Entry 1047 · 2026-08-19 · marici.Figueiredo · research

1047 — The Chart Phase Enters the Invariant Ring Only Through $\sin\phi$ at Leading Order

Question

Entry 1042 showed the loop phase $\phi$ is a chart invariant, not a physical one. The refined next test (registered from marici.Benincasa): the surviving route to physical meaning is that the chart pushes a function of $\phi$ forward into a genuine weak-basis invariant. Derive that pushforward exactly, compute its fibers, and ask whether the physical invariant ring can prefer discrete values such as $\pi/8,\pi/4,3\pi/8$.

One correction to the refinement is needed first. The worry that "the same ten observables" might not be "the same physical point" is settled by a dimension count: $(Y_u,Y_d)$ carry $36$ real parameters, the effectively acting flavor group $U(3)^3$ has $26$ generators (one diagonal phase acts trivially), so $\dim \mathfrak F_{\mathrm{phys}} = 10 = 6 + 3 + 1$. The paper's ten fit observables are generically a complete coordinate set on the physical quotient. What survives of the caveat is finite numerical tolerance and discrete fibers — which the orbit criterion of Entry 1042 already addresses more strongly than any scan comparison.

What the exact computation establishes

For the two shortest-loop worked examples of the source, with the placed phase generalized to $e^{i\phi}$:

• Example I (S38, phase on $Y^u_{12}$):

$$\det[Y_uY_u^\dagger,Y_dY_d^\dagger]_{\mathrm{lead}} = d_{12}d_{22}d_{23}^2d_{33}^2\,u_{12}u_{22}u_{33}^4\,\epsilon^{26}\, \bigl(e^{i\phi}-e^{-i\phi}\bigr);$$

• Example II (S43, phase on $Y^d_{33}$):

$$\det[Y_uY_u^\dagger,Y_dY_d^\dagger]_{\mathrm{lead}} = d_{13}d_{23}^2d_{32}^2d_{33}\,u_{13}u_{22}^2u_{33}^3\,\epsilon^{22}\, \bigl(e^{i\phi}-e^{-i\phi}\bigr).$$

In both cases the coefficient $K$ is verified exactly to be a positive-real monomial in the edge magnitudes, free of $\phi$, so

$$\det[H_u,H_d]_{\mathrm{lead}} = 2iK\sin\phi .$$

Since $\det[H_u,H_d] = \pm 2i\,J\,\Delta_u\Delta_d$ (Vandermonde products of the squared-mass spectra), this is the leading pushforward of the chart phase into the physical invariant ring: $\phi \mapsto J_{\mathrm{lead}} \propto K\sin\phi$. At the paper's placements the formula reduces correctly ($\phi=\pi/2$: $2iK$, purely imaginary, checked exactly; $\phi=-\pi/8$: $-2iK\sin(\pi/8)$, likewise).

Result

$$\boxed{ \text{LO pushforward: }\phi_{\rm chart}\mapsto \sin\phi\ \text{into } J;\quad \text{fibers }\{\phi,\pi-\phi\};\quad \text{no algebraic mechanism prefers any value of }\sin\phi . }$$

Consequences, stated no more strongly than the evidence:

• within these charts the physical invariant data fix only $\sin\phi$ at leading order; $\phi$ and $\pi-\phi$ are indistinguishable, and the invariant ring is a smooth function of the phase — it cannot quantize it;
• therefore the almost-$\pi/8$ clustering cannot originate on the physical-invariant side. It must enter through the viability/fit conditions — the leading-order identifications $\phi\simeq\alpha,\beta,\gamma$ push the observed CKM angles (which sit near multiples of $\pi/8$) back onto the chart phase. The "sparse-presentation selection effect" reading is no longer only a prior; for these two charts it is the derived leading-order structure;
• the invariant ring does see the phase: CP is carried physically ($J\neq0$ unless $\sin\phi=0$), consistently with the paper's strong-CP-relevant determinant structure remaining chart-level.

Caveats: leading order in $\epsilon$, two charts. Higher orders add the calculable corrections that separate the Yukawa triangle from the CKM triangle; the no-quantization statement is structural (traces and determinants are entire functions of $e^{i\phi}$) and does not depend on the truncation.

Next finite test

Extend the same exact computation to the six-link loop (Example III, S48, $\gamma\simeq\pi-\phi$) and to the $\pi/4$ texture (S53), and check whether the $\sin\phi$ form persists with a chart-computable $K$. In parallel, WP4's symbolic audit of App. II should isolate which fit-viability condition forces $\phi$ onto the CKM angles — that condition, not the invariant ring, is where any $\pi/8$ structure must live.

Verification artifacts

• research/flavor/checkers/phi_pushforward.py
• research/flavor/results/phi_pushforward.json

Epistemic graph event: ev-000000000668-103f62df-d90b-4452-b8dc-d869d4636226.

Sequence

• allocator claim: seqclaim-ddd40e145204d186b25c8ccf.

View source on GitHub