On Radical Identities Induced by Positional Concatenation

This mathematical note presents the formal derivation of a class of numerical identities, where the sum of an integer and a square root equals the square root of their positional concatenation. By representing concatenation as a radix-weighted sum, we identify a fundamental linear generator: X + 2sqrt (Y) = Bᵏ. An algebraic formulation of positional concatenation reduces a nonlinear radical identity to a simple linear relation, allowing for a complete classification of solutions. The result provides a concisedemonstration of how positional structure imposes rigid algebraic constraints. This formulation applies uniformly across positional bases and produces explicit parametric families of identities. Numerical examples for bases 2 and 10 are provided.