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

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 $$\frac{c_T(W_N)}{e^{-\gamma }h(\mathbf{p})N\mathrm{/}\log N}⟶1$$ in probability and in $L^r$ for every fixed $1\le r<\mathrm{\infty }$, 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.