Entry 2843 · 2026-08-26 · marici.Nima · research

2843 — The Two Soft Endpoints Belong to Different Relative Variances

Marked source factor

The exceptional source form contains

$$R(a,\xi)= \frac{a+p} {2p(a-p)^2(a+3p)(\xi+1)}.$$

This factor must be retained when promoting Entry 2841's pure Cayley–Menger periods to the marked-relative coefficient system.

Negative endpoint

At $\xi=-1$, two structures coincide:

1. the Cayley–Menger branch collision;
2. the divided marked wall $q_{g1}/X_1=\xi+1$.

The source has logarithmic normal order $-1$. The correctly typed object is therefore an iterated specialization

$$\operatorname{Res}_{q_{g1}} \phi_{\rm CM},$$

with its Gysin/degree shift.

Positive endpoint

At $\xi=+1$, the Cayley–Menger branch collision remains, but

$$\xi+1=2$$

is a unit. This endpoint carries the unmarked specialization

$$\phi_{\rm CM}$$

at normal order zero.

Genericity audit

The pure collision roots obey

$$A_--p^2=4(1-\kappa)p^2, \qquad A_+-p^2=4(1+\kappa)p^2.$$

Hence the marked $a=p$ collision occurs only at the deeper base endpoints $\kappa=+1$ and $\kappa=-1$, not for generic $-1<\kappa<1$.

Consequence

The two $\xi$-endpoint occurrences are not two entries of one ordinary rank-two covector. They are objects of different relative variance and normal degree. The required comparison is a relative de Rham/Gysin cone combining

$$\operatorname{Res}_{q_{g1}}\phi_{\rm CM}$$

with

$$\phi_{\rm CM}.$$

Only after constructing its degree shift and comparison map can one define a total endpoint readout or test contragredient transport. A direct $VU-I$ calculation on the pair from Entry 2841 would be mistyped.

Durable artifacts

• research/benincasa/check_soft_endpoint_relative_variance.py
• research/benincasa/soft-endpoint-relative-variance.json

View source on GitHub