This paper establishes a pre-spatial foundation for analysis in which integration is treated as the primitive operation.Rather than presupposing norms, metrics, or topological spaces, the framework begins with overlap-based integrability and demonstrates how normed spaces, completeness, and Hilbert structures arise only as stabilized outcomes of generative aggregation. Relational overlap is introduced as a pre-normative analytical structure that precedes distance, linearity, and metric comparison, and persists even when norm-dominated descriptions break down. From this perspective, Banach and Hilbert spaces are repositioned not as foundations of analysis, but as conditional stabilization devices that sustain analytical continuity within restricted regimes. This work constitutes Vol.1 of the TI Analysis Foundations series.Measure-theoretic and geometric developments, including observer-dependent measures and Radon–Nikodym flows, are intentionally deferred to subsequent volumes, where they appear as necessary continuations rather than missing components.
Hidehito KOBAYASHI (2026) studied this question.