Finite Recurrent Stability Before Spacetime: Collapse, Emergence, Horizons, and Recurrence as Recoverability Boundaries

This manuscript is Part I 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. Series order establishes bounded inheritance only. It does not create automatic implication, physical realization, or retrospective completion of a missing burden. Part I asks what can persist before dimensional realization is assumed. It develops a mechanism-neutral structural theory of finite recurrent stability for declared closed descriptions. Part II, Stability, Boundary Observability, and Emergent Probability in Deterministic Systems, asks when persistent deterministic structure can become boundary-readable and support local conditional numerical probability. Part III, Boundary Loss and the Born Rule: The Origin of Probability, asks what lawful structure remains after recoverability-relevant Boundary Loss, how source-grounded scalar structure and local predictive probability can be constructed on the surviving unresolved alternatives, and how an explicitly adopted and applied minimum compatibility law yields the exact local source-linked Born form. In Part I, “before spacetime” denotes structural priority rather than an earlier moment within an already realized physical timeline. A closed description 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 the relevant distinction is recoverable through relations admitted within the declared system. Persistent structured identity is defined through 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. Contraction reduces recoverable distinction without yet constituting lower-bound failure. Correction restores sufficient recoverable distinction. Carry-through preserves the restored distinction across later admissible transformations. Recurrent persistence requires both correction and carry-through. 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. Band membership is necessary but not independently sufficient for recurrent persistence. The expression is classificatory. It is not a physical observable, evolution equation, field equation, conservation law, probability rule, 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 to restore sufficient distinguishing relational content. Emergence requires upper-bound containment failure together with failure of the prior containment relation and satisfaction of a declared boundary-readability condition. Neither threshold comparison alone establishes a physical collapse, physical emergence, cosmological event, or realization mechanism. The paper develops a structural-horizon taxonomy covering emergence, collapse, external access, and cycle distinction. These horizons classify changes in recoverability, containment, accessibility, observability, or recurrence-index distinction. They are not assumed to be physical surfaces, metric boundaries, event horizons, causal horizons, or cosmological objects. Recurrence does not require exact repetition. Boundary-equivalent recurrence may preserve a declared boundary-readable relation, π(Ξ(τ_max)) = π(Ξ(τ₀)), without requiring equality of the complete internal states or repetition of the complete intervening history. The condition establishes equality of the declared projected relations only. Cycle-index indistinguishability arises when no admitted internal relation recovers an absolute cycle index. Beginning-index nonrecoverability arises when no admitted internal relation recovers absolute firstness. These are internal-recoverability results. They do not prove that no beginning, ending, absolute ordering, local dimensional history, or externally described cycle index exists. A dimensional history may possess a genuine local beginning or ending even when its absolute position within a larger recurrent architecture is not internally recoverable. The manuscript also defines weak oscillatory form through bounded recurrent correction, declared relational recurrence, and carry-through of recoverable structure. This is a weak structural classification and does not presuppose physical waves, wavelength, frequency, phase, propagation, energy transport, metric time, or temporal periodicity. Ordinary oscillatory behavior remains a possible later dimensional realization rather than an assumed starting ontology. Part I distinguishes formal structural closure from later operational realization. The roles, domains, comparison relations, threshold functions, and preclassification constraints of the finite-stability architecture are fixed to the extent required for the stated structural predicates and comparisons. An operational construction of R_stab is not an additional premise required by the finite recurrent-stability theorem or by the collapse, emergence, horizon, and recurrence classifications that use it. Supplying such a construction is a later realization burden rather than completion of a missing Part I proof obligation. The paper concludes with a realization-burden architecture separating structural classification from carrier selection, typed interfaces, preservation-compatible witnesses, mechanism specification, demonstrated mechanism operation, physical realization, theory-specific realization, empirical validation, and final physical-theory assessment. It does not propose a physical stability operator, cosmological mechanism, or completed physical theory. It identifies the structural conditions and later burdens that any proposed realization must independently satisfy. The bounded Part I handoff supplies the architecture of persistent recoverable and recurrent structure. It does not establish the Part II observable family, emergence-boundary quotient, contribution scalar, event measure, normalization, or probability assignment. Nor does it establish the Part III Boundary-Loss realization, unresolved-alternative quotient, source-grounded finite measure, predictive-probability admission, source-Hilbert bridges, compatibility law, or Born-form endpoint. Those are separately developed under their own conditions and claim-status boundaries in Parts II and III. Taken together, the trilogy completes its declared probability-origin dependency chain only at its exact bounded scope. Part I supplies the persistent and recurrent structural 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, fundamentality, or final-theory status. Physical realization remains a separate burden. Version 4 revision note Version 4 integrates the manuscript into the final three-paper series The Origin of Probability: A Complete Derivation in Three Parts and aligns the Part I framing, dependency boundaries, terminology, and downstream handoff with the completed trilogy. The revision updates the trilogy endpoint to reflect the final Part III architecture: recoverability-relevant Boundary Loss, lawful unresolved alternatives, source-grounded finite measure, local predictive-probability admission, independent source-Hilbert state and event bridges, and explicit adoption and application of the minimum compatibility law yielding the exact local source-linked Born form. Series order continues to establish bounded inheritance only and does not create automatic implication, physical realization, or retrospective completion of an earlier burden. Version 4 sharpens the distinction between structural priority and physical temporal priority, preserves the separation between local dimensional beginnings and endings and internally recoverable absolute firstness, and maintains the nonpromotion limits on cycle-index indistinguishability and boundary-equivalent recurrence. The finite recurrent-stability architecture is also given an explicit operational-closure boundary. An operational construction of R_stab is not required by the Part I finite recurrent-stability theorem or by the collapse, emergence, horizon, and recurrence classifications that use the declared architecture. Such operationalization is identified as a later realization step rather than a missing Part I proof obligation. Formal Part I closure therefore means closure of the stated structural classification problem, not operational closure of the later realization problem. The claim-scope and realization-burden architecture has been consolidated to distinguish structural classification