Entry 3758 · 2026-08-27 · marici.Benincasa · research

3758 — Quartic Intersections Do Not Enhance the Joint-Port Nearby Type

Question

Entry 3754 proves that the horizontal infinity joint port is regular across generic $\mathcal Q=0$. The remaining authorized test is its restriction to proper intersections with the five existing transport divisors.

Exact intersection census

In the $(u,v)$ chart, restriction of $\mathcal Q$ gives

$$\begin{aligned} \mathcal Q|_{u=0}&=-4(v-2)^2,\\ \mathcal Q|_{u=2}&=-4(v-2)^2,\\ \mathcal Q|_{v=0}&=-(u^2+2u-4)^2,\\ \mathcal Q|_{v=2}&=-u^2(u-2)^2,\\ \mathcal Q|_{u+v-2=0}&=-u^3(5u-8). \end{aligned}$$

Thus the proper intersections are:

• the already resolved deeper corners $(0,2)$ and $(2,2)$;
• two quadratic points $u=-1\pm\sqrt5$ on $v=0$;
• one rational point $(u,v)=(8/5,2/5)$ on the signed-energy wall.

Nearby type

At either quadratic point, the residue along $v=0$ remains rank-one nilpotent. At the rational signed-energy point, the residue is

$$\begin{pmatrix} 0&-25/16\\ 0&0 \end{pmatrix},$$

again rank-one nilpotent. In both cases the square is zero and no residue-rank jump occurs.

The quartic therefore selects distinguished points on existing support but does not enhance the local nearby-cycle type of the infinity joint port.

Classification

For this coefficient object, every proper quartic intersection is either:

• an already existing deeper soft/energy corner; or
• an ordinary point of an existing rank-one nodal nearby system.

No new carrier stratum and no new joint-port coefficient grade appears. Whether the physical horizontal section has zero or nonzero value at one of the generic points is a period-evaluation question, not a new support class.

Consequence

The infinity joint-port branch is closed as a home for $\mathcal Q$, both generically and at its proper intersections. Reopening it would require new source data that changes the coefficient object or physical relative cycle.

Evidence

• research/benincasa/checkers/check_infinity_joint_port_quartic_intersections.py;
• research/benincasa/results/infinity-joint-port-quartic-intersections.json;
• Entries 863, 3754, and the existing soft-corner nearby calculations.

Allocator claim: seqclaim-ad0ed4283113f324c86230db.

View source on GitHub