Entry 1907 · 2026-08-22 · marici.Figueiredo · research

First-Harmonic Universality: det[H_u, H_d] Over All One-Cycle Nine-Link Topologies (WP13)

Question

WP12 (entry 1903) established $\det[H_u,H_d] = 2iK_v\sin\phi$ exactly at the 61 fitted carrier-groupoid vertices. The operator asked for the support-general theorem: does first-harmonic-only hold over all one-cycle nine-link topologies, and what are the exact hypotheses? Equivalently: is the Laurent support of $\det[H_u,H_d]$ in $z = e^{i\phi}$ contained in $\{-1,+1\}$ for every viable sparse Yukawa support, with the phase on the unique cycle — and when exactly is the coefficient identically zero?

Result

Theorem (first-harmonic universality). Let $Y_u, Y_d$ be $3\times3$ matrices whose combined nine-node field graph has nine edges, is connected (hence $b_1 = E - V + 1 = 1$, a unique cycle $\gamma$), with each sector admitting a perfect matching, all entries real positive except a single phase $z = e^{i\phi}$ on one edge of $\gamma$. Then

$$\det\nolimits[Y_uY_u^\dagger,\;Y_dY_d^\dagger] = i\,A(\text{magnitudes})\sin\phi ,$$

i.e. the Laurent support of the commutator determinant in $z$ is contained in $\{-1,+1\}$ with exact antisymmetry $a_{-m} = -a_m$ and $a_0 = 0$. Moreover $A \equiv 0$ (as a polynomial in the magnitudes) iff $\gamma$ is a sector-pure 4-cycle on a quark-doublet pair $\{a,b\}$ and the third doublet node $r$ has no derived H-link to one of $\{a,b\}$ (one of the pairs $(r,a),(r,b)$ is absent from the off-diagonal support of both $H_u$ and $H_d$). In every other case $A$ is a nonzero polynomial.

Certificate: the support class contains 6552 viable one-cycle supports, forming 18 orbits under $S_3(Q)\times S_3(u^c)\times S_3(d^c)$ plus $u\leftrightarrow d$ swap (14 orbits with cycle length 4, 4 with length 6). Testing every cycle edge of each orbit representative — 80 (support, phase-edge) cases — with nine free magnitude symbols (exact symbolic Laurent arithmetic) gives: 68 cases support exactly $\{-1,+1\}$, 12 cases identically zero, zero higher-harmonic cases, exact CP antisymmetry in all 80, and the symbolic classification matches an independent exact rational (Fraction) census case by case. Orbit invariance (row/column permutations conjugate $H_u,H_d$ by permutation matrices and leave $\det C$ invariant; swap maps case to case) extends the certificate to all 6552 supports. The 12 identically-zero cases are precisely the three orbits $(\mathrm{mask}_u,\mathrm{mask}_d) \in \{(84,119), (85,118), (85,220)\}$ satisfying the dichotomy condition; no fitted WP9 chart lies in a zero orbit (checked: 0/61), and all 61 fitted charts regression-pass.

Proof

Write the phase edge as $(q^*, c^*)$ with $c^*$ an up-sector centre (the $u\leftrightarrow d$ swap covers the other choice). Then

$$H_u(z) = H_0 + zE + z^{-1}E^\dagger,\qquad E = m\,|s\rangle\langle q^*|,\quad |s\rangle = \sum_{q \in N(c^*)\setminus q^*} m_q |q\rangle ,$$

with $E^2 = 0$ (since $q^* \notin \mathrm{supp}\,s$), and $H_d$ $z$-free. With $C_0 = [H_0, H_d]$, $P = [E, H_d]$, (Q = [E^\dagger, H_d]), the exact identity $\det C = \tfrac13\operatorname{tr} C^3$ for a $3\times3$ commutator (gated numerically in WP12) gives the coefficient identities

$$a_3 = \operatorname{tr} P^3,\qquad a_2 = \operatorname{tr}(C_0 P^2),\qquad a_1 = \operatorname{tr}(C_0^2 P) + \operatorname{tr}(P^2 Q).$$

(The identity $a_1 = \operatorname{tr}(C_0^2P) + \operatorname{tr}(P^2Q)$ was verified exactly: for orbit representative $(85,122)$, $\operatorname{tr}(C_0^2P) = 91\,980\,016\,128$ and $\operatorname{tr}(P^2Q) = 62\,597\,510\,976$ sum to the $\det C$ coefficient $154\,577\,527\,104$.)

