This is a revised preprint version of “A Relational Constraint Framework for Physics from Finite Measurement Structure.” The revision strengthens the formal presentation of refinement domains, set-valued refinement maps, compatible families, and refinement-stable physical realization. It also clarifies the scope of the framework and its relation to classical, quantum-like, and probabilistic directions. This version remains a preprint and has not been peer reviewed.
This is a revised preprint version of “A Relational Constraint Framework for Physics from Finite Measurement Structure.” The revision strengthens the formal presentation of refinement domains, set-valued refinement maps, compatible families, and refinement-stable physical realization. It also clarifies the scope of the framework and its relation to classical, quantum-like, and probabilistic directions. This version remains a preprint and has not been peer reviewed.
This treatise presents a consolidated formulation of the Finite Relational Closure Framework (FRCF), beginning from finite measurement records and developing a common structure for relational consistency, refinement, closure, and empirical sufficiency. It distinguishes structural refinement from physical dynamics and treats classical, relativistic, gravitational, and quantum formalisms as regime-dependent sufficient representations rather than uniquely determined ontologies. The work also develops criteria for model enlargement, refinement stability, and framework-level falsifiability, while making explicit which results are derived within FRCF and which depend on additional physical or mathematical assumptions.
This revised manuscript develops an algebraic formulation of refinement in a finite measurement framework. It treats refinement operations as partially defined, set-valued maps between context-dependent admissible assignments, with emphasis on composition, quotient structure, finite-resolution equivalence, and the conditions under which relational structures remain stable under contextual refinement. The manuscript provides the algebraic machinery supporting the broader refinement-closure program, including admissibility, refinement histories, path dependence, obstruction, and the emergence of effective structure through stable composition. This version clarifies the distinction between admissible generators, uniquely selected generators, and empirically sufficient generators. A local generator constraint is not treated as uniquely forced by finite measurement structure alone. Instead, candidate generators are restricted by admissibility, refinement composition, finite-resolution quotienting, and empirical refinement stress. The revision introduces residual-based language for assessing whether a candidate generator remains sufficient when confronted with additional finite measurement data under refinement. The manuscript therefore frames physical selection not as proof of a uniquely true generator, but as the identification of surviving equivalence classes of generators that remain sufficient within a finite measurement domain. This strengthens the connection between the algebraic refinement structure and later applications to wave-like dynamics, quantum-like contextuality, and empirically constrained model selection.
This preprint develops a refinement–compatibility–stability (RCS) framework in which finite process occurrences generate rational calibration readouts indexed by a directed refinement system, and studies when such readouts determine a stable continuum value. Throughout, a continuum-valued observable means the completion class determined by an admissible stable refinement thread; no operator-algebraic or quantum-mechanical notion of observable is presupposed. Main results. Every stable rational thread determines a unique value in the Cauchy completion of the internal rational field. Equality of completed values is equivalent to asymptotic thread equivalence; realization preserves the induced algebraic operations; and the value is invariant under cofinal changes of refinement schedule. For bounded occurrence ratios, asymptotically negligible perturbations to the numerator and denominator preserve the completed value. Nested rational enclosures of vanishing width provide a sufficient, independently checkable stability certificate. The mathematical completion is distinguished from the physically realizable spectrum selected by a declared class of admissible protocols. Occurrence-ratio generation and stability alone do not determine a unique physical spectrum; additional assumptions about admissible channel composition can induce corresponding algebraic structure. A Pell-type recursion yields a stable irrational completion value using finite positive-integer counts, while an oscillatory model shows that qualitative refinement assumptions alone do not guarantee quantitative stability. Computability is treated conditionally: if the refinement system and protocol count functions are computable, and the stability thresholds are effectively obtainable, the completion value is a computable real. The paper does not assume that every physically admissible protocol is effective, and does not claim that physical observables must be computable. Scope. Apart from standard algebraic and topological facts cited in Sections 2 and 4, the results depend only on the definitions, axioms, postulates and additional conditions stated in this manuscript. No linear, geometric, temporal, or topological realization developed in the broader Observation–Feedback programme is assumed here. The readout is compared with the partial observables of relational physics, and stability, invariance and physical admissibility are treated as logically distinct requirements. The framework originates in the finite-occurrence and refinement-stability sector of the Observation–Feedback (OF) Closure programme (companion manuscript: Observation–Feedback Closure I, v1.5.0, doi:10.5281/zenodo.21859450), which supplies the broader relational motivation. The paper makes no empirical predictions and does not identify the physically realizable spectrum with the full mathematical completion. Version note. Version v1.1 is a clarificatory revision of v1.0. It states the provenance of the “OF”-labelled arithmetic, separates calibration protocols, readout threads and completion values, and sharpens the distinction between quantitative stability and physical admissibility. The only changes inside numbered equations are notational (κ_T → Read_T and N_eff → N_eff,i*); no theorem statement, proof, or mathematical conclusion has been changed. The PDF lists the full set of changes immediately before the Acknowledgments. Status. Preprint; not peer reviewed. The deposit includes the manuscript PDF, LaTeX source and BibTeX bibliography.
The question of how probabilistic systems select individual outcomes remains conceptually open in several areas of physics and complex systems theory. Standard decoherence models successfully describe the emergence of classical behavior from quantum systems, yet they generally treat the selection of specific outcomes as statistically neutral once the probabilistic structure is fixed. The present work is motivated by the possibility that probabilistic selection processes may, under certain conditions, exhibit weak forms of context dependence without requiring any violation of standard probabilistic laws. Rather than proposing a replacement for established physical theory, the paper explores whether minimal and formally controlled modifications of probabilistic weighting could provide a useful phenomenological framework for describing such effects. The approach is intentionally conservative in scope. Its purpose is not to claim evidence for new physics, but to investigate whether concepts such as context, structural compatibility, and informational coupling can be represented within a mathematically bounded probabilistic setting. This preprint proposes a minimal framework for modeling context-dependent selection in probabilistic systems. While standard decoherence theory explains the emergence of classical behavior from quantum systems, it does not specify whether the selection of individual outcomes is entirely neutral or may be weakly influenced by additional constraints. The present work explores the possibility that probabilistic state selection could exhibit small context-dependent biases without modifying the underlying probabilistic structure. To formalize this idea, the model introduces a resonance function that weakly reweights baseline transition probabilities in a controlled and non-disruptive manner. A dynamic extension is considered in which probabilities and resonance co-evolve through simple feedback, damping, and context-dependent source terms. Analytical considerations and numerical simulations indicate that such systems can produce small but systematic and bounded deviations from baseline distributions. Under certain conditions, these deviations may stabilize, leading to attractor-like behavior. The framework is intended as an exploratory and phenomenological model, not as a modification of established physical theory. Its purpose is to provide a formal setting for investigating whether context, information, and structural compatibility can be represented within probabilistic state selection. A possible connection to underlying physical processes was considered in earlier work (https://doi.org/10.5281/zenodo.17246478). In the present paper, however, this line of interpretation is not pursued further and remains speculative. Keywords: decoherence; probability dynamics; attractors; resonance; complex systems; information theory Version note:This version presents a revised and clarified formulation of the proposed framework. In particular, the formal structure has been made more explicit by refining key derivation steps, clarifying the role of the resonance term, and tightening the interpretation of the model within its intended scope. No fundamental change to the core idea has been introduced. The revision primarily sharpens the probabilistic formulation, reduces ambiguity in the definition of the resonance function, and makes the exploratory and phenomenological status of the model more explicit.