Birthday bound for observer convergence: a universal information-budget formula from classical criticality to quantum circuits

**How many independent local observers are required to uniquely identify one of \( n = L^d \) independent critical configurations?**

We establish a conditional birthday-style upper bound \( K^* \leq \lceil 2d \log_2 L / H_{r2}(\infty) \rceil + O(1) \) assuming spatial decorrelation, with a conjectured factor-2 lower bound. For \( S_q \)-symmetric Potts models at continuous critical points we conjecture greedy tightness \( \Delta \in \{0,1\} \).

Using sampling data for \( q=2,3,4 \) Potts and the BEG tricritical model, we verify the formula across four universality classes. For Ising (\( q=2 \)), a Temperley-Lieb algebra decomposition gives an exact orbit count verified to \( \leq 0.01 \) bits at \( w=3,7,8,9 \). A quantum extension to measurement-induced phase transitions empirically gives finite \( K^* \) near the area-law phase.