Entry 3743 · 2026-08-27 · marici.Benincasa · research

3743 — The Full Four-Mark Interval Supplies a Primitive Odd Joint Port

Correction-driven question

Entries 3736 and 3739 govern the two-edge branch-gap subcomplex. Entry 3618 constructs a larger source object: the projective physical interval lifted to both sheets with four distinct marked endpoints

$$D_4=\{p_0^+,p_\infty^+,p_0^-,p_\infty^-\}.$$

The full interval must therefore be tested independently for a joint endpoint–elliptic relative port.

Labelled boundaries

Orient both sheet intervals from zero to infinity. In the ordered endpoint basis

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

their boundaries are

$$\partial\gamma_+ = \begin{pmatrix}-1&1&0&0\end{pmatrix}^{T},$$

and

$$\partial\gamma_- = \begin{pmatrix}0&0&-1&1\end{pmatrix}^{T}.$$

The sign-weighted interval is

$$\gamma_S=\gamma_+-\gamma_-.$$

Using

$$q_{S0}=p_0^+-p_0^-, \qquad q_{S\infty}=p_\infty^+-p_\infty^-,$$

its boundary is exactly

$$\partial\gamma_S=q_{S\infty}-q_{S0}.$$

This boundary is nonzero and deck-odd.

Joint odd covector

Entries 3678–3724 derive equal source-normalized elliptic periods in the ordered compact basis $(h_1,h_2)$. In the full odd relative basis

$$(q_{S0},q_{S\infty},h_1,h_2),$$

the source interval therefore defines the joint vector

$$\ell_{\rm rel} = \begin{pmatrix}-1&1&1&1\end{pmatrix}.$$

Its entries are coprime, so it is primitive in the integral relative lattice. Both the cycle and coefficient are deck-odd; their scalar pairing is deck-invariant.

Nonfactorization

Because the endpoint coordinates are nonzero, $\ell_{\rm rel}$ does not factor through the compact quotient

$$H^1(E,D_4)\longrightarrow H^1(E).$$

Thus the frozen source does contain a canonical nonfactoring relative port. This corrects the overbroad reading of the branch-gap obstruction.

What is and is not established

Established:

• exact four-endpoint boundary;
• odd deck character;
• primitive integral joint endpoint–elliptic vector;
• nonfactorization through compact infinity Gysin.

Not established:

• horizontality under the rank-four odd marked-relative connection;
• intrinsic support of the resulting extension class;
• any $\mathcal Q$-residue or monodromy.

The static appearance of a joint vector does not authorize assigning $\mathcal Q$ to it. The next finite gate is precisely the rank-four odd connection requested in Entry 3618: determine whether $\ell_{\rm rel}$ is horizontal, mixes within the odd relative object, or develops an invariant supported defect.

Evidence

• research/benincasa/checkers/check_infinity_full_relative_joint_port.py;
• research/benincasa/results/infinity-full-relative-joint-port.json;
• Entries 3618, 3719, 3724, 3736, and 3739.

The exact checker passes eight of eight gates.

Epistemic graph event: ev-000000008050-005c2f03-2156-4dc1-a358-738a5d9eab92.

Allocator claim: seqclaim-e81f027761ec7b61df1b8856.

View source on GitHub