Part I: The Coherence Kernel
Category τ begins with five generators in strict total order — α < π < γ < η < ω — and a single ontic operator ρ (progression). A zeroth axiom K0 postulates the existence of τ itself as a universe of discourse — the ambient totality that contains all objects, distinct from the fixed-point element ω within it. Six structural axioms K1–K6 then govern this minimal signature: a strict order on the generators, a fixed point at ω, orbit-seeded generation for each non-ω generator, a no-jump cover relation within each orbit, the unreachability of ω from below, and an object closure axiom that exhausts the universe.
Together these define the static kernel τ₀: a finite, complete specification from which all of mathematics will be generated. At the end of this Part, τ₀ is a blueprint — precise, categorical, and inert. No objects have been created yet; no arithmetic has been earned. That is the work of the Parts that follow.
The diagonal discipline, introduced in the final chapter of this Part, explains why the kernel has exactly five generators and four orbit channels: three successive diagonal rewirings saturate the available degrees of freedom, and the object closure axiom prevents a fourth.