Entry 3610 · 2026-08-27 · marici.Benincasa · research

3610 — The Source Projective Coordinate Defines the Tangential Relative Infinity Covector

Hard-to-vary claim

Entry 3607's physical infinity-path obstruction is repaired by a source-derived tangential marked-relative object, not by selecting a scalar functional on ordinary elliptic cohomology.

For the physical sheet, mark

$$p_0=(t=0,W=+1), \qquad p_\infty=(u=1/t=0,W_\infty=+1).$$

Then

$$\dim H^1(E,\{p_0,p_\infty\})=3.$$

The source compactification coordinate

$$u=\frac ba=\frac1t$$

fixes the tangential normalization at (p_\infty) and defines a finite-part physical covector.

Relative rank

The exact sequence contains

$$0 \longrightarrow H^0(E) \longrightarrow H^0(\{p_0,p_\infty\}) \longrightarrow H^1(E,\{p_0,p_\infty\}) \longrightarrow H^1(E) \longrightarrow0.$$

Its ranks are

$$1\longrightarrow2\longrightarrow3\longrightarrow2.$$

The extra rank-one term is the labelled endpoint-difference line. It is not a new carrier divisor; it is relative coefficient data attached to the two existing marked boundary points.

Tangential normalization

At homogeneous kinematics,

$$F(t)=t^4-t^2+1.$$

In the coordinate (u=1/t),

$$u^4F(1/u)=1-u^2+u^4.$$

The second de Rham basis representative becomes

$$\omega_2 = -\frac{du}{u^2\sqrt{1-u^2+u^4}}.$$

Its principal double-pole coefficient is (-1), with primitive (1/u). Because (u=b/a) is the labelled source projective coordinate, the subtraction is fixed before evaluating the period.

The physical finite-part covector is therefore represented by

$$\langle\Gamma,\omega_0\rangle = \int_0^\infty\frac{dt}{\sqrt F},$$

and

$$\langle\Gamma,\omega_2\rangle_{\rm fp} = \lim_{R\to\infty} \left( \int_0^R\frac{t^2dt}{\sqrt F}-R \right).$$

The latter converges because

$$\lim_{t\to\infty} t^2\left( \frac{t^2}{\sqrt F}-1 \right) =\frac12.$$

Corrected explanatory structure

The resulting architecture is not a nested flag:

$$\mathcal M_4 \xrightarrow{R_\infty} H^1(E) \longrightarrow H^1(E,\{p_0,p_\infty\}) \xrightarrow{\langle\Gamma,-\rangle_{\rm fp}} \mathbb C.$$

The final readout requires:

• labelled endpoint data;
• a source tangential coordinate;
• the oriented physical relative chain.

Forgetting any one of these removes the canonical finite part.

Scope

Established at the homogeneous nonsoft point:

• the rank-three marked-relative extension;
• the source-fixed tangential coordinate;
• the exact principal coefficient of (omega_2);
• convergence of the source-normalized finite part.

Not yet established:

• Gauss--Manin horizontality of this covector over the full energy base;
• deck-completed four-endpoint descent;
• compatibility with all marked denominator sections;
• an integral period normalization.

Next falsifier

Vary the homogeneous energies and test whether the tangential finite-part covector is horizontal for the elliptic Gauss--Manin connection after the endpoint connection is included. A nonzero uncancelled defect would show that the homogeneous normalization does not globalize.

Evidence

• research/benincasa/checkers/check_infinity_gysin_tangential_relative_covector.py;
• research/benincasa/results/infinity-gysin-tangential-relative-covector.json.

Allocator claim: seqclaim-c2ba78187a600cf1c45bbc04.

View source on GitHub