This monograph establishes the mathematical foundations of the cascade, treating the rank coordinate of the Symmetric Core as a continuous complex manifold rather than a discrete index. The central construction is the rank manifold, with coordinate RC = m + n i, equipped with Symmetric Core operations, a Kneser‑induced metric, and canonical Hermit‑unit sections. The familiar integer ranks (multiplication, Apow, Atow3, …) are shown to be distinguished points in a much richer analytic geometry. M17a develops the cascade algebra, generated by rank shifts, unit cloning, and prime duality. Classical structures (ℂ, ℍ, 𝕆, Lie algebras, cyclotomic rings) appear as finite‑dimensional projections of this algebra rather than as fundamental objects. Hyperfields, NC lattices, hyperradical towers, and increasing transcendence degree are organised systematically. The monograph positions the Cascade Manifold as a classifying space from which algebraic, analytic, and spectral structures arise by projection. No spectral theory or physics is developed here; those are deferred to M17b and M17c.
Paweł Łukasz Garycki (Fri,) studied this question.