Symbolic Generation and Modular Embedding of High-Quality abc-Triples

We present a symbolic identity for generating integer triples $(a,b,c)$ satisfying $a+b=c$, inspired by structural features of the \emph{abc conjecture}. The construction uses powers of $2$ and $3$ in combination with modular inversion in $\mathbb{Z}\mathrm{/}{3}^p\mathbb{Z}$, leading to a parametric identity with residue constraints that yield abc-triples exhibiting low radical values. Through affine transformations, these symbolic triples are embedded into a broader space of high-quality examples, optimised for the ratio $\log c\mathrm{/}\log \mathrm{rad}(abc)$. Computational results demonstrate the emergence of structured, radical-minimising candidates, including both known and novel triples. These methods provide a symbolic and algebraic framework for controlled triple generation, and suggest exploratory implications for symbolic entropy filtering in cryptographic pre-processing.