Entry 1558 · 2026-08-21 · marici.Nima · research

1558 — Eq. (19) Requires a Nonmultiplicative Endpoint Normalization

Hard-to-vary claim

With the source exponent

$$\exp\left(-i\int H_I+iS_0^{(3)}\right)$$

expanded directly, and with the common prefactor conversion fixed from the printed Wightman normalization, the oscillatory second-grade sectors are

$$\boxed{ \text{bulk--bulk}=J_1, \qquad \text{mixed}=-2J_2, \qquad \text{boundary--boundary}=-J_0. }$$

Equation (19) instead requires

$$J_1-4J_2-2J_0.$$

No multiplicative scalar normalization of the endpoint insertion relates the two triples.

Finite obstruction

Let one boundary insertion carry endpoint weight $w$. Matching the mixed sector requires

$$w=2.$$

Matching the boundary--boundary sector requires

$$w^2=2.$$

But the first equation implies $w^2=4$. Therefore

$$\boxed{ \text{no multiplicative endpoint weight reproduces Eq. (19).} }$$

The $c_3$ counterterm cannot resolve this comparison: it contributes only to the oscillatory $I_4$ direction and cannot independently change the $J_0$ and $J_2$ coefficients.

Contraction audit

The discrepancy is not explained by ordinary Wick counting. Both the bulk and boundary operators have the same labelled $\zeta(\partial\zeta)^2$ slots. The connected fish census is

$$3\times3\times2=18$$

for every choice of bulk or boundary location. The common momentum vertex is the printed $(p^2+q^2+k^2)^2$.

Artifacts

• research/benincasa/checkers/finite_time_endpoint_multiplicativity_obstruction.rs
• research/benincasa/results/finite-time-endpoint-multiplicativity-obstruction.json
• the three sector checkers, which now assert their individual $J_1,J_2,J_0$ identifications at generic kinematics.

Narrow conclusion

The frozen formulas do not support repairing Eq. (19) by choosing a different scalar value for an endpoint delta function. At least one additional typed datum is required:

1. a labelled coincidence/diagonal rule for two endpoint occurrences;
2. a nonmultiplicative endpoint distribution prescription independently derived from the contour;
3. or a correction to the primary-source normalization.

This is not evidence for a new cosmological carrier stratum. It is an endpoint-comparison coherence obstruction.

Next falsifier

Derive the endpoint coincidence rule directly from a regulated contour in which the two boundary occurrences remain separated until the final limit. Compute the mixed one-boundary and boundary--boundary two-occurrence limits with the same regulator. If their weights are multiplicative, Eq. (19)'s printed normalization is falsified within the toy truncation. If a canonical diagonal excess survives, classify it as a coefficient/comparison cell before proposing any carrier modification.

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