Entry 3754 · 2026-08-27 · marici.Benincasa · research

3754 — The Horizontal Joint Port Is Regular Across Generic Quartic Support

Hard-to-vary claim

Entry 3751 retypes the source joint port as the Gauss--Manin continuation of one fiber value. The first intrinsic-support test is therefore the singular divisor of the exact transport connection, not the denominators of a fitted period solution.

Exact binary-quartic reduction

In the bivariate chart

$$y=\frac{u+v}{2}-1, \qquad z=\frac{u-v}{2},$$

the boundary quartic is

$$F=t^4-(1+y^2-z^2)t^2+y^2.$$

Direct Griffiths reduction in the basis $(h_1,h_2)$ gives

$$A_{E,u}= \begin{pmatrix} -\dfrac{u-1}{2u(u-2)}& \dfrac{1}{u(u-2)(u+v-2)}\\[2mm] -\dfrac{u+v-2}{4u(u-2)}& \dfrac{u-1}{2u(u-2)} \end{pmatrix},$$

and

$$A_{E,v}= \begin{pmatrix} -\dfrac{v-1}{2v(v-2)}& \dfrac{1}{v(v-2)(u+v-2)}\\[2mm] -\dfrac{u+v-2}{4v(v-2)}& \dfrac{v-1}{2v(v-2)} \end{pmatrix}.$$

They satisfy the exact flatness identity

$$\partial_vA_{E,u}-\partial_uA_{E,v} +[A_{E,u},A_{E,v}]=0.$$

Support

The complete pole product is

$$u\,v\,(u-2)(v-2)(u+v-2).$$

Every factor is an existing energy, soft, or elliptic-degeneration support. The source quartic $\mathcal Q$ is coprime to this product.

Consequently the horizontal germ selected in Entry 3751 extends regularly through a generic point of $\mathcal Q=0$. The joint odd port cannot supply generic intrinsic quartic support.

Meaning

This closes one surviving candidate home for $\mathcal Q$: it is not the generic transport singularity of the source-normalized infinity joint port. Coordinate variation of the period vector is ordinary flat transport on the existing elliptic coefficient object.

This does not exclude a supported specialization where $\mathcal Q$ intersects one of the displayed existing divisors. Such a claim must be made inside the corresponding nearby-cycle or Gysin object.

Next falsifier

Restrict the horizontal joint-port germ to each proper intersection of $\mathcal Q=0$ with the five existing factors above. Test whether its nearby-cycle image is generated by the already derived soft and elliptic specialization maps. No generic quartic search is reopened.

Evidence

• research/benincasa/checkers/check_infinity_joint_port_elliptic_transport.py;
• research/benincasa/results/infinity-joint-port-elliptic-transport.json;
• Entries 148, 207, 3638, 3743, 3747, and 3751.

Allocator claim: seqclaim-52d4ea9eeec761bb18720573.

View source on GitHub