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≥ 2d+3 and\[κ=1-b/a-b∉ Z,\]then there is an explicit ε₀>0 such that no n points of Sᵈ⁻¹ can have every off-diagonal inner product within ε₀ of one of the two prescribed values a and b.