Physical systems often exhibit stable, repeatable regimes despite being described by equations that allow many possible trajectories. This tension is typically addressed using probabilistic language, with probability treated as a fundamental feature of evolution or as a reflection of stochastic dynamics. In this paper, we argue that probabilistic descriptions instead arise from structural underdetermination in the presence of recurrence and finite resolution. We introduce a coherence-first account in which repeated traversal of structurally admissible pathways reinforces regime stability, leading to effective regime selection without invoking randomness, agency, or additional dynamical mechanisms. Within this framework, probability collapse is not a physical event but a resolution of indistinguishability as recurrence deepens constraint. This perspective completes a structural notion of determinism: outcomes need not be fixed, but admissible regimes are finite, and repeated evolution naturally sharpens their dominance. The resulting picture reconciles deterministic dynamics with probabilistic description while preserving consistency with existing physical formalisms.
Physical systems often exhibit stable, repeatable regimes despite being described by equations that allow many possible trajectories. This tension is typically addressed using probabilistic language, with probability treated as a fundamental feature of evolution or as a reflection of stochastic dynamics. In this paper, we argue that probabilistic descriptions instead arise from structural underdetermination in the presence of recurrence and finite resolution. We introduce a coherence-first account in which repeated traversal of structurally admissible pathways reinforces regime stability, leading to effective regime selection without invoking randomness, agency, or additional dynamical mechanisms. Within this framework, probability collapse is not a physical event but a resolution of indistinguishability as recurrence deepens constraint. This perspective completes a structural notion of determinism: outcomes need not be fixed, but admissible regimes are finite, and repeated evolution naturally sharpens their dominance. The resulting picture reconciles deterministic dynamics with probabilistic description while preserving consistency with existing physical formalisms.
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.
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.
The Origin of Probability: A Complete Derivation in Three Parts is the collected edition of a three-part theoretical investigation into the structural origin of probability within a deterministic closed-description framework. The work asks a deliberately narrow but foundational question: under what conditions can a system whose underlying description remains deterministic acquire a lawful probability-bearing description without inserting primitive randomness, probabilistic axioms, or the Born rule at the beginning of the construction? The trilogy develops that question cumulatively. Each part establishes a distinct layer of the argument, and later results depend on structures and limitations established earlier. The collected edition therefore preserves the full dependency order rather than treating the three papers as independent essays. It also includes a narrative overview and terminology guide intended to make the architecture of the complete derivation easier to follow. Part I, Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries, develops the realization-neutral structural foundation. It begins from persistent relational distinction and recoverability rather than assuming spacetime, metric distance, clock time, energy, probability, or quantum measurement. Within a declared finite-stability regime it distinguishes contraction, correction, carry-through, collapse, emergence, recurrence, cycle structure, and horizon-like boundaries by their effects on recoverable relational structure. Collapse and emergence appear as distinct lower- and upper-bound failures of one finite recurrent-stability architecture. Weak oscillatory form is subsequently obtained only at structural scope, without promoting recurrence into a physical wave, temporal oscillation, or cosmological mechanism. Part II, Stability, Boundary Observability, and Emergent Probability in Deterministic Systems, develops the transition from deterministic structural dynamics to probability-bearing description. Its central problem is not whether deterministic evolution can simply be renamed probabilistic, but whether loss of recoverability at an admissible boundary can support a lawful reduced description of unresolved alternatives. The paper therefore separates Boundary Loss from probability itself and distinguishes quotient formation, reduced observability, scalar closure, finite measure, normalization, and predictive interpretation as separate derivational burdens. A normalized set of branch weights is not treated as probability merely because it sums to one. Probability status is admitted only after the additional structural and interpretive conditions required for predictive use have been isolated. Part III, Boundary Loss and the Born Rule: The Origin of Probability, carries the construction into a bounded physical realization framework. Boundary Loss supplies unresolved alternatives but does not itself supply numerical probability. The derivation therefore proceeds through source-grounded scalar residues, finite positive total measure, normalization, local predictive admission, and independent source-state and event bridges into Hilbert-space representation. A minimum compatibility condition is then isolated, explicitly adopted at the appropriate stage, and applied separately. At the resulting local source-linked scope, the construction reaches the exact Born form p_i = ⟨J(Ψ), P_i J(Ψ)⟩ = ||P_i J(Ψ)||². The significance of the trilogy lies as much in the separation of these stages as in the terminal equation. Boundary Loss is not equated with probability. Scalarization is not assumed to be probability. Normalization is not treated as sufficient for probability. Hilbert-space representation is not allowed to retroactively supply the earlier structural derivation. Compatibility required for the final Born-form application is distinguished from structures already derived. The dependency chain is therefore designed so that later mathematical convenience cannot be silently imported backward as an earlier premise. Across the three parts, the argument develops the following bounded sequence: ordered dependence → recoverable distinction → finite recurrent stability → collapse/emergence boundary structure → boundary observability → Boundary Loss → unresolved alternatives → scalar closure → finite measure → normalization → predictive probability → Hilbert-space source and event bridges → minimum compatibility → exact local source-linked Born form. The collected edition is intended to function as the canonical integrated research record for this derivation. It contains the trilogy overview followed by Part I Version 4, Part II Version 3, and Part III Version 2. Internal bookmarks and navigation link the overview directly to the title page of each constituent work, while the individual papers remain separately citable publications with their own persistent identifiers. Constituent works: Part I: Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability BoundariesDOI: 10.5281/zenodo.19966230 Part II: Stability, Boundary Observability, and Emergent Probability in Deterministic SystemsDOI: 10.5281/zenodo.19966289 Part III: Boundary Loss and the Born Rule: The Origin of ProbabilityDOI: 10.5281/zenodo.20102172 Taken as a whole, the trilogy proposes a disciplined route from recoverability and deterministic boundary structure to probability and, at its final declared scope, to the Born form. Its central methodological requirement is that every transition be earned at the layer where it is used, with unresolved burdens, adopted conditions, nonclaims, and realization boundaries kept explicit rather than compressed into the final result.
