Exact Kakeya minima over ℤ/Nℤ: a transfer principle, multiplicativity in the plane, and K(6,3) = 65

We develop a transfer principle for exact Kakeya minima over the rings ℤ/Nℤ: a single finite, machine-checkable "fractional cost" inequality on one factor of a coprime splitting forces the lower bound K(qM,n) ≥ K(q,n)·K(M,n) for every coprime co-factor at once, and at the true price it makes the Kakeya minimum exactly multiplicative. We prove three theorems: the transfer principle itself; a direction-price theorem for 𝔽₃³ (any s points contain full lines in at most s of the 13 direction classes — strictly stronger than the known minimum value, and false over 𝔽₂³); and a plane schema showing the certificate holds at the true price for every prime, whose only nontrivial input is a classical construction. Together these yield — to our knowledge — the first exact Kakeya minima over composite moduli in any dimension, including K(6,3) = 65 = K(2,3)·K(3,3) and the closed formula K(N,2) = ∏p|N K(p,2) for every squarefree N, along with infinite families such as K(3M,3) = 13·K(M,3) for M coprime to 3, and two sixteen-point certificates extending exactness beyond squarefree moduli (K(4,2) = 10, hence K(4M,2) = 10·K(M,2) for odd M) and into dimension four (K(2M,4) = 6·K(M,4) for odd M). Exhaustive search computations, reported alongside but kept outside the formal development, further show the certificate fails at ℤ/8 and ℤ/9 — where the rings are strictly cheaper than the fields of the same order — locating K(72,2) as the smallest unknown plane cell. All theorems, certificates, and exact multiplicative values are formally verified in Lean 4 (kernel-checked, no native evaluation), with the complete development and independent Python verification scripts published in a companion repository.