Proposition: Calibration and dictionary nonuniqueness

With distinct time labels, exact event-time calibration at a fixed event $e$,

has at most one supported outcome. For a normalized input it therefore has one deterministic outcome, or is impossible.

The clock and payload channels do not uniquely determine the history map, rod frame, or conical deficit.

Proof. Distinctness permits at most one index with ${T_k}^{\mathrm{hist}}=T(e)$, and probabilities sum to one. If the corresponding clock effect has probability one, positivity implies that the input is supported in that clock subspace.

For nonuniqueness, leave every finite matrix unchanged and reassign the history times, or rotate the developing rod frame and corresponding port coordinates. The abstract instrument is unchanged. Likewise different prescribed deficits admit isometric small regular laboratories carrying the same finite matrices and schedule. Their global completions differ.