Finiteness of planar central configurations of six bodies for all positive masses: Part I, exceptional-mass first limits

Abstract: We separate the generic positive-mass finiteness problem for planar six-body central configurations from finiteness on exceptional mass loci. The Chang--Chen diagram enumeration has 117 graphs. The arguments below confine 40 first-limit diagram obligations to proper necessary mass loci. In particular, a direct function-field argument establishes the No.~66 mass condition for both zero and nonzero drift. We establish the algebraic reduction from an infinite positive family to a germ with an unbounded seven-product, and use the stated diagram rules and computational certificates to confine its mass vector to the union of these loci. Finiteness on the exceptional positive mass loci, the proposed subject of Part II, will be discussed in another paper.