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From Structural Persistence to Quantum Probability: An SN Derivation of Hilbert-Space Measurement

Fiona Mcgeough

Published May 17, 2026• 0 citations
Publisher

Abstract

From Structural Persistence to Quantum Probability: An SN Derivation of Hilbert-Space Measurement
DOI
10.5281/zenodo.20260293

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This work investigates the minimal structural conditions required for persistence under transformation. Rather than assuming time, dynamics, or probabilistic laws, the analysis begins from two basic principles: triadic closure (the requirement that transformations be internally verifiable) and structural equilibrium (the requirement that identity-preserving and identity-changing contributions remain simultaneously operative). From these principles, the paper shows that any system capable of persistence must admit a representation of deviation with specific structural properties. Composability of transformations leads to a cocycle structure, which induces a vector space of deviations. The equilibrium condition then constrains this space to carry an inner product, ultimately selecting a complex Hilbert space as the minimal representational framework. Within this setting, deviation admits a canonical operator formulation. A unique scalar measure of deviation is derived from structural requirements of positivity, invariance, and additivity. This measure is necessarily quadratic. The Born rule then emerges as the probabilistic interpretation of this quadratic structural density. The result provides a unified structural account of both the Hilbert space framework of quantum theory and its probability rule, derived from a single chain of reasoning grounded in the conditions required for persistence. This version is a revised preprint currently under review.
The mathematical structure of quantum theory is usually postulated rather than explained—Hilbert spaces, operators, and probabilistic structure—without explaining why this arena should exist in the first place. In this work we address a more primitive question: starting from the absence of any prior structure, what conditions must be satisfied for coherent persistence to be possible at all? We begin from a state of zero determination (the Void), understood not as physical emptiness but as the absence of admissibility constraints. From this starting point, we show that the possibility of persistence forces a minimal triadic structure consisting of reference, transformation, and return. This structure enables self-verification without external supervision and distinguishes genuine continuity from accidental resemblance. The requirement of self-verifying persistence forces deviation to be representable quantitatively and to compose across histories. This necessity leads uniquely to a linear vector space structure. Stability further imposes a knife-edge viability condition: identity-preserving and identity-violating contributions must equalize at closure. From this condition, an inner product and norm are forced, yielding Hilbert space geometry and operator representations as consequences of viability rather than axioms. Within this framework, familiar features of quantum kinematics—such as uncertainty, unitary evolution, and probabilistic weighting—arise as structural necessities rather than postulated rules. The present work establishes the kinematic foundations of coherence, deriving the mathematical arena of quantum theory from the requirement that something persist at all. Applications to mass relations, symmetry, temporal ordering, and composite systems are developed separately.Version 1.0. This preprint establishes the structural foundations of Coherence Algebra and derives Hilbert space from self-verification and persistence.
This paper establishes a minimal formal foundation for structure under irreversibledistinguishability. Starting from a single primitive—distinction—we derive equivalence,refinement, irreversible loss, persistence, probability, Hilbert space structure, unitaryadmissible transformations, counting, and intrinsic temporal ordering. No physical,probabilistic, or dynamical axioms are assumed. Probability, time, and reversible evolution arise necessarily as constrained structures induced by unresolved refinement andirreversible elimination. The formal core is shown to be minimal, irreducible, andcomplete with respect to admissible structure under loss. This paper constitutes thefoundational core of a multi-paper series deriving physics, chemistry, biology, cognition,and society as successive constrained extensions of irreversible distinguishability.