Exclusion of Axially Weighted Regular-Tetrahedral Four-Filter Configurations in a Five-Source Quantum Minimax Problem

For five pure-qubit Bloch sources and the ratio between optimal pure-filter and relaxed Bloch-ball product yields, we rule out every positive four-filter Karush–Kuhn–Tucker configuration whose filter directions form a regular tetrahedron with one distinguished axial weight and three equal remaining weights. The proof classifies ten admissible source-stationary directions, exhaustively enumerates 252 five-source subsets, and excludes the remaining transverse-balance cases through exact polynomial resultants and rational sign certificates. Independent symbolic and integer verification scripts accompany the manuscript. This excludes one continuous family of necessary stationarity configurations; it does not settle the global five-source filtering optimum or exclude arbitrary four-active geometries. Scholarly priority and independent peer review remain unconfirmed.