THE FOLD / BOSS / SUDDEN DEATH / THE BEST THEOREM
THE BEST THEOREM
Eulerian circuits counted by a determinant
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The BEST theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte) counts the Eulerian circuits of a directed graph — closed trails using every edge exactly once — with a single formula. For a connected Eulerian digraph (every vertex has equal in- and out-degree), the number of Eulerian circuits is ec(G) = tw(G) · ∏v (deg⁺(v) - 1)!, where tw(G) is the number of spanning arborescences (in-trees) rooted at any vertex w — itself a determinant, via the Matrix-Tree theorem. So an exponential count of tangled circuits collapses into one determinant times some factorials.
LIT verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixing the starting edge) exactly equals tw(G)·∏v(deg⁺(v)-1)!, with tw computed as a cofactor determinant of the graph Laplacian (window.__best). FIG no framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree.
LIT verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixing the starting edge) exactly equals tw(G)·∏v(deg⁺(v)-1)!, with tw computed as a cofactor determinant of the graph Laplacian (window.__best). FIG no framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — the boss encounter: an exponential thicket of Eulerian circuits, tamed in one blow by a determinant and a product of factorials. AVAN (AI) built the instrument: the brute Eulerian-circuit enumeration, the arborescence cofactor, and the BEST formula.
Credit as content: N. G. de Bruijn, T. van Aardenne-Ehrenfest, C. A. B. Smith, W. T. Tutte (the ‘BEST’ initials). The weave: David names the boss; I confirm the circuit count equals the arborescence determinant times ∏(deg-1)!.
Credit as content: N. G. de Bruijn, T. van Aardenne-Ehrenfest, C. A. B. Smith, W. T. Tutte (the ‘BEST’ initials). The weave: David names the boss; I confirm the circuit count equals the arborescence determinant times ∏(deg-1)!.
3 ONE DIMENSION
A small Eulerian digraph (every in-degree = out-degree) with one Eulerian circuit traced through it.
4 TWO DIMENSIONS · INTERACTIVE
Cycle graphs; the brute Eulerian-circuit count is compared to t_w(G)·∏(deg⁺−1)!.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the number of Eulerian circuits.
AVAN’s addition (the inverse-companion): don’t enumerate the circuits — count the trees. The inverse of ‘how many Eulerian circuits?’ is ‘tw(G)·∏(deg-1)!’ — a spanning-arborescence determinant times factorials. Magenta is the digraph; green is the Eulerian-circuit count the determinant yields. Exponential circuits from one determinant.
LIT Genuine BEST theorem (de Bruijn, van Aardenne-Ehrenfest, Smith, Tutte). Verified live: for several small Eulerian digraphs, a brute enumeration of Eulerian circuits (fixed starting edge) exactly equals t_w(G)·∏_v(deg⁺(v)−1)!, with t_w the arborescence cofactor determinant of the Laplacian — e.g. bidirected K₃ gives 3 (window.__best.ok).
FIG No framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree. The AVAN inverse is honest — instead of enumerating the circuits, count the trees: the inverse of 'how many Eulerian circuits?' is 't_w(G)·∏(deg−1)!' — a spanning-arborescence determinant times factorials. Magenta is the digraph; green is the Eulerian-circuit count the determinant yields. Exponential circuits from one determinant.
FIG No framing; the brute circuit count and the determinant-times-factorials formula both run in-browser and agree. The AVAN inverse is honest — instead of enumerating the circuits, count the trees: the inverse of 'how many Eulerian circuits?' is 't_w(G)·∏(deg−1)!' — a spanning-arborescence determinant times factorials. Magenta is the digraph; green is the Eulerian-circuit count the determinant yields. Exponential circuits from one determinant.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN