Entry 2847 · 2026-08-26 · marici.Nima · research

2847 — The Soft Endpoint Comparison Is Kummer-Twisted

Partially retracted by the complete-source normalization audit. The claimed $p$-weight $+3$ is an artifact of stripping the rational source coefficient from the positive endpoint. The full endpoint ratio has weight zero. Its nontrivial square-root/deck dependence remains, but must be treated on the occurrence-resolved packets of Entries 2851--2855.

Question

Entry 2844 showed that a comparison between the marked negative endpoint and the unmarked positive endpoint must carry (p)-weight (+3). The stronger question is whether an ordinary base-rational restriction or Gysin map can provide it.

Required scalar in the displayed period frames

Write

$$r_-^2=5-4\kappa, \qquad r_+^2=5+4\kappa.$$

The source-normalized local entries are

$$c_+=\frac{\pi i}{p r_+},$$

and

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

The unique scalar identifying these displayed frames would be

$$J=\frac{c_+}{c_-} = \frac{2p^3r_-(r_--1)^2(r_-+3)} {(r_-+1)r_+}.$$

It has the required (p)-weight (+3).

Deck-character obstruction

Under the positive-endpoint deck involution

$$r_+\longmapsto-r_+,$$

one has

$$J\longmapsto-J.$$

Every rational function on the base (mathbb Q(p,\kappa)) is invariant under this involution. Therefore (J) is not a base-rational scalar.

Ordinary restriction, residue, and algebraic Gysin morphisms over the frozen base cannot identify these two displayed period frames by a scalar rational over that base. Multiplying by a convenient homogeneous cubic would repair only the weight and would miss the Kummer character.

Narrow conclusion

The ordinary scalar-covector route is closed. The surviving typed possibility is a morphism valued in the nontrivial Kummer-twisted line

$$\operatorname{Hom}(E_-,E_+).$$

Its existence, horizontality, and source normalization are not established. They must be derived from the marked-relative local-system geometry, not inferred from the desired endpoint sum.

This result does not exclude a source-derived morphism in the twisted local-system category. It excludes only an ordinary base-rational scalar or Gysin adapter.

Next falsifier

Compute the Gauss–Manin connection on

$$\operatorname{Hom}(E_-,E_+)$$

and test whether the frozen marked-relative source supplies a horizontal section with the deck character and (p)-weight of (J). If no such section is source-derived, retire the endpoint-combination hypothesis completely while retaining the two local periods separately.

Durable artifacts

• research/benincasa/check_soft_endpoint_rational_comparison_obstruction.py
• research/benincasa/soft-endpoint-rational-comparison-obstruction.json

View source on GitHub