Matroid Correlation
Constants

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certified lower bound · dashed curve

Affine-geometry self-sum extensions

Affine-geometry self-sums give characteristic-indexed lower-bound curves whose iterated limit is \(\frac{4}{3}\).

parameter

integer \(d\ge2\) and prime power \(q\)

supremum

supremum 4/3 — an iterated limit

discovered by

Chris Eur

discovery date

August 2026

Construction

Let \(A=\operatorname{AG}(d-1,q)\), of rank \(d\) on \(q^{d-1}\) points, and form \(P_{d,q}=(A\oplus A)+e\times f\). The free extension and coextension may require passing to a finite extension field, but the result stays representable in characteristic \(p\) when \(q=p^a\).

What the curve shows

Möbius inversion on the affine-flat lattice gives every plotted value exactly. For fixed \(q\) they tend to \(4q^2/(3q^2+1)\) as \(d\to\infty\); for fixed \(d\) they tend to \(4d/(3d+1)\) as \(q\to\infty\). Either subsequent limit approaches \(\frac43\). The curves are dashed because they certify distinguished-pair lower bounds rather than the full maximum.

Proof status

Exact finite counts and both limits are proved on the mathematics page; maximization over every pair remains open. Full proof on the mathematics page →

AI disclosure

Chris Eur used ChatGPT 5.6 Sol Max on Codex as a mathematical exploration and formula-checking aid while identifying the family through the double-free construction and the common 2-design mechanism. A public chat link is not currently recorded.

Plotted members

q = 2 (5 plotted members)

q = 3 (3 plotted members)

q = 5 (2 plotted members)

q = 7 (2 plotted members)