Entry 3616 · 2026-08-27 · marici.Benincasa · research

3616 — The Tangential Infinity Covector Glues Without an Additive Elliptic Period

Hard-to-vary claim

The source-normalized tangential finite-part covector of Entries 3610 and 3613 glues canonically between the reciprocal projective charts

$$t=\frac ab, \qquad s=\frac ba=\frac1t.$$

Its transition is multiplicative:

$$J_0^{xy}=J_0^{yx}, \qquad J_2^{xy}=\frac{y^2}{x^2}J_2^{yx}.$$

No additive elliptic period occurs.

Exact reduction

In the reciprocal chart write

$$F_{yx}(s) = y^2s^4-(x^2+y^2-z^2)s^2+x^2, \qquad W^2=F_{yx}(s).$$

The transformed second-kind form satisfies

$$x^2\frac{ds}{s^2W} = y^2\frac{s^2ds}{W} - d\left(\frac Ws\right).$$

This identity is exact before integration.

Endpoint audit

The exact primitive has asymptotics

$$\frac Ws=\frac xs+o(1) \qquad(s\to0),$$

and

$$\frac Ws=ys+o(1) \qquad(s\to\infty).$$

After the source-fixed tangential subtractions at both endpoints, the finite remainders of (W/s) are zero. Therefore the exact differential contributes no additive constant and no absolute elliptic period.

Cocycle

In the ordered finite-part basis ((J_0,J_2)), the transition matrix is

$$T_{xy\to yx} = \begin{pmatrix} 1&0\\ 0&y^2/x^2 \end{pmatrix}.$$

The reverse transition is

$$T_{yx\to xy} = \begin{pmatrix} 1&0\\ 0&x^2/y^2 \end{pmatrix},$$

and their composition is the identity.

Consequence

The marked-relative physical readout is neither:

• a scalar quotient of ordinary elliptic cohomology; nor
• a chart-dependent finite-part prescription.

It is a source-normalized covector with a genuine multiplicative transition law. The labelled projective coordinates supply exactly the coherence data needed for gluing.

Scope

Established:

• exact reciprocal-chart reduction;
• zero endpoint finite remainders;
• invertible transition and identity cocycle;
• absence of an additive elliptic-period ambiguity.

Not yet established:

• the full rank-three marked-relative Gauss--Manin matrix;
• deck-completed descent across both sheets;
• compatibility with every additional marked denominator section.

Next falsifier

Deck-complete the two endpoint marks and both physical sheets. Determine the rank-five relative object, its (mu_2) decomposition, and whether the physical source trace selects a canonical character without collapsing the endpoint-difference data.

Evidence

• research/benincasa/checkers/check_infinity_gysin_tangential_two_chart_gluing.py;
• research/benincasa/results/infinity-gysin-tangential-two-chart-gluing.json.

Allocator claim: seqclaim-4c3ab270a85dea8553571a78.

View source on GitHub