A Structure for the Beal Conjecture
Year: 2026
Type: preprint
Abstract: This manuscript records a structural program for the Beal conjecture whose guiding principle is that infinity is a direction, not a destination: computation can verify finite windows, while any statement about infinitely many cases must come from a structure. We prove an exact \emph{seed theorem}: every solution of $A^x+B^y=C^z$, coprime or not, is a coprime primitive triple $u+v=w$ scaled by a common factor, and the exponents determine which terms of the seed are constrained. We connect this to the geometry of the exponent triple (spherical, flat, hyperbolic), to a ``cube--cell'' model in which an exponent below $3$ collapses one dimension to the lattice unit (a \emph{sheet}), and to a flat-case descent read as a lattice scaling. We develop the author's \emph{start} approach: fix $A^x$ and ask for all hits. We define \emph{channels}, prove that a start has at most one hit per channel, derive closed forms for the main channels, and give a complete classification of collisions that share two terms (the \emph{Chain Lemma}), together with data on all other collisions. A map of exponent triples shows where known theorems have leverage and where the structure is silent (the pairwise coprime triples such as $(3,4,5)$). A final section introduces a \emph{phase number} of an exponent triple and sets out the architecture that an argument by contradiction would have to follow, including the point where it is incomplete. Every statement is labelled as known, derived, empirical, heuristic or open; the manuscript asserts no verdict on the conjecture and is offered for assessment.
Author: Tantisukarom, Chaiya
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225144666