Causal-Algebraic Geometry: A Grothendieck-Type Framework for Partial Orders with Applications to Manifold Detection and Arithmetic

Paper and formal verification for "Causal-Algebraic Geometry: A Grothendieck-Type Framework for Partial Orders, with Applications to Manifold Detection and Arithmetic." The deposit includes the manuscript and the complete Lean 4 / Mathlib codebase verifying the theorems therein: the causal spectrum CSpec and its universal property as terminal object in the category of causal topologies; the sheaf-γ bridge (causally convex = ring homomorphism open); the separation theorem; Dilworth's theorem via Hall's marriage theorem; the width scaling law and polynomial/exponential dichotomy; the Wilson loop trace theorem; the arithmetic bridge recovering Spec(Z); supermultiplicativity and the dimension law log|CC([m]^d)| = Θ(m^{d−1}) for all d ≥ 2; and the exact growth constant ρ = 16 for square grids (upper bound fully verified; lower bound verified except for one auxiliary lemma). This work is Paper I of a four-paper series. Paper II: "Order-Convex Subsets of Grid Posets: A New Exponential Combinatorial Class." Paper III: "Black Hole Thermodynamics from Counting Convex Subsets of a Lattice." Paper IV: "Jackiw-Teitelboim Gravity and Hagedorn Density of States from Counting Causally Convex Subsets."