Multi-adic Approximation Prohibition and the Structural Closure of UNIFORM: A Qualitative Route to the abc Conjecture

We present a structural proof of the uniform finiteness (UNIFORM) property underlying the abc conjecture. The core mechanism is a purely qualitative obstruction called Multi-adic Approximation Prohibition (MAPN): elements generated by a fixed finite set of primes cannot penetrate arbitrarily deeply into principal unit filtrations simultaneously for all primes dividing a fixed modulus. Our approach replaces quantitative lower bounds from the theory of linear forms in logarithms by a lattice-theoretic argument based on p-adic linearization and covolume blow-up. The key lemma shows that the index [Z^d : L_n] grows exponentially as depth n increases, forcing rapid thinning of the solution lattice incompatible with infinite depth-ascent sequences. As consequences, UNIFORM holds, and hence the abc conjecture is structurally closed. No use is made of Baker–Matveev bounds, the Leopoldt conjecture, or the Subspace Theorem.