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

This article consolidates a long working discussion of the Beal conjecture (if $A^x+B^y=C^z$ with $x,y,z\ge3$, then $A,B,C$ share a prime factor) organised around one idea: \emph{assume that every base $A\ge2$ has at least one hit} $A^x+B^y=C^z$ ($x,y,z\ge3$), 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 $3^3+6^3=3^5$ 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 $5^{16}+9375^3=250^5$); (iv) the boundary table: every base $2\le A\le13$ has a hit, and \textbf{$A=14$ 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 $\gcd(y,z)=1$ and one sub-case. Every statement carries a status tag. A corrections log lists everything retracted. \emph{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.