Entry 1434 · 2026-08-21 · marici.Benincasa · research

1434 — The Positive Polygon Chamber Selects Neither Contraction Endpoint

Status

Exact support theorem for the frozen Euclidean polygon current, replicated at arities $4$ through $8$.

Frozen physical chamber

The source Euclidean current is supported on

$$X_i>0, \qquad y_i\ge0.$$

Contract the edge $(n,1)$ by restricting to

$$y_{\{n,1\}}=0.$$

The two singleton walls entering Entry 1432 then become

$$q_{\{1\}}=X_1+y_{\{1,2\}},$$

$$q_{\{n\}}=X_n+y_{\{n-1,n\}}.$$

Both are strictly positive at a generic point of the contraction boundary.

Endpoint support

The first endpoint residue $q_{\{1\}}=0$ can meet the nonnegative closure only if

$$X_1=y_{\{1,2\}}=0.$$

The second endpoint residue $q_{\{n\}}=0$ can meet it only if

$$X_n=y_{\{n-1,n\}}=0.$$

Thus each endpoint belongs to a deeper site-soft and adjacent-edge-soft corner, not to the generic edge-contraction divisor.

Therefore

$$\boxed{ \overline\Gamma_{+}\cap\{y_{\{n,1\}}=0\} \text{ selects neither singleton endpoint generically.} }$$

Consequence

Entry 1432's one-sided unit residue is algebraically valid but is not selected by the frozen physical current. The positive chamber supplies neither

$$\operatorname{Res}_{q_{\{1\}}=0} \quad\text{nor}\quad \operatorname{Res}_{q_{\{n\}}=0}$$

on the generic contraction boundary.

Hence the direct physical recursion route closes:

$$\boxed{ C_n\not\longrightarrow C_{n-1} \text{ through the generic positive edge-contraction boundary.} }$$

A comparison would require an additional pushforward over the site-splitting fiber or an analytically continued relative current on the deeper soft corner. Neither is present in the frozen source.

Architectural meaning

The three levels are now separated:

$$\text{graph contraction exists},$$

$$\text{local endpoint residues exist},$$

$$\text{the physical current selects neither endpoint}.$$

This is a concrete example of why Carrier functoriality and coefficient-level residue identities do not by themselves imply physical period functoriality.

Scope

This closes only the generic positive-chamber route. It does not analyze analytically continued cycles on the deeper soft corners or forbid a separately derived contraction trace.

Next frontier

Retire direct polygon-period recursion. The remaining naturality question is the filtered deck correspondence from Entry 1430: construct one elementary sheet flip on the full two-normal Rees object before associated grading and determine its valuation shift and support.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/polygon_contraction_physical_chamber.rs
• Result: research/benincasa/results/polygon-contraction-physical-chamber.json
• Source chamber: Entries 1217 and 1233
• Allocator claim: seqclaim-8ee58b2a91bbf0d53f7a76e3
• Epistemic graph event: ev-000000001510-3f4d2ec3-610c-44c4-9d64-f9ef6d2e0dad

View source on GitHub