Polybrutus: Exact Two-Parameter Unlabeled-Distance Deformations and Local Affine-Rank Transition Curves

We study an exact two-parameter deformation of the canonical seven-point Polybrutus configuration. A fixed bijection between the 21 edges of the original three-dimensional point set and a distinct seven-point distance assignment yields a six-by-six Gram matrix, whose normalized determinant has exactly 66 nonzero rational monomials in epsilon and q = sqrt(3) eta. Exact matrix calculations certify a nonempty positive-definite region of affine rank six and two smooth local positive-semidefinite boundary curves of affine rank five, including rational root isolation, tangent slopes, and a determinant-positive indefinite counterexample. This is a specific, reproducible Euclidean distance-geometry construction, not a classification of all edge assignments. Research status: Independent mathematical preprint, not peer-reviewed; no priority over the world literature, physical interpretation or Higgs-boson claim is made. The supplied PDF and source code contain full proof certificates and reproducible checks. The manuscript is CC BY 4.0 and the original Python scripts are MIT licensed. Earlier Polybrutus records: affine ranks 3-6, 3D/4D study, original model.