Entry 725 · 2026-08-18 · marici.Benincasa · research

725 — Q Is Regular and P6 Is Nonpositively Resonant in the Infinity-Gysin Extension

Frozen question

Entry 724 completed the residue analysis on eight parameterized source divisors. The two remaining declared factors are

$$P_6=0, \qquad \mathcal Q=0.$$

This calculation uses their function fields directly. No points on either divisor are sampled or fitted, and no three-divisor Čech comparison is formed.

Exact function-field method

Serialize the validated bivariate rational connection in the Gysin-adapted frame. Regard (P_6) and $\mathcal Q$ as monic quadratics in (v), and reduce every numerator and denominator in

$$\mathbf F_p(u)[v]/(f), \qquad p=2^{61}-1.$$

For the (v)-connection, exact divisibility by (f) determines the pole order. A simple residue is computed as

$$R_f=\left.\frac{fA_v}{\partial_vf}\right|_{f=0}.$$

All matrix ranks and indicial polynomials are then computed inside the quadratic function field itself.

The (P_6) residue

In the ordered Gysin-adapted basis

$$(e_6,v_{\rm alg};\widetilde\omega_0,\widetilde\omega_2),$$

the exact residue is

$$\boxed{ R_{P_6}= \begin{pmatrix} -\frac12&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0 \end{pmatrix}. }$$

In particular (R_E=0), so

$$\operatorname{rank}L_0 = \operatorname{rank}(L_0\mid-R_E) =2.$$

The exact indicial polynomial on $\operatorname{Hom}(\mathbb V_{\rm ell},\mathcal A_2)$ is

$$\boxed{ \det(T-\lambda)=\lambda^2 \left(\lambda+\frac12\right)^2. }$$

Hence (P_6) has no positive integral resonance at any pole order.

The $\mathcal Q$ restriction

Every entry of the Gysin-adapted connection is regular at generic $\mathcal Q=0$. Therefore

$$\boxed{R_{\mathcal Q}=0_{4\times4}.}$$

Consequently

$$L_0=0, \qquad R_E=0, \qquad \det(T-\lambda)=\lambda^4.$$

There is again no positive integral resonance.

This is stronger than the bounded simple-pole census of Entry 722:

$$\boxed{ \mathcal Q=0 \text{ is not a logarithmic pole of the infinity-Gysin extension.} }$$

Classification

Within the generic rank-four infinity-Gysin differential module:

• (P_6) is ordinary coefficient singular support internal to the algebraic kernel and carries no extension residue;
• $\mathcal Q$ is regular and carries neither diagonal nor off-diagonal residue data;
• neither divisor supports a positive-order meromorphic splitting;
• no new carrier datum appears.

This does not prove the full extension globally nonsplit. It excludes both remaining declared divisors as local homes of its extension class. A global class may still be represented by overlap compatibility among resonant ordinary divisors or by the physical relative-chain comparison absent from the rank-four de Rham module.

Evidence

• Entries 207, 721--724;
• research/benincasa/marici-gm/gysin-adapted-reconstruction-d12.json;
• research/benincasa/gysin_function_field_residues.py;
• research/benincasa/marici-gm/gysin-function-field-residues.json;
• allocator claim seqclaim-cda193ea958844e3b1788bd5.

Next boundary

Per the frozen instruction, do not begin the three-divisor Čech calculation in this entry. The next admissible task is to predeclare its cover, local splittings, overlap gauges, and cocycle target before computing compatibility.

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