This paper defines structural availability on a path space Σ through primitive pairwise mappings. Two mappings, and defined on Σ × Σ, take binary values indicating the presence or absence of order-type and proximity-type relational support and thereby determine which relational distinctions are structurally available. Each mapping induces polarity classes on Σ × Σ, and their joint elements generate a finite availability differentiation space AD consisting of four and only four elements. This space is uniquely determined, admits a canonical identification with 0, 1 × 0, 1, and exhausts all structurally admissible elements. The framework starts from pre-availability structural conditions and proceeds to the determination of availability itself. The resulting structure provides a complete classification of relational availability on the path space and introduces no topology, metric, measure, or dynamical assumption. This version incorporates structural revisions, formatting standardization, and consistency improvements across the series.
Sean X. Tan (Mon,) studied this question.
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