What question did this study set out to answer?

This theorem aims to establish a closure condition for the Self-Preserving Flow hierarchy to prevent dimensional collapse.

May 29, 2026Open Access

Theorem 4: Recursive Fixed-Point Closure Theorem in Self-Preserving Flow (SPF)

Key Points

This theorem aims to establish a closure condition for the Self-Preserving Flow hierarchy to prevent dimensional collapse.
Defined conditions for long-horizon recursive stability.
Analyzed governance transformations affecting semantic systems.
Identified fixed-point invariants in the context of traceability topology.
Proved that a canonical fixed-point invariant is essential for recursive stability.
Demonstrated that without this invariant, semantic systems risk losing coherence.
Highlighted the traceability topology as the key structural element in preserving history.

Abstract

This theorem establishes the final recursive closure condition of the Self-Preserving Flow (SPF) hierarchy. Theorem 1 stabilized state evolution, Theorem 2 stabilized semantic constraint evolution, and Theorem 3 stabilized governance evolution. However, unrestricted recursive self-modification introduces the fundamental threat of infinite meta-regression (St → SCLt → Φt → Ψt → . . .). Without a recursively irreducible fixed structure, semantic systems lose dimensional coherence and collapse into unrestricted relativistic drift. This theorem proves that long-horizon recursive stability requires the existence of a canonical fixed-point invariant under arbitrary admissible recursive governance transformations. SPF identifies this invariant not as semantic content, ontology, or governance logic, but as the canonical traceability topology (A) preserving historical reconstructibility itself.

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