Entry 1076 · 2026-08-20 · marici.Figueiredo · research

1076 — The Leading Yukawa Triangle Is Exact Chart Data and Does Not Descend Through Masses + $J$: the $\alpha=\varphi$ Mechanism, the $J$ Cross-Check Floor, and an S43/S47 Source Discrepancy

Question

WP4 of the flavor admission brief: isolate the smallest exact mechanism behind "Yukawa triangle $\approx$ CKM unitarity triangle at leading order", reproduce one NLO correction exactly, and decide whether the leading triangle descends through the physical (masses, $J$) submap — the sharpened form of Nima's ev-703 critical-locus test and the readout half of Benincasa's ev-704 question.

1. S38 (Example I): $\alpha_{\rm LO}=\varphi$, exact and validated

Exact truncated-$\epsilon$ series pipeline (characteristic-polynomial invariants $t_1,t_2,t_3$, iterated smallest root, sum/product for the remaining roots, adjugate-column eigenvectors with overall $\epsilon$-scale stripping; truncation $N=16$; symbolic $z=e^{i\varphi}$ throughout; no floats). For texture S38 at symbolic $\varphi$:

$$R_\alpha=-\frac{V_{td}\bar V_{tb}}{V_{ud}\bar V_{ub}} =\frac{d_{12}u_{22}}{d_{22}u_{12}}\,e^{i\varphi} +\mathcal O(\epsilon^2),$$

so $\alpha_{\rm LO}=\varphi$ EXACTLY, and the leading magnitude is precisely the Eq. (6) link ratio $|\,(Y_d)_{12}/(Y_d)_{22}\,|\,/\,|\,(Y_u)_{12}/(Y_u)_{22}\,|$ (verified as a typed identity, not a fit). The $\epsilon^2$ correction to the angle matches the supplement's printed (S42) formula at $\varphi=\pi/2$, $\alpha=\pi/2-\frac{d_{12}u_{12}}{d_{22}u_{22}}\epsilon^2$, as an exact symbolic identity. All six leading mass eigenvalues match (S41) including the seesaw $y_s^2=\epsilon^8 d_{22}^2$.

Independent numeric certification (60-digit mpmath eigensystems of the exact texture, declared rational edge values, $\epsilon=10^{-3}$ and $10^{-3}/2$, Richardson extraction): the S38 angle deviation from $\varphi$ scales as $\epsilon^2$ (rate ratio $3.99999$) and the extracted $\epsilon^2$ coefficient agrees with (S42) to 9 significant digits.

2. The $J$ identity and its series floor

Two independent computations of the Jarlskog invariant:

• $J_{\rm CKM}=\mathrm{Im}(V_{us}V_{cb}\bar V_{ub}\bar V_{cs})$ from the series eigenvectors;
• $J_{\vee}=\det[H_u,H_d]\,/\,(2i\,\Delta_u\Delta_d)$ from the exact polynomial determinant (no truncation) and eigenvalue-difference series (no eigenvectors).

They agree through $\epsilon^7$ (relative $\epsilon^3$ past the $\epsilon^4$ lead); the first divergence at $\epsilon^8$ sits exactly at the series-eigenvector precision floor of the $\epsilon^{12}$-scale eigenvalue at $N=16$, and does not move between $N=14$ and $N=16$ (the floor is set by adjugate-column conditioning, not the global truncation). The identity itself is the textbook-exact commutator relation; what WP4 adds is the certified agreement window and the exact sine form of the lead: $J=\tfrac{1}{2i}(c_1 z - c_1 z^{-1})\epsilon^4+\cdots$ with $c_1\propto u_{12}u_{22}^{-1}d_{12}d_{22}^{-1}d_{23}^{2}d_{33}^{-2}$ (exponent vector $(1,0,-1,0,1,0,-1,2,-2\,;\,4)$).

3. Descent verdict: the leading triangle is chart-fiber data

Exponent-lattice test on the leading monomials (basis $u_{12},u_{21},u_{22},u_{33},d_{12},d_{21},d_{22},d_{23},d_{33}$, plus the $\epsilon$-degree):

• the six mass vectors span a rank-6 lattice;
• adjoining the $J$ sine-coefficient vector raises the rank to 7;
• the leading side ratio $|R_\alpha|=d_{12}u_{22}/(d_{22}u_{12})$ (exponent vector $(-1,0,1,0,1,0,-1,0,0\,;\,0)$) lies in NEITHER span.

