PRH | Essay | 7.22 • The Blur–Equivalence Principle

We formulate a general blur-equivalence principle for mathematical models. Once a finite blur budget is fixed, all blur-compatible descriptions of the same universe are equivalent with respect to blur-local properties, including finite-time blow-up and the need to pass to a larger "universe" of objects (poles). The formal setting is that of blurred universes, where each state space is equipped with a family of blur operators and a notion of blur-local observables. A morphism between such universes is admissible if it commutes with blur up to the prescribed budget. We show that blur-local, stable and monotone properties are invariant along any chain of admissible descriptions.

From this we deduce that, at fixed blur budget, events such as blow-up, the creation of a pole, or the existence of a Lyapunov-type certificate are intrinsic to the blur universe rather than artefacts of a particular coordinate system or parametrization. Either every blur-compatible description sees the event, or none of them do.

The principle is formulated abstractly and proved using a minimal fragment of the general category of blur. Concrete applications - to evolution equations, analytic continuation, randomness, and complexity - are developed elsewhere; here we concentrate on the structural statement itself, with only schematic illustrations via poles and certificates.