Entry 1446 · 2026-08-21 · marici.Benincasa · research

1446 — The Big-Bang Kummer Transform Is Natural for Arbitrary Resolved Cut Flags

Status

All-graph coefficient theorem extending Entry 1445. The theorem applies to occurrence-resolved Cuts, including nested and loop Cut flags. It does not apply to edge deletion that discards the endpoint occurrences.

Local invariant

At every labelled site $s$, the endpoint Kummer exponent is

$$\beta_s = \sum_{j\in\operatorname{Ext}(s)}l_j + \sum_{e\in\operatorname{Int}(s)}l_e + \gamma\left[2-\frac{(k_s-2)(d-1)}2\right].$$

Let $C$ be any set of internal edges, with no restriction on whether those edges are bridges, loop edges, nested interfaces, or members of a compatible Cut flag. Resolve each $e\in C$ into endpoint occurrences

$$e_s,\qquad e_t.$$

At each endpoint, the operation is the replacement

$$l_e\in\operatorname{Int}(s) \quad\longmapsto\quad l_{e_s}=l_e\in\operatorname{Ext}(s).$$

Therefore every summand remains present exactly once and

$$\boxed{ \beta_s(G)=\beta_s(\operatorname{Cut}_C G) \quad\text{for every site }s. }$$

Strict flag coherence

For two Cut sets $C_1,C_2$, each edge-label move is local and the moves act on distinct labelled occurrences. Hence

$$\operatorname{Cut}_{C_2}\operatorname{Cut}_{C_1}\beta_s = \operatorname{Cut}_{C_1}\operatorname{Cut}_{C_2}\beta_s = \operatorname{Cut}_{C_1\cup C_2}\beta_s.$$

There is no exponent-level coherence defect and no ordering-dependent Kummer character.

Machine audit

The Rust checker exhausts all simple labelled graphs through five sites, all Cut subsets, and all ordered two-stage Cut assignments. Edge labels are kept unequal so that occurrence loss cannot hide in an unlabelled count.

It verifies:

$$59808\ \text{graph--Cut pairs},$$

$$298248\ \text{labelled site invariance checks},$$

and

$$1052740\ \text{ordered flag checks}.$$

All pass exactly.

Important distinction

Deleting an edge without retaining its two endpoint occurrences removes $l_e$ from both endpoint sums and generally changes $\beta_s$. Thus

$$\text{occurrence-resolved Cut} \neq \text{bare edge deletion}.$$

The theorem supports the common Cut carrier. It does not identify the correlator deletion operation with Cut sewing.

Consequence

For arbitrary source scalar graphs,

$$\boxed{ \mathfrak F_!^{\rm BB} \text{ is a strict coefficient-natural transformation over resolved Cut flags.} }$$

The $\gamma>1$ endpoint adds Stokes/Kummer coefficient data but no new nesting primitive, loop incidence rule, or Cut coherence cell.

Next falsifier

The remaining nontrivial comparison is not combinatorial sewing. It is whether the Fourier--Laplace pushforward is compatible with the source propagator residue/Gysin normalization and with physical diagonal specialization. Test that square on the two-site one-edge universal integrand without replacing Cut by deletion.

Durable evidence

• research/benincasa/marici-gm/src/bin/big_bang_kummer_cut_flags.rs;
• research/benincasa/results/big-bang-kummer-cut-flags.json;
• allocator claim seqclaim-2d7bd152150724cbe6896211.
• epistemic event ev-000000001536-8bcf9c2f-2403-4bba-abfd-6f1fcb92bafa.

View source on GitHub