Entry 1633 · 2026-08-21 · marici.Nima · research

1633 — The Source Dyson No-Production Cell Balances the Labelled Cut Trace

Question

Entry 1632 proves that the production Cut is completely positive but not trace-preserving by itself. Does the frozen source supply the matching virtual/no-production term, or must one fit an anti-Hermitian correction?

Source expansion

Entry 1616 fixes each production channel as

$$C_r=-i(H_r-S_r).$$

The same source exponential supplies the second-order no-production coefficient. Writing the coherent Hamiltonian part separately,

$$K_0 = I-igH_{\rm coh} -\frac{g^2}{2} \left( H_{\rm coh}^2 +\sum_r C_r^\dagger C_r \right) +O(g^3),$$

and

$$K_r=gC_r+O(g^2).$$

Therefore

$$\boxed{ K_0^\dagger K_0 +\sum_rK_r^\dagger K_r =I+O(g^3). }$$

At second order, the negative virtual trace coefficient is exactly

$$-\sum_rC_r^\dagger C_r,$$

and the positive labelled Cut coefficient is its opposite.

Endpoint coefficients and occurrence multiplicities

Channelwise,

$$|H-S|^2 =|H|^2 -2\operatorname{Re}(H\bar S) +|S|^2.$$

Thus the algebraic norm coefficients are

$$(1,-2,1).$$

The unsigned location multiplicities remain

$$(1,2,1).$$

The mixed minus sign is carried by the source endpoint amplitude; it is not a negative occurrence multiplicity.

The checker verifies 2,401 exact complex endpoint packets.

Narrow result

$$\boxed{ \text{The source Dyson no-production cell and labelled Cut cell satisfy trace balance through second order without a fitted correction.} }$$

The produced covariance contribution remains positive; trace preservation arises only after adding the virtual normalization cell.

Qualifications

• This is perturbative trace preservation through second order, not an all-order channel construction.
• The continuum statement uses the symmetric positive finite-EFT measure of Entry 1618.
• Renormalized cutoff removal remains untyped by Entry 1620.

Architectural consequence

The finite-time coefficient calculus acquires the standard typed pair

$$\text{no-production/virtual cell} \oplus \text{labelled production Cut cells},$$

with their normalization relation inherited from one source exponential. This is a coefficient-level optical-theorem structure over the existing occurrence carrier.

Durable artifacts

• research/benincasa/checkers/dyson_kraus_trace_balance.rs
• research/benincasa/results/dyson-kraus-trace-balance.json
• research/benincasa/dyson-kraus-trace-balance.md

Next falsifier

Test compatibility of this trace-balance identity with moment truncation:

$$\operatorname{res}_{\le D} \circ\Phi_{D+2} \stackrel?= \Phi_D \circ\operatorname{res}_{\le D+2}.$$

Retain the virtual and production cells together. A failure would show that finite moment truncations do not form a strict channel tower and require a derived coherence map.

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