Realizable Circuit Complexity: Embedding Computation in Space-Time

Classical circuit complexity characterizes parallel computation in purely combinatorial terms, ignoring the physical constraints that govern real hardware. The standard classes $\mathbf{NC}$, $\mathbf{AC}$, and $\mathbf{TC}$ treat unlimited fan-in, free interconnection, and polynomial gate counts as feasible -- assumptions that conflict with geometric, energetic, and thermodynamic realities. We introduce the family of realizable circuit classes $\mathbf{RC}_d$, which model computation embedded in physical $d$-dimensional space. Each circuit in $\mathbf{RC}_d$ obeys conservative realizability laws: volume scales as $\mathcal{O}(t^d)$, cross-boundary information flux is bounded by $\mathcal{O}(t^{d-1})$ per unit time, and growth occurs through local, physically constructible edits. These bounds apply to all causal systems, classical or quantum. Within this framework, we show that algorithms with runtime $\omega(n^{d/(d-1)})$ cannot scale to inputs of maximal entropy, and that any $d$-dimensional parallel implementation offers at most a polynomial speed-up of degree $(d-1)$ over its optimal sequential counterpart. In the limit $d\to\infty$, $\mathbf{RC}_\infty(\mathrm{polylog})=\mathbf{NC}$, recovering classical parallelism as a non-physical idealization. By unifying geometry, causality, and information flow, $\mathbf{RC}_d$ extends circuit complexity into the physical domain, revealing universal scaling laws for computation.