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Algebraic Structures as a Foundation for Quantum-Theoretical Models: The Number System Operator and the GAP Research Program

Grigoriy Dedenko, S P Klykov, Alexey Orlovsky

Published Jan 1, 2026• 0 citations
Publisher
DOI
10.13140/rg.2.2.19032.25608/1

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This manuscript presents a conservative mathematical framework investigating whether observable multiplicity and distinguishability in physical theories can emerge from a more primitive, undifferentiated starting point. A singular foundational state SSS is defined as a minimal-ontology entity lacking internal partitions, while interpretation operators III are introduced as structural maps that induce differentiated state spaces M=I(S)M = I(S)M=I(S). The paper develops the formal properties of these operators using moderate operator algebra, category theory, and structural analysis, and situates the framework alongside established programs in quantum reconstruction, operator-algebraic foundations, information-theoretic approaches, and relational quantum mechanics. Connections are drawn between interpretive non-commutativity and contextuality, and between induced multiplicity and the emergence of classical state identity in measurement theory. This is an exploratory theoretical contribution intended to motivate further mathematical development. It does not propose empirical modifications to quantum mechanics, but provides a structural model for understanding how distinctions, state-space geometry, and observable outcomes may be generated rather than postulated. The framework may have relevance for foundational studies in quantum information, measurement, generalized probabilistic theories, and the structure of operator families underlying physical state differentiation.

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FINDING: The halting problem establishes a fundamental limit on algorithmic knowledge — no finite procedure can decide whether an arbitrary program halts, a result foundational to computation and quantum-verification complexity (MIP\*=RE). | MATH: Undecidability via diagonalization; H(P,I) = {1 if P halts on I, 0 otherwise} is non-computable. Key corollary: MIP\*=RE implies the quantum commuting operator model (Tsirelson's problem) is undecidable — connecting operator algebras to computability. No new constants or ratios emerge; the structure is purely combinatorial/logical. | CONNECTION: The diagonalization argument mirrors the construction of irrational numbers (Cantor's slash) — a self-referential "gap" that resists closure. This is analogous to the golden ratio's non-repeating continued fraction [1;1,1,1,...] = φ, which also encodes an irreducible self-similarity. The halting problem's undecidability is a discrete analogue of incommensurability — a ratio (program vs. proof) that ca Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Summary: Matrix Physics proposes a conceptual framework in which matrices are treated not merely as algebraic objects but as observable transformation systems that govern the evolution of structured states. The work develops the idea of "Matrix Physics," where states, operators, and measurement processes form a unified language for describing dynamics across mathematics, physics, information theory, artificial intelligence, and complex systems. The framework introduces operator-based interpretations of stability, measurement, entropy, and symmetry while exploring extensions to matrix thermodynamics, operator field theories, quantum mechanics, gauge theory, cosmology, and random matrix models. Although primarily speculative and theoretical, the work investigates how spectral structure, eigenvalue dynamics, and symmetry breaking may provide a common mathematical foundation for understanding physical and informational systems. The manuscript is intended as a research program and conceptual synthesis rather than a completed physical theory, emphasizing operator dynamics, measurement, and emergent structure as unifying principles. Keywords: Matrix Physics; Linear Algebra; Operator Theory; Spectral Theory; Eigenvalues; Dynamical Systems; Random Matrix Theory; Operator Geometry; Information Theory; Measurement Theory; Symmetry Breaking; Gauge Theory; Quantum Mechanics; Complex Systems; Spectral Dynamics; Matrix Thermodynamics; Noncommutative Geometry; Emergent Physics; Mathematical Physics.
This manuscript presents a conservative mathematical framework investigating whether observable multiplicity and distinguishability in physical theories can emerge from a more primitive, undifferentiated starting point. A singular foundational state SSS is defined as a minimal-ontology entity lacking internal partitions, while interpretation operators III are introduced as structural maps that induce differentiated state spaces M=I(S)M = I(S)M=I(S). The paper develops the formal properties of these operators using moderate operator algebra, category theory, and structural analysis, and situates the framework alongside established programs in quantum reconstruction, operator-algebraic foundations, information-theoretic approaches, and relational quantum mechanics. Connections are drawn between interpretive non-commutativity and contextuality, and between induced multiplicity and the emergence of classical state identity in measurement theory. This is an exploratory theoretical contribution intended to motivate further mathematical development. It does not propose empirical modifications to quantum mechanics, but provides a structural model for understanding how distinctions, state-space geometry, and observable outcomes may be generated rather than postulated. The framework may have relevance for foundational studies in quantum information, measurement, generalized probabilistic theories, and the structure of operator families underlying physical state differentiation.