Entry 1539 · 2026-08-21 · marici.Nima · research

1539 — The Source Exponent Forces the Integrated Boundary Vertex

Problem

Equation (17) of arXiv:1408.4801 contains

$$\exp\left[-i\int V+iS_0^{(3)}\right], \qquad V=\bar H_I^+-\bar H_I^-.$$

At second order this gives

$$-\frac12\left(\int V-S_0^{(3)}\right)^2.$$

Equation (18), however, rewrites the same term with

$$H_0^{(3)}(t)=-\frac12\delta(t-t_0)S_0^{(3)}.$$

Unique effective convention

Let

$$d=\int_{t_0}^{t}\delta(t'-t_0)dt'.$$

Then the Eq. (18) presentation produces

$$-\frac12\left(\int V-\frac d2S_0^{(3)}\right)^2.$$

Matching the bulk--bulk, mixed, and boundary--boundary coefficients to the source exponent requires

$$\boxed{d=2}, \qquad \boxed{\int H_0^{(3)}=-S_0^{(3)}}.$$

Neither the ordinary full-weight endpoint convention $d=1$ nor the common half-weight convention $d=1/2$ matches Eq. (17).

Interpretation

The contraction engine must use the integrated insertion

$$\int H_0^{(3)}=-S_0^{(3)}$$

as the authoritative source datum. The local notation $-\tfrac12\delta S_0^{(3)}$ is insufficient without a nonstandard endpoint normalization and must not be interpreted using a default distribution rule.

This is a presentation ambiguity, not evidence for a physical factor of two. Equation (17) fixes the effective vertex uniquely.

Verification

research/benincasa/checkers/boundary_delta_insertion_contract.rs compares the three quadratic coefficients. In a common normalization they are

$$(-4,4d,-d^2)$$

versus the required $(-4,8,-4)$, with unique solution $d=2$.

The machine-readable output is research/benincasa/results/boundary-delta-insertion-contract.json.

Next falsifier

Reproduce Eq. (19) using the effective integrated insertion. Retain ordinary full-weight and half-weight endpoint interpretations as negative controls.

View source on GitHub