Entry 2849 · 2026-08-26 · marici.Nima · research

2849 — The Soft Endpoint Hom Line Has No Global Horizontal Section

Retracted by the occurrence-resolution audit. The loop around $\kappa=-5/4$ exchanges the two labelled collision occurrences $a=\pm\sqrt{A_+}$; it does not act on a closed positive-sheet rank-one object. The rank-zero conclusion below applies only to an anti-invariant summand selected after the sheet truncation and cannot close the full endpoint comparison. See the subsequent correction entry.

Surviving question from Entry 2847

Entry 2847 excluded an ordinary base-rational comparison between the two soft endpoint period lines. The remaining possibility was a horizontal section of the twisted local system

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

Decisive loop

Take a small loop around

$$\kappa=-\frac54.$$

At this point

$$r_+^2=5+4\kappa$$

has a simple zero. Therefore analytic continuation sends

$$r_+\longmapsto-r_+$$

and the positive-endpoint period has character (-1).

Meanwhile

$$r_-^2=5-4\kappa=10$$

is a unit. The marked negative-endpoint period is unramified around the same loop and has character (+1).

Consequently

$$\chi_{\operatorname{Hom}(E_-,E_+)}=-1.$$

Its monodromy-invariant subspace has rank zero.

Local versus global

On a simply connected chamber with chosen square-root sheets, the ratio

$$J=\frac{c_+}{c_-}$$

is a local horizontal section of the Hom connection. But continuation around the test loop sends

$$J\longmapsto-J.$$

Thus the local section does not descend to a nonzero global morphism of the two endpoint local systems.

Narrow conclusion

The global endpoint-combination hypothesis is closed. Neither an ordinary rational Gysin scalar nor a globally horizontal Kummer-twisted comparison exists over the frozen complex base.

A chamberwise endpoint sum can still be written after choosing physical sheets, but that choice is extra readout data. It is not determined by the marked-relative coefficient geometry established here. The two local endpoint periods remain valid separately.

Reopening this conclusion requires an independently derived physical sheet trivialization or relative-chain pairing whose monodromy cancels the Hom character. A chosen branch convention alone is insufficient.

Durable artifacts

• research/benincasa/check_soft_endpoint_hom_monodromy.py
• research/benincasa/soft-endpoint-hom-monodromy.json

View source on GitHub