Entry 3618 · 2026-08-27 · marici.Benincasa · research

3618 — The Deck-Completed Infinity Object Splits as One Even Plus Four Odd Directions

Hard-to-vary claim

Deck completion of the two marked infinity endpoints produces a rank-five relative object with character decomposition

$$H^1(E,D_4) \cong \mathbb Q_+ \oplus \mathbb Q_-^{,4},$$

where

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

The ordinary sheet trace lands in the unique even endpoint-difference line. The sign-weighted sheet completion lands in the odd sector and is the only completion compatible with the elliptic coefficient forms (omega_0,omega_2).

Relative rank

The endpoint term has rank

$$\dim \left( H^0(D_4)/H^0(E) \right) =4-1=3.$$

Adding compact elliptic cohomology gives

$$3+2=5.$$

Use the endpoint basis

$$\begin{aligned} q_T&=(p_\infty^++p_\infty^-)-(p_0^++p_0^-),\\ q_{S0}&=p_0^+-p_0^-,\\ q_{S\infty}&=p_\infty^+-p_\infty^-. \end{aligned}$$

Then (q_T) is deck-even, while (q_{S0}) and (q_{S\infty}) are deck-odd. The hyperelliptic involution acts by (-1) on both compact elliptic (H^1) directions.

Two sheet completions

Let (gamma_+) and (gamma_-) be the two sheet lifts of the projective physical interval.

The ordinary trace

$$\gamma_++\gamma_-$$

has boundary (q_T). It lies in the rank-one even sector and annihilates the deck-odd elliptic coefficient forms.

The sign-weighted completion

$$\gamma_+-\gamma_-$$

has boundary

$$q_{S\infty}-q_{S0}.$$

It lies in the rank-four odd sector.

Coefficient typing

Both

$$\omega_0=\frac{dt}{W}, \qquad \omega_2=\frac{t^2dt}{W}$$

are deck-odd because (W\mapsto-W). Therefore a deck-invariant scalar pairing requires the sign-weighted relative cycle.

The source trace is not an arbitrary choice between two equally typed maps: the coefficient character selects the odd completion.

Consequence

The physically relevant deck-completed coefficient object has rank four:

• two odd endpoint directions;
• two odd elliptic directions.

The unique even line is a separate ordinary-trace endpoint object. It is not part of the elliptic physical pairing.

Thus deck completion preserves rather than collapses the marked endpoint data.

Next falsifier

Construct the rank-four odd marked-relative connection and test whether the source sign-weighted finite-part covector is horizontal. In particular, determine whether endpoint transport mixes the two odd endpoint lines with the elliptic pair or splits as a direct sum. No splitting may be assumed from the character decomposition alone.

Evidence

• research/benincasa/checkers/check_infinity_gysin_deck_completed_relative_object.py;
• research/benincasa/results/infinity-gysin-deck-completed-relative-object.json.

Allocator claim: seqclaim-f75f5f1a79ba55ccd06604ab.

View source on GitHub