{"id":"palomar/PALOMAR-2026-10-28-000025","url":"https://github.com/jfairfaxball-348/Pan-Disciplinary-Invariance-Conjecture/tree/54cb32b0400e000990db37b83e805a3f44cb13cc","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-28-000025-v1.json","repo":"https://github.com/jfairfaxball-348/Pan-Disciplinary-Invariance-Conjecture","tags":["math.CO","msc:05C69","msc:05C80"],"score":0,"title":"Zero-Sum Quartet Incidence and Sharp Maximal-Independent-Set Witnesses on Eight Vertices","trust":"high","domain":"math.CO","source":"palomar","authors":["John Fairfax-Ball"],"reviews":0,"subject":"math.CO","summary":"We prove an exact four-term probability law for zero-sum maximal-independent-set witnesses. For every connected simple graph G on eight vertices and every admissible eight-point cap U in F_3^3, sample uniformly from all 40,320 bijections of the vertices onto U. Let N0 count zero-sum maximal independent four-sets, m count the size-four maximal independent sets, n0 and n2 count their unordered pairs intersecting in zero and two vertices, and h count compatible unordered triples. Put M1 = 2m/35, M2 = 2n0/35 + n2/315, and M3 = h/315. Then P(N0=0) = 1-M1+M2-M3, P(N0=1) = M1-2M2+3M3, P(N0=2) =…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-28","reactions":{},"formalized":true,"formalizes":[{"title":"Zero-Sum Quartet Incidence and Sharp Maximal-Independent-Set Witnesses on Eight Vertices (working manuscript)","identifier":"","relationship":"formalizes"}]}