PRH | Inter | 5.11 • A Category of Blur and the Grand Lemma

We axiomatize blur-a tunable neighbourhood/relaxation-as a categorical construction. A blurred object is a diagram $\mathcal{B}_X: I \rightarrow \mathcal{C}$ indexed by blur scales with a reading map $\rho_X: \lim _I \mathcal{B}_X \rightarrow X$ (sharp limit) and exhaustion $\operatorname{colim}_I \mathcal{B}_X \simeq 1$. An uncertainty window is a lax retract pair $j_X^{\varepsilon}: X \rightarrow B_{\varepsilon} X, r_X^{\varepsilon}: B_{\varepsilon} X \rightarrow X$. We prove a property-transport principle: for properties stable under retracts and filtered limits and monotone for $\precsim, \mathrm{P}(X)$ holds iff $\mathrm{P}_{\varepsilon}\left(B_{\varepsilon} X\right)$ holds eventually. Blur-morphisms are natural transformations; the Grand Lemma shows sending sharp equals sending blurred then reading, yielding a simple string diagram calculus and graded (co)monad scale composition. Examples include analytic blur (Markov/convolution semigroups) and logical blur (quantale-enriched nuclei).