Timing inaccessibility and the projection bound: Resolving Maxwell’s demon for continuous biological substrates

We show that the thermodynamic advantage of biological, continuous substrates over digital simulators arises from timing inaccessibility: below the Landauer threshold, temporal order cannot be irreversibly registered without dissipating ≥k_BTln2 per binary decision. Consequently, exponentially many micro-trajectories map to the same observable outcome (path degeneracy). Continuous high-dimensional substrates exploit this by integrating sub-Landauer couplings during evolution and paying only at projection (dimensional collapse to a low-dimensional output). We derive a Projection Bound for quasistatic projection at effective temperature T_eff: [Formula: see text] which reduces under typical-set conditions to E_collapse≥k_BT_effln(N_ɛ,pre/N_ɛ,post). Combined with a Temporal Registration Bound (order over M bins needs log₂M! bits), we quantify the gap: enumerative digital tracking scales exponentially with dimension, whereas projection cost scales like lnG∼D. For biologically plausible parameters, we estimate degeneracies of 10⁴²-10⁹⁴ (protein folding) and 10⁵⁰-10¹⁰⁰ (neural dynamics) as upper bounds under independence assumptions. The framework reconciles stochastic resonance (amplitude detection) with order inaccessibility, and clarifies why analog/neuromorphic systems gain efficiency by deferring projection.