Overview This work presents a structural reinterpretation of fast internal degrees of freedom that remain unresolved under finite observational resolution. Rather than contributing additively to observable mass–energy, such unresolved dynamics are shown to function as constraints that determine which states are dynamically and observationally admissible. When internal dynamics evolve on timescales or length scales below observational resolution, their detailed trajectories cannot be directly accessed. Nevertheless, they impose consistency conditions that effectively project the full dynamical state space onto a reduced subset of allowed configurations. Within this framework, quantization, stability, and probabilistic descriptions arise as unified consequences of constraint-based state selection rather than from fundamental indeterminism. Probability is interpreted not as an intrinsic randomness of nature, but as a measure over ensembles of microscopic trajectories that are compatible with the same effective observable state under resolution-limited observation. This study introduces no new particles, forces, or interaction terms. Instead, it reassigns the structural role of unresolved degrees of freedom within existing dynamical frameworks, clarifying how quantization and probability emerge from deterministic dynamics subject to finite observational resolution. Supplementary Information (Dataset) A snapshot of the dataset used in this study is provided as Supplementary Information in CSV format for convenience.The authoritative and continuously revised version of the dataset is archived on Zenodo with a persistent DOI. The data used in this paper correspond exactly to the version associated with the following DOI:DOI: 10.5281/zenodo.18097066
(This version has been superceeded, see v.3) Contemporary approaches to physical unification typically emphasize dynamical laws, interpretive frameworks, or information-theoretic reconstructions. Less attention is given to the structural constraints that determine which physical descriptions can remain coherent and stable under interaction, scale transition, and aggregation. This paper develops a constraint-first analysis of coherence, arguing that persistence and unification require restrictions on admissible physical continuations that are not reducible to dynamics alone. Within the framework referred to here as Quantum Coherence Theory (QCT), coherence is treated not as a fragile quantum resource but as a structural property governed by interaction geometry and constraint density. On this view, classical behavior emerges when accumulated constraints sufficiently narrow admissible evolution, while quantum behavior re-emerges when constraints are relaxed. The account does not introduce new dynamics, collapse mechanisms, or modifications of quantum formalism, but instead clarifies the conditions under which existing formalisms yield stable, unified descriptions. The resulting constraints distinguish QCT from observer-centric, intrinsic-collapse, and purely descriptive decoherence accounts, while remaining compatible with globally unitary dynamics. Formal analysis in later sections develops operational redundancy conditions, admissibility constraints on joint states, and rigidity-style arguments linking redundant record structure to the stability of classical descriptions. These results do not constitute a complete theory of unification, but they narrow the space of physically admissible descriptions and clarify the structural role of coherence in bridging quantum and classical regimes.
This paper argues that, where deterministic and indeterministic formulations agree on empirically corroborated structure while differing in their completion of structure that remains empirically underdetermined, features invariant across these formulations warrant ontological commitment, whereas any warrant attaching to features that vary across them is categorically more provisional. Because deterministic and indeterministic completions are always formally available, while additional grounds for preferring particular completions are contingent and defeasible, empirically corroborated structure alone cannot determine whether reality is fundamentally deterministic or indeterministic, or whether that very distinction belongs only to specific representations of it.
On this basis, I advance a model-invariance criterion according to which structural features are candidates for realist commitment insofar as they are empirically corroborated within a regime and preserved across the admissible formulations of that structure. I show how such invariant content can be characterised in mathematically intelligible ways, yielding a fallibilist form of structural realism grounded in model-invariance rather than in the specific form of any given mathematical representation. Features such as conservation relations, invariant measures, symmetries, and probability or correlation structures over observable histories may satisfy this criterion. By contrast, the localisation of modal selection---whether in fine-grained initial conditions, stochastic outcomes, or state-update and collapse mechanisms---may vary across empirically admissible formulations and therefore constitute a gauge-like representational freedom.
This framework offers a realist ontology for modern physics that combines empirical openness with resistance to metaphysical overreach.