Recursive Temporal Expansion and Admissible Convergence: An Expanded Mathematical Scaffold for Recursive Cognition, Environmental Modulation, and Multi-Layer Continuity Traversal

This paper presents an expanded mathematical scaffold for modelling recursive cognition as temporal expansion, weighted traversal, admissible convergence, and multi-layer continuity reconstruction under uncertainty. The framework consolidates developments across the Paton System cognitive branch, including recursive temporal expansion, recursive continuity re-entry, fragmented continuity, recursive field interruption, sensory admissibility switching, environmental modulation, latent continuity persistence, recursive housekeeping, symbolic continuity anchoring, recursive overload, convergence pressure, and distributed recursive exposure. The framework introduces mathematical structures for recursive weighting, linkage continuity, recursive expansion, admissibility conditions, traversal density, fragmentation pressure, convergence regulation, interruption operators, continuity persistence coefficients, environmental modulation, sensory verification switching, and recursive field stabilisation. The paper proposes that cognition may be interpreted as a dynamically weighted continuity traversal field whose stability depends upon recursive admissibility conditions, continuity inheritance, environmental modulation, processing margin, recursive load distribution, and convergence regulation. The framework does not claim neurological finality or predictive completeness. Its purpose is structural preservation, recursive continuity modelling, mathematical articulation, and future refinement within the Paton System cognitive branch. Related conceptual and visual topology interpretation is developed separately in: Recursive Continuity Topology: A Structural Bridge Between Stable and Fragmented Cognitive Traversal Within the Paton System.