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 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 record provides an index and dependency declaration for a one-way trilogy of works: (1) Minimal Ontological Foundation (MOF v3)(2) Phase–Flow Coherence (PFC v4)(3) Alpha Locking (v6) The trilogy establishes a finite-resolution rigidity chain, proceeding strictly in one direction. MOF v3 fixes the minimal ontological conditions for world-formation, from which irreversible internal update order (time), internal projection (measurement), and structural probability follow as necessary consequences. PFC v4 fixes the unique internally coherent representation of dynamics and probability under those conditions, eliminating all alternatives except Schrödinger-type bookkeeping with Born-rule weighting. Alpha Locking v6 completes the chain by showing that finite-resolution update structures cannot terminate arbitrarily. A unique minimal admissible depth is forced under purely discrete accept/reject predicates, leaving the fine-structure constant as a structural residue. No result depends on downstream assumptions.No fitting, tuning, or optimization is introduced at any stage. This overview introduces no new claims and contains no additional results.It serves solely to fix reading order, dependency, and scope.
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 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
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 record provides the overview and organizing entry point for a research program examining several long-standing “unsolved problems” in fundamental physics. Rather than proposing new particles, interactions, or dynamical laws, this series demonstrates that a number of canonical problems arise from a shared category error. Under finite resolution and internal description, these problems are shown to dissolve structurally and independently. The series establishes a common set of minimal premises—finite resolution, irreversible internal projection, and non-privileged description—and applies them uniformly to multiple domains. Each associated volume isolates a single problem and demonstrates its elimination without cross-dependence on other topics. The following volumes form part of this research program and are linked as related works: Vacuum Energy Hierarchy Problem Quantum Gravity Arrow of Time Dark Matter Each volume is designed as an entry-point paper and may be read independently. Detailed mathematical, conceptual, and phenomenological developments are archived in the cited records under persistent identifiers. Disagreement with the conclusions of this program requires explicit rejection of one or more stated premises. This research program consists of the following volumes: Vol.1 — Quantum Measurement (foundational layer)Vol.2 — Hierarchy ProblemVol.3 — Vacuum EnergyVol.4 — Quantum GravityVol.5 — Arrow of TimeVol.6 — Dark Matter Each volume is archived as an independent record with its own DOI and can be accessed via the related works listed above.
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.