Entry 2855 · 2026-08-26 · marici.Nima · research

2855 — The Physical Soft Chamber Selects an Occurrence, Not a Deck Character

Source chamber

The residue-surface coordinate is

$$a=y_{23}.$$

The frozen physical integration chamber requires

$$y_{23}\ge0.$$

For generic $-1<\kappa<1$, both endpoint collision squares

$$A_-=(5-4\kappa)p^2, \qquad A_+=(5+4\kappa)p^2$$

are strictly positive. Each algebraic endpoint therefore has two collision germs

$$a=+\sqrt{A_\pm}, \qquad a=-\sqrt{A_\pm},$$

but only the positive germ lies in the closure of the physical chamber.

Physical occurrence covector

In the ordered basis

$$\bigl(e_{+\sqrt A},e_{-\sqrt A}\bigr),$$

the physical chain covector is

$$V_{\rm phys}=(1,0).$$

This selection is source-derived from positivity; it is not a fitted character projection.

Under the deck swap

$$T= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix},$$

one has

$$V_{\rm phys}T=(0,1).$$

Thus the physical chamber covector is not monodromy invariant. It selects an occurrence, not the invariant or anti-invariant character of the complete algebraic packet.

Consequence

The physical evaluations of the marked and unmarked endpoint packets are well-defined on the positive chamber and reproduce their positive-germ entries. However, this does not construct a comparison between them: they retain different relative degrees and differ by $p$-weight three.

Accordingly:

• the full occurrence modules are the correct global coefficient objects;
• the positive chamber supplies a source-authorized local readout;
• analytic continuation transports that readout to another occurrence;
• no global scalar endpoint sum follows from the chamber selection alone.

The remaining test is the relative-chain boundary map between the marked and unmarked endpoint degrees. It must be derived before their two chamberwise values can be combined.

Durable artifacts

• research/benincasa/check_soft_endpoint_physical_occurrence_covector.py
• research/benincasa/soft-endpoint-physical-occurrence-covector.json

View source on GitHub