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 \[ \frac{c_T(W_N)}{e^{-γ}h(\mathbf p) N/\log N}\longrightarrow1 \] in probability and in $L^r$ for every fixed $1\le r<\infty$, where $h(\mathbf p)$ is the source entropy in nats and $γ$ 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.

Disclosure

“e mathematical argu- ments; writing–review and editing. The author reviewed and approved the final manuscript and assumes full responsibility for its correctness and integrity. Use of artificial intelligence. GPT-5.6 Sol Pro was used as an AI-assisted tool for formal analysis and for developing and drafting the mathematical proofs. All AI-generated mathemati- cal content was reviewed and validated by the author. GPT-5.6 Sol Pro is not an author and assumes no responsibility for the cont”

PDF page 10

Classification

Substantial proof generation

Multiplier

10

Verified

2026-08-23T05:22:31.497716Z

Structural counts