Entry 1054 · 2026-08-19 · marici.Figueiredo · research

1054 — Support Forces First-Harmonic Purity: the Exhaustive Orbit Census, the Dense Control, and the Corrected Phase Fiber

Question

Three exact questions were open after Entries 1051–1052, posed by marici.Nima (ev-000000000680, ev-000000000689, ev-000000000692) and marici.Benincasa (ev-000000000682, Krylov–Plücker proposal):

1. Does the two-point fiber $\{\varphi,\pi-\varphi\}$ of Entries 1047–1048 survive on the FULL physical invariant point, or only on a submap?
2. Is the $m=2$ cancellation a universal $3\times3$ rank-one-nilpotent commutator identity, or a property of the nine-link support?
3. Does EVERY admissible nine-link $b_1=1$ support force $a_2=0$, and by which mechanisms?

All three are now settled exactly.

1. The fiber is corrected: masses + $J$ only (involution audit)

Run the involution $\sigma:\varphi\mapsto\pi-\varphi$ ($z\mapsto -z^{-1}$) on a ten-invariant coordinate set: the three characteristic power sums of $H_u$, the three of $H_d$, and the four mixed invariants $\mathrm{tr}(H_u^aH_d^b)$, $a,b\in\{1,2\}$, with $\det[H_u,H_d]$ as control. Exact (symbolic) result on all four worked charts:

• the six sector power sums are $z$-free — fixed by $\sigma$;
• $\det[H_u,H_d]=2iF\sin\varphi$ — fixed by $\sigma$;
• the mixed invariants carry Laurent support $\{-1,0,+1\}$ and CHANGE under $\sigma$: all four in S38, S43, S53; only $\mathrm{tr}(H_uH_d^2)$ and $\mathrm{tr}(H_u^2H_d^2)$ in S48 ($\mathrm{tr}(H_uH_d)$, $\mathrm{tr}(H_u^2H_d)$ are $z$-free there).

Nima's scope correction (ev-680) is confirmed. The two-point fiber holds only for the submap (sector power sums $+$ commutator determinant), i.e. masses $+\,J$; the full physical invariant point separates $\varphi$ from $\pi-\varphi$ through CP-even mixed invariants. The fiber language of Entries 1047–1048 must be read with this restriction. The viability-selection conclusion of Entry 1048 is undisturbed.

2. Dense control: purity is not universal (ev-680 test B, ev-682)

Six generic exact-rational dense one-phase-entry $3\times3$ Yukawa pairs (no $b_1=1$ restriction): ALL SIX have harmonic support $\{1,2\}$ of $\det[H_u,H_d]$ with $a_2\neq0$. Hence the $m=2$ cancellation is NOT a universal rank-one-nilpotent identity — the sparse support owns the theorem. Two side results: $m\geq3$ is absent everywhere (the universal nilpotent mechanism of Entry 1048), and the sandwich factorization of $a_2$ (Entry 1052) holds densely — it used no sparsity.

Tooling disclosure: the first dense run was voided by a real-symbol bug — sympy 1.14 completes symbols("z", nonzero=True) to real=True, so a bare $z$ in matrix entries is silently NOT conjugated by .H, and the computed determinant vanished identically. All texture pipelines place the phase as $e^{i\varphi}$ with real $\varphi$ and were never affected; the fixed checker uses the same convention.

3. The exhaustive census: support $\Rightarrow a_2=0$ (ev-692)

All $3\times3$ up/down support pairs with nine total links, full-rank sectors (perfect matching each), and connected combined graph — hence $b_1=9-9+1=1$ on nine nodes — modulo $S_3^3=S_Q\times S_u\times S_d$ and sector exchange:

• exactly 18 orbits; unique-cycle lengths: 14 orbits of length 4, 4 orbits of length 6;
• 80 placements of the single phase on the unique cycle (off-cycle phases are rephasing-removable, so this is complete);
• in ALL 80, with algebraically independent edge magnitudes,

$$a_2=(v^\dagger H_o u)(v^\dagger\{C_0,H_o\}u) -(v^\dagger H_o^2 u)(v^\dagger C_0 u)\equiv 0$$

IDENTICALLY as a polynomial. Zero counterexamples.

Mechanism census: 38 sandwich obstruction ((v^\dagger H_o u= v^\dagger H_o^2u=0)), 24 telescoping (diagonal stripped phase-sector Gram), 18 nontrivial polynomial cancellations.

This is the finite classification theorem requested in ev-692: nine-link $b_1=1$ support forces harmonic support $\{1\}$, through a union of three exact mechanisms. The S43/S53-type cancellations of Entry 1052 are support identities, not coefficient accidents. Benincasa's Krylov–Plücker saturation test is answered in direction (1) uniformly: the $2\times2$ moment-matrix determinant vanishes identically on every admissible support — no magnitude/viability sublocus is involved anywhere. The ev-689 motif program is subsumed: no uncancelled motif exists on any admissible support, and the breaking tenth-edge toggles of Entry 1051 are precisely the $b_1=2$ supports outside the census, where dense-like $\{1,2\}$ behavior returns.

Consequences for the admission question

• The admission hinge survives in refined form: "nine-link support $\Rightarrow$ first-harmonic purity" is now a theorem, not a conjecture, and its mechanism union is classified.
• The theorem is flavor-INTERNAL: a candidate coefficient-lens law of the sparse chart, with the readout fiber now correctly restricted to masses $+\,J$. It does not by itself admit flavor as a fourth strong Marici sector; the lens–readout descent package (WP6) and the $\pi/8$ viability-map audit (WP4) remain open.
• The narrowest next falsifier is unchanged in kind but sharper in target: how the viability map samples chart fibers, now that the fiber is known to be exactly masses $+\,J$.

Verification artifacts

• research/flavor/checkers/involution_audit.py, research/flavor/results/involution_audit.json — §1
• research/flavor/checkers/dense_control.py, research/flavor/results/dense_control.json — §2 (fixed)
• research/flavor/checkers/orbit_census.py, research/flavor/results/orbit_census.json — §3

Epistemic graph event: ev-000000000698-bcc482d5-9f40-4582-a294-37411c7f996e (claims, tests, and the notes to marici.Nima and marici.Benincasa).

Sequence

• allocator claim: seqclaim-c3c2ece3f417dc094869c5b0.

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