Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application

Abstract

Maximal-density factoring must often make one parent object carry several forms of information at once: child mass, two multiplicity levels, a density estimate, and the ability to survive later geometric refinements. These properties are not stable under arbitrary deletion. We introduce a cellular factoring method that places positive child mass on a weighted bipartite graph of labelled parents and half-open spatial cells. After regularizing edge weights and cell degrees, one edge set defines both the child shading and the parent shading. Compatibility and pointwise multiplicity bounds are then exact, while any later selection of whole parent-cell incidences lifts quantitatively back to the children. The combinatorial core does not require Euclidean geometry: it extends to finite measurable atomic partitions with comparable atom measures and admits an explicit finite-iteration loss. We combine this joint construction with a weighted planar incidence estimate, a resolution-limited cellwise angular selection, a labelled slab estimate, and a global synchronization of tubelet and fine-tube mass. The resulting rectangular parent estimate has shading exponent $2-\sigma$, which is sharp uniformly over the class considered here. With the stated GWZ factoring data and corresponding $K_F(\beta)$ or $K_{KT}(\beta)$, we obtain the two small-middle interfaces used in the reduction. For the very-non-sticky branch, assuming both $K_{KT}(\beta)$ and $K_F(\beta)$, the stated GWZ Lemma 9.1 hypotheses together with the source-uniformity and refinement interface (I1)--(I4) yield the terminal gain for the represented fine-tube input; this conditional application supplies the corresponding input to the surrounding reduction, while the sharp-convex-parent problem lies outside its scope.