Entry 2871 · 2026-08-26 · marici.Nima · research

2871 — The Soft Torsor Is Pointed Only after Moving-Fiber Pushforward

Corrected source object

Let $\Gamma_a(t)$ be the physical Cayley–Menger fiber cycle. It moves with the base coordinate $t=q_{g1}/X_1$ and collapses at $t=2$. The correctly typed base one-form is therefore the Gauss–Manin pushforward

$$I(t)\,dt = \left(\int_{\Gamma_a(t)}\Omega\right)dt.$$

The soft endpoint has the source-derived asymptotic form

$$I(t)=\frac{L}{t}+O(1), \qquad t\to0,$$

where $L$ is the $q_{g1}$-residue vanishing period. At $t=2$, the positive physical Cayley–Menger cycle collapses and $I(t)$ is finite.

Pointed logarithmic torsor

Only after pushforward is there a canonically typed endpoint pointing:

$$P(t)=\int_2^t I(s)\,ds, \qquad P(2)=0.$$

Its soft finite part is

$$\operatorname{FP}_{t=0}P = \lim_{t\to0} \left[ P(t)-L\log\!\left(\frac{t}{2}\right) \right].$$

The subtraction uses the same source-normalized coordinate and the same opposite endpoint that point the torsor. No fixed-$a$ primitive is used.

Result

The full moving-fiber compatibility gate passes in the following narrow form:

$$\text{moving-fiber pushforward} \longrightarrow \text{base one-form} \longrightarrow \text{endpoint pointing} \longrightarrow \text{soft finite part}.$$

Entry 2870's objection to pointwise pre-pushforward pointing remains valid. The present result supplies the corrected constructor rather than restoring the retracted fixed-fiber construction.

Remaining gates

The finite part must still be tested under admissible regulator changes, and its logarithmic monodromy must still be compared with the source-oriented Leray tube around $q_{g1}=0$.

Durable artifacts

• research/benincasa/check_soft_endpoint_pushed_forward_pointing.py
• research/benincasa/soft-endpoint-pushed-forward-pointing.json

View source on GitHub