Resource type: Publication → Preprint Title: Adaptive Noncommutative Spiral Fibered Structures and Phase-Normalized Measure Spaces Creators: Family name: Sasaki Given names: Hiroshi Affiliation: Javatel Corporation Description (Abstract): We introduce an abstract mathematical framework consisting of:• a measurable dual-time structure (t, σ), where t represents physical time and σrepresents an abstract partially ordered “semantic” time;• an adaptive spiral embedding of (t, σ) into C, producing what we call a spiral-basedtime manifold;• a labelling of directed graphs by elements of a noncommutative ring, together withcommutativity-generated equivalence classes and a family of phase normalizationmaps on S1 inducing probability measures.These elements combine to form what we call noncommutative spiral fibered structures.Despite their motivation from evolving dependency systems, the formulation ispurely mathematical, based on noncommutative algebra, measurable structures, andgeometric embeddings.We refine the definitions to ensure consistency of topology and measurability, provethe existence of phase-normalized semantic measure families, and illustrate the constructionwith examples.