Stability, Boundary Observability, and Emergent Probability in Deterministic Systems

This paper is Part II of the three-paper series The Origin of Probability: A Complete Derivation in Three Parts. The trilogy develops a bounded structural route from persistent recoverable organization, through boundary observability and conditional numerical probability, to recoverability-relevant Boundary Loss, lawful unresolved alternatives, source-grounded finite measure, local predictive probability, independent source-Hilbert state and event bridges, and the exact local source-linked Born form obtained after explicit adoption and application of a minimum compatibility law. Part I, Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries, asks what can persist before dimensional realization is assumed. Part II asks when persistent deterministic structure can become boundary-readable and support numerical probability. Part III, Boundary Loss and the Born Rule: The Origin of Probability, asks what lawful unresolved structure remains when recoverability-relevant Boundary Loss removes guarantee, how source-grounded scalar structure and local predictive probability can be constructed on those alternatives, and how an explicitly adopted and applied minimum compatibility law yields the exact local source-linked Born form. Series order establishes bounded inheritance and handoff only. It does not create automatic implication, physical realization, or retrospective completion of a missing premise, condition, scalar, measure, mechanism, or proof. The paper addresses an antecedent question to ordinary probability theory: under what conditions can a closed deterministic description support a normalized measure with probability-measure status without assuming that status at the outset? The difficulty lies in the difference between assigning numbers and establishing probability. A nonnegative weight may exist without identifying the relevant alternatives. A normalized map may exist without an event measure. A measure may exist without agreeing with the proposed scalar weights. Agreement may be imposed after an outcome, supplied by an external weighting source, or embedded in a description that retains additional independent weight-bearing coordinates. None of those partial constructions independently establishes probability within the deterministic description itself. The analysis begins with the requirements for persistent structured identity. Persistent identity is treated through recoverable cross-state relational distinction rather than exact sameness, repeated naming, scalar aggregation, or externally preserved memory. Within the paper’s explicitly bounded structural registries, the route identifies ordered dependence, intrinsic structure-sensitive stabilization, distributed relational recurrence, bounded recoverability, and preservation of the distinctions required for persistent refinable identity. Bounded recurrent relational correction is classified locally 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, physical recurrence, or a physical carrier. It identifies an oscillatory structural form whose stronger realization requires additional correspondence and realization conditions. The paper then develops the transition from persistent structure to boundary observability. For a fixed admitted carrier and observable family, states that are indistinguishable under every admitted observable are organized by an emergence-boundary quotient. Every admitted observable factors through that quotient, so the quotient retains exactly the distinctions readable through the fixed admitted family. The quotient is canonical relative to the admitted carrier and observable family. It does not independently derive, uniquely select, complete, or physically privilege that observable family. Nor does it establish a physical boundary, measurement interaction, transport process, probability space, or physical realization. Boundary readability does not supply numerical weight. A boundary-readable alternative carrier, a nonempty class of total nonnegative candidate weight maps, and a nontrivial candidate contribution scalar are therefore introduced separately. These structures establish a candidate numerical architecture only. They do not select a preferred scalar, establish scalar closure or uniqueness, provide an event measure, authorize probability interpretation, or identify a physical scalar. Within an expressly narrowed scalar-probability regime, complete scalar closure requires every admitted observable-weight-bearing map to factor through the same contribution scalar, together with exclusion of every independent observable-weight-bearing coordinate. Where an admitted invariant changes observable weight while the contribution scalar remains fixed, general scalar closure and scalar uniqueness fail locally. Such conditioned failures do not establish universal impossibility, and removal of the displayed obstruction does not by itself establish positive closure. The scalar and measure branches remain separate. A normalized scalar assignment does not independently establish an event measure, and an admitted event measure does not independently supply the contribution scalar or establish scalar closure. The branches join only on the same exact carrier and under exact singleton compatibility. Numerical probability-measure status becomes available only under a cumulative condition package. The complete declared regime requires the exact finite carrier and event measure, a normalized scalar component on that same carrier, scalar-measure compatibility, one-scalar factorization, exclusion of independent observable-weight-bearing coordinates, pre-outcome assignment, absence of an external weighting source, and an expressly narrowed predictive regime. Additivity, commensurability, regularity, positive scale, finite nonzero total, and normalization remain separately controlled conditions within that route. Inside the complete regime, the normalized measure receives local, conditional, regime-relative, pre-outcome probability-measure status, and its singleton values agree with the normalized contribution-scalar assignment. The equality is the endpoint of a joined construction, not its starting assumption. This is not an unconditional derivation of probability from determinism. Probability does not follow from deterministic closure, persistence, boundary readability, scalar content, measure realization, or normalization alone. The paper identifies the precise cumulative threshold at which a deterministic closed description can support probability without having imported probability into its premises. The result also preserves a strict distinction between probability-measure status≠physical probability mechanism≠empirically validated physical theory. A finite abstract witness establishes formal nonvacuity and joint satisfiability of the complete package. It does not establish that an independently specified deterministic system, physical theory, or empirical model enters that regime. The paper therefore delegates nine later burdens: mechanism satisfaction, physical scalar selection, boundary enforcement, branch generation, refinement implementation, physical realization, theory-specific realization, empirical validation, and final physical-theory status. Delegating those burdens does not satisfy them. No local obstruction is promoted into universal impossibility. Objective physical chance, stochastic dynamics, measurement outcomes, collapse, a physical probability mechanism, unrestricted probability, and final physical-theory status are not established. Within the trilogy, Part II supplies the middle structural layer: persistent recoverable structure↓boundary-readable alternatives↓candidate scalar content↓event-measure compatibility and normalization↓conditional numerical probability-measure status. Part III begins from a different question: what lawful unresolved structure remains when a shared boundary-readable residue no longer guarantees which admissible alternative obtains? Part III independently establishes recoverability-relevant Boundary Loss, a scalar-neutral resolution relation, and a finite nontrivial representative-invariant unresolved context. It separately grounds nonnegative source-linked scalar residues on those alternatives, constructs a finite scalar measure, establishes strict positivity before normalization, and admits the normalized measure as local predictive probability only under an explicit finite predictive criterion. It then independently establishes source-state and source-event Hilbert bridges, isolates the remaining compatibility burden, explicitly adopts the minimum substantive compatibility law at the declared scope, and applies that law to obtain the exact local source-linked Born form. None of those downstream constructions retrospectively supplies a missing Part II scalar, carrier identity, event measure, scalar-measure compatibility condition, normalization condition, or probability interpretation. Taken together, the trilogy completes its declared probability-origin dependency chain only in a bounded and explicitly qualified sense. Part I supplies the persistence and recurrence foundation. Part II supplies the conditional boundary-readable numerical-probability architecture. Part III carries the route through recoverability-relevant Boundary Loss, lawful unresolved alternatives, source-grounded finite measure, local predictive admission, independent source-Hilbert bridges, and explicit compatibility adoption and application to the exact local source-linked Born form. This completion does not establish a complete theory of quantum mechanics, an unrestricted global Born rule, measurement or collapse dynamics, empirical confirmation, physical quantization, necessity, f