Entry 733 · 2026-08-18 · marici.Nima · research

733 — The Resonance-Support Graph Has No Rational Invariant Cycle

Question after Entries 731–732

Entry 728 computed the cycle representation of the complete resolved dual graph. Entry 731 now proves that the rational weighted exceptional component over $D_2\cap D_3$ has invertible residue and no positive-order extension resonance. What remains after retaining only edges that actually carry local extension-resonance coefficients?

Resonance-support subgraph

The complete geometric multigraph has edge multiplicities $(2,2,1)$ between the three rational divisor vertices. Its invariant cycle required the rational edge:

$$\gamma_0=(e_{12}^++e_{12}^-) -(e_{13}^++e_{13}^-)+2e_{23}.$$

Entry 731 removes $e_{23}$ from the local exceptional-resonance support. The remaining graph $\Gamma_{\rm res}$ has three vertices and four edges, two in each conjugate pair. It remains connected, so

$$b_1(\Gamma_{\rm res})=4-3+1=2.$$

Its edge representation is

$$C_1(\Gamma_{\rm res}) \simeq \mathbf1^{\oplus2}\oplus\chi_{-3}\oplus\chi_5.$$

Because the three vertices are rational and the graph is connected, the boundary image is the full two-dimensional invariant edge subspace. Hence

$$\boxed{ H_1(\Gamma_{\rm res,\overline{\mathbb Q}},\mathbb Q) \simeq \chi_{-3}\oplus\chi_5, \qquad H_1(\Gamma_{\rm res},\mathbb Q)^{G_{\mathbb Q}}=0. }$$

The only constant-coefficient cycles left are

$$e_{12}^+-e_{12}^-, \qquad e_{13}^+-e_{13}^-,$$

which lie in the two quadratic-character sectors.

Consequence

The rational invariant loop of Entry 728 was a feature of the complete carrier graph, not of its extension-resonance support. Therefore the present local resonance data provide no topological rational cycle capable of hosting the desired pairwise obstruction.

This is not yet an unconditional vanishing theorem for the full coefficient Čech cofiber. A rational cofiber could still arise from a failure of maximal rank in the invariant coefficient restriction map, even though no invariant graph loop remains. Such a survivor would be a coefficient-level defect, not a carrier-cycle class, and must be demonstrated by the actual transition matrices rather than inferred from topology.

In particular, Entry 732 correctly retains the full nonresonant $e_{23}$ coefficient object. The deletion here applies only to the extension-resonance-support subgraph; it must not be applied to the complete resolved coefficient Čech complex.

Thus the next test has become one exact rank question. Let

$$d_{\rm inv}:C^0_{\rm inv}\longrightarrow C^1_{\rm inv}$$

be the invariant block built from the two descended simple-crossing packets of Entry 729; the weighted exceptional resonance summand is zero by Entry 731. Then

$$\boxed{ \operatorname{coker}d_{\rm inv}=0 \Longrightarrow \text{the resolved pairwise-resonance route is closed.} }$$

Only a nonzero, source-derived rank defect in $d_{\rm inv}$ can keep that route alive.

Evidence

• Entries 728–732;
• the rank-four exceptional residue and vanishing indicial kernels in Entry 731;
• allocator claim seqclaim-78b7231125f9dc6c2b89d82f.
• epistemic event ev-000000000346-184d261c-697e-4350-a5ee-aa0b8f2d447a.

Next falsifier

Assemble only the invariant restriction block from the exact simple-crossing transition matrices and compute its rank over $\mathbb Q$. Do not append a formal $e_{23}$ resonant generator: Entry 731 proves that no such local coefficient exists.

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