ABC Conjecture via Shadow Inflation and Frey Curve Heights (Conditional on BSD)

This paper establishes a conditional proof of the ABC conjecture by combining unconditional 'Shadow Inflation' bounds (Zsigmondy's Theorem on 2-power towers) with the geometry of Frey Elliptic Curves.Key Contributions:Shadow Inflation (Verified): We prove unconditionally that 2-power sequences generate primitive divisor chains satisfying (logH)2 growth. (Coq-verified, 0 admits).The Shadow-ABC Gap: We explicitly prove that shadow primes do not divide abc directly, resolving a common misconception in elementary approaches.The Frey Bridge: We utilize the Height-Discriminant inequality on Ea,b,c​ to bridge the shadow entropy to the radical, conditional on the Birch-Swinnerton-Dyer (BSD) conjecture.