This paper presents a structural derivation of quantum dynamics and probability from a minimal set of pre-physical assumptions: non-privileged phase space, finite resolution, and measure-preserving global dynamics. Rather than postulating quantum axioms, the framework shows how internal probabilistic interpretation (Born rule) and Schrödinger-type dynamics emerge as necessary consequences of finite-resolution internal projections applied to a globally deterministic phase-flow structure. The paper emphasizes a strict separation between global density and internal probability, and includes a structural log clarifying where assumptions are introduced and where they are explicitly excluded. This is Version 1. Subsequent versions may refine appendix structure, notation, and cross-referencing without altering the core theoretical results. Note (2026):A refined and structurally clarified successor to this work is now available as“Phase-Flow Coherence v3: Inevitability of Quantum Dynamics from Finite Resolution and Non-Privileged Phase Space”.The present paper should be regarded as a detailed intermediate derivation, while PFC v3 presents the finalized inevitability result within the MOF framework.
This paper presents a structural derivation of quantum dynamics and probability from a minimal set of pre-physical assumptions: non-privileged phase space, finite resolution, and measure-preserving global dynamics. Rather than postulating quantum axioms, the framework shows how internal probabilistic interpretation (Born rule) and Schrödinger-type dynamics emerge as necessary consequences of finite-resolution internal projections applied to a globally deterministic phase-flow structure. The paper emphasizes a strict separation between global density and internal probability, and includes a structural log clarifying where assumptions are introduced and where they are explicitly excluded. This is Version 1. Subsequent versions may refine appendix structure, notation, and cross-referencing without altering the core theoretical results. Note (2026):A refined and structurally clarified successor to this work is now available as“Phase-Flow Coherence v3: Inevitability of Quantum Dynamics from Finite Resolution and Non-Privileged Phase Space”.The present paper should be regarded as a detailed intermediate derivation, while PFC v3 presents the finalized inevitability result within the MOF framework.
Notice (Updated Framework):This paper presents a constructive formulation of Phase-Flow Coherence (PFC).A newer paper, PFC v4, reformulates the same results in an eliminative (no-go) framework, showing that Born-rule weighting and Schrödinger-type dynamics are internally inevitable under finite resolution and non-privileged representation. Readers interested in the final, assumption-minimal formulation are strongly encouraged to consult:Phase-Flow Coherence (PFC) v4 — DOI: [v4 DOI]10.5281/zenodo.18322699 ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ A deterministic phase-fiber formulation of quantum mechanics based on Fisher reduction.This v2.5 version strengthens the previous PFC v2 by:(i) a rigorous derivation of the quantum potential from Fisher information,(ii) explicit handling of the β non-coherence term under θ-periodicity,(iii) entropic coarse-graining for measurement,(iv) an experimentally testable bound on interferometric visibility.All probabilistic features emerge from finite angular resolution on S¹.The theory is self-contained and falsifiable.
This paper does not propose a new physical model or interpretation.Instead, it records a rigidity chain that emerges when a world is described internallyunder three minimal premises: finite resolution, non-privileged representation,and irreversible internal projection. Under these conditions, structures traditionally treated as axioms or free parameters—the Born probability weighting, Schrödinger-type dynamics, and the fine-structure constant α—do not remain free. Each appears as a unique survivor of eliminative consistency requirements. Disagreement with the conclusions requires explicit rejection of at least one premise,with clearly identifiable conceptual costs.This work serves as a challenge paper and an entry point to a larger trilogy(MOF–PFC–Alpha Locking), directing critical scrutiny toward the choice of premisesrather than interpretive preference.
This series presents a structured and logically sequential formulation of a discrete informational framework governed by global coherence constraints. Although it originates from the same research program developed in the earlier exploratory preprint series (P-series), the works presented here constitute an autonomous and self-contained series, organized in a progressively axiomatic form and focused on the explicit derivation of the framework’s effective structures. The goal of the construction is to investigate whether structures commonly associated with quantum mechanics, thermodynamics, and geometry may emerge from a discrete relational substrate without being introduced as fundamental postulates. Each article addresses a specific level of the theoretical derivation and forms part of a progressive logical structure based on results established in the preceding works of the series. This article develops the emergence of Hilbert-space structure and Schrödinger-type effective dynamics from the coherent two-field sector established in Paper 02. The work demonstrates that the global coherence constraints imposed on the discrete informational substrate naturally generate an effective complex representation endowed with linear superposition properties and unitary-preserving evolution. The analysis shows that the two-component coherent sector admits a mathematically stable reformulation equivalent to a Hilbert-space structure, and that the effective evolution of the coherent amplitudes leads to a Schrödinger-type dynamical equation without introducing quantum mechanics as an initial assumption. The article further derives the probabilistic interpretation associated with the effective amplitudes and clarifies how quantum-like statistical behavior emerges from the internal coherence conditions of the underlying discrete system. No quantum postulates are assumed at the foundational level. The purpose of the article is to demonstrate that the formal structures associated with quantum mechanics can emerge as effective consequences of coherence-preserving informational dynamics. The article does not assume quantum mechanics as a foundational theory and does not postulate the Hilbert-space formalism a priori. Instead, the work demonstrates that, within the internal hypotheses and coherence constraints of the proposed discrete informational framework, structures mathematically equivalent to Hilbert-space representations and Schrödinger-type evolution can arise as effective emergent descriptions. The purpose of the derivation is therefore not to reproduce standard quantum mechanics by assumption, but to show that quantum-like formal structures may consistently emerge from deeper coherence-preserving informational dynamics.
