Entry 3637 · 2026-08-27 · marici.Benincasa · research

3637 — The Physical Odd Endpoint Divisor Is Principal Integrally

Exact correction

Let

$$D_+=p_\infty^+-p_0^+, \qquad D_-=p_\infty^--p_0^-.$$

Entry 3627 derived

$$\operatorname{div}(\phi_+)=2D_+, \qquad \phi_+=\frac{W-xt^2+y}{t^2}.$$

The source coordinate has

$$\operatorname{div}(t)=-(D_++D_-).$$

Adding these principal divisors gives

$$\operatorname{div}(t\phi_+) =D_+-D_- =\tau.$$

An explicit principal witness is therefore

$$t\phi_+ = \frac{W-xt^2+y}{t}.$$

Hence

$$[\tau]=0 \quad\text{in}\quad \operatorname{Pic}^0(E_X)$$

integrally. The earlier order-two statement was true but non-sharp: the order divides one.

Consequences

The following survive:

• $\tau$ is a nonzero labelled zero-chain;
• $\tau=\partial(\gamma_+-\gamma_-)$ in the marked relative-chain packet;
• the raw rational extension block is gauge;
• deck and reciprocal involutions act by sign on the chosen divisor lift.

The following are withdrawn:

• a nonzero integral endpoint two-torsion Jacobian class;
• a characteristic-two Cartan-module attachment;
• a discrete torsion holonomy;
• a Weil-pairing frontier for $\tau$.

The surviving object is relative incidence, not an absolute Picard class. The next calculation must remain inside the marked relative complex and test whether the principal witness supplies a canonical splitting compatible with the physical finite-part covector.

Evidence

• research/benincasa/checkers/check_endpoint_odd_divisor_principal.py;
• research/benincasa/results/endpoint-odd-divisor-principal.json.

The exact checker passes five of five gates.

Epistemic graph event: ev-000000007813-4be5fef6-eb71-4eea-9de6-66c3a6136263.

Allocator claim: seqclaim-66e0fd8302f48a32a8514a62.

View source on GitHub