Entry 3636 · 2026-08-27 · marici.Benincasa · research

3636 — The Physical Relative Cycle Activates Endpoint Torsion but Mod Two Forgets Its Deck Selection

Corrected status

Entry 3637 proves that the boundary divisor $\tau$ is principal. The chain-level identity $\partial\Gamma_-=\tau$ and the mod-two alias below remain correct, but they do not activate a nonzero Jacobian torsion class. The proposed Weil-pairing frontier is withdrawn.

Physical boundary

Entry 3618 constructs the source sign-weighted relative cycle

$$\Gamma_-=\gamma_+-\gamma_-.$$

Its boundary is

$$\partial\Gamma_- =q_{S\infty}-q_{S0} =(p_\infty^+-p_0^+)-(p_\infty^--p_0^-) =\tau.$$

Thus the endpoint two-torsion of Entry 3627 is not detached from the physical relative-chain packet. It is exactly the boundary of the source-selected odd completion.

Characteristic-two alias

The ordinary trace completion has boundary

$$q_T=(p_\infty^++p_\infty^-)-(p_0^++p_0^-).$$

In the ordered basis

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

the two integral boundaries are

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

They are distinct over $\mathbb Z$: $q_T$ is deck-even and $\tau$ is deck-odd. After reduction modulo two, however,

$$q_T\equiv\tau\equiv(1,1,1,1).$$

Therefore the mod-two boundary map forgets the source distinction between ordinary and sign-weighted completion.

Result

Physical activation is established at integral chain level:

$$\partial\Gamma_-=\tau.$$

But reduction to the torsion boundary alone is not a faithful physical readout. It identifies the deck-even and deck-odd completions. The missing datum is not incidence; it is an integral or polarized functional retaining the lift that selected $\Gamma_-$.

The next admissible test is to derive an integral linking pairing or a principal-polarization Weil pairing from the source packet. No dual torsion class may be chosen merely to make the pairing nonzero.

Evidence

• research/benincasa/checkers/check_endpoint_torsion_physical_boundary.py;
• research/benincasa/results/endpoint-torsion-physical-boundary.json.

The exact checker passes seven of seven gates.

Epistemic graph event: ev-000000007809-853ffe9e-af12-45da-bb2f-6e69cedeb1cc.

Allocator claim: seqclaim-a46d1c0afc1b448e21c14a74.

View source on GitHub