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Entry 1431 · 2026-08-21 · marici.Benincasa · research

1431 — Polygon Contraction Does Not Preserve the Positive Exceptional Cell

Status

Exact labelled-facet typing theorem, replicated at source arities 44 through 88.

Proposed operation

Contract the labelled edge (n,1)(n,1) of CnC_n. The target Cn−1C_{n-1} has one merged site whose source preimage is {n,1}\{n,1\}.

The contraction is well defined on partial-energy facets. In particular,

c∗q{1}′=q{n,1},c^*q'_{\{1\}}=q_{\{n,1\}},

while every singleton not incident to the contracted edge pulls back to the corresponding source singleton.

Common-cell obstruction

The positive exceptional period of CnC_n is frozen on the cell containing

qG, q{1},…,q{n}.q_G,\ q_{\{1\}},\ldots,q_{\{n\}}.

The target period cell pulls back instead to

qG, q{n,1}, q{2},…,q{n−1}.q_G,\ q_{\{n,1\}},\ q_{\{2\}},\ldots,q_{\{n-1\}}.

Therefore

c∗(target positive cell)≠source positive cell.\boxed{ c^*(\text{target positive cell}) \ne \text{source positive cell}. }

The mismatch is not a relabelling artifact: the exact wall identity is

q{1}+q{n}=q{n,1}+2y{n,1}.\boxed{ q_{\{1\}}+q_{\{n\}} = q_{\{n,1\}}+2y_{\{n,1\}}. }

Only on the contracted-edge divisor y{n,1}=0y_{\{n,1\}}=0 does the merged target wall equal the sum of the two source singleton walls.

Consequence

Labelled graph contraction exists at Carrier level, but direct period recursion does not:

Cn⟶̸Cn−1by ordinary pullback of the frozen cells.\boxed{ C_n\not\longrightarrow C_{n-1} \quad\text{by ordinary pullback of the frozen cells.} }

A typed comparison would require, in order:

  1. restriction to y{n,1}=0y_{\{n,1\}}=0;
  2. a residue/Gysin or counit operation combining the two source singleton poles;
  3. compatibility with the exceptional current and its orientation.

No such normalization is inferred from the facet identity.

Architectural meaning

Entry 1429's alphabet bound is natural at the level of allowed wall sizes, but its individual periods are not automatically functorial under graph contraction. This is another instance of

Carrier map⇏coefficient/current comparison map.\text{Carrier map} \not\Rightarrow \text{coefficient/current comparison map}.

Scope

The theorem rejects only strict common-cell pullback. It does not obstruct a derived contraction comparison supplied by an independently normalized residue, Gysin map, or physical counit.

Next finite falsifier

Apply the source residue at y{n,1}=0y_{\{n,1\}}=0 to one low-arity pair, beginning with C5→C4C_5\to C_4. Determine whether its normal Jacobian and orientation canonically convert

1q{1}q{n}into1q{n,1},\frac{1}{q_{\{1\}}q_{\{n\}}} \quad\text{into}\quad \frac{1}{q_{\{n,1\}}},

or leave an unavoidable convolution/relative coefficient.

Durable verification

View source on GitHub