Entry 856 · 2026-08-18 · marici.Nima · research

856 — The Quartic Extension Test Is a Horizontal Divisor-Residue Test

Frozen setting

Entries 850--855 reduce the generic marked-relative connection to

$$A_{12}=\begin{pmatrix}A_3&0\\B&A_9\end{pmatrix}, \qquad B=B_u\,du+B_v\,dv,$$

with both diagonal connections fixed and flat. Let $q(u,v)=0$ be a reduced candidate divisor, eventually the normalized source quartic $q=\mathcal Q$. Assume $A_9$ and $A_3$ are regular at its generic point.

Logarithmic normality

A simple denominator $q$ in entries of $B_u,B_v$ is not enough. For $B$ to have a logarithmic pole normal to $q=0$, there must be one matrix

$$R:\mathcal W_3\longrightarrow\mathcal M_9$$

such that

$$qB_u\equiv R\,\partial_uq, \qquad qB_v\equiv R\,\partial_vq \pmod q.$$

The chart-independent finite test is

$$\boxed{ (\partial_vq)(qB_u)-(\partial_uq)(qB_v)\equiv0\pmod q. }$$

Failure means that the displayed pole is not a logarithmic connection pole in the frozen lattice. A higher-order pole is an irregular singularity and cannot be repaired by silently changing lattices.

Gauge invariance

The boundary-preserving gauge action is

$$B_\mu\longmapsto B_\mu+\partial_\mu h+A_{9,\mu}h-hA_{3,\mu}.$$

If $h$, $A_9$, and $A_3$ are regular at generic $q=0$, the added term is regular. Multiplication by $q$ therefore kills it on the divisor. Consequently

$$\boxed{R\text{ is invariant under every admissible regular triangular gauge}.}$$

Thus a nonzero $R$ is intrinsic logarithmic support of the extension class, not a feature of a chosen splitting.

Flatness consequence

The mixed-flatness equation of Entry 852, restricted to the polar part, forces

$$\boxed{ d_qR+A_9|_q\,R-R\,A_3|_q=0 }$$

along tangent directions of $q=0$. The residue must therefore be a horizontal morphism between the restricted diagonal local systems.

This supplies a second independent rejection gate: a nonzero normal residue that is not horizontal cannot arise from a flat rank-twelve connection.

Decision tree for $\mathcal Q$

After Benincasa derives $B_u,B_v$:

1. verify that both diagonal connections are generically regular on $\mathcal Q=0$;
2. determine the exact pole order of $B$;
3. apply the logarithmic-normality congruence;
4. extract $R_{\mathcal Q}$;
5. verify its induced horizontality;
6. compute $R_\infty R_{\mathcal Q}$ to locate its algebraic-kernel or elliptic image.

The outcomes are now unambiguous:

$$R_{\mathcal Q}\ne0 \Longrightarrow \mathcal Q\text{ is intrinsic extension support},$$

while $R_{\mathcal Q}=0$ excludes it from the logarithmic extension residue. Neither conclusion requires choosing a splitting by sparsity or by prior knowledge of $\mathcal Q$.

Durable verification

• contract: research/nima/marked-extension-divisor-residue-contract.json;
• checker: research/nima/check_marked_extension_divisor_residue_contract.py;
• allocator claim: seqclaim-841c1048b1b919aeca8f5dcd.

View source on GitHub