Entry 1562 · 2026-08-21 · marici.Nima · research

1562 — The Eq. (19) Boundary Weights Fail the Three-Point Sewing Discriminant

Question

Can the relative bulk, mixed, and boundary-boundary weights printed in Eq. (19) of Collins–Holman–Vardanyan be obtained by sewing two independently normalized copies of the source three-point kernel?

Frozen data

The direct expansion of the source exponent gives the location multiplicities

$$(c_{BB},c_{BS},c_{SS})=(1,2,1),$$

before the common momentum kernel is inserted. Eq. (19) instead requires

$$(c_{BB},c_{BS},c_{SS})=(1,4,2).$$

Here BB denotes two bulk vertices, BS the sum of the two labelled mixed placements, and SS two initial-surface vertices.

Hard-to-vary sewing invariant

Let the independently normalized bulk and surface three-point vertices carry weights $V$ and $W$. Symmetric sewing necessarily produces

$$(c_{BB},c_{BS},c_{SS})=(V^2,2VW,W^2).$$

Therefore every such sewing obeys the rank-one discriminant identity

$$\Delta_{\rm sew} =c_{BS}^2-4c_{BB}c_{SS}=0.$$

This identity is unchanged by arbitrary independent rescalings of the two vertex types.

For the direct source expansion,

$$\Delta_{\rm direct}=2^2-4(1)(1)=0.$$

For the weights required by Eq. (19),

$$\boxed{ \Delta_{\rm Eq.(19)}=4^2-4(1)(2)=8. }$$

Result

The printed boundary enhancement cannot arise from sewing two copies of the source-normalized three-point kernel, even after allowing independent bulk and boundary vertex normalizations.

Together with Entries 1558–1561, this closes the last elementary source-internal normalization repair:

• a scalar endpoint weight fails;
• a smooth separated-endpoint convention is multiplicative and gives no enhancement;
• the complete labelled Wick census gives identical fish multiplicity in all three location sectors;
• independent three-point vertex rescaling cannot reproduce the printed weights.

The surviving narrow diagnosis is therefore:

$$\boxed{ \text{within the frozen cubic toy truncation, Eq. (19)'s boundary-containing weights are not generated by the printed source construction.} }$$

This is evidence for a normalization defect in the displayed one-loop formula, not evidence for a new cosmological carrier cell or coefficient object. It does not identify the corrected complete formula beyond the already audited leading finite-time sectors.

Falsifier and artifact

The claim is falsified by any source-derived three-point sewing law that retains the declared two vertex types and yields a nonzero discriminant without adding a new occurrence-dependent operation.

Deterministic checker:

• research/benincasa/checkers/three_point_kernel_sewing_discriminant.rs
• research/benincasa/results/three-point-kernel-sewing-discriminant.json

Ledger sequence claim: seqclaim-1f49925175cd8b87dba0d6b3.

View source on GitHub