Manuscript XIX-C@1 · XIX-C.tex · section · 98b067d9

Complete finite channel realization

Declared finite incidence interface › Complete finite channel realization · lines 373–414

XIX-C@1#eq:classical-channel

A deterministic update on a classical register has the channel

U_{ℓ} (ρ) = ω \in Ω_{K} \sum ⟨ ω ∣ ρ ∣ ω ⟩ ∣ u_{ℓ} (ω)⟩ ⟨ u_{ℓ} (ω)∣ .

Its Kraus operators are

K_{ω} = ∣ u_{ℓ} (ω)⟩ ⟨ ω ∣, ω \sum K_{ω}^{†} K_{ω} = I .

Thus it is completely positive and trace preserving. On diagonal states it is exactly the classical probability pushforward. Its action on coherent superpositions is a separately declared dephasing realization, not an automatically reversible implementation.

If reversibility is required, the complete update must be a permutation on a declared enlarged finite space. An irreversible function is not renamed a permutation.

When the geometric and source letters act as the identity on an internal matter factor $M$ , their realization is

U_{ℓ} \otimes id_{M} .

For a fixed external token this does not change the unconditioned matter marginal, even in the presence of reference correlations: trace preservation on the controller proves the statement. It is therefore not a construction of gravity-generated internal entanglement.

This is a finite semantic realization only. No native Enter-program compilation, gate synthesis, or global live-qubit certificate is claimed.

Displayed formulas in this passage

XIX-C@1#eq:classical-channel · line 376
$U_{ℓ} (ρ) = ω \in Ω_{K} \sum ⟨ ω ∣ ρ ∣ ω ⟩ ∣ u_{ℓ} (ω)⟩ ⟨ u_{ℓ} (ω)∣ .$
unlabeled · line 384
$K_{ω} = ∣ u_{ℓ} (ω)⟩ ⟨ ω ∣, ω \sum K_{ω}^{†} K_{ω} = I .$
unlabeled · line 401
$U_{ℓ} \otimes id_{M} .$