A Cyclotomic Quadrichotomy for Ramanujan's Mock Theta Functions: Block Ratios, Finite Identities, and False-Theta Representations

An exact-arithmetic census of the twenty-four canonical mock theta functions of orders 3, 5, 7 and 10 at roots of unity, in two complementary channels: odd prime order, and order congruent to 2 modulo 4. A block-ratio lemma sorts every boundary series into one of four mutually exclusive classes — regular, terminating, divergent, or singular — identifying exactly which finite blocks admit coordinate arithmetic. Principal results: (i) a finite identity valid at every odd level, giving phi^(3)(-zeta) = f(zeta)/2; (ii) five functions are singular at every root of unity, a hard obstruction; (iii) an exact discriminant channel theorem with the sharp nondegeneracy condition p does not divide a(a-2)(a+2); (iv) two new half-range false-theta representations, for phi^(3) and for Watson's nu, verified exactly at every prime tested, from which a four-fibre count reproduces the Legendre-symbol coordinate law for phi^(3). All computational assertions use exact power-basis arithmetic and integer resultants; no floating-point calculation enters any claim. The package contains the full Lean 4 / Mathlib development (no custom axioms; axiom audit reports only propext, Classical.choice and Quot.sound), the complete exact-arithmetic Python reproduction code, and a single command that regenerates every table and re-checks every displayed identity. This version corrects several results announced in an earlier preprint; see CORRECTIONS.md and Section 11 of the manuscript.