From Observation to General Law

The A-C Coupling Theorem presents a general law for n-dimensional linear Diophantine systems of the form N = Σ cᵢ · xᵢ. Two structural conditions are identified as jointly sufficient for a direct, single-step determination of the minimum starting value A₀ = N mod c₂: Condition one (Foundation): the top coefficient must satisfy c₁ ≡ 1 (mod c₂). This renders the top layer transparent modulo c₂, making the steering value independent of the highest coefficient. Condition two (Chain): the remaining coefficients must form an exact divisibility chain cₙ | cₙ₋₁ | ... | c₂. This guarantees that every remainder at every layer is absorbable without gaps. Two families of instances are examined in detail: the euro family (modulus 9, coefficients 19, 9, 3, 3, 1) where A₀ equals the digital root, and the clock family (modulus 12, coefficients 25, 12, 4, 1) where A₀ = N mod 12. Concrete examples from 2D through 6D and beyond verify both conditions through modular arithmetic. The law is general: for any valid coefficients and any N, A₀ is determinable in O(1) time. The digital root is not a property of any specific system, it is a consequence of the structure itself. The paper includes a formal proof of the general theorem, with explicit inductive steps for both the Foundation Lemma and the Attainability Lemma.