A First-Order Entropy Law for Canonical T-Complexity of Finite-Alphabet i.i.d. Sources

Abstract

Let $W_N$ be an exact length-$N$ block from a strictly positive i.i.d. source $\mathbf p$ on a fixed finite alphabet. We prove that the canonical T-complexity $c_T$ satisfies \[ \frac{c_T(W_N)}{e^{-\gamma}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\mathbf p)$ is the source entropy in nats and $\gamma$ is the Euler-Mascheroni constant. The proof combines an exact length budget for canonical recovery, a critical-scale $E_1$ estimate for an ideal backward chain, and an exact finite-block boundary representation. An exact Doob-transform identity expresses the finite-boundary law relative to the ideal law conditioned at each step to avoid the current history-dependent successor codeword. A history-uniform renewal estimate then makes the telescoping endpoint density uniformly asymptotic to one, so no one-step approximation errors accumulate.