Translator This work addresses a foundational but rarely examined question in physics: under what conditions does an observable exist at all? Treating finite resolution as a structural feature rather than a technical limitation, the paper shows that observable existence is not guaranteed and must be assessed relative to the refinement structure induced by realistic probes. Within the framework of Finite Resolution Physics (FRP), a refinement-stability requirement is introduced as a necessary condition for observable existence. This requirement is independent of statistical typicality, dynamical modeling, or spacetime and field ontologies. Applying it to a realistic interferometric probe specified strictly at the acquisition level, the paper proves that commonly assumed scalar quantities such as interferometric phase and visibility fail to remain stable under admissible refinement paths and therefore do not exist as probe-invariant observables within FRP. In addition to the structural result, the paper discusses reported interferometric experiments in which extracted phase- or visibility-like quantities exhibit dependence on detector resolution, spatial averaging, or reconstruction procedures. Without modeling these experiments or predicting their detailed behavior, the FRP analysis clarifies why such non-stabilization is not paradoxical: once an observable fails to exist structurally, finite resolution imposes no requirement that numerical procedures applied to experimental data should converge. The analysis is complemented by a conceptual connection to Operational Reflexive Systems, which provide an algebraic account of operational realizability. Overall, the paper establishes a principled distinction between calculable quantities and observables, and shows how finite resolution alone can rule out observables rather than merely limit their precision.
Translator This work addresses a foundational but rarely examined question in physics: under what conditions does an observable exist at all? Treating finite resolution as a structural feature rather than a technical limitation, the paper shows that observable existence is not guaranteed and must be assessed relative to the refinement structure induced by realistic probes. Within the framework of Finite Resolution Physics (FRP), a refinement-stability requirement is introduced as a necessary condition for observable existence. This requirement is independent of statistical typicality, dynamical modeling, or spacetime and field ontologies. Applying it to a realistic interferometric probe specified strictly at the acquisition level, the paper proves that commonly assumed scalar quantities such as interferometric phase and visibility fail to remain stable under admissible refinement paths and therefore do not exist as probe-invariant observables within FRP. In addition to the structural result, the paper discusses reported interferometric experiments in which extracted phase- or visibility-like quantities exhibit dependence on detector resolution, spatial averaging, or reconstruction procedures. Without modeling these experiments or predicting their detailed behavior, the FRP analysis clarifies why such non-stabilization is not paradoxical: once an observable fails to exist structurally, finite resolution imposes no requirement that numerical procedures applied to experimental data should converge. The analysis is complemented by a conceptual connection to Operational Reflexive Systems, which provide an algebraic account of operational realizability. Overall, the paper establishes a principled distinction between calculable quantities and observables, and shows how finite resolution alone can rule out observables rather than merely limit their precision.
Translator This work provides a concrete, example-driven exposition of the core concepts underlying Finite Resolution Physics (FRP) and related closure-based frameworks. Rather than introducing new axioms or formal constructions, the paper focuses on operational realizations of probes, admissible composition, closure, stabilization, and dynamical description in physically meaningful contexts. The guiding perspective is closure-first: physical description is treated as emerging from probe-conditioned operational access and the propagation of admissible consequences, rather than from assumed state spaces or universal laws. Stabilization is not presupposed and is analyzed as a contingent, resolution-dependent property. When stabilization occurs, effective descriptions—such as Hamiltonian, stochastic, or geometric encodings—may be introduced; when it fails, the absence of such descriptions is treated as a meaningful structural outcome rather than a pathology. Through a series of illustrative physical examples (including clock probes, signal probes, and mixed probing contexts), the paper clarifies: what constitutes a probe in practice, how admissible composition is realized physically, how closure propagates operational consequences, when stabilization succeeds, partially succeeds, or fails, and how different representational regimes (quantum, statistical, non-dynamical) arise without being assumed a priori. The paper is intentionally non-axiomatic and does not fix the semantics of FRP terminology once and for all. Its role is clarificatory and diagnostic: to anchor interpretation by use, while leaving the framework structurally open to further development. It is intended as a companion to more formal FRP and closure-theoretic works, and as an accessible entry point for readers seeking operational intuition rather than foundational postulates. Keywords
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.
Quantum field theory unifies special relativity and quantum mechanics yet relies on externally imposed prescriptions—quantization, renormalization, and probabilistic interpretation—that lack a common algebraic foundation. This work derives these features from a single structural principle: the finite-resolution realization of the additive group [Formula: see text] under four internal consistency constraints. These constraints—finite operational resolution (FR1), multiplicative linearity (FR2), phase coherence (FR3), and statistical typicality (FR4)—generate, without additional postulates, the Hilbert-space formalism of quantum theory, the Born probability rule, and the Kolmogorov structure of statistical mechanics as limiting projections of one additive algebra. When extended to spacetime translations and their automorphisms, the same framework reproduces the kinematic and dynamical structure of quantum field theory while resolving ultraviolet divergences through intrinsic finiteness rather than external renormalization. Quantization and unitarity appear as consequences of additive symmetry, while the gravitational field emerges as the algebraic kernel that transports boundary information into bulk observables without requiring a separate quantization of the metric. The result is a unified, finite, and strictly algebraic foundation for physical law.
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.
Translator This paper reformulates quantum entanglement within the framework of Finite-Resolution Physics (FRP), a finite-coherence version of quantum theory derived from the Reflexive Monoid Algebra (RMA) hierarchy.All experimental predictions of entanglement—joint probabilities, Bell-inequality violations, and Born frequencies—remain identical to those of standard quantum mechanics.The difference lies in structure: entanglement is no longer a separate postulate about tensor-product Hilbert spaces but an algebraic phase-consistency relation (FR3) and typicality constraint (FR4) linking additive subsystems under finite resolution.This reinterpretation explains nonlocal correlations as boundary-level coherence, not superluminal influence, and describes decoherence as the loss of FR3 coherence, marking the transition from quantum (Qμ) to statistical (Sμ) regimes.The work provides a unified algebraic account of entanglement, nonlocality, and decoherence within a finite-resolution operational framework.
Quantum mechanics employs probabilistic formalisms that are often interpreted as reflecting intrinsic indeterminacy of physical reality. Despite its empirical success, this interpretation leaves unresolved why deterministic behavior re-emerges at macroscopic scales. This paper argues that quantum probability arises instead from finite observational resolution imposed by spacetime granularity. When physical processes are subdivided beyond the Planck-scale limits of temporal and spatial resolution, individual micro-events become unobservable in principle. In this regime, observation collapses continuous deterministic dynamics into ensemble-averaged and probabilistic descriptions. The infinite subdivision principle is introduced to formalize this transition: subdivision beyond observational bandwidth irreversibly removes phase information at the observer level while leaving underlying dynamics intact. Probability amplitudes are shown to encode compressed residual phase relationships among unresolved micro-configurations, while expectation values represent structurally stable quantities accessible above the resolution threshold. Within this framework, quantum mechanics is reinterpreted as a resolution-limited projection of deterministic structure rather than as a fundamentally indeterminate theory. This perspective aligns naturally with Existence-Quantized Geometric Theory (EQGT), in which existence and observability are governed by threshold conditions and event-biased observation.