Entry 3633 · 2026-08-27 · marici.Benincasa · research

3633 — Deck and Reciprocal Transport Fix the Endpoint Two-Torsion Class

Superseded object

Entry 3637 proves that $\tau$ is principal integrally. The permutation calculation below remains correct for the labelled divisor lift, but it is not transport of a nonzero torsion line.

Source divisor

In the ordered marked-divisor basis

$$(p_\infty^+,p_0^+,p_\infty^-,p_0^-),$$

an integral lift of the physical odd endpoint class is

$$\tau=(1,-1,-1,1).$$

Entry 3627 proves $2\tau=0$ in the endpoint Jacobian.

Native transport

The deck involution $W\mapsto-W$ exchanges the two sheets at each endpoint. It sends

$$\tau\longmapsto-\tau.$$

The reciprocal chart transition $t\mapsto1/t$, accompanied by the source exchange $x\leftrightarrow y$, exchanges zero and infinity on each sheet. It also sends

$$\tau\longmapsto-\tau.$$

Their composite fixes the integral lift. More importantly, because $2\tau=0$, each involution individually fixes the actual torsion class:

$$-\tau=\tau \quad\text{in}\quad \langle\tau\rangle\cong\mathbb Z/2.$$

Result

The source deck and reciprocal generators induce the trivial representation on the endpoint two-torsion line. They do not supply a nontrivial discrete holonomy or the labelled Cartan action excluded by Entry 3632.

The surviving datum is therefore a transport-invariant $\mathbb Z/2$ class on the generic locus. Its remaining possible physical content is a mod-two pairing with a source-derived integral relative cycle or a supported specialization on an existing discriminant locus.

Scope

This computes the subgroup generated by sheet exchange and reciprocal endpoint exchange. It does not determine transport around every discriminant component of the full parameter base.

Evidence

• research/benincasa/checkers/check_endpoint_torsion_native_transport.py;
• research/benincasa/results/endpoint-torsion-native-transport.json.

The exact checker passes six of six gates.

Epistemic graph event: ev-000000007806-572d26b5-0e1c-4ff5-aed4-831d6e363844.

Allocator claim: seqclaim-eaeb0623d13ee754c44f2b3a.

View source on GitHub