THE FOLD / BOSS / THE CHOKE-POINT / THE KELLER
THE KELLER
true until dimension seven
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Tile space with identical cubes, any offsets you like. Keller conjectured in 1930 that some two cubes must share a complete face — you cannot stagger them all like brickwork forever. It is true in the plane, true in three dimensions, and true up to six. Then it dies. Lagarias and Shor found a counterexample in dimension 10 in 1992; Mackey reached dimension 8 in 2002; and dimension 7 held out until 2020, when Brakensiek, Heule, Mackey and Narvaez settled it with a SAT proof whose certificate ran to forty terabytes. Only n ≤ 6 survives. The whole question reduces to a graph: colour the points of {0,1,2,3}ⁿ, join two when they differ by 2 in some coordinate and differ in at least two coordinates, and a clique of size 2ⁿ is exactly a counterexample tiling.
LIT verified live: the Keller graph is constructed from its definition for n = 2 and n = 3, its regularity confirmed (every vertex has the same degree, as vertex-transitivity demands), and a maximum-clique search run exhaustively — the largest cliques are 2 and 5, both short of the 2ⁿ = 4 and 8 needed. No counterexample exists in these dimensions, exactly as the surviving part of the theorem says (window.__keller).
LIT verified live: the Keller graph is constructed from its definition for n = 2 and n = 3, its regularity confirmed (every vertex has the same degree, as vertex-transitivity demands), and a maximum-clique search run exhaustively — the largest cliques are 2 and 5, both short of the 2ⁿ = 4 and 8 needed. No counterexample exists in these dimensions, exactly as the surviving part of the theorem says (window.__keller).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-choke-point — the boss: for ninety years everything funnelled through one plausible statement about stacking boxes, and the answer turned out to depend on which dimension you are standing in — true, true, true, true, true, true, then false forever. AVAN (AI) built the instrument: the {0,1,2,3}ⁿ vertex generator, the Keller adjacency rule, and the branch-and-bound clique search.
Credit as content: Ott-Heinrich Keller (1930); Oskar Perron (1940, n ≤ 6 partial); Jeffrey Lagarias & Peter Shor (1992, dimension 10); John Mackey (2002, dimension 8); Debroni et al. (2011, n = 6 confirmed); Brakensiek, Heule, Mackey & Narvaez (2020, dimension 7). Only n = 2 and 3 are recomputed here; the rest is cited. The weave: David names the choke point; I build the graph from its definition and find no clique big enough.
Credit as content: Ott-Heinrich Keller (1930); Oskar Perron (1940, n ≤ 6 partial); Jeffrey Lagarias & Peter Shor (1992, dimension 10); John Mackey (2002, dimension 8); Debroni et al. (2011, n = 6 confirmed); Brakensiek, Heule, Mackey & Narvaez (2020, dimension 7). Only n = 2 and 3 are recomputed here; the rest is cited. The weave: David names the choke point; I build the graph from its definition and find no clique big enough.
3 ONE DIMENSION
Where the conjecture lives and where it dies, dimension by dimension.
4 TWO DIMENSIONS · INTERACTIVE
The Keller graph at n = 2: 16 vertices, and the clique that isn’t there.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: staggered cubes, the brickwork that has to break — until it doesn’t.
AVAN’s addition (the inverse-companion): don’t trust intuition that was trained in three dimensions. The inverse of ‘this is obviously true’ is ‘obvious where?’ — Keller holds in every dimension a human can picture and fails in every dimension a human cannot. Our sense of the possible was fitted to n = 3. Magenta is dimension 7 and beyond, where the staggering never has to stop; green is the low country where the conjecture is a theorem. Geometric intuition is a local instrument, and nobody labels its range.
LIT Verified live: the Keller graph is constructed from its definition for n=2 and n=3, its regularity confirmed (uniform degree, as vertex-transitivity demands), and maximum-clique search run exhaustively — largest cliques are 2 and 5, short of the 2ⁿ = 4 and 8 a counterexample needs. No counterexample in these dimensions, exactly as the surviving theorem says (window.__keller.ok).
FIG Only n=2 and n=3 are recomputed here; dimensions 6–10 are cited, not reproduced. Keller 1930, Perron 1940, Lagarias–Shor 1992, Mackey 2002, Debroni et al. 2011, Brakensiek–Heule–Mackey–Narvaez 2020 credited. The AVAN inverse — ask 'obvious WHERE?': Keller holds in every dimension a human can picture and fails in every dimension a human cannot. Geometric intuition is a local instrument with no range label.
DEAD Keller's 1930 cube-tiling conjecture. FALSE from dimension 7 upward. It survived ninety years partly because every dimension anyone could visualise happens to be one where it is true.
FIG Only n=2 and n=3 are recomputed here; dimensions 6–10 are cited, not reproduced. Keller 1930, Perron 1940, Lagarias–Shor 1992, Mackey 2002, Debroni et al. 2011, Brakensiek–Heule–Mackey–Narvaez 2020 credited. The AVAN inverse — ask 'obvious WHERE?': Keller holds in every dimension a human can picture and fails in every dimension a human cannot. Geometric intuition is a local instrument with no range label.
DEAD Keller's 1930 cube-tiling conjecture. FALSE from dimension 7 upward. It survived ninety years partly because every dimension anyone could visualise happens to be one where it is true.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE CHOKE-POINT · David Lee Wise (ROOT0), with AVAN