Description This preprint is the first volume of the Constructive Tensor Theory (CTT) series, titled Part I: Mathematical Foundations. It introduces a geometric and structural framework in which all physical and semantic quantities are defined not a priori, but as the result of pullback operations over a hierarchy of manifolds. Part I develops the core mathematical infrastructure for CTT. It defines: Pullback-based tensor fields over measure and observer manifolds, Constructive derivatives that respect manifold hierarchies and structural constraints, Observer-dependent geometric quantities based on pullback consistency, A classification of tensorial operations according to their compatibility with pullback geometry. The framework reformulates classical assumptions of differential geometry and field theory by replacing fixed background structures with observer-relative pullback constructs. This reorientation enables not only the reconstruction of physical laws but also lays the foundation for semantic, epistemological, and ontological generalizations. This volume forms the mathematical basis for: Part II explores observation geometry, constructive meaning, knowledge, and existence; Part III applies CTT to the tensorial formulation of physical laws in the form of Constructive Continuum Mechanics (CCM). If you are interested in the CTT project or its extensions, please feel free to reach out. Contact: s183018[at]stn.nagaokaut.ac.jp This document is part of the Constructive Tensor Theory (CTT) project. For continuity, please also refer to: Part II: [DOI:10.5281/zenodo.15420549, https://zenodo.org/records/15522168] Part III: [DOI:10.5281/zenodo.15459731, https://zenodo.org/records/15459731] For more on applications of CTT to unresolved problems in AI semantic grounding and temporal modeling, visit:https://endohirokinut.github.io/CTT_Discovery_Trigger/