Projective-geometry self-sum extensions

certified lower bound · dashed curve

Projective-geometry self-sums give characteristic-indexed lower-bound curves with the same two limiting regimes as the affine geometries.

\[ \lim_{d\to\infty} R_{ef}(P_{d,q}) = \frac{4q^2}{3q^2+1} \] \[ |E(P_{d,q})| = 2\tfrac{q^d-1}{q-1}+2, \qquad \operatorname{rk}(P_{d,q}) = 2d+1 \]

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 \(G=\operatorname{PG}(d-1,q)\), of rank \(d\) on \((q^d-1)/(q-1)\) points, and form \(P_{d,q}=(G\oplus G)+e\times f\). A field extension may be necessary to carry out the free extension and coextension.

What the curve shows

Möbius inversion on the projective-flat lattice gives every plotted value exactly; for fixed \(q\) the values tend to \(4q^2/(3q^2+1)\), for fixed \(d\) to \(4d/(3d+1)\), and a subsequent limit approaches \(\frac43\).

Proof status

The displayed pair is certified exactly; full maximization 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)

parameter	\|E\|	rank	R_ef	≈
rank(PG) = 3	16	7	10/9	1.11111
rank(PG) = 4	32	9	128/109	1.17431
rank(PG) = 5	64	11	70/59	1.18644
rank(PG) = 6	128	13	297/250	1.18800
rank(PG) = 8	512	17	7456/6275	1.18821

q = 3 (4 plotted members)

parameter	\|E\|	rank	R_ef	≈
rank(PG) = 3	28	7	36/31	1.16129
rank(PG) = 4	82	9	256/213	1.20188
rank(PG) = 5	244	11	1640/1351	1.21392
rank(PG) = 6	730	13	12528/10261	1.22093

q = 5 (3 plotted members)

parameter	\|E\|	rank	R_ef	≈
rank(PG) = 3	64	7	32/27	1.18519
rank(PG) = 4	314	9	2944/2417	1.21804
rank(PG) = 5	1564	11	3200/2593	1.23409

q = 7 (2 plotted members)

parameter	\|E\|	rank	R_ef	≈
rank(PG) = 3	116	7	180/151	1.19205
rank(PG) = 4	802	9	7296/5963	1.22355