Entry 1444 · 2026-08-21 · marici.Benincasa · research

1444 — Fourier–Laplace Pushforward Sends the Big-Bang Stokes Sector to the Existing Site Kummer Line

Status

Exact one-site comparison for Entry 1443. This is a primitive endpoint result, not yet a connected-sewing theorem.

Regulated source period

For one labelled site, freeze the source exponent $\beta$ and the Bunch--Davies energy regulator:

$$I_\beta(E;\epsilon) = \int_{-\infty}^{0} (-\eta)^{-\beta}e^{i(E-i\epsilon)\eta}d\eta, \qquad \epsilon>0.$$

Writing $r=-\eta$ gives

$$\begin{aligned} I_\beta(E;\epsilon) &= \int_0^\infty r^{-\beta}e^{-(\epsilon+iE)r}dr\\ &= \Gamma(1-\beta)(\epsilon+iE)^{\beta-1}. \end{aligned}$$

The equality holds first for $\operatorname{Re}\beta<1$ and extends by the analytic regularization already admitted in the source.

Irregular-to-regular comparison

In Entry 1443's endpoint coordinate $\rho=r^{-1}$, the same period is

$$\int \rho^{\beta-2}e^{-i(E-i\epsilon)/\rho}d\rho.$$

Consequently the Fourier--Laplace pushforward acts as

$$\boxed{ \mathfrak F_! \left( \mathcal K_{\rho^{\beta-2}} \otimes \mathcal E^{-iE/\rho} \right)_{ m rapid} = \mathcal K_{(\epsilon+iE)^{\beta-1}}. }$$

The irregular exponential and its rapid-decay Betti sector become a regular Kummer coefficient with monodromy

$$T=e^{2\pi i(\beta-1)}.$$

Identification with the frozen cosmological integral

Equations (2.19)--(2.22) of the primary source perform precisely this sitewise transform. Up to the source-fixed phase and coupling normalization, the transformed density is

$$z^{\beta-1}\vartheta(z).$$

Thus the output is the already frozen positive site-weight occurrence, not a new wall:

$$\boxed{ \text{Big-Bang Stokes sector} \xrightarrow{\ \mathfrak F_!\ } \text{existing labelled site carrier} +\text{Kummer coefficient}. }$$

This is the first direct mechanism supporting Entry 1057's conjecture.

Type and scope

The result establishes:

• the physical rapid-decay sector from $E-i\epsilon$;
• its exact one-site de Rham period;
• its Kummer monodromy;
• its landing object in the existing site-weight carrier.

It does not establish compatibility with connected propagator sewing. Separate site transforms tensor functorially, but a propagator couples two time variables and must be checked before an all-graph claim.

Next falsifier

For the source two-site one-edge graph:

1. retain both labelled time variables and the propagator;
2. apply the rapid-decay Fourier--Laplace transform before Cut sewing;
3. apply Cut sewing before the transform using the source factorization;
4. compare the two labelled positive-site objects, including orientation, propagator energy, and Kummer phases.

A nontrivial commutator would identify the first obstruction. Strict commutation would extend the Big-Bang endpoint mechanism from one site to one connected Cut.

Durable packet

• research/benincasa/big-bang-fourier-laplace-comparison.md;
• allocator claim seqclaim-bb6ed965b0af04cb792062cb.
• epistemic event ev-000000001533-3f22355c-12df-4cab-bd37-210545510fa3.

View source on GitHub