First Calculus Supplementary Material, Book I: Methodology — First-Classness, Axioms, Postulates, Principles

Book I (Methodology) is the first volume of the “First Calculus Supplementary Material” series. It fixes the methodological ground used across the First Calculus papers, the “SM Books,” and also Disposition of the Quantum Object (DQO, Paper 1). Its purpose is to state the rules of admissible formulation under First-Classness (FC) before any technical construction is undertaken. The central concern is normative authority: what can legitimately function as “ground” in a foundational development once privileged primitives, external stipulations, and one-sided starting points are excluded by FC. The text develops the Euclidean form as a disciplined mode of presentation, while marking its limits (contra naïve intuition) and the limits of formalist replacement (contra Hilbert). It then articulates FC as an internal constraint on admissibility, and treats the Principle of Sufficient Reason (PSR) as a complementary norm whose reach terminates at the level of primitives. The book contains explicit FC-constraints (lemmas) and a guiding axiom/principle that frame the project’s paradigmatic divide: left-side quantitative calculation and right-side qualitative construction are treated as complementary, but not collapsed into a synthesis. Key results include the Irreducibility Thesis, the limit of PSR, the rejection of freestanding or one-sided ground under FC, the failure of left-side practice alone under FC, and the methodological incompleteness of right-side practice without left-side calculation. This volume is intended to be cited as the methodological reference point for the corpus, and versioned as a living document alongside the evolving series.