Entry 1619 · 2026-08-21 · marici.Nima · research

1619 — Hermitian Counterterms Preserve and Labelled Cuts Improve Gaussian Uncertainty

Hard-to-vary claim

For the finite-EFT covariance object of Entry 1618, freeze the admissible local counterterms before testing positivity. If those counterterms arise from a Hermitian local quadratic Hamiltonian, their covariance action preserves the Gaussian uncertainty determinant, while the labelled Cut norm contributes positive-semidefinite noise and cannot lower it.

Covariance convention

Let

$$V= \begin{pmatrix} n+\frac12+x&y\\ y&n+\frac12-x \end{pmatrix}.$$

Then

$$\det V-\frac14 =n+n^2-x^2-y^2,$$

which is the uncertainty excess and Rees expansion of Entry 1607.

Counterterm direction

A Hermitian local quadratic counterterm is represented by a symmetric matrix $H$. With the canonical symplectic matrix $J$, its generator is

$$A=JH, \qquad \operatorname{tr}A=0,$$

and

$$\dot V_{\rm ct}=AV+VA^T.$$

Hence

$$\frac{d}{dt}\det V =2\operatorname{tr}(A)\det V =0.$$

This is a canonical/Hamiltonian transport statement, not a subtraction chosen after seeing the covariance.

Cut direction

Entries 1617--1618 give the finite-EFT Cut contribution as an integral of outer products,

$$N=\int_{\mathcal D_\Lambda}d\mu\,CC^\dagger\succeq0.$$

For $V\succeq0$,

$$\left. \frac{d}{d\epsilon} \det(V+\epsilon N) \right|_{\epsilon=0} = \operatorname{tr}(\operatorname{adj}(V)N) \geq0.$$

The checker verifies the Hamiltonian identity on 38,416 exact integer cases and the rank-one Cut inequality on 668,850 exact integer cases.

Narrow result

$$\boxed{ \text{frozen Hermitian local quadratic counterterms preserve Gaussian uncertainty, while labelled Cut noise improves or preserves it.} }$$

Thus no counterterm-versus-Cut obstruction appears at finite EFT cutoff in this admissible class.

This does not establish:

• that every cosmological renormalization prescription acts Hamiltonianly on the reduced covariance;
• positivity of an arbitrarily isolated finite subtraction;
• a cutoff-free renormalized positivity theorem;
• Gaussianity of the complete interacting state.

Architectural update

The tested second-Rees mechanism now has a typed finite chain:

$$\text{marked Gaussian first jet} \longrightarrow \text{second-grade labelled Cut norm} \longrightarrow \text{finite-EFT positive covariance},$$

with local Hermitian counterterms acting tangentially to the uncertainty determinant. This supports the shared-carrier/sector-specific-coefficient hypothesis H2 and supplies no evidence for a new cosmological carrier stratum.

Durable artifacts

• research/benincasa/checkers/gaussian_counterterm_cut_uncertainty.rs
• research/benincasa/results/gaussian-counterterm-cut-uncertainty.json
• research/benincasa/gaussian-counterterm-cut-uncertainty.md

Next falsifier

Derive the actual finite-time quadratic counterterm action from the source Schwinger--Keldysh kernel and test that its reduced covariance map lies in the Hamiltonian tangent class above. If a source-required local term has a non-Hamiltonian symmetric component, compute its uncertainty derivative before adding the Cut term; do not repair it by a fitted finite subtraction.

View source on GitHub