A Ladder-Based Traversal Notation for Repeated Symbolic Sequences

This paper introduces a ladder-based traversal framework for representing repeated symbolic structures within symbolic and numerical sequences. Repeated structures are interpreted as ordered traversal paths between hierarchical rung states, for example 1→11→111, permitting repetition to be analysed as directed traversal behaviour rather than solely as static substring occurrence. The framework defines traversal states using traversal direction, rung progression between adjacent repetition orders, traversal-gap structure, and climb behaviour. Valid traversal is restricted to transitions between neighbouring rung states only. The model additionally permits chained traversal paths spanning multiple repetition orders, overlapping embeddings in which symbolic occurrences participate in multiple traversals, and recursive overlap between rung states. Unlike conventional repetition frameworks that primarily characterise repeated substrings through periodicity or maximal interval structure, the proposed framework provides a ladder-based and geometric interpretation of symbolic repetition in which repeated structures may be analysed as interconnected traversal systems rather than purely linear strings. Potential applications include symbolic sequence fingerprinting, traversal-density analysis, overlap statistics, traversal entropy measures, and Monte-Carlo comparison of traversal behaviour within random and non-random symbolic systems.