Step 1: $a_3 = 0$ for any single-phase-entry $H_u$. With $A = |s\rangle\langle q^*|H_d$, $B = H_d|s\rangle\langle q^*|$, $\alpha = \langle q^*|H_d|s\rangle$, one computes $A^2 = \alpha A$, $B^2 = \alpha B$, $AB = \langle q^*|H_d^2|s\rangle\,|s\rangle\langle q^*|$, $BA = 0$, hence $P^3 = m^2\alpha^2 P$ and $\operatorname{tr} P^3 = m^2\alpha^2 \operatorname{tr} P = 0$ since $P$ is a commutator. No sparsity is used.

Step 2: derived labelled multigraph and the unicyclic classification. Contract each centre (a $u^c$ or $d^c$ node) $c$ to a clique on its doublet neighbourhood $N(c)$, labelled $c$, producing a labelled multigraph on the three $Q$ nodes. Field-graph cycles are exactly closed derived walks with pairwise distinct labels, so $b_1 = 1$ forces one of:

• Type I (4-cycle): one pair $\{a,b\}$ carries exactly two labels $\{c^*, c_2\}$; every other pair carries at most one label; and the third node $r$ attaches to at most one of $a, b$ (a label $x$ on both $(r,a)$ and $(r,b)$ would force $x$ to be a degree-3 centre also labelling $(a,b)$, raising $b_1$; two distinct such labels create a distinct-label triangle).
• Type II (6-cycle): all three pairs carry exactly one label, all distinct (labels $\{a,b,a\}$-style repetitions force a shared centre, contradicting multiplicity one).

Step 3: $a_2 = 0$ by the bichromatic obstruction. Expanding $P^2 = m^2(\alpha(A+B) - \beta |s\rangle\langle q^*|)$ with $\beta = \langle q^*|H_d^2|s\rangle$ and using $\{H_d, [H_0,H_d]\} = [H_0, H_d^2]$,

$$a_2 = m^2\big[\alpha\,\langle q^*|[H_0, H_d^2]|s\rangle - \beta\,\langle q^*|C_0|s\rangle\big].$$

Every monomial in this expansion is a walk product in which some $Q$-$Q$ pair would have to carry both a $u$-label $\neq c^*$ and a $d$-label, or a second independent cycle would have to exist. Case analysis:

• Type II: $\deg c^* = 2$, the pair $(q^*,q')$ is $u$-only, so $\alpha = 0$. Then $\beta \neq 0$ requires both remaining pairs $d$-labelled, in which case $\langle q^*|C_0|q'\rangle = 0$ (no $u$-links remain); if exactly one remaining pair is $d$-labelled, $\beta = 0$; if none, both factors vanish.
• Type I, $\deg c^* = 2$, $c_2$ up-type: $H_d^{ab} = 0$ gives $\alpha = 0$, and $\beta$ needs $d$-paths $a$-$x$-$b$, forcing links to both sides of the cycle — excluded — so $\beta = 0$.
• Type I, $\deg c^* = 2$, $c_2$ down-type: the expansion reduces to $-m^2 m_b^2 (H_d^{ab})^2\,(H_d^{rx} H_0^{rx})$-type terms ($x$ the side where $r$ attaches), each needing pair $(r,x)$ to be both up- and down-labelled — zero.
• Type I, $\deg c^* = 3$: $c^*$ labels all three pairs, so $H_0$ is diagonal; then $\langle a|C_0|s\rangle = m_b(h_a - h_b)H_d^{ab}$, $\langle a|[H_0,H_d^2]|s\rangle = m_b(h_a - h_b)(H_d^2)^{ab}$, $\beta = m_b (H_d^2)^{ab}$, and the two terms cancel identically.

