I introduce Scale Calculus, a domain-general mathematical framework built on four axioms (M1–M4) governing spectral vessel aggregation. The central result, T-H1, establishes the Spectral-Coupling Correspondence: α = ln(κ) ⟺ κ = e^α, where κ is the coupling coefficient of a vessel and α its spectral exponent. I derive integer arithmetic as a tight instantiation, proving 1+1=2 from vessel aggregation and the Fundamental Theorem of Arithmetic from spectral irreducibility. I further derive rings and fields via dual aggregation, p-adic media, the Adèle ring, the Riemann Hypothesis as prime self-duality, Langlands correspondences as adèlic T-H1, and quantum mechanics as Scale Calculus. A forbidden zone κ ∈ (e^{−1}, e^{−1/2}) is identified, populated exclusively by biological and neural systems. Anti-probability vessels in the complex-κ extension are proposed as candidates for matter-antimatter asymmetry, and a resigned medium is introduced as a dark energy candidate. The paper closes with a conjectural description of the person as a complete vessel and four open problems at the boundary of current mathematics.