Computational Complexity of Physical Counting

Abstract

We characterize which coordinates of a factored state space determine optimal actions. For $\mathcal{D}=(A,S,U)$ with $S=X_1\times\cdots\times X_n$, coordinate set $I$ is sufficient if $s_I=s'_I\Rightarrow\operatorname{Opt}(s)=\operatorname{Opt}(s')$. The decision quotient $Q=S/{\sim}$ ($s\sim s'\Leftrightarrow\operatorname{Opt}(s)=\operatorname{Opt}(s')$) is the minimal abstraction: any abstraction preserving optimal actions factors uniquely through $Q$. We prove fourteen first-principles theorems (thirteen from pure mathematics, one empirical). The chain from counting measure to probability to Bayes' theorem to $Q$ follows from finite set cardinality. Fisher information, entropy, optimal transport, rate-distortion, and thermodynamics each independently recover $\mathrm{srank}$ as decision complexity measure. From $\log x\leq x-1$ alone, Bayesian updating uniquely minimizes expected log loss. Complexity: SUFFICIENCY-CHECK and MINIMUM-SUFFICIENT-SET are coNP-complete; ANCHOR-SUFFICIENCY is $\Sigma_2^P$-complete; stochastic and sequential variants PP- and PSPACE-complete with strict separation. Six subcases admit polynomial algorithms. Under ETH, succinct encodings carry $2^{\Omega(n)}$ lower bounds. Verification requires $\geq 2^{n-1}$ witness pairs. Two results carry empirical conditions. Conditional on Landauer's principle ($k_BT\ln 2$ per bit erasure; experimentally confirmed 2012), $dU\geq\lambda\,dC$ follows by composition with bit-operation bounds; rejecting it requires rejecting Landauer. Conditional on stochastic thermodynamics (Barato--Seifert 2015), $\mathrm{Var}(J)/\langle J\rangle^2\geq 2/\sigma$ bounds decision precision by entropy production, minimal $\sigma$ scaling with $\mathrm{srank}$.