Hits, Misses and Cores: A Structural Study of the Beal Conjecture ...

This article continues an earlier compilation on the Beal conjecture (if $A^x+B^y=C^z$ with $x,y,z\ge3$ then $A,B,C$ share a prime). Its organising idea is to replace density arguments by \emph{local} (vertical) structure: for a fixed base, a fixed triple, or a fixed coprime identity (\emph{core}), decide exactly when a hit exists and why not. We record: (i) the hit/miss partition of triples, the failure of the converse of Beal, a finiteness theorem for the exponent set of a fixed triple (infinite only for powers of $2$), the prime-support law, and root reduction; (ii) a complete case analysis of three prime bases; (iii) a mechanical \emph{placement sieve} for the primes of a base, with the count $2^{r+1}-1$ of live placements for squarefree bases and its dependence on valuations otherwise; (iv) the walls for $14,20,21,22,23$, now with an explicit valuation-gcd criterion that kills every candidate core found; (v) the inversion of the problem into a \emph{list of cores}, its out-of-sample behaviour, closed forms for the two main families (differences of powers, $M^s+1$), the exponent-pair attribution of the tail and the growth law $N^{1/z+1/y}$; (vi) a Euclid-type theorem: no finite list of hit-hosting cores is final, with a closed-form extension step, and the observation that the same generators cannot extend a list of \emph{rigid} cores (all faces exponent $\ge3$). Every statement carries a status tag. \emph{No proof of the Beal conjecture is claimed; the final list of rigid cores is conjecturally empty, and this is exactly Beal.} A corrections log lists everything retracted.