Extension of the two-tier schema to all natural numbers

Ascertain whether the two-tier algebraic schema of first-tier nests A, B1, B2 and second-tier shifted-square nests accounts for all natural numbers a by yielding solutions to A^4 + a B^4 = C^4 + a D^4 for every a ∈ N.

Background

Beyond empirical coverage for small a, the authors question whether the schema might, in principle, generate solutions for every natural number a.

This extends the bounded-interval coverage question to a global one over N, linking their constructions to longstanding conjectural universality.