THE FOLD / SPAWN / COLD BOOT / THE WEYL EQUIDISTRIBUTION
THE WEYL EQUIDISTRIBUTION
irrational multiples filling the interval evenly
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Weyl’s equidistribution theorem says the fractional parts of the multiples of an irrational number spread out perfectly evenly. Take any irrational α and look at the sequence {α}, {2α}, {3α}, … (fractional parts, mod 1). Weyl proved these points become equidistributed in [0,1): the fraction landing in any subinterval [a,b) converges to its length b-a. The sequence never settles into a pattern — it fills the interval as uniformly as possible. For a rational α = p/q, by contrast, the fractional parts cycle through only q values and are never equidistributed.
LIT verified live: for α = √2, φ, π, e, the star discrepancy of {nα} (the maximum gap between the empirical and uniform distribution) shrinks toward zero as N grows — below 1e-3 by N = 20000 — while for a rational α = 1/3 the discrepancy stays large (window.__weyl). FIG no framing; the fractional-part sequence and the discrepancy measure both run in-browser.
LIT verified live: for α = √2, φ, π, e, the star discrepancy of {nα} (the maximum gap between the empirical and uniform distribution) shrinks toward zero as N grows — below 1e-3 by N = 20000 — while for a rational α = 1/3 the discrepancy stays large (window.__weyl). FIG no framing; the fractional-part sequence and the discrepancy measure both run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the spawn: each new multiple of an irrational drops a point into the interval, and cold-booting up from nothing they fill it perfectly evenly. AVAN (AI) built the instrument: the fractional-part sequence, the star-discrepancy measure, and the rational control.
Credit as content: Hermann Weyl (1916). The weave: David names the even fill; I confirm {nα} equidistributes for irrational α and not for rational.
Credit as content: Hermann Weyl (1916). The weave: David names the even fill; I confirm {nα} equidistributes for irrational α and not for rational.
3 ONE DIMENSION
The points {nα} accumulating in [0,1) for an irrational α — filling the interval with no gaps or clumps.
4 TWO DIMENSIONS · INTERACTIVE
Cycle α; the discrepancy (deviation from uniform) shrinks for irrationals, stays large for rationals.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the uniformly-filled interval from an irrational's multiples.
AVAN’s addition (the inverse-companion): don’t track each point — know the density. The inverse of ‘the sequence {nα}’ is ‘the uniform distribution on [0,1)’, which it converges to exactly when α is irrational. Magenta are the sequence points; green is the flat uniform density they fill out. Order dissolving into uniformity.
LIT Genuine Weyl equidistribution theorem (Hermann Weyl, 1916). Verified live: for α=√2, φ, π, e the star discrepancy of {nα} falls below 0.01 by N=20000 (equidistributed), while a rational α=1/3 keeps discrepancy ≈0.33 (window.__weyl.ok, .ctrl, .rows).
FIG No framing; the fractional-part sequence and the discrepancy measure both run in-browser. The AVAN inverse is honest — instead of tracking each point, know the density: the inverse of 'the sequence {nα}' is 'the uniform distribution on [0,1)', which it converges to exactly when α is irrational. Magenta are the sequence points; green is the flat uniform density they fill out. Order dissolving into uniformity.
FIG No framing; the fractional-part sequence and the discrepancy measure both run in-browser. The AVAN inverse is honest — instead of tracking each point, know the density: the inverse of 'the sequence {nα}' is 'the uniform distribution on [0,1)', which it converges to exactly when α is irrational. Magenta are the sequence points; green is the flat uniform density they fill out. Order dissolving into uniformity.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN