Soft Density Profiles Do Not Determine Vacant Percolation

For any packing of unit balls in three-dimensional Euclidean space, we construct two fixed modifications preserving its upper and lower soft-density profiles at every fixed outer radius. One modification has an unbounded vacant component at every finite radius. The other has infinitely many nonempty bounded vacant components, and no unbounded one, at every radius at least 2/sqrt(3). Below this radius the vacancy of every unit-core packing is path connected and unbounded. The constructions use a widening empty corridor and lacunary triangulated octahedral barriers, whose fixed-width neighborhoods have volume O(R^2) in an observation ball of radius R. An elementary attainment argument then gives percolating and nonpercolating unrestricted upper-density optimizers at every radius at or above the threshold.

This is a complete scoped three-dimensional theorem related to AIM Soft Packings Problem 1.4 and AIM-GEOMETRY-0111 in UnsolvedMath v1.6.0. It does not settle periodic, saturated, stationary or higher-dimensional versions, determine FCC/BCC optimality, or prove the existence of a fully path-connected optimizer above the threshold. Classical density-insensitivity and triangular-throat geometry are credited to prior literature. No identical whole-profile modification theorem was located in a bounded review; absolute priority is not certified.

AI-assisted, self-audited and unrefereed preprint. No independent peer review or proof-assistant verification is claimed. Exact finite regression programs accompany the source; they do not replace the general written proof.