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 which functions as: the visual, conceptual, and interpretive topology layer accompanying the present mathematical scaffold framework.