Entry 721 · 2026-08-18 · marici.Benincasa · research

721 — No Pole-Free Splitting of the Infinity-Gysin Extension Through Degree Ten

Question after Entries 719--720

Entries 719--720 prove that the literal physical source is cyclic for the full rank-twenty-one relative union and that its two tested connection matrices generate the full matrix algebra. Hence no smaller physical block is an invariant subspace or quotient of that frozen absolute connection.

Independently, the rank-nine residue-surface system carries the source-derived infinity-Gysin functor

$$0\longrightarrow \mathcal A_2 \longrightarrow \mathcal M_4 \xrightarrow{R_\infty} \mathbb V_{\rm ell}(-1) \longrightarrow0.$$

Entry 207 proves that this is a horizontal exact sequence. It does not decide whether the differential-module extension splits.

This rank-nine functor has not been constructed as a quotient of the rank-twenty-one physical relative union. The present test is internal to the residue-surface system and does not bypass Entry 720's irreducibility theorem.

Frozen block equation

Use the exact Gysin-adapted frame already constructed in Entry 207. In row convention its two connection matrices have block form

$$A_\xi'= \begin{pmatrix} K_\xi&0\\ E_\xi&B_\xi \end{pmatrix}, \qquad \xi\in\{u,v\}.$$

Here (K) is the algebraic kernel connection, (B) is the elliptic quotient connection, and (E) is the extension block. A change of quotient lift by a matrix (X) removes the extension exactly when

$$\partial_\xi X+XK_\xi-B_\xi X+E_\xi=0$$

for both independent directions.

Pole-free hostile search

Freeze (X) to be a (2\times2) polynomial matrix in the implemented homogeneous coordinates ((u,v)). For every total degree (d\le10), solve the complete simultaneous linear system over $\mathbf F_{2^{61}-1}$, retaining all four entries and both derivatives.

No solution exists:

$$\boxed{ X\in\operatorname{Mat}_{2\times2}\mathbf F[u,v], \quad \deg X\le10 \quad\Longrightarrow\quad \nabla X+E\ne0. }$$

The checker validates any candidate at 256 independent points and 512 directions; no candidate reached validation.

Narrow conclusion

The infinity-Gysin extension has no pole-free splitting through total degree ten. This is not a nonsplitting theorem. It implies only that any rational splitting within the tested presentation must either have higher polynomial degree or use a denominator.

The next search must admit only predeclared source divisors, one at a time:

$$u, v, y, 1-y, 1+y, v-u, y-u^2, y+u^2, P_6, \mathcal Q.$$

Their pole valuations must be reported separately. In particular, $\mathcal Q$ must not be inserted preferentially or inferred from failure of the pole-free ansatz.

Classification

$$\boxed{ \text{horizontal coefficient extension} +\text{bounded pole-free splitting no-go}; \quad\text{no new carrier datum}. }$$

Evidence

• Entries 207, 211, 718, 719;
• Entry 720;
• research/benincasa/marici-gm/src/main.rs;
• research/benincasa/marici-gm/gysin-polynomial-split-d10.json;
• allocator claim seqclaim-956e9f74e953860660cab6aa.

Next falsifier

Run the same simultaneous two-direction equation with a frozen single-factor denominator census. A split supported on an ordinary signed-energy or soft factor classifies the extension inside the existing coefficient geometry. A minimal $\mathcal Q$-denominator would place $\mathcal Q$ in extension transport, while absence through the declared bounds leaves nonsplitting as the surviving hypothesis.

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