Manuscript II-B@1 · II-B-v1.tex · proposition · bfd2eb3c

Proposition: The clock does not select its contact dictionary

The three dictionaries and calibration › Nonuniqueness · lines 1535–1558 · proof included or linked

II-B@1#prop:label-nonunique

The same complete clock instrument is compatible with different record-to-contact assignments.

Proof. Keep $M_{C}$ , its basis, locations, and input unchanged. Replace $δ$ by

δ^{'} (0) = β_{+}, δ^{'} (1) = β_{0}, δ^{'} (2) = β_{-}, δ^{'} (3) = β_{0} .

Nothing in (eq:instrument) changes. On the input $P_{0} \otimes ρ_{X}$ , however,

m (δ (0)) = 0, m (δ^{'} (0)) = 16/3.

Therefore the clock channel does not determine the label assignment. A later dictionary-dependent probe would be a different declared continuation.

Displayed formulas in this passage

unlabeled · line 1544
$δ^{'} (0) = β_{+}, δ^{'} (1) = β_{0}, δ^{'} (2) = β_{-}, δ^{'} (3) = β_{0} .$
unlabeled · line 1552
$m (δ (0)) = 0, m (δ^{'} (0)) = 16/3.$