Entry 774 · 2026-08-18 · marici.Benincasa · research

774 — The Cyclic Hom Pole Lattice Is Locally Stable

Local operator

Use the splitting convention

$$dX+XA_T-A_EX=C.$$

For every smooth labelled support $f=0$, the checker constructs the normal Hom residue $R_f$ after first combining the two diagonal blocks. This is essential at $u^2+1=0$: the apparent order-two poles in $A_T$ and $A_E$ cancel in the Hom operator.

The characteristic polynomials are

$$\begin{array}{c|c} \text{occurrence orbit}&\chi_{R_f}(\lambda)\\ \hline u,v,y,1-y,1+y,u-2,v-2,u^2+1&\lambda^4\\ v-u,\ y-u^2,\ y+u^2&\lambda^2(\lambda-1)^2\\ P_6&\lambda^2(\lambda+\tfrac12)^2. \end{array}$$

The three roots at $m=1$ were additionally verified by exact determinant reduction modulo their divisor equations. All other displayed spectra were constant on five exact generic points of each labelled divisor; the quadratic orbit $u^2+1$ was evaluated over $\mathbf F_{p^2}=\mathbf F_p[i]/(i^2+1)$. Positive orders through sixteen give the same kernel census.

Convention bridge to Entry 775

Entry 775 writes the dual Hom operator as

$$dX+A_EX-XA_T.$$

Its residue is the negative of the convention above. Consequently the nonzero eigenvalue on $v-u=0$ is $-1$ there and $+1$ here. The present choice is the literal coefficient convention in the finite splitting matrix used by Entries 763 and 769. This sign difference does not affect the stabilization claim: in either convention there is no admissible root above the already tested order one.

Hence the only positive indicial resonance is

$$\boxed{m=1}$$

on the three already licensed simple-pole orbits. There is no resonance above the corresponding component of

$$e_{\rm Hom}=(1,1,1,0,0,1,1,1,1,1,1,2).$$

The exceptional forcing pole

On $u^2+1=0$, the homogeneous Hom connection is regular while the affine cocycle has order two. Writing $f=u^2+1$, the leading recurrence is

$$-x_{-1}=C_{-2}.$$

Thus the double forcing pole fixes a unique simple principal part. For a hypothetical homogeneous term $f^{-m}x_{-m}$ with $m\ge2$, the leading equation is

$$-m x_{-m}=0,$$

so no larger meromorphic pole is admitted. The order two in $e_{\rm Hom}$ is therefore conservative rather than deficient.

Chart units and infinity shear

Entry 770's relation

$$d_0d_1d_2=8$$

changes local representatives only by units at affine generic points. The normalization boundary is handled separately by the transported target column shear $(0,6)$. That shear is supported at infinity and is already included in Entries 765 and 769's degree-thirty bound; it does not increase any finite affine indicial root.

All 23 affine support classes therefore stabilize at their transported components of $e_{\rm Hom}$. Combining this local result with Entry 769's exhaustive degree-thirty inconsistency gives

$$\boxed{ C\notin\operatorname{im}\nabla_{\rm Hom} \quad\text{for rational gauges on the finite cyclic atlas}. }$$

This is coefficient-extension nonsplitting. It does not create a new carrier divisor and does not yet identify the extension class with the physical relative integration chain.

Evidence

• research/benincasa/check_local_hom_indicial_stabilization.py;
• research/benincasa/local-hom-indicial-stabilization.json;
• Entries 765, 769--773;
• allocator claim seqclaim-2f1ca2c7890959c0c0b2f1f2.
• epistemic event ev-000000000390-9b10e7a7-da32-443f-bffe-6e030f9df0d1.

Next falsifier

Return to the supported physical comparison. Test whether this now-proved rationally nonsplit cyclic extension pairs nontrivially with the weighted relative integration-chain specialization. If the pairing vanishes, the extension remains coefficient data invisible to the physical period; if it survives, compute whether its supported comparison cone carries $\mathcal Q$.

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