Hiroshi Kohashiguchi

This paper presents a synthesis of four interconnected research programs that together establish a quantum-native interpretation of Chaitin's halting probability Omega and Teilhard de Chardin's Omega Point. We begin with the Unified Omega Hypothesis, which proposed that existence itself might be understood as a computation whose completion corresponds to the determination of Omega. However, Minimal Axioms for Quantum Structure demonstrated that classical computation cannot derive quantum structure (Axiom A1: superposition), establishing a no-go theorem formally verified in Coq. This negative result necessitated a fundamental revision. Two independent lines of evidence support the conclusion that a quantum substrate is overwhelmingly favored: (1) Algorithmic Naturalness shows that quantum mechanics has minimal description length on a quantum substrate (K_UQ << K_UC), and (2) Artificial Physics demonstrates through evolutionary simulation that matrix operations (quantum-like structures) dominate scalar operations under selection pressure, achieving >90% dominance within 3.8 ± 1.7 generations, while spontaneous emergence of matrix operations from scalar primitives never occurred (0/5 runs over 500 generations). We therefore propose the Quantum Omega Hypothesis: if the universe is computational and the principle of algorithmic naturalness holds, then a quantum-native substrate is the most parsimonious explanation. Under this interpretation, Chaitin's Omega becomes a quantum state vector |Omega_Q> = Σ_p α_p |p| — the wavefunction of the algorithmic multiverse. This synthesis provides a coherent framework connecting algorithmic information theory, quantum foundations, and evolutionary dynamics, while clearly distinguishing between established facts and interpretive proposals.

We present a computational framework for understanding the emergence of quantum-like structures through evolutionary competition of Domain-Specific Languages (DSLs) under resource constraints. Using dynamic task evaluation---where graph size N ~ U(3,10) and steps k ~ U(2,6) vary randomly---we prevent scalar DSLs from "memorizing" fixed solutions. Our experiments with N=100 population and 10 runs show that Matrix DSLs achieve 100% dominance within 3.8 ± 1.7 generations (95% CI: [2.6, 5.0]), providing evidence for the Substrate Hypothesis. We further investigate spontaneous emergence of matrix operations from scalar-only DSLs across 500 generations (Experiment 2A), finding that matrix operations never emerged (0/5 runs) despite scalar DSLs achieving only 60--80% success on dynamic tasks. When matrix operations are injected (Experiment 2B), success rates jump from 70--80% to 100% (+24.0% ± 5.5%), demonstrating the decisive advantage of matrix structure for generalization.

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.

We present a comprehensive investigation into the minimal axioms required to derive quantum structure from classical computation. Through systematic analysis of multiple computational models—SK combinatory logic, reversible logic gates (Toffoli, Fredkin), reversible cellular automata, and lambda calculus—we establish that no form of computation, whether irreversible or reversible, can generate quantum structure. Our main results are: 1. The No-Go Theorem: Reversible n-bit gates are 2^n × 2^n permutation matrices that embed into the classical symplectic group Sp(2·2^n, R), not the unitary group U(2^n). This theorem has been formally verified in Coq. 2. Axiom Identification: Using Generalized Probabilistic Theories (GPTs), we identify A1 (state space extension/superposition) as the unique primitive axiom that cannot be derived from computation. 3. Universality: Results hold across all Turing-complete computation models tested, establishing computation-model independence. These findings definitively answer the question "Can quantum structure be derived from computation?" with: No, without the superposition axiom A1. The gap between classical and quantum is precisely A1—and this gap is mathematically unbridgeable by computation alone.

We investigate whether complex number structure, fundamental to quantum mechanics, can be derived from SK combinatory logic—a minimal, Turing-complete computational system. Through systematic exploration of four distinct approaches (Sorkin's quantum measure theory, algebraic structure of reduction operators, path space holonomy, and information-theoretic derivation), we find that complex structure does not automatically emerge from SK computation. While we discovered that phase differences can be computed from information-theoretic quantities (specifically, the difference in S-reduction counts between paths), the choice of computation formula remains arbitrary. This constitutes a negative result with significant implications: it establishes constraints on the relationship between computational substrates and quantum structure, clarifying what additional assumptions are necessary to bridge the gap between computation and quantum mechanics.

Following our previous work establishing that complex structure does not automatically emerge from SK combinatory logic, we investigate whether reversible computation provides the missing ingredient for quantum structure. Through systematic analysis of four computational models—reversible logic gates (Toffoli, Fredkin), continuous-time quantum walks, reversible cellular automata, and the non-commutativity of SK operators—we establish a hierarchy of quantum-like behaviors: Level 0 (Irreversible): SK computation—classical, deterministic Level 1 (Discrete Reversible): Toffoli/Fredkin gates, RCA—classical, embeddable in Sp(2N,R) where N = 2^n for n-bit gates Level 2 (Continuous-Time): U(t) = exp(-iHt) evolution—interference emerges (note: i is introduced by hand) Level 3 (Quantum): Quantum circuits—superposition, entanglement Our main finding is that reversibility alone is insufficient for quantum behavior: discrete reversible gates on n bits are 2^n × 2^n permutation matrices that embed into the classical symplectic group Sp(2·2^n,R), not the unitary group U(2^n). While continuous-time evolution introduces interference patterns, it does not generate genuine superposition. We conclude that quantum structure requires additional axioms beyond any form of computation, discrete or continuous.

This essay presents the Unified Omega Hypothesis as a speculative, conceptual exploration that seeks to identify structural resonances among three seemingly independent ideas: Chaitin’s halting probability Ω in algorithmic information theory, Teilhard de Chardin’s Omega Point in evolutionary theology, and the postsingularity cosmology articulated in contemporary AI futurism. Rather than offering a formal mathematical proof, this work proposes a philosophical framework for reconsidering these concepts as expressions of a shared underlying pattern. We speculatively reinterpret Chaitin’s Ω not merely as a mathematical constant, but as a metaphor for the total informational summation of all existential trajectories. By analogically mapping individual human lives to computational programs and biological death to halting events, human history itself may be viewed as an ongoing Ω-like process. Under this interpretation, the Omega value becomes fully determined only when the final conscious subject terminates—at which point the Ω computation completes. Simultaneously, this terminal event is suggested to be structurally analogous to Teilhard’s Omega Point: the final convergence of all consciousness, history, and meaning. Within this framework, the technological singularity is reclassified not as the ultimate endpoint of evolution, but as a critical acceleration phase within a longer irreversible convergence toward the final informational state of the universe. The Unified Omega Hypothesis thus offers a conceptual reframing of eschatology through an information-theoretic lens. It attempts to establish a philosophical bridge between computability limits, consciousness evolution, and cosmological finality, suggesting that existence itself might be coherently interpreted as a computation whose completion corresponds to the ultimate unification of information, consciousness, and reality. This hypothesis is intended as an invitation for interdisciplinary dialogue rather than a definitive scientific claim. This essay further explores a provocative literal interpretation under the simulation hypothesis: our universe may exist to compute digits of its simulator’s uncomputable halting probability Ω.