Local Witnessed Descent and the Nonlocality of Strong Regularity

This paper studies the limits of local refinement methods in discrete structure theorems through a precise local-information refinement model. We formalize a no-silent-interval principle, under which global structure emerges via forced local descent, and show that this principle governs efficient results such as weak (spectral) graph regularity. We then prove a complementary negative result for strong (Szemerédi-type) regularity. Although strong irregularity witnesses exist globally, they are not extractable from the block-density interface available to local refinement algorithms until exponential resolution is reached. Under nested constructions, this leads to unavoidable tower-type complexity along any such refinement path. The contribution is not a new regularity bound, but a structural explanation of the polynomial–tower dichotomy in regularity theory in terms of witness extractability under a natural algorithmic interface. The results are proved using explicit constructions and an operational blindness argument, and do not contradict classical lower bounds, but rather explain their inevitability. This work is self-contained and field-internal. The abstract notion of witness-based descent used here was introduced independently in an earlier framework paper, cited below for provenance only.