matroid #8 (Uniform self-sum extension P_5) — Matroid Correlation Constants

matroid #8 — Uniform self-sum extension P_5

n: 22
rank: 11
bases: 426,888
α(M): 50/43 ≈ 1.1628

status: current record for this ground-set size
discovered by: Family: Alexander Divoux and Shouda Wang; explicit realization for this site
discovery date: 2026-08
AI use: yes — ChatGPT 5.6 Sol Ultra; Claude Fable 5 (via Claude Code): ChatGPT 5.6 Sol Ultra assisted the discovery of the family; Claude Fable 5 constructed and certified the explicit GF(999983) realization.
submitted by: David Renshaw
submitted at: 2026-08-05 18:26:38 UTC

Description (matrix) GF(999983)

[[1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,846092,0],
 [1,2,3,4,5,6,7,8,9,10,0,0,0,0,0,0,0,0,0,0,377526,0],
 [1,4,9,16,25,36,49,64,81,100,0,0,0,0,0,0,0,0,0,0,941861,0],
 [1,8,27,64,125,216,343,512,729,1000,0,0,0,0,0,0,0,0,0,0,834328,0],
 [1,16,81,256,625,1296,2401,4096,6561,10000,0,0,0,0,0,0,0,0,0,0,808370,0],
 [0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,210943,0],
 [0,0,0,0,0,0,0,0,0,0,1,2,3,4,5,6,7,8,9,10,10913,0],
 [0,0,0,0,0,0,0,0,0,0,1,4,9,16,25,36,49,64,81,100,851404,0],
 [0,0,0,0,0,0,0,0,0,0,1,8,27,64,125,216,343,512,729,1000,133900,0],
 [0,0,0,0,0,0,0,0,0,0,1,16,81,256,625,1296,2401,4096,6561,10000,392180,0],
 [150143,432992,481556,380945,429741,747649,610445,194433,430170,276094,65440,570291,733318,406539,462041,917250,8050,407824,673913,960238,244547,1]]

The full basis list is too large to store; recompute it from the description above.

Maximizing pair

(20, 21); basis counts at (i, j) = (20, 21):

	`j` ∈ `B`	`j` ∉ `B`
`i` ∈ `B`	105,840	151,704
`i` ∉ `B`	63,504	105,840

Notes

P_5 = (U(5,10) + U(5,10)) + e x f: the r=5 member of the uniform self-sum extension family (R_ef = 4r^2/(3r^2+2r+1) -> 4/3), found by Alexander Divoux and Shouda Wang with the aid of ChatGPT 5.6 Sol Ultra, August 2026; its reinterpretation as a double free extension is due to humans. Explicit realization over GF(999983): two Reed-Solomon blocks for the uniform summands, a free-extension column e (element 20) and a free-coextension row with unit column f (element 21) chosen at random, with genericity certified post hoc: the verified 426,888 bases and the cell counts (105840, 63504, 151704, 105840) at the unique maximizing pair (e, f) match the construction's counting identity exactly, so alpha(P_5) = R_ef(P_5) = 50/43 on the nose. Constructed and verified for this site with AI assistance (Claude Fable 5 via Claude Code).

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