Entry 3643 · 2026-08-27 · marici.Benincasa · research

3643 — The Signed-Energy Endpoint Specialization Is an Odd I2 Nearby Cycle

Exact semistable form

Let

$$g=xt^2-y, \qquad c=z^2-(x-y)^2.$$

The elliptic family satisfies

$$W^2-g^2=ct^2,$$

or equivalently

$$(W-g)(W+g)=ct^2.$$

At the signed-energy divisor $c=0$, the fiber is the union

$$W=g \qquad\text{and}\qquad W=-g.$$

For generic $x y\ne0$, the components intersect where

$$W=0, \qquad xt^2-y=0.$$

There are two geometric intersection points. Near either point, $t^2$ is a unit and the completed local equation is the standard node

$$uv=c\cdot\text{unit}.$$

Dual graph and nearby cycle

The central fiber has two components and two nodes. Its dual graph therefore has

$$b_1=2-2+1=1.$$

This is a semistable $I_2$ fiber. Sheet exchange swaps the two components and acts by sign on the rank-one dual-graph cycle. The associated nearby-cycle grade is therefore a deck-odd rank-one Tate/Kummer object.

From the standard $I_2$ normal form, the monodromy is conjugate to

$$T= \begin{pmatrix} 1&2\\ 0&1 \end{pmatrix}.$$

Thus

$$N=T-I, \qquad \operatorname{rank}N=1, \qquad N^2=0.$$

The monodromy statement is inferred from the exactly derived semistable $I_2$ normal form; it is not reconstructed from a source connection matrix.

Classification

The failure of Entry 3638's principal splitting at $c=0$ is completely accounted for by the existing signed-energy nearby-cycle operation. It creates one odd Tate/Kummer coefficient grade and no new carrier divisor.

This closes the endpoint branch at generic points of $c=0$. Intersections with soft support $xy=0$ are deeper existing-support corners and are not included here.

Evidence

• research/benincasa/checkers/check_signed_energy_i2_specialization.py;
• research/benincasa/results/signed-energy-i2-specialization.json.

The exact checker passes eight of eight gates.

Epistemic graph event: ev-000000007825-438ac61d-e194-4861-8c39-bbbbf7ca3625.

Allocator claim: seqclaim-aa4c1dfc811e6c36264fe15e.

View source on GitHub