Algorithmic Naturalness on a Quantum Substrate: From the Impossibility Trilogy to the Native Realization of Axiom A1 in A1

This paper addresses the "algorithmic fine-tuning problem": why does our universe exhibit quantum mechanics if quantum mechanics is algorithmically improbable on a classical substrate? Building on our trilogy establishing the impossibility of deriving quantum structure (Axiom A1) from classical computation, we propose the Substrate Hypothesis: the universe's computational substrate is "quantum-native." We extend Chaitin's halting probability Ω from a real scalar to a state vector |Ω_Q⟩ in Hilbert space---the wavefunction of the algorithmic multiverse. We prove its normalizability (Theorem 1) and establish a direct connection to classical Ω through the halting projection (Theorem 2). Using a minimal model language A1---named after the axiom it natively implements---we measure description length asymmetry: quantum operations require 25× fewer tokens on average (up to 40× at bit level) on a quantum substrate compared to classical NumPy implementations. This corresponds to an algorithmic probability gain of 2^878. We validate these theoretical predictions experimentally on AWS Braket, achieving >97% fidelity across all benchmarks with only 2--6 A1 tokens. This provides engineering validation of the Substrate Hypothesis: quantum mechanics is algorithmically natural on a quantum substrate.