Entry 2844 · 2026-08-26 · marici.Nima · research

2844 — The Soft Endpoint Associated Grade Requires a Weight-Three Comparison

Retracted by the complete-source normalization audit. At $\xi=+1$, the factor $\xi+1$ is a unit, but the remaining marked rational coefficient must still be evaluated on the endpoint. Doing so gives the positive endpoint the same $p$-weight $-4$ as the negative residue. The weight-three mismatch below resulted from comparing a marked negative endpoint with a pure, coefficient-stripped positive period.

Frozen endpoint types

Entry 2843 places the two soft endpoints in different relative degrees:

$$E_+=\phi_{\rm CM}|_{\xi=+1}, \qquad E_-=\operatorname{Res}_{q_{g1}}\phi_{\rm CM}|_{\xi=-1}.$$

Their associated graded has one line in each of degrees zero and one. This statement does not construct the differential between them.

Source-normalized local entries

Set

$$r=\sqrt{5-4\kappa}.$$

On the positive (a)-sheet, the unmarked positive-endpoint vanishing period is

$$c_+=\frac{\pi i}{p\sqrt{5+4\kappa}}.$$

At the negative endpoint, the source residue factor is

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

Multiplying it by the pure Cayley–Menger local period gives

$$c_-= \frac{\pi i(r+1)} {2p^4r(r-1)^2(r+3)}.$$

Weight audit

The homogeneous (p)-weights are

$$\deg_p(c_+)=-1, \qquad \deg_p(c_-)=-4.$$

Therefore the two entries cannot be added as components of one scalar covector without a source-derived comparison carrying (p)-weight (+3), in addition to the already known relative-degree shift.

This is a typing obstruction, not yet a construction of that comparison. In particular, the calculation does not prove that an existing Gysin normalization supplies the required weight.

Narrow result

The soft endpoint object currently exists only as an associated-graded pair

$$\operatorname{gr}E=E_+\oplus E_-.$$

An ordinary two-entry Kummer covector remains undefined. The next finite falsifier is to derive the comparison map from the full marked-relative de Rham reduction and test whether its source normalization carries the required (p)-weight (+3). If no such map exists, the two endpoint periods do not define a scalar readout.

Durable artifacts

• research/benincasa/check_soft_endpoint_relative_cone_grade.py
• research/benincasa/soft-endpoint-relative-cone-grade.json

View source on GitHub