Abstract

We introduce Remainder Dynamics: the complete theory of physical reality as the dynamics of remainder propagation across sovereign domain hierarchies. Classical mechanics, quantum mechanics, and general relativity are not unified by Remainder Dynamics — they are contained within it as limiting cases corresponding to different suppressions of the full categorical structure. The foundational object is the category C_SRNUDT, whose objects are sovereign domains (triples of cell, sheaf stalk, cosheaf costalk), whose morphisms are boundary-compatible remainder-preserving causal maps, and whose central structure is the pairing ⟨[R],[γ]⟩: H¹(F) ⊗ H₁(F̆) → ℝ between remainder obstruction classes and propagation path classes. The fundamental equation of Remainder Dynamics is this pairing. The SRNUDT Boundary Theorem — that remainder localizes to the fixed-point set of domain symmetry — is the master theorem of Remainder Dynamics. The Riemann Hypothesis and Navier-Stokes global regularity follow as conditional corollaries. The theory makes new predictions at every scale from quantum gravity to cosmology.