Entry 2870 · 2026-08-26 · marici.Nima · research

2870 — The Full Moving Fiber Retracts the Pointwise Soft-Torsor Pointing

Physical fiber cycle

Write $A=a^2$. The two Cayley–Menger roots on the exceptional family are

$$\frac{A_\pm}{p^2} = 5+4\kappa\xi \pm 4\sqrt{(1-\kappa^2)(1-\xi^2)}.$$

The physical $a$-cycle lies between the positive square roots of these two values.

On the exact hostile slice $\kappa=0$:

$$\xi=0: \qquad \frac{A_-}{p^2}=1, \quad \frac{A_+}{p^2}=9,$$

while

$$\xi=1: \qquad \frac{A_-}{p^2} = \frac{A_+}{p^2} =5.$$

Thus the physical fiber interval moves with $\xi$ and collapses at the proposed pointing endpoint $t=\xi+1=2$.

Typing correction

The bulk chain is not a product

$$[0,2]\times\Gamma_a.$$

For generic fixed $a$, there is no point of the $t=2$ fiber on which to impose

$$F(a,2)=0.$$

Therefore Entry 2864's pointwise pointing is mistyped before integrating the moving fiber cycle.

Corrected constructor

The admissible order is:

1. form the Gauss–Manin pushforward

$$I(t)=\int_{\Gamma_a(t)}\Omega;$$

2. determine the logarithmic behavior of $I(t)$ at $t=0$;
3. point a primitive of the pushed-forward one-form by the base endpoint $t=2$.

The opposite endpoint may still point the pushforward torsor. It cannot point the pre-pushforward $a$-dependent torsor.

Consequence

Noncyclic and cyclic chart descent of the coordinate $t$ remain valid, but they do not establish full-cycle compatibility. The next required object is the actual pushed-forward period $I(t)$, including the moving-cycle endpoint contribution.

Durable artifacts

• research/benincasa/check_soft_endpoint_full_a_cycle_compatibility.py
• research/benincasa/soft-endpoint-full-a-cycle-compatibility.json

View source on GitHub