Universal Entropy Ceiling in Finite-Depth Quantum Circuits: Numerical Evidence and a Modular-Forms Conjecture

We present large-scale numerical evidence for a universal entropy ceiling in finite-depth quantum circuits. Using GPU-accelerated simulation of up to 10^6random quantum states across local dimensions d=2 through d=30, we demonstratethat bipartite entanglement entropy saturates at a dimension-dependent ceiling following a logarithmic scaling law S/S_max ≈ 0.55 + 0.082·ln(D). The ceiling depends only on total Hilbert space dimension, not local structure. Weconnect this to the weight-12 transition in modular forms and conjecture anupper bound of 2,048 logical qubits for pairwise-coupled architectures, falsifiable against current industry roadmaps.