The Beal Conjecture: A Structure-First Investigation, ...

This document consolidates a long working discussion of the Beal conjecture: if $A^x+B^y=C^z$ with $A,B,C$ positive integers and $x,y,z\ge 3$, then $A,B,C$ share a prime factor. The discussion followed one methodological principle: treat the six free quantities $(A,B,C,x,y,z)$ through a single structure, and explain the smallest instance ($3^3+6^3=3^5$) reproducibly instead of searching. We record (i) exact reductions and equivalent formulations, (ii) case studies ($(3,3,3)$, $(3,3,4)$, $(3,3,r)$, the square term, three distinct odd primes), (iii) the Catalan--Mihailescu proof as a model of free exponents tamed by structure, (iv) Frey-curve face-control, and (v) the current state of Darmon's program, with literature citations. Every statement carries a status tag (proved in the literature, derived here, computed here, heuristic, or recalled from memory). A corrections log lists every mistake made and retracted during the discussion. \emph{No new proof of the Beal conjecture is claimed.} The document is written so that a new discussion can resume without re-deriving or re-verifying anything.