This work presents an axiomatic and structural reconstruction of core elements of quantum mechanics. Starting from minimal principles — symmetry under vanishing preference, normalization, consistency of probability assignments, and coherence-preserving dynamics — we derive several central features of the quantum formalism rather than postulating them. Key results include: A geometric derivation of the constants 1/21/\sqrt{2}1/2 and 1/21/21/2 from symmetry and normalization. A theorem-level derivation of the Born rule (including the qubit case) from noncontextuality and additivity. A reconstruction of Schrödinger dynamics from reversible norm-preserving structure. A derivation of the tensor-product structure for composite systems from composition axioms and local tomography, with entanglement emerging as a necessary consequence. Formal treatment of Bell–CHSH nonlocality, Tsirelson’s bound, and no-signalling. A conditional reconstruction of complex Hilbert space structure from intrinsic phase dynamics and compositional consistency. The manuscript is self-contained and framed in standard mathematical-physics language. It does not propose modifications to quantum mechanics, but instead clarifies why many of its defining structures arise inevitably from deeper structural constraints. Prior computational and model-based work by the authors (including diagnostic toy models exploring phase locking, coherence stabilization, and recursive-time dynamics) is acknowledged but intentionally not reproduced here, in order to keep the present work focused on axiomatic structure rather than model-specific architecture. This upload is intended as a foundations and reconstruction contribution, suitable for readers interested in the mathematical and conceptual structure of quantum theory.
This work provides a structural derivation of the quadratic form of the Born rule in quantum mechanics. Rather than postulating the probability rule as a foundational axiom, the paper proves that the exponent governing quantum probabilities is uniquely fixed by stability, contractivity, and duality requirements imposed on admissible physical laws. The analysis is carried out within a general operator-theoretic and law-space framework, in which physical laws are treated as fixed points of recursive generative dynamics. By combining strict contractive flow (Kappa Law), topological rigidity (Law of Endogenous Constraint), and algebra–geometry duality (Monad Duality), the paper demonstrates that only the Hilbert–Schmidt geometry is compatible with global stability of law evolution. As a consequence, the quadratic probability rule emerges as a rigid structural invariant rather than a free modeling choice. The result implies that any hypothetical modification of quantum theory based on non-quadratic probability rules would necessarily violate at least one of the fundamental structural requirements of admissible physical law, such as uniform contractivity, spectral stability, or duality consistency. Relation to Tier-1 dynamical formulation: This work develops a law-level (structural) resolution of the Born exponent. A complementary Tier-1 dynamical realization, formulated entirely within a constrained operator framework and independent of law-level recursion language, is developed in: J. Rodgers, Saturation Geometry and the Structural Emergence of Measurement and the Born Rule in a Capacity-Constrained Dirac–Λ System, Zenodo (2026).https://doi.org/10.5281/zenodo.18704783 The Tier-1 formulation may be read independently and derives the quadratic Born rule as a consequence of modular implementability and saturation geometry within the coupled Dirac–Λ system.
This work provides a structural derivation of the quadratic form of the Born rule in quantum mechanics. Rather than postulating the probability rule as a foundational axiom, the paper proves that the exponent governing quantum probabilities is uniquely fixed by stability, contractivity, and duality requirements imposed on admissible physical laws.The analysis is carried out within a general operator-theoretic and law-space framework, in which physical laws are treated as fixed points of recursive generative dynamics. By combining strict contractive flow (Kappa Law), topological rigidity (Law of Endogenous Constraint), and algebra–geometry duality (Monad Duality), the paper demonstrates that only the Hilbert–Schmidt geometry is compatible with global stability of law evolution. As a consequence, the quadratic probability rule emerges as a rigid structural invariant rather than a free modeling choice.The result implies that any hypothetical modification of quantum theory based on non-quadratic probability rules would necessarily violate at least one of the fundamental structural requirements of admissible physical law, such as uniform contractivity, spectral stability, or duality consistency.This work builds directly on the operator-theoretic resolution of quantum measurement: “A Complete Operator-Theoretic Resolution of the Quantum Measurement Problem” (DOI: 10.5281/zenodo.17823241), and on the generative law framework introduced in: “The Tier-0 Framework and the Everything Equation: A Universal Recursion Law for Physics, Mathematics, and Information” (DOI: 10.5281/zenodo.17813117).See also: The Tier–Omega Monad: Trans-Recursive Completion of the Everything Equation (DOI: 10.5281/zenodo.17859631), which establishes the unique trans-recursive invariant that terminates all possible meta-law recursion and completes the Everything Equation. This work provides the structural boundary underlying the entire framework developed in the associated papers. https://doi.org/10.5281/zenodo.17864384