Entry 724 · 2026-08-18 · marici.Benincasa · research

724 — The Infinity-Gysin Extension Is Locally Residue-Split on Eight Source Divisors

Question after Entry 722

Entry 722 excludes a global splitting with one declared simple-pole denominator and numerator degree at most ten. A stronger possible explanation would be a local logarithmic extension obstruction on one of the source divisors.

Frozen residue equation

In the Gysin-adapted frame write the residue along a smooth divisor (f=0) as

$$R_f= \begin{pmatrix} R_K&0\\ R_E&R_B \end{pmatrix}.$$

The logarithmic extension class vanishes at the residue level exactly when

$$L_0(X):=XR_K-R_BX=-R_E$$

is solvable. A pole of order (m>0) can begin only in the kernel of the indicial operator

$$L_m(X)=-mX+XR_K-R_BX.$$

Parameterized divisors

Take transverse Laurent residues on

$$u, v, y, 1-y, 1+y, v-u, y-u^2, y+u^2.$$

For every divisor, evaluate sixteen independent generic points over $\mathbf F_{2^{61}-1}$. Reconstruct every entry of the (4\times4) residue matrix before testing the extension equation.

Result

The residue equation is solvable at every tested point on every divisor:

$$\boxed{ R_E\in\operatorname{im}L_0 \quad\text{on all eight parameterized source divisors.} }$$

Thus the global failure in Entries 721--722 is not explained by a local logarithmic residue class on these divisors.

The indicial census through pole order ten separates two groups:

$$\begin{array}{c|c} \text{divisor}&\ker L_m, 1\le m\le10\\ \hline u,v,y,1-y,1+y&0\text{ for every tested }m\\ v-u,y-u^2,y+u^2&\dim\ker L_1=2,\quad\ker L_m=0\ (2\le m\le10). \end{array}$$

Hence the first five divisors cannot support a meromorphic splitting with pole order at most ten. The final three admit a local order-one resonant leading term, but Entry 722 proves that none extends to the tested global polynomial numerator.

Narrow interpretation

The obstruction is now constrained to global compatibility, mixed-divisor gluing, the unparameterized (P_6) or $\mathcal Q$ divisors, higher-degree numerators, or genuine differential-module nonsplitting. No new carrier stratum is indicated.

In particular, residue solvability must not be promoted to a global splitting, and the three order-one resonances must not be fitted into a mixed denominator without deriving their compatibility.

Evidence

• Entries 207, 721--722;
• research/benincasa/marici-gm/src/main.rs;
• research/benincasa/marici-gm/gysin-local-residue-obstruction-16.json;
• allocator claim seqclaim-4e67d8188640f98c24e207e3.

Next falsifier

Compute the local residue equations on (P_6=0) and $\mathcal Q=0$ using their function fields rather than fitted rational points. Then compute the Čech compatibility of the local residue splittings on pairwise intersections of the three resonant divisors. A nonzero overlap cocycle would locate the extension globally without introducing a new carrier component.

View source on GitHub