Published June 12, 2026 | Version v1
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Description
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.
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General_Law_for_n-Dimensional.pdf
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Additional details
- A General Law for n-Dimensional Linear Diophantine Systems