Structural Closure of UNIFORM via Multi-adic Incompatibility: A Conditional Framework for the abc Conjecture

We present a structural research program aimed at closing the UNIFORM barrier in the abc conjecture without recourse to Baker–Matveev transcendental bounds. Building on Trace Theory, we isolate the uniformity requirement for bounded-radical points in arithmetic progressions and propose a purely p-adic, multi-adic incompatibility mechanism as a candidate closure. The core construction defines subgroups H_{q,n} generated by a finite prime set S in (Z/q^n Z)× and their intersection with principal unit filtrations U_{q,n}. We introduce the multi-adic depth invariant κ(u) and formulate two central lemmas: (1) Multi-adic incompatibility—no infinite sequence of S-units can have κ → ∞; (2) Arithmetic capacity—some prime q|M must act as a bottleneck. We prove that if these lemmas hold, UNIFORM follows, which implies the abc conjecture. Computational verification (Appendix A) demonstrates that κ remains bounded in tested cases, supporting the structural prediction. This paper does not claim a proof of abc. It identifies the precise structural mechanism by which UNIFORM could be closed and provides a bridge between elementary reduction, IUT's abstract height gap, and explicit arithmetic invariants.