Parameterized constructions
Families
The curves on the Explore chart, as objects: each family is a parameterized construction with a proved formula. A solid curve is a proved value of the full invariant; a dashed curve certifies a distinguished-pair lower bound. Full proofs live on the mathematics page. New families need human proof review and are contributed by pull request, not by the submission form.
proved exact · solid curve
Huh–Schröter–Wang finite-field spikes
\(\overline{\alpha}(M_p^5) = \frac{8}{7}\)
A rank-5 family over \(\operatorname{GF}(p)\) whose correlation constant is exactly \(\frac{8}{7}\) for every displayed prime \(p\).
certified lower bound · dashed curve
Uniform self-sum extensions
\(R_{ef}(P_r) = \frac{4r^2}{3r^2+2r+1}\)
A finite matroid for every integer \(r\ge2\), with a certified distinguished-pair value tending to \(\frac{4}{3}\) from below.
certified lower bound · dashed curve
Affine-geometry self-sum extensions
\(\lim_{d\to\infty} R_{ef}(P_{d,q}) = \frac{4q^2}{3q^2+1}\)
Affine-geometry self-sums give characteristic-indexed lower-bound curves whose iterated limit is \(\frac{4}{3}\).
certified lower bound · dashed curve
Projective-geometry self-sum extensions
\(\lim_{d\to\infty} R_{ef}(P_{d,q}) = \frac{4q^2}{3q^2+1}\)
Projective-geometry self-sums give characteristic-indexed lower-bound curves with the same two limiting regimes as the affine geometries.
certified lower bound · dashed curve
Steiner-system sparse-paving extensions
\(R_{ef}(P_{r,n}) = \frac{4r(n-r+1)}{(3r+1)(n-r)+4r+2}\)
Sparse-paving matroids from Steiner systems give nonuniform 2-design bases and distinguished-pair values approaching \(\frac43\).