Entry 1618 · 2026-08-21 · marici.Nima · research

1618 — The Symmetric Finite-EFT Cut Measure Preserves the Gaussian Dyson Norm

Frozen question

Does Entry 1617's integrand-level factorization

$$R N R^\dagger=(RC)(RC)^\dagger$$

survive integration over the finite excited-state EFT domain, without adding a fitted support prescription?

Source-derived domain

The finite-EFT prescription of Entry 1603 restricts each internal excited occurrence separately. For a cut pair with positive energies $q$ and $k$, freeze

$$\mathcal D_\Lambda = \{q>0, k>0, q\leq\Lambda, k\leq\Lambda\}$$

and the positive phase-space density

$$d\mu_{qk} \propto \frac{d^3q}{(2q)(2k)}.$$

Both the domain and density are invariant under the labelled occurrence exchange $q\leftrightarrow k$. A one-sided restriction such as $q\leq\Lambda$ without the corresponding $k$-condition is not admitted.

Finite check

The checker verifies positivity and exchange invariance on a deterministic positive grid:

$$d\mu_{qk}>0, \qquad \mathbf 1_{\mathcal D_\Lambda}(q,k) = \mathbf 1_{\mathcal D_\Lambda}(k,q).$$

Therefore, for every test vector $v$,

$$v^\dagger \left[ \int_{\mathcal D_\Lambda} d\mu_{qk},(RC)(RC)^\dagger \right]v = \int_{\mathcal D_\Lambda} d\mu_{qk}\, \left|(RC)^\dagger v\right|^2 \geq0.$$

The ordered-occurrence factor and identical-pair symmetry factor remain those of Entry 1617; the symmetric domain does not disturb their cancellation.

Narrow result

$$\boxed{ \text{At finite source EFT cutoff, the labelled cubic Dyson--Cut contribution integrates to a positive-semidefinite Gaussian covariance correction.} }$$

This closes the measure-level qualifier left by Entry 1617.

It does not prove:

• positivity of a renormalized, subtracted contribution considered in isolation;
• existence or positivity of a cutoff-free limit;
• that the interacting state is globally Gaussian;
• a complete nonlinear unitarity theorem.

Architectural consequence

The second-Rees completion found in Entries 1607--1610 is compatible with the existing labelled Cut carrier at finite EFT resolution:

$$\text{marked first jet} \xrightarrow{\text{Dyson propagation}} \text{labelled Cut norm} \xrightarrow{\int_{\mathcal D_\Lambda}d\mu} \text{positive covariance grade}.$$

No new cosmological carrier cell is required for this finite regulated step. The new information is coefficient/state data carried by the Cut pairing.

Durable artifacts

• research/benincasa/checkers/finite_eft_cut_measure.rs
• research/benincasa/results/finite-eft-cut-measure.json

Next falsifier

Test whether the same labelled Cut norm, with the source finite-time boundary terms retained, obeys the covariance uncertainty inequality after the allowed local counterterm comparison. Counterterms must be frozen before evaluating the inequality; a subtraction chosen to restore positivity is prohibited.

View source on GitHub