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.
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.
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 paper is Part II of The Pre-Emergence Origin of Probability trilogy. Part I develops the structural requirements for finite recurrent stability before spacetime. Part II asks when persistent boundary-readable structure can support numerical probability. Part III examines what probability-status remains after boundary loss, before numerical weighting, and under what narrower conditions Born-form weighting may follow. The paper analyzes how probability-measure status may arise within a closed deterministic description rather than being assumed as a primitive numerical rule. A closed description cannot appeal to an external observer, hidden weighting convention, retained label, post-outcome selection, or independently supplied probability measure to explain its own probability assignments. Numerical probability must instead be supported by structures and relations admitted within the described system. The analysis begins with the requirements for persistent structured identity. Persistent identity is treated as recoverable cross-state relational distinction rather than exact sameness, repeated naming, scalar aggregation, or externally preserved memory. Within the tested structural registry, persistent refinable identity requires ordered dependence, a sufficient stabilization architecture, distributed relational structure, recurrence, bounded recoverability, and noncollapse of the distinctions being preserved. Bounded recurrent relational correction is classified as weak oscillatory form in a deliberately structural sense. The classification does not establish a physical wave, sinusoidal motion, exact periodicity, frequency, propagation, dynamics, mechanism operation, or physical carrier. It identifies a recurrent recoverability architecture that a later realization would have to preserve. The paper then develops the transition from persistent structure to boundary observability. Representation-sensitive bulk descriptions may depend on phase origin, refinement history, decomposition, internal labels, subpart identity, or other distinctions that do not survive admissible boundary readout. Under admissible-observable indistinguishability, boundary-readable observables factor through an emergence-boundary quotient that preserves only the distinctions carried by the admitted observable family and surviving the declared transport relation. The boundary quotient does not itself provide probability. It supplies a local architecture of boundary-readable alternatives and permits the admission of candidate nonnegative contribution scalars. Neither the alternative carrier nor a candidate scalar establishes scalar uniqueness, scalar closure, additivity, measure realization, normalization, or probability interpretation. Numerical probability-measure status becomes available only under a narrower cumulative condition package. The package requires an exact finite carrier, a finite positive event measure, a normalized scalar component on the same carrier, exact scalar-measure compatibility, one-scalar factorization of every admitted weight-bearing map, exclusion of independent weight-bearing invariants, pre-outcome assignment, absence of an external weighting source, and a declared scalar-only predictive regime. Additivity, commensurability, regularity, positive scale, finite nonzero total, and normalization remain separately controlled conditions within that route. Under those conditions, normalized scalar weights may receive conditional probability-measure status inside the declared local regime. This is not an unconditional derivation of probability from determinism. The result does not establish unrestricted probability, universal scalar representation, a global probability law, objective chance, stochastic dynamics, collapse, measurement theory, physical realization, or a final physical theory. Within the trilogy, Part II therefore supplies the middle structural layer: persistent recurrent structuretoboundary-readable alternativestocandidate scalar contenttoconditional numerical probability-measure status. Part III begins from a different question. It examines what remains when a boundary-readable description loses the ability to guarantee which admissible alternative obtains, before any numerical scalarization is supplied. Part II provides the scalar, measure, normalization, compatibility, and realization-burden architecture inherited by that later analysis, but it does not pre-establish the boundary-loss results of Part III. ### Revision note This revision is a substantial reconstruction and trilogy integration rather than a minor editorial update. The derivational architecture, definitions, condition packages, failure boundaries, dependency order, and claim-scope controls have been revised throughout. The manuscript has also been rewritten in new prose to present the argument as Part II of The Pre-Emergence Origin of Probability trilogy. Major changes include a clearer separation of persistent identity, ordered dependence, stabilization, distributed relational recurrence, weak oscillatory form, boundary observability, candidate scalar status, scalar closure, measure realization, normalization, and conditional probability-measure status. The emergence-boundary quotient and transport-survival architecture have been clarified, and the numerical-probability result has been narrowed to an expressly declared scalar-only predictive regime satisfying the complete carrier, measure, compatibility, factorization, pre-outcome, and independent-invariant exclusion conditions. The revision also strengthens the distinction between candidate weights, normalized weights, conditional measures, and probability interpretation. It adds more precise failure localization, preserves the distinction between conditioned failure and universal impossibility, and expands the realization-burden architecture separating mechanism satisfaction, physical scalar selection, boundary enforcement, branch generation, refinement implementation, physical realization, theory-specific realization, empirical validation, and final physical-theory status. The paper is now explicitly positioned between Part I, which establishes the structural requirements for finite recurrent stability before spacetime, and Part III, which develops boundary-unresolved and pre-numerical probability-status after recoverability loss. The central research program remains mechanism-neutral. No physical mechanism, collapse process, specific realization theory, unrestricted probability law, or empirical validation is claimed.
