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) = ∏₏|₍ 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.