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 \( K^* ≤ 2d log_2 L / Hᵣ₂(∞) + O(1) \) assuming spatial decorrelation, with a conjectured factor-2 lower bound. For \( S_q \)-symmetric Potts models at continuous critical points we conjecture greedy tightness \( Δ ∈ \{0,1\} \). 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.