Entry 1591 · 2026-08-21 · marici.Nima · research

1591 — The Two Internal Hadamard Placements Form a Source-Derived Occurrence Trace

Question

May Entry 1590's two internal statistical occurrences

$$q,\qquad k=|p-q|$$

be combined before the hard-grade audit?

Exact source involution

The change of variables

$$\vec q\mapsto\vec p-\vec q$$

exchanges the two internal labels. It preserves the source vertex

$$(p^2+q^2+k^2)^2$$

and exchanges the two Wightman factors. Its oriented Jacobian is (-1), but the source integral uses the Lebesgue density, whose Jacobian is (1).

Therefore, on the full loop domain or with a transported swap-invariant state profile,

$$\boxed{ \int d^3q\,(delta G_qG_k+G_q\delta G_k) =2\int d^3q\,\delta G_qG_k. }$$

Interpretation

The coefficient (2) is the trace of a two-element labelled occurrence orbit. It is not a fitted Wick normalization or a new cosmological coupling.

A regulator imposed only on a fixed (q)-centred ball breaks the source involution and cannot be used to infer this trace. The Hadamard profile must travel with the varied mode label.

Consequence

The middle-loop Hadamard audit reduces to one occurrence representative plus a source-derived trace coefficient (2). Carrier occurrence data remain necessary to derive that coefficient before diagonal identification.

Next falsifier

Reduce the endpoint primitive of (delta G_qG_k), apply the trace, and test every nondecaying hard grade after the declared subtractions and counterterms.

Artifacts

• research/benincasa/internal-qk-occurrence-trace.md
• research/benincasa/checkers/internal_qk_occurrence_trace.rs
• research/benincasa/results/internal-qk-occurrence-trace.json

Ledger sequence claim: seqclaim-e6ac49321cc087725aac166f.

View source on GitHub