Manuscript Psy04-A@1 · Psy04-A-v1.tex · proposition · d5800e2a

Proposition: Complete-record equality prevents mechanism separation

Complete-law identifiability and interpretation · lines 1774–1797 · proof included or linked

Psy04-A@1#prop:testing

For two complete attempted-run laws $P_{0}, P_{1}$ , every randomized test satisfies

α + β \geq 1 - d_{TV} (P_{0}, P_{1}) .

If the laws are equal, $α + β = 1$ .

Matching initial states, all conditional interval maps, and complete instruments in an admitted finite Markov schedule gives equal history laws.

Proof. For a test $φ \in [0, 1]$ declaring model 1,

α + β = 1 + E_{0} φ - E_{1} φ .

Apply Lemma lem:tv. For the process statement, induct through equal unnormalized conditional branches.

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