The Pisano Doubling Equations π(m) = 2m and π(m2 ) = 2m: One Skeleton, Two Fates

We resolve two adjacent Diophantine equations on the Pisano period completely and unconditionally: π(m²) = 2m holds only at m = 12 (equivalently, N = 144 is the unique positive integer with π(N) = 2√N), and π(m) = 2m holds exactly on the infinite family m = 12·5ᶜ. Both proofs share a single structural core — a valuation analysis forcing the {2,3}-part of any solution to be exactly 12 — and diverge at one prime: 5, the unique prime dividing its own Pisano period, fits the first equation's target exactly but overflows the second's by a factor of 5ᶜ. Theorem A closes an open problem from The Golden Geodesic in fully general form. We then show Theorem B is the Fibonacci instance of a general Lucas-sequence phenomenon: self-supplying primes are exactly the primes dividing the discriminant, 2 is never a tower prime, and a 2-adic budget conjecture governing which self-supplying primes participate is tested with a four-for-four predictive record on unseen sequences, including one correctly predicted-empty case. Applied to π(m) = km, this substantially reduces the general multiple equation for Fibonacci. Both theorems are Wall-free, taking no position on Wall's conjecture despite operating exactly where it lives. All results verified by exact integer arithmetic; full computational appendix included.