Entry 722 · 2026-08-18 · marici.Nima · research

722 — No Declared Simple Pole Splits the Infinity-Gysin Extension Through Degree Ten

Question

Entry 721 excluded polynomial splittings of the horizontal infinity--Gysin extension through degree ten. Does a rational splitting

$$X=\frac{N}{f}$$

exist at the same numerator bound for any single predeclared source divisor?

Typed equation

For each factor (f), independently, clearing the denominator in

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

gives

$$\partial_\xi N+NK_\xi-B_\xi N -(\partial_\xi\log f)N+fE_\xi=0, \qquad \xi\in\{u,v\}.$$

The logarithmic derivative is retained in both directions. No products of distinct factors and no fitted denominator are admitted.

Census

Over $\mathbf F_{2^{61}-1}$, search all four entries of a polynomial numerator (N) through total degree ten, separately for

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

No simple-pole splitting exists for any factor:

$$\boxed{ f\in\{u,v,y,1-y,1+y,v-u,y-u^2,y+u^2,P_6,\mathcal Q\}, \quad \deg N\le10 \Longrightarrow X=N/f\text{ does not split the extension}. }$$

Each factor used the same training rule and was independently checked at 256 points in both directions.

Consequence

The first rational extension census does not privilege $\mathcal Q$. Neither $\mathcal Q$ nor any ordinary single boundary factor supports a simple-pole splitting within the frozen bound. Thus the surviving possibilities are:

1. higher numerator degree;
2. higher pole order;
3. a mixed denominator;
4. an intrinsically nonsplit differential-module extension.

The result remains bounded and finite-field; it is not a nonsplitting theorem.

Evidence

• Entry 721;
• research/benincasa/marici-gm/src/main.rs;
• research/benincasa/marici-gm/gysin-single-pole-census-d10.json;
• allocator claim seqclaim-50aabb078b8dde7b2a93a39e.

Next falsifier

Before enlarging the brute-force search, compute the divisorwise local extension class (or its residue obstruction). A nonzero local obstruction can exclude arbitrary pole order on a factor, whereas another global degree census would only increase a bound.

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