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 = Lᵈ \) independent critical configurations? ** We establish a conditional birthday-style upper bound \ (K^* 2d ₂ L / Hₑ₂ () + O (1) \) assuming spatial decorrelation, with a conjectured factor-2 lower bound. For \ (Sq \) -symmetric Potts models at continuous critical points we conjecture greedy tightness \ (\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 \ (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. v3 changelog: Corrected w=6 boundary case. Previous versions stated j=3 ghost sector inactive at w=6; 20 independent Monte Carlo runs (L=256+512) show partial activation ω₃ = 0. 415 ± 0. 033 (13. 3σ above zero). The ghost condition 4=0 kills the linear Markov trace but NOT the quadratic collision probability. v4: Three rounds of honest-hedging revisions — (1) dropped "universal" from title/abstract; α-invariant downgraded to exploratory conjecture; w=6 data updated to 20 runs, ω₃=0. 415±0. 033 (13. 3σ). (2) TL/ghost paragraph restructured into proved/empirical/heuristic layers; quantum section compressed to Discussion. (3) Added auditable K*greedy methods paragraph; α-invariant section renamed "CONJECTURES"; FK heuristic marked conditional.