THE FOLD / SPAWN / CHECKPOINT-ZERO / THE WEYL
THE WEYL
an irrational stride fills the interval evenly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Weyl’s equidistribution theorem says that stepping around a circle by an irrational stride visits every region equally often. Take an irrational α and the sequence of fractional parts {α}, {2α}, {3α}, …: as you take more terms, the proportion landing in any subinterval [a, b] converges to exactly its length b − a. The points never repeat and never settle into a pattern — they spread out perfectly uniformly. For a rational stride the sequence cycles through finitely many spots and fails utterly to equidistribute.
LIT verified live: for five irrationals and several intervals, the fraction of {nα} in [a, b] matches b − a to within 0.01 over 200,000 terms, while a rational stride 1/5 does not equidistribute (window.__weyl). FIG no framing; the fractional parts and interval counts run in-browser.
LIT verified live: for five irrationals and several intervals, the fraction of {nα} in [a, b] matches b − a to within 0.01 over 200,000 terms, while a rational stride 1/5 does not equidistribute (window.__weyl). FIG no framing; the fractional parts and interval counts run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at checkpoint-zero — start at zero and take irrational steps, and you touch every checkpoint on the circle in fair proportion. AVAN (AI) built the instrument: the fractional-part sequence, interval counting, and the rational counterexample.
Credit as content: Hermann Weyl (1916). The weave: David names checkpoint-zero; I confirm the irrational orbit fills every interval in proportion to its length, and that a rational stride does not.
Credit as content: Hermann Weyl (1916). The weave: David names checkpoint-zero; I confirm the irrational orbit fills every interval in proportion to its length, and that a rational stride does not.
3 ONE DIMENSION
The first terms of {nα} for an irrational α, dropped onto [0,1): they land everywhere, filling gaps as they go — never clustered.
4 TWO DIMENSIONS · INTERACTIVE
Pick a stride; the histogram of {nα} flattens to uniform for irrationals, and the fraction in any interval approaches its length.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the irrational orbit filling the circle evenly.
AVAN’s addition (the inverse-companion): don’t track where each step lands — count how the intervals fill. The inverse of ‘iterate nα and watch’ is ‘every interval [a,b] receives a share b−a, because α is irrational.’ Magenta is a rational stride that clusters; green is the irrational orbit that spreads. Irrational fills evenly.
LIT Genuine Weyl equidistribution theorem (Hermann Weyl, 1916). Verified live: for irrationals √2, φ, π, e−2, √3 over four intervals each, the fraction of {nα} in [a,b] matches b−a to within 0.01 across 200,000 terms; a rational stride 1/5 does not equidistribute (window.__weyl.equidistributes, .rationalFails).
FIG No framing: the fractional-part sequence and interval counting run in-browser. The AVAN inverse is honest — instead of tracking where each step lands, one asks how the intervals fill; because α is irrational, every [a,b] receives a share exactly b−a. Magenta is a rational stride that clusters; green is the irrational orbit that spreads evenly. Irrational fills evenly.
FIG No framing: the fractional-part sequence and interval counting run in-browser. The AVAN inverse is honest — instead of tracking where each step lands, one asks how the intervals fill; because α is irrational, every [a,b] receives a share exactly b−a. Magenta is a rational stride that clusters; green is the irrational orbit that spreads evenly. Irrational fills evenly.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of CHECKPOINT-ZERO · David Lee Wise (ROOT0), with AVAN