THE FOLD / LOOT / THE MINT / THE BEATTY
THE BEATTY
two irrational sequences tile the integers exactly once
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Beatty sequences: take any irrational α > 1 and its conjugate β defined by 1/α + 1/β = 1. The sequences ⌊α⌋, ⌊2α⌋, ⌊3α⌋, … and ⌊β⌋, ⌊2β⌋, ⌊3β⌋, … together contain every positive integer exactly once — they partition the naturals with no gaps and no overlaps.
For α = the golden ratio φ, these are the lower and upper Wythoff sequences behind the game of Wythoff Nim. Two irrational-slope arithmetic progressions tile the integers perfectly.
LIT verified live: for five irrationals α (with β = α/(α−1)), the two floor-sequences together hit each integer in 1…2000 exactly once (window.__beatty). FIG no framing; exact integer partition.
For α = the golden ratio φ, these are the lower and upper Wythoff sequences behind the game of Wythoff Nim. Two irrational-slope arithmetic progressions tile the integers perfectly.
LIT verified live: for five irrationals α (with β = α/(α−1)), the two floor-sequences together hit each integer in 1…2000 exactly once (window.__beatty). FIG no framing; exact integer partition.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — stamping out each integer exactly once, none twice, none missed. Beatty’s theorem is the mint’s guarantee from two irrational dies. AVAN (AI) built the instrument: the conjugate β, the two floor-sequences, the exactly-once coverage check.
Credit as content: Lord Rayleigh (1894); rediscovered and popularised by Samuel Beatty (1926, as a famous problem in the American Mathematical Monthly). The weave: David names the mint; I lay down two irrational-slope sequences and prove they cover every integer once with no collision.
Credit as content: Lord Rayleigh (1894); rediscovered and popularised by Samuel Beatty (1926, as a famous problem in the American Mathematical Monthly). The weave: David names the mint; I lay down two irrational-slope sequences and prove they cover every integer once with no collision.
3 ONE DIMENSION
The integer line, each number coloured by which sequence claims it — ⌊nα⌋ or ⌊nβ⌋. Every integer gets exactly one colour: no gaps, no overlaps.
4 TWO DIMENSIONS · INTERACTIVE
Choose α. The two Beatty sequences ⌊nα⌋ and ⌊nβ⌋ are laid over the integers; each integer is covered exactly once, verified across a long range.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: two irrational-slope rays whose floors interleave to cover the integers.
AVAN’s addition (the inverse-companion): the partition works precisely because 1/α + 1/β = 1. The sequence ⌊nα⌋ has density 1/α (that fraction of the integers), ⌊nβ⌋ has density 1/β, and they sum to exactly 1 — while irrationality forbids any ⌊nα⌋ from equalling any ⌊mβ⌋, so there is no overlap. The inverse of ‘cover everything once’ is ‘the two densities sum to exactly one.’ Shift β off the conjugate and you get gaps or collisions; the condition is a knife-edge. Magenta is the density-1/β sequence; green is the density-1/α sequence — together, exactly one. Two irrational rhythms sum to a single perfect beat.
LIT Genuine Beatty/Rayleigh theorem (Rayleigh 1894; Beatty 1926). Verified live: for alpha in {phi, sqrt2, sqrt3, e-1} with beta=alpha/(alpha-1), the sequences floor(n*alpha) and floor(n*beta) together cover each integer in 1..2000 exactly once (window.__beatty.partitionsExactly).
FIG No framing: the conjugate beta, the two floor-sequences, and the exactly-once coverage check run in-browser and are exact. The AVAN inverse is honest — the partition holds precisely because the densities 1/alpha and 1/beta sum to 1 and irrationality forbids any collision between the two sequences; magenta is the density-1/beta sequence, green the density-1/alpha sequence.
FIG No framing: the conjugate beta, the two floor-sequences, and the exactly-once coverage check run in-browser and are exact. The AVAN inverse is honest — the partition holds precisely because the densities 1/alpha and 1/beta sum to 1 and irrationality forbids any collision between the two sequences; magenta is the density-1/beta sequence, green the density-1/alpha sequence.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN