Does Every Base Have a Hit? The Beal Conjecture ...

This article consolidates a long working discussion of the Beal conjecture (if Aˣ+Bʸ=Cᶻ with x,y,z≥3, then $A,B,C$ share a prime factor) organised around one idea: assume that every base A≥2 has at least one hit Aˣ+Bʸ=Cᶻ (x,y,z≥3), and use the least base without a hit as a boundary. We record: (i) the framework of coprime cores and the defect lattice that explains why 3³+6³=3⁵ behaves as it does; (ii) the radical bound and the valuation structure (carry law) of a core; (iii) fixed-term experiments and the proof that, for a prime base, hits correspond to three explicit types of coprime cores, giving each prime a set of exponents x that is a union of arithmetic progressions (example 5¹⁶+9375³=250⁵); (iv) the boundary table: every base 2≤ A≤13 has a hit, and $A=14$ is the first base for which no hit was found; (v) a detailed analysis of $A=14$: hits with a common prime reduce to seven explicit equations, and hits with no common prime are excluded in all cases except (y,z)=1 and one sub-case. Every statement carries a status tag. A corrections log lists everything retracted. No proof of the Beal conjecture is claimed, and the hit assumption is an exploration, not a theorem. The text is written so that a new discussion can resume without re-deriving anything.