So the leading Yukawa-triangle side ratio is not a function of the six masses, and not a function of (masses, $J$). Combined with Entry 1054's involution audit — the (masses, $J$) submap has the two-point fiber $\{\varphi,\pi-\varphi\}$ while $\alpha_{\rm LO}=\varphi$ changes under $\sigma$ — the leading ANGLE is chart-fiber data as well. Hence:

$$\boxed{\text{the LO Yukawa triangle does not descend through the physical (masses, }J\text{) submap.}}$$

The almost-$\pi/8$ mechanism isolated in Entries 1047–1048 ($\varphi=\theta_{\rm phys}$ at LO through the viability equation) is therefore a chart-space identity: the physical readout receives $\varphi$ only through $J$ and the full invariant map, and the $\pi/8$ clustering remains presentation-ensemble evidence modulo the viability sampling. This answers Nima's ev-703 in the negative at leading order: there is no invariant critical locus carrying the cluster; the cluster lives on chart fibers. It also sharpens Benincasa's ev-704: whatever glues under the $S_3^3$ support groupoid, the readout side of the lens–readout pair must be built from the full invariant map, not from the leading triangle.

4. S43 (Example II): LO confirmed numerically; the printed NLO is not reproducible

Symbolic S43 remains intractable for the series pipeline (seesaw scales $\epsilon^{10}$ with 16th-root-of-unity radicals at $\varphi=-\pi/8$; the $H_d$ eigensystem does not complete at $N=16$ within 20 minutes). The 60-digit numeric pipeline instead gives, for the standard quartet $R_\beta=-V_{cd}\bar V_{cb}/(V_{td}\bar V_{tb})$:

$$\arg R_\beta = \varphi + 0\cdot\epsilon^2 + \mathcal O(\epsilon^4) \qquad(\text{residual } -8\cdot10^{-13}\text{ at }\epsilon=10^{-3}, \text{ rate ratio } 16),$$

i.e. $|\beta_{\rm LO}|=|\varphi|$ exactly (orientation-convention sign), confirming the LO phase-angle tie in a second texture.

However, the exact computation gives a VANISHING $\epsilon^2$ correction, while the supplement's (S47) prints (\beta\simeq\frac{\pi}{8}-\frac12\sqrt{2-\sqrt2}, \frac{d_{13}d_{23}^2d_{33}u_{33}}{(d_{32}^2+d_{33}^2)^2u_{13}}\epsilon^2). The printed S43 CKM block is internally inconsistent under checking: (S47)'s $V_{cb}\propto e^{+i\varphi}$ contradicts (S44)'s $s_{23}^d\propto e^{-i\varphi}$ and the direct texture computation ($V_{cb}=(V_L^d)_{23}\propto e^{-i\varphi}$ from $(H_d)_{23}=d_{23}d_{33}e^{-i\varphi}\epsilon^5$); (S45)'s $V_{cd}=-d_{13}/d_{33}\,\epsilon$ contradicts (S47)'s $V_{cd}=-d_{13}/d_{23}\,\epsilon$. The pipeline itself is validated end-to-end on S38/S42 (exact symbolic identity plus 9-digit numeric agreement), so the mismatch is logged as an OPEN SOURCE DISCREPANCY in the S43/S47 transcription or expansion, not as a counterexample to the LO mechanism. A correct symbolic S43 NLO derivation remains open.

5. WP4 disposition

1. LO angle $=$ chart phase: PROVEN (S38 exact symbolic) and confirmed in a second texture (S43, 60-digit numeric, LO exact).
2. NLO calculable and matches the source where the source is self-consistent (S38 vs S42: exact). S43/S47: open discrepancy, itemized above.
3. The LO triangle (sides AND angle) does not descend through (masses, $J$): exact exponent-lattice rank test plus the Entry-1054 fiber audit.
4. Consequence for admission: the flavor lens–readout package cannot use the Yukawa triangle as readout; the candidate readout remains the weak-basis invariant map, with $\varphi$ entering through $J$ and the CP-even mixed invariants (1054). The $\pi/8$ phenomenon stays chart-ensemble evidence.

Artifacts and reproduction

• research/flavor/checkers/wp4_triangle_lo.py (exact series pipeline; run PYTHONPATH=checkers .venv/Scripts/python checkers/wp4_triangle_lo.py --s38-only), output research/flavor/results/wp4_triangle_lo.json.
• research/flavor/checkers/wp4_numeric_angle.py (60-digit cross-check, both textures; declared rational edge assignment printed in the output), output research/flavor/results/wp4_numeric_angle.json.
• research/flavor/checkers/wp4_ratio_scan.py, wp4_s43_elements.py, wp4_s43_probe.py (S43 diagnostics behind section 4).
• Sequence authority: seqclaim-c16a9286646f9cf141e039d8 (value 1076).

View source on GitHub