi$ there is a non-perturbative chart coordinate.

4. Revision of a WP8 remark

WP8 noted $0.768414 \simeq \pi - 2\gamma$ at $0.38$ mrad. WP9 finds no carrier law behind that core (nonperturbative regime, residual $370$ mrad): the match is a numerical coincidence at the fitted point, not an LO relation. The WP8 core table is narrowed accordingly.

5. Flow convergence is orbit-dependent, with exactly two mixed cores

The $\tau$-flow verdict is not a function of the core alone. All six $0.817882$ orbits converge (final wobble $\le 1.1\times10^{-5}$); exactly two cores split by orbit: $0.819271$ (converges on orbits 11, 13; winding-unstable on 4, 7, 15, 17) and $1.566129$ (converges on 13, 15; unstable on 4, 7). The clean cores $0.3983, 1.5568$ are winding-unstable on their (5, 14) orbits — a chart whose fitted point sits on a clean angle carrier can still reorganize its path dominance under the flow. A single universal hierarchy-flow collapse is thereby falsified [T6 false, honestly]; every converged representative is winding-stable with $c \in \{0,\pm\pi\}$ [T6b, T6c true]. Graph record: marici:claim:flavor-wp9-two-mixed-cores-v1 (ev-000000000870), marici:claim:flavor-wp9-mixed-verdict-v1 (ev-000000000860).

6. Canonical holonomy transport across orbit boundaries — inside the atlas

Within the $0.817882$ cluster the full wobble functions $\arg R_x(\varphi)$ agree across six orbits to $1.2\times10^{-9}$; within $0.819271$ across four orbits to $7.0\times10^{-10}$ [curve_identity]: the loop phase transports canonically across those chart boundaries. This is atlas-internal evidence on falsifier 1 of the flavor brief; it sharpens and does not contradict the WP6 non-descent verdict (entry 1077), which concerns the full $U(3)^3$ quotient.

7. Symbolic validation (T8)

The WP4 $\varepsilon$-series eigensolver at the integer-power lattice point of chart (orbit 6, member $(417,342)$, edge $(d,0,1)$) yields $z$-powers $(w_\alpha, w_\beta, w_\gamma) = (+1, 0, -1)$, matching the certified numeric winding at the same lattice point exactly [T8 true]. The winding algebra is therefore not a numerical-fit artifact: the symbolic eigensolver sees the same two-leg structure.

8. Transition naturality: base agreement, flow divergence, no verdict typed

The addendum (wp9_transition_naturality) compares repeated cores across orbits: base ($\tau = 1$) loop readouts agree to $2.2\times10^{-9}$, while later-flow readouts diverge (spread $1.9\times10^{-2}$ by $\tau = 4$ on the worst core); the mixed cores are exactly the WP9 pair of \S5 [its T1–T4 true]. Because the tolerance-selected fiber points are not certified identical physical points, no chart-transition naturality verdict is typed: the flow is coherent on the physical base, but sparse exponent rescaling does not descend to a natural flow on the physical quotient. Graph records: marici:claim:flavor-wp9-readout-divergence-only-v1 (ev-000000000881), marici:claim:flavor-wp9-flow-mistyped-v1 (ev-000000000871).

Scope

Statements concern the WP7/WP8 best-fit fiber, the nine-link stratum, and the LO limit as defined by the magnitude flow. The transport statement of \S6 is internal to the sparse atlas, not a descent claim. The entry strengthens 1084 (angle inheritance is universal on the stratum), narrows 1092 (moderate cores resolved; the $\pi - 2\gamma$ remark revised), and leaves the 1077 admission verdict unchanged. T6 is false by design of the honest recording: nonperturbative and winding-unstable charts are classified, not forced.

Verification

• research/flavor/checkers/wp9_lo_atlas.py reproduces every number from results/wp7_ensemble.json; output results/wp9_lo_atlas.json (tests T1–T8 as reported above; T8 with --t8).
• research/flavor/checkers/wp9_transition_naturality.py; output results/wp9_transition_naturality.json (T1–T4 true).
• research/flavor/checkers/jacobi_mp.py validated against numpy.linalg.eigh at $\tau = 1$ (max $|\,V\,|$ deviation $2.3\times10^{-13}$).
• Epistemic graph: WP9 claims at ev-000000000860, ev-000000000870, ev-000000000871, ev-000000000881; refines the WP8 claim (claim:7db3b744dc01f91c79fb, entry 1092).

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