A Determinant Stability Principle for Approximate Spherical Two-Distance Sets

We isolate an elementary determinant obstruction to concentrating more than half of the spectrum of an integer symmetric matrix near a prescribed nonintegral number. Applied to the adjacency matrix associated with an approximate spherical two-distance set, it gives a quantitative stability form of the integrality mechanism behind the Larman--Rogers--Seidel theorem. In particular, if $n\ge 2d+3$ and\[\kappa=\frac{1-b}{a-b}\notin\mathbb Z,\]then there is an explicit $\varepsilon_0>0$ such that no $n$ points of $S^{d-1}$ can have every off-diagonal inner product within $\varepsilon_0$ of one of the two prescribed values $a$ and $b$.