Entry 1432 · 2026-08-21 · marici.Benincasa · research

1432 — Polygon Contraction Has an Occurrence-Two Residue Obstruction

Status

Exact local Poincaré-residue theorem on the contracted-edge divisor.

Local model

Entry 1431 gives, on $y_{\{n,1\}}=0$,

$$q_{\{1\}}+q_{\{n\}}=q_{\{n,1\}}.$$

Set

$$q=q_{\{n,1\}}, \qquad s=q_{\{1\}}, \qquad t=q_{\{n\}}=q-s.$$

The two source singleton poles have the local logarithmic form

$$\Omega=\frac{ds\wedge dq}{s(q-s)}.$$

Endpoint residues

At the first occurrence $s=0$,

$$\operatorname{Res}_{s=0}\Omega=\frac{dq}{q}.$$

At the second occurrence $t=0$, use $ds=dq-dt$, so

$$ds\wedge dq=-dt\wedge dq.$$

Therefore

$$\operatorname{Res}_{t=0}\Omega=-\frac{dq}{q}.$$

Consequently,

$$\boxed{ \operatorname{Res}_{s=0}\Omega + \operatorname{Res}_{t=0}\Omega=0, }$$

while the occurrence-oriented difference is

$$\boxed{ \operatorname{Res}_{s=0}\Omega - \operatorname{Res}_{t=0}\Omega =2\frac{dq}{q}. }$$

Consequence

A one-sided occurrence residue gives the target merged-wall form with unit coefficient. It breaks exchange symmetry between the two endpoint occurrences.

The exchange-resolved oriented construction is canonical over the labelled occurrence complex, but produces a factor $2$. Hence there is no simultaneously

$$\text{integral} + \text{exchange-symmetric} + \text{unit-normalized}$$

contraction map in this local model without additional data.

The permitted repairs are sharply typed:

1. a physical current selects one occurrence;
2. an independently normalized trace/counit supplies the factor $1/2$;
3. the target retains the occurrence-two coefficient.

None is inferred merely from graph contraction.

Relation to cosmological partial energies

This is the same structural factor $2$ already seen when two resolved interface occurrences are identified on the physical diagonal. It is therefore not evidence for a new cosmological coupling or carrier cell. It is an occurrence-identification coefficient.

Scope

The calculation is local and logarithmic. It does not construct the global physical contraction current, establish a period recursion, or authorize division by two.

Next finite falsifier

Inspect the frozen five-cycle Betti chamber under $C_5\to C_4$. Determine whether its oriented boundary selects one endpoint occurrence or the exchange-resolved difference. This decides between unit normalization and the occurrence-two coefficient without fitting the answer to $C_4$.

Durable verification

• Checker: research/benincasa/marici-gm/src/bin/polygon_contraction_local_residue.rs
• Result: research/benincasa/results/polygon-contraction-local-residue.json
• Allocator claim: seqclaim-678641e806f7073ceb4b7b6f
• Epistemic graph event: ev-000000001508-a3088fc5-4949-4bfc-9e35-ba7a5d02ef67

View source on GitHub