We present a structural research program aimed at closing the UNIFORM barrier in
the abc conjecture without recourse to quantitative lower bounds from the theory of linear
forms in logarithms. Building on Trace Theory, we isolate the uniformity requirement for
bounded-radical points in a single arithmetic progression and propose a purely p-adic, multi-
adic incompatibility mechanism as a candidate closure.
The core construction involves subgroups Hq,n generated by S in (Z/qnZ)× and their
intersection with principal unit ltrations. We formulate two central lemmasmulti-adic in-
compatibility and arithmetic capacityand show how, if established, they imply UNIFORM
by a nite-state/nite-depth argument.
This paper does not claim a proof of the abc conjecture. It identies a precise structural
mechanism by which UNIFORM could be closed and formulates the exact lemmas required.