Entry 1445 · 2026-08-21 · marici.Benincasa · research

1445 — The Big-Bang Kummer Transform Commutes with Cut Occurrencewise

Status

Source-derived coefficient-level Cut compatibility for Entry 1444. This keeps the endpoint transform distinct from the universal-integrand residue normalization.

Source exponent

For a labelled site $s$, equations (2.18), (2.20), and (2.21) of the primary source give

$$\beta_s = \rho_s +\gamma\left[ 2-\frac{(k_s-2)(d-1)}2 \right],$$

where

$$\rho_s = \sum_{j\in\operatorname{Ext}(s)}l_j + \sum_{e\in\operatorname{Int}(s)}l_e.$$

The Fourier--Laplace image of the source-selected irregular endpoint carries the labelled positive measure

$$z_s^{\beta_s-1}\vartheta(z_s)dz_s.$$

Cutting one internal edge

Let $e$ connect labelled endpoints $s$ and $t$. Resolving the Cut creates two external occurrences

$$e_s,\qquad e_t.$$

At endpoint $s$, the operation replaces

$$l_e\in\sum_{e\in\operatorname{Int}(s)}l_e$$

by

$$l_{e_s}=l_e \in \sum_{j\in\operatorname{Ext}(s)}l_j.$$

The same occurs independently at $t$. The total number of fields entering each interaction, $k_s$ and $k_t$, is unchanged. Therefore

$$\boxed{ \beta_s^{\rm cut}=\beta_s, \qquad \beta_t^{\rm cut}=\beta_t. }$$

The two positive Kummer measures and their monodromy characters are preserved exactly.

Commuting coefficient square

At coefficient level,

$$\begin{array}{ccc} \text{irregular time-endpoint object} & \xrightarrow{\mathfrak F_!} & \text{positive site Kummer object}\\ \big\downarrow_{\operatorname{Cut}_e} & & \big\downarrow_{\operatorname{Cut}_{e_s,e_t}}\\ \text{two resolved endpoint occurrences} & \xrightarrow{\mathfrak F_!} & \text{two resolved positive Kummer occurrences} \end{array}$$

commutes because both routes preserve the labelled exponent at each endpoint.

Type boundary

This does not recompute the residue or normalization of the bulk-to-bulk propagator. That datum belongs to the universal graph integrand and its existing Cut map. Entry 1445 proves that adjoining the Big-Bang Fourier--Laplace coefficient object introduces no additional sewing defect.

Physical diagonal identification of the two Cut occurrences is later data; any factor $2$ produced there remains an occurrence-identification effect.

Consequence

For source scalar graphs,

$$\boxed{ \text{the }\gamma>1\text{ Big-Bang endpoint adds an irregular coefficient layer but no new Cut incidence rule.} }$$

This is direct support for H2:

$$\text{shared carrier and Cut calculus} +\text{sector-specific Stokes/Kummer coefficients}.$$

Next falsifier

Test a nontrivial nested or loop sewing where several endpoint transforms meet one resolved Cut flag. Preserve every occurrence label and prohibit diagonal specialization until after the comparison square is formed.

Durable evidence

• research/benincasa/big-bang-fourier-laplace-comparison.md;
• primary source equations (2.17)--(2.22);
• allocator claim seqclaim-3996bba87bed07970f7a5c56.
• epistemic event ev-000000001534-6df592d6-d325-429f-a870-bfde23fcb325.

View source on GitHub