The Parrot's Theorems - An invariance classification of operational measurement reports

Abstract

An observer who cannot see a system directly -- only a dial and a reading -- reports numbers, but not every number is a property of the system. We sort operational readouts into three recurring invariance classes: integer counts of exposed degrees of freedom, reparametrization-invariant spectral rates, and locations on the dial that are fixed only up to a power law. Class membership follows the invariance group rather than the notation, and it decides interpretation. A count or a spectral rate may be read as a physical quantity; a characteristic dial location -- a susceptibility-peak scale among them -- is spectrum times a declared convention, $\sigma_c=\rho_\star\tau$, and is physical only once its gauge is fixed. The technical core is a completed characterization of when a peak location is invariant (exactly under power laws) together with this decomposition; a second, dial-free channel (correlated fluctuations at a clamped dial) recovers the same intrinsic rate and supplies a strong consistency test. We work each statement on a hand-checkable example and read one physical anchor through the scheme -- a peer-reviewed quantum-hardware susceptibility measurement -- showing which of its readouts survive relabeling and which move with the noise model. A documented blind-test failure marks the boundary between what the method can identify against a declared target and what it cannot identify blind. The payoff is a discipline that prevents a concrete, recurring error: mistaking a convention-carrying readout for a physical invariant.