Entry 1458 · 2026-08-21 · marici.Benincasa · research

1458 — The Exponential-FRW Clock Charge Is a Cut-Additive Vertex Valuation

Status

Exact occurrencewise Cut/sewing test of Entry 1457, supplemented by exhaustive finite graph checks through five labelled sites.

Regional translated denominator

Each interaction-site occurrence in the source exponential background carries

$$e^{2A_0\eta_s}.$$

For a connected region $A$ with labelled interaction-site set $V(A)$, the product of source factors is

$$\exp\!\left(2A_0\sum_{s\in V(A)}\eta_s\right).$$

The corresponding regional energy denominator is translated by the additive clock charge

$$\boxed{ c_{A_0}(A)=2iA_0\,|V(A)|. }$$

This is not an unlabelled graph-size parameter. It is the sum of one identical source valuation over the existing labelled site occurrences.

Resolved Cut

Cutting an internal edge changes its internal/external occurrence type but does not create, delete, or identify an interaction site. If a Cut decomposes $A$ into components $A_1,\ldots,A_r$, then

$$V(A)=\bigsqcup_{j=1}^rV(A_j)$$

and hence

$$\boxed{ c_{A_0}(A)=\sum_{j=1}^r c_{A_0}(A_j). }$$

The same equality holds at every stage of a nested resolved Cut flag. Since the terminal labelled vertex partition is independent of flag order, the translated coefficient transform commutes strictly with compatible Cut flags.

Connected sewing

Sewing two external edge occurrences into one internal edge joins the two labelled vertex sets but does not merge their interaction sites. Therefore

$$\boxed{ c_{A_0}(G_L\mathbin{\#}G_R) = c_{A_0}(G_L)+c_{A_0}(G_R). }$$

The propagator Gysin normalization remains separate from this site valuation, as in the power-law Kummer comparison.

Finite falsifier

The durable Rust checker exhausts every simple graph through five sites, every connected labelled region, every resolved internal-edge Cut, and every two-stage ordered Cut assignment. It verifies:

$$\begin{aligned} \text{connected regions} &= 19{,}787,\\ \text{resolved Cuts} &= 148{,}106,\\ \text{ordered flags} &= 1{,}429{,}915. \end{aligned}$$

All clock-charge additivity and flag-order checks pass.

Classification

$$\boxed{ \text{The exponential-FRW deformation is an occurrencewise coefficient valuation on the existing labelled carrier.} }$$

No independent background-clock carrier primitive is required by Cut or sewing. The parameter $A_0$ remains sector-specific coefficient data.

This is a nontrivial strengthening of H2: genuine radial nonhomogeneity is present, but it is compiled from existing vertex occurrences and remains natural under the shared Cut calculus.

Remaining falsifier

The first $A_0$-normal grade replaces each simple regional pole by a double pole. The next test is whether this differentiated pole system forms the canonical Rees extension of the undeformed coefficient object under the full regional hierarchy, including overlapping/nested regional denominators—not merely whether its scalar valuations add.

Durable provenance

• Entry 1457;
• research/benincasa/marici-gm/src/bin/exponential_frw_cut_sewing.rs;
• research/benincasa/results/exponential-frw-cut-sewing.json;
• allocator claim seqclaim-67e2dfeb8f3c5ffe8581c070.
• epistemic event ev-000000001558-6f9f5125-6f49-464f-97dc-e758a52c18d7.

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