Hidden Norms in Nested Radicals

This work develops a unified arithmetic–geometric framework for the exact denesting of nested radicals. Rather than treating radical identities as isolated symbolic coincidences, the paper interprets denesting as a structured power-membership problem in radical extensions. For an inner sum u=a+b, with a^m=A and b^m=B, the framework begins with the sign-sensitive relative norm [ C=A-(-1)^mB, ] and the corresponding full norm [ NL/K(a+b)=C^m. ] The paper proves that if a+b is a k-th power in the relevant radical field, then every relative norm to an intermediate field must itself be a k-th power. This separates a necessary norm obstruction from the stronger question of exact power membership in L^×/(L^×)^k. The framework is then developed through several increasingly restrictive stages: [ norm compatibility ;⊇; elimination compatibility ;⊇; lifting ;⊇; genuine denesting. ] For explicit cubic families, the paper derives a rational-square parametrization and an associated normalized elimination quartic [ 48p^4+64Rp^3+24Bp^2-B^2=0. ] A rational root of this quartic is shown to be only an elimination-visible candidate: reverse square lifting, parameter reconstruction, branch conditions, and substitution into the original radical identity are required before a genuine denesting certificate is obtained. This leads to the distinction between genuine and phantom certificates. The paper further develops a square-root interface valid for arbitrary positive outer exponent k, investigates higher-index Galois-orbit obstructions under explicit Kummer-independence hypotheses, and incorporates exceptional fifth-root identities into the same structural perspective. The resulting viewpoint connects symbolic radical denesting with Kummer theory, arithmetic norm equations, Diophantine elimination, rational lifting, Galois-orbit structure, and arithmetic geometry. The work also proposes a reproducible computational certificate programme and formulates a distinct-direction uniqueness conjecture for future investigation. The central principle is that denesting is not established merely by factorization or norm compatibility: each candidate must survive the successive arithmetic, algebraic, lifting, and exact identity tests required by the framework.