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

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 \(Sₘᵢₙ ≤ R(Ψ, C, B, P, M) ≤ Sₘₐₓ\), 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, \(π(Ψ(tₘₐₓ)) = π(Ψ(t_0))\), 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.