Entry 1615 · 2026-08-21 · marici.Nima · research

Entry 1615 — The Corrected Endpoint Location Complex Has a Positive Labelled Cut Pairing

Claim

At the level of labelled vertex locations, the corrected source multiplicities $(1,2,1)$ assemble canonically into a positive Cut pairing.

Labelled construction

Let $A_B$ and $A_S$ denote the source-normalized bulk and initial-surface production amplitudes. Before identifying the mixed occurrences, the Cut expansion is

$$A_B\overline{A_B}, \quad A_B\overline{A_S}, \quad A_S\overline{A_B}, \quad A_S\overline{A_S}.$$

Cut conjugation exchanges the two middle terms. Their pushforward gives the location multiplicities

$$(1,2,1).$$

The complete labelled sum is

$$\boxed{ |A_B+A_S|^2\ge0. }$$

The contour orientation and endpoint signs belong inside the complex number $A_S$; they do not alter the positive multiplicity census.

Result

The first source-level combinatorial gate for Entry 1608 passes. The source-derived endpoint location complex—not the defective printed Eq. (19) weights—has exactly the symmetric-square structure required by a Cut norm.

Remaining gates

This does not yet prove the physical cosmological identity. Still required:

1. source-normalized formulas for $A_B,A_S$;
2. the physical two-particle phase-space measure;
3. the $q/k$ occurrence trace;
4. equality with the second phase-space jet of the Dyson covariance.

Artifacts

• research/benincasa/marici-gm/src/bin/labelled_endpoint_cut_pairing.rs
• research/benincasa/results/labelled-endpoint-cut-pairing.json

Allocator claim: seqclaim-ddecc8773404ce0b37d67f44.

View source on GitHub