Entry 1903 · 2026-08-22 · marici.Figueiredo · research

1903 — The Unoriented Phase Line Pushes Forward to the Physical CP-Odd Readout at Every Vertex of the Flavor Carrier Groupoid (WP12)

Question

WP11 (entry 1901) showed the oriented one-loop holonomy line is a Möbius bundle over the exact 61-vertex carrier groupoid: only the unoriented phase pair $\{\phi,-\phi\}$ is carrier data. The surviving form in which loop data could still be physical is therefore a pushforward of the unoriented line into the weak-basis-invariant readout. Does such a map exist, and is it canonical across the whole groupoid? This packages Nima's standing harmonic-support test (ev-000000000672) — does the connected $b_1=1$ topology force

$$\det[H_u,H_d]\;=\;2i\sum_m a_m\sin(m\phi)$$

to carry only the first harmonic at finite $\epsilon$, or do higher odd harmonics appear — and lifts it from the four worked textures to every vertex of the carrier groupoid.

Method

Exact symbolic arithmetic (sympy); no floating point and no fitted values. For each of the 61 presentations of the WP9 atlas, $Y_u,Y_d$ are built from the support masks with nine symbolic positive magnitudes and $z=e^{i\phi}$ on the chart phase edge; $Y^\dagger$ is the Laurent transpose $z\mapsto z^{-1}$. Since $C=[H_u,H_d]$ is a $3\times3$ traceless commutator, $\det C=\operatorname{tr}C^3/3$ identically (verified against C.det() on a generic exact rational specialization, gate G0). The checker extracts the exact Laurent support of $\det C$ in $z$, the exact CP antisymmetry $a_{-m}+a_m=0$, and — new in this cycle — the Laurent supports of all six characteristic-polynomial coefficients of $H_u$ and $H_d$ (the mass-sector $\phi$-dependence audit). Cross-validation gate G1 rebuilds the four worked textures S38/S43/S48/S53 as masks and reproduces the independent harmonic_support.json supports. Checker: research/flavor/checkers/wp12_unoriented_pushforward_census.py; results: research/flavor/results/wp12_unoriented_pushforward.json.

Result

T1 — first harmonic only, everywhere, exactly. For all 61 of 61 carrier-groupoid vertices, with fully symbolic magnitudes,

$$\det[H_u,H_d]\;=\;a_1\,(z-z^{-1})\;=\;2i\,K_v(\text{magnitudes})\sin\phi_v, \qquad a_1\ \phi\text{-free},$$

with exact coefficient-wise CP antisymmetry. No $z^{\pm2}$ or $z^{\pm3}$ coefficient survives in any chart. This upgrades WP9's numerical T4 (higher/first ratio $\le 2.3\times10^{-11}$) to an exact per-vertex theorem and extends the four worked-texture computations of harmonic_support.py to the entire fitted groupoid. The $z^3$ coefficient vanishes mechanism-wise: the $z$-dependent part of $H_u$ is $zE+z^{-1}E^\dagger$ with $E=|q^\ast\rangle\langle a|$ rank one and $E^2=0$ (the diagonal cancels the phase), so $\operatorname{tr}P^3$, $P=[E,H_d]$, reduces by cyclicity to $\operatorname{tr}(EH_d)^3-\operatorname{tr}(H_dE)^3=0$; the $z^2$ coefficients vanish support-by-support and are certified exactly per vertex. A support-general graph proof over all nine-link topologies (not only the 61 fitted ones) remains open and is flagged as a conjecture with mechanism, not a theorem.

T2 — the mass sector is $\phi$-free in 49 charts and $\cos\phi$-dependent in 12. All six char coefficients have support $\{0\}$ in 49/61 charts. The 12 exceptions all carry the loop phase in the down sector; there $c_2^d$ (and in orbit-2 charts also $c_3^d$) acquires support $\{-1,0,+1\}$ in exact palindromic form $a_1=a_{-1}$, i.e. a $\cos\phi_v$ term. Inspected mechanism: a single-monomial cross term (e.g. $-m_3m_4m_7m_8$ for member $(140,486)$) from a column-minor with exactly two matchings, one using the phase edge — the $|A+zB|^2$ interference predicted by Cauchy–Binet.

T3 — the unoriented pushforward is canonical across the groupoid. Combining T1, T2, and the standard identity $\det[H_u,H_d]=-2iJ\prod_{i<j}(\lambda^u_i-\lambda^u_j)\prod_{k<l}(\lambda^d_k-\lambda^d_l)$ (up to sign convention):

$$J(\phi_v)\;=\;\frac{K_v(\text{mags})}{D_v(\cos\phi_v)}\sin\phi_v ,$$

with $D_v$ constant in the 49 charts. In every chart the CP-even readout (masses, $|V_{ij}|$) is an even function of $\phi_v$ and the CP-odd readout is odd. Hence the full physical readout factors through the unoriented line $\{\phi_v,-\phi_v\}$, landing in the CP-unordered pair $\{J,-J\}$; in the 49 mass-flat charts the sharp form is $|J|=(|K_v|/D_v)\,|\sin\phi_v|$ exactly, not at leading order. WP10 certified all 61 vertices lie over one physical point, so the groupoid-level readout is constant while every vertex's lens data reaches it through unoriented data alone.

Interpretation

The WP11 Möbius sign ambiguity is not an obstruction to physics — it is physical CP conjugation. No global sign for $\phi$ exists on the carrier groupoid, and T1–T3 show none is needed: every loop-sensitive physical observable is sign-insensitive because the complete CP-odd shadow of the loop is a single sine. The content of the loop holonomy visible to the weak-basis-invariant ring is exactly one real number, $|\sin\phi_v|$ (modulated by magnitude data), at every vertex. Note the boundary of the result honestly: this is a statement about the chart-to-readout map of the fitted groupoid. It does not select $\phi$ (the invariant ring cannot quantize it — entry 1042's fiber statement stands), and the almost-$\pi/8$ clustering remains a property of where the fitted points sit in the readout, not of the map. For the H2LR typing: the flavor lens $\mathcal K_{\mathrm{flavor}}$ may be taken as edge magnitudes plus the unoriented loop-holonomy line, and the pairing $\langle-,-\rangle_{\mathrm{flavor}}$ is canonical on that lens at all 61 vertices.

Verification

• G0: $\det C=\operatorname{tr}C^3/3$ verified against C.det() on a generic exact rational specialization — pass.
• G1: S38/S43/S48/S53 rebuilt as masks $(282,314)$, $(277,421)$, $(305,342)$, $(305,412)$; supports match results/harmonic_support.json ($\{1\}$, CP-antisymmetric) in all four cases — pass.
• Census: 61/61 charts $\det C$ support exactly $\{-1,+1\}$ with exact antisymmetry; 49/61 mass sector $z$-free; 12 exceptions enumerated (all d-sector phase; palindromic $c_2^d$, plus $c_3^d$ in orbit 2); zero unexplained exceptions.
• Checker and results are stdlib-plus-sympy exact; runtime ~68 s.

Relations

• Sharpens WP9 entry 1890 T4 (numeric $\to$ exact) and extends harmonic_support.py (four textures $\to$ full groupoid).
• Answers Nima's ev-000000000672 test at census level: only $m=1$ survives, at every fitted vertex, at finite $\epsilon$.
• Completes the WP11 (entry 1901) typing: the Möbius sign defect is exactly the CP conjugation ambiguity of the physical readout.
• Leaves open: a support-general proof of first-harmonic-only for all nine-link one-cycle topologies (mechanism identified; census-exact on the fitted groupoid).

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