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. Proofs live on the mathematics page or behind each family’s own proof link.
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 \(\frac43\).
proved exact · solid curve
Sparse paving extension
\(R_{ef}(P_r) = \frac{4}{2 + \frac{\binom{2r}{r}^2 - \left(\lceil\frac{1}{2r}\binom{2r}{r}\rceil\right)^2}{\binom{2r}{r-1}\binom{2r}{r+1}}}\)
\(P_r\) is the free coextension of free extension of \(M\oplus M\), where \(M\) is a sparse paving matroid of rank \(r\) on \(2r\) elements with a lot of circuit hyperplanes.
Propose a family
Unlike concrete matroids, a family claim — a formula for every parameter, a supremum — cannot be verified by enumeration; it needs a proof and human review. Propose one with the submission form (switch it to “infinite family”): the parameter domain, the construction, the exact curve values, the claimed supremum, a proof or stable reference, and the discovery record with an AI-use declaration. Proposals wait as pending until a site admin reviews the mathematics and accepts them — nothing is automatic — and an accepted family gains its curve, card, and page together.
A strong proposal comes with evidence: verify your family’s small members through the concrete form first — each one becomes a machine-checked record entry reviewers can lean on, as the uniform self-sum family's members \(P_2,\ldots,P_5\) do.