TI Analysis Foundations, Vol.1: Pre-Spatial Structures and Overlap-Based Integrability

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 shows 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. Its measure‑theoretic continuation appears in TI03: Observer‑Dependent Measure (https://doi.org/10.5281/zenodo.17671551), where the overlap‑based generative framework developed here becomes the basis for observer‑dependent coverings, assembler time, and the emergence of classical Lebesgue measure as a Dirac‑limit of evolving observer distributions. In this sense, Vol.1 provides the pre‑spatial analytical foundations upon which TI03 builds: integration precedes space, and measure itself emerges as a stabilized observational regime. Measure‑theoretic and geometric developments—including observer‑dependent measures, Radon–Nikodym flows, and dynamic σ‑algebras—are therefore not missing components, but necessary continuations that follow naturally from the generative principles established in this volume.