Entry 1590 · 2026-08-21 · marici.Nima · research

1590 — Only the Middle Keldysh Placement Carries State Dependence Inside the Hard Loop

Type refinement of Entry 1589

The three statistical placements

$$F_LA_MA_R,\qquad R_LF_MA_R,\qquad R_LR_MF_R$$

do not have the same momentum variance in the source one-loop graph.

The left and right factors are external Dyson legs at observed momentum (p). The middle factor is the self-energy containing loop momenta

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

Hence only (F_M) places state dependence inside the hard loop. The outer placements multiply the ordinary Bunch--Davies self-energy by an external state variation and belong to vacuum renormalization acting on that leg.

Internal occurrence resolution

The middle statistical variation has two labelled internal occurrences,

$$q\quad\text{and}\quad k.$$

They may be identified only after deriving the source (q\leftrightarrow k) symmetry and preserving the integration orientation.

Raw ultraviolet grade

For simultaneous (q\sim k\sim Q), the source data give

$$d^3q:\ +2, \qquad (p^2+q^2+k^2)^2:\ +4, \qquad G_qG_k:\ -2.$$

Therefore the raw radial integrand has degree

$$\boxed{4.}$$

The source Hadamard condition (\beta_Q=o(Q^{-2})) lowers the middle state-dependent grade strictly below (2).

No convergence claim follows yet: time primitives, endpoint terms, subtractions, and counterterms remain to be included.

Consequence

The next Hadamard audit is one middle-self-energy calculation with two internal occurrence labels, not three symmetric Dyson-placement audits.

This is a coefficient-filtration question over the existing contour carrier.

Artifacts

• research/benincasa/keldysh-uv-responsibility-split.md
• research/benincasa/checkers/keldysh_uv_responsibility_split.rs
• research/benincasa/results/keldysh-uv-responsibility-split.json

Ledger sequence claim: seqclaim-de80afe4fc079a4cc0441afc.

View source on GitHub