Manuscript XXII-E@1 · XXII-E.tex · section · 51c99d2f

The frozen class and its limitations

Conditional operational input › The frozen class and its limitations · lines 212–250

The XXII-C class is

P_{3q} = (W_{3})^{N} .

Each member $W_{e} \in W_{3}$ has three parties

J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2} .

With the transposed unnormalized Choi convention, validity means

W_{e} \geq 0, Tr [W_{e} (M_{A} \otimes M_{B} \otimes M_{C})] = 1

for all positive channel Choi matrices satisfying $Tr_{a_{O}} M_{a} = I_{a_{I}}$ . The access $A_{0}$ consists of the separately tested, one-use local instruments and unconditional classical receiver records declared in XXII-C CheungXXIIC (def:process, spec:access, def:class).

The direct influence relation has no cross-experiment edges. Its strongly connected component quotient, ordered by reachability, is $(E_{D}, ⪯_{D})$ . Every component has at most three elements in its reflexive future, and every chain has at most three distinct components CheungXXIIC (thm:order) CheungXXIID (lem:finite-futures).

An experiment identifier is not a spacetime coordinate. The data contain no derivative operation with respect to those identifiers, no Hamiltonian assignment, and no continuum metric field. The absence of cross-experiment tests is a declaration of access, not a geometric separation measurement.

Displayed formulas in this passage

unlabeled · line 215
$P_{3q} = (W_{3})^{N} .$
unlabeled · line 219
$J = {A, B, C}, H_{a_{I}} = H_{a_{O}} = C^{2} .$
unlabeled · line 225
$W_{e} \geq 0, Tr [W_{e} (M_{A} \otimes M_{B} \otimes M_{C})] = 1$