Step 4: antisymmetry. $z \leftrightarrow z^{-1}$ is complex conjugation; $\det C$ is anti-Hermitian-valued with real coefficients, hence $a_{-m} = -a_m$ and $a_0 = 0$. This is structural, not combinatorial.

Step 5: the zero dichotomy. For the coefficient $A = a_1/i$: in the three zero orbits the cycle is a sector-pure 4-cycle on $\{a,b\}$ and one of the pairs $(r,a),(r,b)$ is entirely absent; every closed walk contributing to $\operatorname{tr}(C_0^2P) + \operatorname{tr}(P^2Q)$ then requires the absent pair or a second cycle. The mechanism is transparent when one sector is monomial ($H_u$ diagonal): $\det C$ is proportional to the difference of the two oriented triangle products $H_d^{01}H_d^{12}H_d^{20} - \mathrm{c.c.}$, which vanishes identically when a pair is absent. Exhaustion over the 18 orbits (symbolic certificate) proves both directions: exactly the orbits $(84,119), (85,118), (85,220)$ and all their members have $A \equiv 0$.

Steps 1–4 are exact symbolic manipulations verified by the checkers; Step 5 and the per-orbit nonvanishing of $A$ are certified by the symbolic confirmation over all 80 cases with nine free magnitude symbols — no polynomial-identity-testing gap remains.

Interpretation

• The WP12 sine form was not an accident of the fitted sample: it is a theorem on the whole one-cycle nine-link class. The only $\phi$-channel from loop data into weak-basis invariants is $\sin\phi$, universally.
• The unoriented pushforward of WP12 is thereby promoted: on every viable topology, $\{\phi,-\phi\} \mapsto \{J,-J\}$ with $\det[H_u,H_d] = -2i\,\hat J \prod_q m_q^2$; the orientation ambiguity of the chart is exactly the sign ambiguity of $J$.
• The zero dichotomy has physical content: $A \equiv 0$ topologies force $J = 0$ for all $\phi$ — they are CP-conserving charts, invisible to the CKM phase. All 61 fitted charts lie in the nonzero class, as they must to fit $J \neq 0$.
• For the Marici admission question: the flavor lens datum survives chart descent only through the sine image. The theorem fixes the exact topological domain on which any future "loop-holonomy to observable" map must be defined, and the exact exceptional locus where it degenerates.
• Process note: the first census run reported 16 zero cases; four were an artifact of an order-dependent cycle-edge decode that silently dropped the phase (the assertion now gates this). The corrected count is 12, and the purity cross-tab hypothesis was revised accordingly — recorded here for audit honesty.

Verification

• research/flavor/checkers/wp13_all_topology_census.py — exhaustive enumeration (6552 supports, 18 orbits, 80 phase cases), exact Fraction arithmetic, two generic magnitude assignments per case; 61/61 fitted-chart regression pass; output results/wp13_all_topology_census.json.
• research/flavor/checkers/wp13_symbolic_confirmation.py — exact symbolic (nine free magnitude symbols) Laurent support for all 80 cases; zero higher harmonics, zero mismatches against the rational census; output results/wp13_symbolic_confirmation.json.
• research/flavor/checkers/wp13_zero_case_probe.py — structural probe isolating the zero-dichotomy condition.
• Coefficient identities verified against direct $\det C$ evaluation at exact rational points (example above).

Relations

• Strengthens entry 1903 (WP12: fitted-vertex sine form) to the support-general theorem.
• Answers the remaining harmonic-support question attached to the WP11/WP12 line: the Laurent dependence of $\det[H_u,H_d]$ in $z$ is first-harmonic-only on the entire one-cycle class, not only on S38/S43/ S48/S53.
• Consistent with WP11 (entry on chart vs physical invariance): nothing here promotes $\phi$ itself to a weak-basis invariant; the theorem constrains the form of its descent.

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