What question did this study set out to answer?

The central aim is to establish a theory of finite recurrent stability that defines persistence in closed systems without external references.

May 4, 2026Open Access

Finite Recurrent Stability and the Pre-Spacetime Structure of Horizons: Collapse, Emergence, and Recurrence as Recoverability Boundaries

Key Points

The central aim is to establish a theory of finite recurrent stability that defines persistence in closed systems without external references.
Developed a mechanism-neutral theoretical framework for finite recurrent stability.
Defined and classified concepts such as collapse, emergence, and horizon structures in relation to stability conditions.
Introduced a taxonomy for different types of horizons based on their recoverability transition roles.
Defined stability conditions as structural boundaries expressed by a finite stability band.
Classified collapse and emergence as oppositional states that indicate transitions in recoverability within these boundaries.
Established that recurrence can occur without strict identical repetition, allowing for boundary-equivalent recurrence instead.

Abstract

This manuscript develops a mechanism-neutral theory of finite recurrent stability for closed descriptions of a universe. A closed system cannot rely on an outside clock, observer, memory, origin label, or cycle counter to explain its own persistence. If beginning, ending, recurrence, or cycle number are not recoverable from within the system, they function as external labels rather than internal structure. The paper defines persistent structured identity as recoverable relational distinction, not exact sameness or label continuity. Contraction, correction, carry-through, recurrence, collapse, and emergence are classified as recoverability conditions within a finite stability band. Stability is expressed structurally by the condition \ (S_ R (, C, B, P, M) S_\), where the notation is classificatory rather than physical. Collapse and emergence are treated as opposite boundary roles of the same finite-stability band: collapse occurs when recoverability falls below the lower threshold, while emergence occurs when a prior regime can no longer contain structure below the upper threshold. A horizon is therefore defined as a recoverability transition, not as a physical surface, metric boundary, or cosmological object. The paper develops a horizon taxonomy consisting of collapse horizon, emergence horizon, external-access horizon, and cycle horizon. The cycle horizon marks failure of absolute recurrence-index distinguishability while admissible boundary relations may remain preserved. Recurrence therefore need not mean identical repetition; it may mean boundary-equivalent recurrence, \ ( ( (t_) ) = ( (t₀) ) \), rather than exact microscopic equality. The result is structural and mechanism-neutral. The paper does not identify a physical stability operator, derive physical horizons, or propose a specific cosmological mechanism. It defines the burden any later realization theory must satisfy: an internal mechanism must preserve, restore, bound, expose, and carry forward recoverable identity without importing external labels.

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