A dimension-independent box bound from dual total correlation

Let S be a finite nonempty set in a product of d coordinate spaces, and let US be its uniform law. We prove that some nonempty Cartesian box B satisfies |S B|≤ 10DTC(US)|S|, where dual total correlation is computed using natural logarithms. The constant does not depend on the dimension or on the coordinate alphabets. The proof follows actual coordinate slices: their average disagreement is bounded linearly by mutual information, while an exact entropy identity retains the deficit available within each slice. Selecting one slice with both costs controlled makes the induction close with the same constant. The resulting Loomis--Whitney stability estimate has coefficient $20(d-1)$ under the usual normalized near-equality assumption. A two-point example shows that linear dependence on d is necessary in that normalization. The constants are not claimed sharp. This deposit contains the preprint and its LaTeX source package, including one explanatory vector figure generated within the source. The manuscript gives the full proof and an appendix comparing the estimate with an earlier argument. AI assistance is disclosed in the manuscript. Version 1.0 is prepared for journal submission; it is not a peer-reviewed or accepted publication.