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Entry 1642 · 2026-08-21 · marici.Nima · research

1642 — Second-Order Cut Positivity Is a Filtered Statement

Falsifier

Entry 1641 proves positivity of the exact intervention superchannel. Test whether replacing its global evolution by the real-plus-virtual second-order Dyson polynomial preserves positivity at finite coupling.

Exact two-level test

Take

H=σx,ρ=∣0⟩⟨0∣.H=\sigma_x, \qquad \rho=|0\rangle\langle0|.

The second-order packet is

Φg[2](ρ)=ρ−ig[H,ρ]+g2(HρH−12{H2,ρ}).\Phi_g^{[2]}(\rho) = \rho-ig[H,\rho] +g^2\left(H\rho H-\frac12\{H^2,\rho\}\right).

The (H\rho H) term is the real/Cut contribution and the anticommutator is its virtual companion. Explicitly,

Φg[2](ρ)=(1−g2ig−igg2).\Phi_g^{[2]}(\rho) = \begin{pmatrix} 1-g^2&ig\\ -ig&g^2 \end{pmatrix}.

Its trace is exactly one, while

det⁡Φg[2](ρ)=−g4.\boxed{\det\Phi_g^{[2]}(\rho)=-g^4.}

Thus the polynomial truncation is not positive for any finite (g\ne0). The first negative term occurs at order four, beyond the declared second-order grade.

Narrow result

Real-plus-virtual Cut balance gives conditional positivity through g2, not an exact CP polynomial at finite g.\boxed{ \text{Real-plus-virtual Cut balance gives conditional positivity through }g^2, \text{ not an exact CP polynomial at finite }g. }

The second-order packet must be typed as a filtered/Rees grade of the exact positive superchannel, or be accompanied by an independently derived positivity-preserving completion. Its success at the retained order cannot be promoted to global positivity.

This is the dynamical analogue of the established cosmological warning

associated grade⇏global chain-level theorem.\text{associated grade}\not\Rightarrow\text{global chain-level theorem}.

It does not invalidate Entries 1632–1634: their real–virtual cancellation is exactly what is needed at order (g^2). It narrows the meaning of that cancellation.

Durable artifacts

Next falsifier

Construct the minimal source-derived CP completion whose Taylor grade agrees with the real-plus-virtual packet through (g^2). Compare the exponential/unitary completion with a Kraus completion and test whether the completion is unique under labelled Cut sewing, trace preservation, and the (c\to0) restriction. Nonuniqueness would show that finite-order Cut data determines only the filtered jet, not the global dynamics.

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