A Structural Classification Framework for 3-Digit Fixed Points of a Hybrid Reversal-Plus-Digit-Square Operator in Arbitrary Bases

We study fixed points of the hybrid operator FB(N)=revB(N)+ΣB(N) in arbitrary positional bases. The 2-digit case admits a complete closed parametric classification. For 3-digit states, we prove the structural reduction a−c=2, derive the quadratic relation B2−x2−2y2=2, and obtain the branch equation 2rs−1=y2 together with the modular condition y2≡−1(mod2r) and an exact digit-admissibility threshold. Every admissible residue branch yields infinitely many distinct fixed points in distinct bases. The manuscript also defines collision fibers for multiple fixed points in a single base. Computational results are clearly separated from the mathematical proofs, and the remaining collision and higher-digit questions are stated as open problems.