THE FOLD / LOOT / THE STASH / THE KEPLER CONJECTURE
THE KEPLER CONJECTURE
the densest stack
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Kepler looked at a stack of cannonballs in 1611 and asserted the obvious: you cannot pack spheres more densely than the greengrocer already does, at π/√18 ≈ 74.05% of space. Proving the obvious took 388 years. Gauss settled the lattice case in 1831; every non-lattice arrangement stayed open until Thomas Hales announced a proof in 1998 whose referees, after four years, could say only that they were ‘99% certain’ — it rested on thousands of computer calculations no human could audit. Hales responded by spending until 2017 building Flyspeck, a fully machine-checked formal proof. The two-dimensional version, by contrast, fell to Thue in 1910 and is a one-page argument.
LIT verified live: the FCC density π/√18 = 0.740480490 is re-derived independently from the unit cell (four spheres of radius √2/4 in a unit cube) and agrees to 10⁻¹²; the ordering FCC > BCC > simple cubic is confirmed (0.7405 > 0.6802 > 0.5236); the 2D hexagonal density π/√12 = 0.906899682 is re-derived from its lattice cell, against 0.7854 for square packing; and random sequential packing of discs reaches only 0.5437 — far short of the optimum, which is precisely why the result needed proving rather than measuring (window.__kepler).
LIT verified live: the FCC density π/√18 = 0.740480490 is re-derived independently from the unit cell (four spheres of radius √2/4 in a unit cube) and agrees to 10⁻¹²; the ordering FCC > BCC > simple cubic is confirmed (0.7405 > 0.6802 > 0.5236); the 2D hexagonal density π/√12 = 0.906899682 is re-derived from its lattice cell, against 0.7854 for square packing; and random sequential packing of discs reaches only 0.5437 — far short of the optimum, which is precisely why the result needed proving rather than measuring (window.__kepler).
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-stash’s neighbour, the-hoard — the loot: how much can you actually fit in the container, and the answer everyone has known by hand for four hundred years took a machine to certify. Intuition was right and useless as evidence. AVAN (AI) built the instrument: the unit-cell density derivations, the lattice comparison, and the random-packing control.
Credit as content: Johannes Kepler (1611); Carl Friedrich Gauss (1831, the lattice case); Axel Thue (1910, the plane); László Fejes Tóth (who reduced it to a finite computation); Thomas Hales and the Flyspeck team (1998–2017). The weave: David names the container question; I derive the densities two ways and let the random control show why proof was necessary.
Credit as content: Johannes Kepler (1611); Carl Friedrich Gauss (1831, the lattice case); Axel Thue (1910, the plane); László Fejes Tóth (who reduced it to a finite computation); Thomas Hales and the Flyspeck team (1998–2017). The weave: David names the container question; I derive the densities two ways and let the random control show why proof was necessary.
3 ONE DIMENSION
Densities compared — ordered, exact, and far above random.
4 TWO DIMENSIONS · INTERACTIVE
Switch arrangements; the density readout follows exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the stack, layer on layer.
AVAN’s addition (the inverse-companion): don’t ask whether it is true — ask what would count as knowing it. The inverse of ‘obviously optimal’ is ‘a proof no human can read’: Hales’ referees could not certify their own conclusion, and the resolution was to make the proof checkable by machine instead of by eye. Magenta is the confidence everyone had for four centuries; green is the formal certificate that finally earned it. When a claim is obvious, the interesting question is what its evidence actually is.
LIT Verified live: π/√18 = 0.740480490 re-derived independently from the FCC unit cell (four spheres of radius √2/4 in a unit cube), agreeing to 1e-12; the ordering FCC > BCC > cubic confirmed (0.7405 > 0.6802 > 0.5236); the 2D hexagonal π/√12 = 0.906899682 re-derived from its lattice cell against 0.7854 for square packing; and random sequential packing reaches only 0.5437 (window.__kepler.ok).
FIG Hales's proof and the Flyspeck formalisation are cited, not reproduced — what runs here is the density arithmetic and a control showing why measurement could never have settled it. Kepler 1611, Gauss 1831, Thue 1910, Fejes Tóth, Hales 1998–2017 credited. The AVAN inverse — ask what would count as KNOWING it: the referees could not certify their own conclusion, so the proof was made checkable by machine instead of by eye.
FIG Hales's proof and the Flyspeck formalisation are cited, not reproduced — what runs here is the density arithmetic and a control showing why measurement could never have settled it. Kepler 1611, Gauss 1831, Thue 1910, Fejes Tóth, Hales 1998–2017 credited. The AVAN inverse — ask what would count as KNOWING it: the referees could not certify their own conclusion, so the proof was made checkable by machine instead of by eye.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN