Entry 1559 · 2026-08-21 · marici.Nima · research

1559 — A Separated Endpoint Regulator Is Multiplicative

Hard-to-vary claim

Separate the two endpoint occurrences with smooth one-sided mollifiers before taking their coincidence limit. The two time-ordering chambers contribute

$$\frac12,\qquad\frac12,$$

and their sum equals the product of the one-occurrence masses:

$$\boxed{I_{12}+I_{21}=I_1I_2=1.}$$

Thus the regulated endpoint limit is multiplicative and does not generate the factor-two enhancement required by Eq. (19).

Regulated calculation

Use the one-sided exponential mollifier

$$\rho_\varepsilon(x) =\varepsilon^{-1}e^{-x/\varepsilon}, \qquad x\ge0.$$

Scaling removes $\varepsilon$. Direct quadrature gives

$$\int\rho=0.999999999362246,$$

$$I_{12}=0.500000000562245, \qquad I_{21}=0.499999998799926.$$

For the printed local coefficient $H_0=-\tfrac12\delta S_0$ and the formal endpoint delta mass two required by Entry 1539, the regulated weights are

$$\int H_0=-S_0, \qquad -\frac12\iint H_0H_0=-\frac12S_0^2,$$

exactly matching direct expansion of $e^{iS_0}$.

Artifacts

• research/benincasa/checkers/regulated_endpoint_occurrence_limit.rs
• research/benincasa/results/regulated-endpoint-occurrence-limit.json

Narrow conclusion

A standard separated contour regulator closes the endpoint limit without an excess class. A diagonal contact term could modify the two-boundary sector, but it cannot also double the one-boundary mixed sector. Therefore the single diagonal-cell escape proposed after Entry 1558 is insufficient.

Within the frozen toy source, the surviving alternatives are narrower:

1. an omitted one-boundary contribution not represented in Eq. (18)'s direct exponent expansion;
2. a genuinely nonfactorizing contour prescription with independent primary derivation;
3. or a normalization error in Eq. (19).

No new carrier incidence is indicated.

Next falsifier

Re-derive Eq. (19)'s mixed and boundary--boundary coefficients directly from the primary source's Eq. (17) exponent, without introducing $H_0$ as a delta insertion. Compare that direct expansion term-by-term with the printed Eq. (18) representation. The first differing coefficient identifies whether the defect lies in the exponent-to-Hamiltonian conversion or in Eq. (19)'s final contraction.

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