This manuscript presents a synthesis and methodology paper for the Finite Relational Closure Framework (FRCF). Rather than reconstructing a single physical theory in detail, it clarifies how FRCF organizes effective physical regimes from finite measurement records, admissible relational structures, finite-resolution equivalence, and refinement stability. The manuscript defines an effective physical regime as a refinement-stable class of finite relational measurement structures that supports reproducible comparison laws, stable invariants, and an effective mathematical representation within a specified domain. It distinguishes finite fit, refinement stability, sufficiency, and uniqueness, and proposes explicit criteria for evaluating future FRCF applications. Examples from the broader series, including wave-like propagation, quantum kinematics, measurement conditioning, quantum dynamical kernels, phase laws, and empirical generator stability, are used to illustrate how different regimes may require different stable constraints while sharing a common finite-relational architecture. The manuscript is methodological and programmatic. It does not claim that all physical theories have been reconstructed, but presents FRCF as a framework for studying how effective physical structures may arise, persist, be compared, or fail under admissible refinement.
Modern theoretical physics faces an abundance of mathematically consistent frameworks and a scarcity of decisive empirical constraints. This work proposes a principled response to that situation by identifying stability under intervention as a primary selection criterion for viable physical theories. Adopting an explicitly operational perspective, the paper formulates stability as the non-amplification of operational distinguishability under physical processes, a requirement motivated by the finite precision and reproducibility of experimental practice. When combined with compositional consistency, purification, and reversible symmetry, this principle strongly constrains admissible statistical frameworks. Within this setting, quantum theory emerges not as a fundamental ontology or a historically contingent choice, but as the uniquely stable non-classical fixed point compatible with reproducible physical control by finite agents embedded within the systems they study. The defining structural features of quantum mechanics, including convex state spaces, affine dynamics, state–effect duality, and a Born-type probability rule, arise as consequences rather than postulates. The framework further suggests that spacetime locality and geometry need not be assumed a priori, but can be understood as constraints on the propagation of distinguishability in composite systems. Potential deviations from quantum theory are predicted to manifest as failures of stability, such as breakdown of dilation, violation of distinguishability monotonicity, or loss of local tomography. This repository contains the final submission-ready manuscript together with its LaTeX source and bibliography, deposited on Zenodo prior to journal submission.This version (v1.1) incorporates clarifications on scope, logical dependencies between principles, worked examples within the Jordan-algebra classification, strengthened hedging of speculative material, and refined statements of falsifiability, in response to detailed referee feedback.
his paper develops the sixth and final foundational module of the BCLayers branch of the Boundary Compensation programme. BC-Layers Iintroduced finite-resolution readout channels. BC-Layers II formalized layermorphisms, preserved invariants, loss ledgers and evolution drift. BC-LayersIII developed task-relative multi-channel sufficiency. BC-Layers IV treatedhidden-sector elimination as a certified Schur–Feshbach response morphism.BC-Layers V organized parameter-dependent reductions into local responseatlases with pole walls, recalibration maps, transition budgets and mandatorycertificate reset.The present paper asks the limiting question: which losses can be repairedby admissible refinement, and which distinctions remain inaccessible under every protocol allowed by a declared observation class? Let Xb be an upper realization class and let P be a registered family of admissible passive observationprotocolsΠ : X −→ D b Π.The protocol envelope isΠP(X) =Π(X)Π∈P,and it induces the persistent indistinguishability relationX ∼P Y ⇐⇒ Π(X) = Π(Y ) for every Π ∈ P.A distinction is irrecoverable relative to P when it survives this complete admissible envelope.The central No-Resurrection Principle states that no deterministic postprocessing, finite channel aggregation or passive adaptive decision treeconstructed solely from protocols in P can distinguish a persistently indistinguishable pair. The result also extends to randomized post-processing:identical descriptor inputs produce identical output probability laws when theauxiliary randomness is independent of the tested realization. State-updatingor destructive protocols require a stronger history-indistinguishability semantics and are not covered by initial-state equality alone.For a task Q : X → Y b , the paper distinguishes single-protocol, finite-family,envelope, approximate and asymptotic recoverability. Along a refinement family Πr, the exact task oscillationEr(Q) = sup{dY (Q(X), Q(Y )) : Πr(X) = Πr(Y )}is monotone nonincreasing. For finite-resolution data, the robust two-scale oscillationEr,δ(Q) = sup{dY (Q(X), Q(Y )) : dr(Πr(X), Πr(Y )) ≤ δ}separates descriptor tolerance δ from admissible task uncertainty ε. Its monotonicity requires calibrated transport of metrics and tolerances across refinement maps.The paper classifies recoverable, resolution-induced, task-irrelevant,equivalence-calibrated, protocol-structural and probe-class-relative losses.It separates information-theoretic recoverability from computational tractability, and it treats the full protocol envelope as a set-theoretic information upperbound whose topological, measurable, computable and operational structuresrequire separate declarations.The principal statuses include RECOVERABLE_BY_REFINEMENT, FINITE_SUFFICIENCY_THRESHOLD, ROBUST_SUFFICIENCY_THRESHOLD, PROTOCOL_STRUCTURAL_LOSS, NO_RESURRECTION_CERTIFIED, HISTORY_SEMANTICS_REQUIRED, andREADY_FOR_BCM.The paper does not claim an ontologically absolute information limit, doesnot identify deterministic task factorization with the Fisher–Neyman factorization the
This manuscript develops a mechanism-neutral structural theory of finite recurrent stability for closed descriptions of a universe. A closed system cannot rely on an external clock, observer, memory, origin label, or cycle counter to explain its own persistence. Beginning, ending, recurrence, and cycle number count as internal structure only when they are recoverable through relations admitted within the system itself. Persistent structured identity is defined as recoverable relational distinction rather than exact sameness, retained naming, or externally recognized continuity. The paper separates contraction, correction, carry-through, recurrence, collapse, and emergence as distinct recoverability conditions. Finite recurrent stability is classified by the structural condition S_min ≤ R_stab(Ξ, Γ, B_cond, P_stress, M_carry) ≤ S_max, together with the declared identity, correction, transformation, boundary, containment, comparison, recurrence, and carry-through conditions. The expression is classificatory, not a physical observable, evolution equation, or dynamical law. Collapse and emergence are treated as distinct exits from the same finite-stability architecture. Collapse requires lower-bound recoverability failure together with failure of the admitted correction and carry-through conditions. Emergence requires upper-bound containment failure together with loss of the prior containment relation and satisfaction of a declared boundary-readability condition. Neither threshold inequality alone establishes a physical collapse, physical emergence, or cosmological event. The paper develops a structural-horizon taxonomy covering collapse, emergence, external access, and cycle distinction. These horizons classify transitions in recoverability, containment, accessibility, readability, or recurrence-index distinction. They are not assumed to be physical surfaces, metric boundaries, event horizons, or cosmological objects. Recurrence need not imply exact repetition. Boundary-equivalent recurrence may preserve a declared boundary relation, π(Ξ(τ_max)) = π(Ξ(τ_0)), while the complete internal states remain unequal. This permits recurrent governing structure without requiring identical histories. Cycle-index indistinguishability arises when no admitted internal relation recovers an absolute cycle index. Beginning-index nonrecoverability arises when no admitted internal relation recovers firstness. Neither result proves that no beginning, ending, absolute ordering, or distinct history exists. The manuscript also defines weak oscillatory form as bounded recurrent correction, declared relational recurrence, and carry-through of recoverable structure without presupposing physical waves, frequency, propagation, energy transport, or temporal periodicity. It concludes with a realization-burden architecture separating structural classification from carrier selection, interface preservation, mechanism specification, mechanism operation, physical realization, theory-specific realization, and empirical validation. The paper therefore does not propose a physical stability operator or cosmological mechanism. It identifies the structural conditions and additional burdens that any later realization theory must satisfy. Version 2 revision note Version 2 is a substantial reconstruction rather than a minor editorial update. The derivational route was rebuilt and rechecked in full, with revised definitions, dependency ordering, governing conditions, failure conditions, and claim-scope boundaries. The manuscript was then rewritten throughout in new prose to reflect the reconstructed argument. Major changes include the formalization of finite recurrent stability through the revised RstabR_{\mathrm{stab}}Rstab framework; sharper separation of contraction, correction, carry-through, collapse, and emergence; a revised structural-horizon taxonomy; clearer treatment of boundary-equivalent recurrence, cycle-index indistinguishability, and beginning-index nonrecoverability; a more precise definition of weak oscillatory form; and an expanded realization-burden architecture separating structural classification from mechanisms, physical realization, theory-specific interpretation, and empirical validation. The revision also adds stronger nonpromotion controls, appendix-based claim and condition ledgers, and clearer limits on what the structural results do and do not establish. The central research program remains mechanism-neutral, but the argument, notation, scope discipline, and presentation have all been substantially revised.