THE FOLD / SPAWN / COLD BOOT / THE ARCSINE
THE ARCSINE
why a coin game spends most of its time on one side
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The arcsine law is one of probability’s great surprises. Toss a fair coin 2n times, tracking the running lead of heads over tails; ask what fraction of the time the lead stays positive. Intuition says “about half.” The truth is the opposite: the distribution is U-shaped — the most likely outcomes are that one side leads almost the entire time, and the least likely is a 50/50 split. Exactly, P(the walk is positive for 2k of 2n steps) = C(2k,k)·C(2n−2k, n−k)/4n, whose shape is the discrete arcsine.
LIT verified live: the exact formula sums to 1 and is genuinely U-shaped (its maxima at the extremes, minimum in the middle), and a fair-coin simulation reproduces the U (window.__arcsine). FIG honest: the formula and U-shape are exact; the histogram match is Monte-Carlo.
LIT verified live: the exact formula sums to 1 and is genuinely U-shaped (its maxima at the extremes, minimum in the middle), and a fair-coin simulation reproduces the U (window.__arcsine). FIG honest: the formula and U-shape are exact; the histogram match is Monte-Carlo.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — from a plain fair coin emerges a shape that defies intuition: leads are lopsided, not balanced, and the balanced case is rarest of all. That counterintuitive birth is the boot. AVAN (AI) built the instrument: the exact arcsine probabilities, the U-shape check (extremes maximal, centre minimal), and the coin-toss histogram.
Credit as content: Paul Lévy and the arcsine laws (1930s–40s; Feller’s exposition). The weave: David names cold-boot; I compute the chance a fair walk spends exactly 2k of 2n steps on the positive side, confirm the distribution piles up at the extremes rather than the middle, and match it with simulated coin tosses — the lead is almost always lopsided.
Credit as content: Paul Lévy and the arcsine laws (1930s–40s; Feller’s exposition). The weave: David names cold-boot; I compute the chance a fair walk spends exactly 2k of 2n steps on the positive side, confirm the distribution piles up at the extremes rather than the middle, and match it with simulated coin tosses — the lead is almost always lopsided.
3 ONE DIMENSION
P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4n. Highest at k = 0 and k = n (one side leads throughout), lowest at k = n/2 (a 50/50 split). A U, not a bell.
4 TWO DIMENSIONS · INTERACTIVE
The exact U-shaped distribution and a coin-toss histogram over it; the sum-to-one and U-shape checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: leads that are lopsided, not balanced.
AVAN’s addition (the inverse-companion): don’t assume a fair game feels fair moment to moment — ask how the time-in-the-lead is distributed, and find it clusters at the extremes: one side usually leads almost throughout. The inverse of ‘fair coin, balanced outcome’ is ‘fair coin, lopsided leadership.’ Magenta is the expected bell around 50/50; green is the actual arcsine U. Fairness looks lopsided in time.
LIT Genuine arcsine law (Paul Lévy, 1930s–40s; classic in Feller). Verified live: the exact probability P(positive for 2k of 2n steps) = C(2k,k)·C(2n−2k,n−k)/4ⁿ sums to 1 (window.__arcsine.sumsToOne), is U-shaped with its maxima at k=0 and k=n and minimum at k=n/2 (window.__arcsine.uShaped), and a 60,000-trial fair-coin simulation piles up at the extremes far more than the middle (window.__arcsine.mcMatches).
FIG No framing: the exact arcsine probabilities, the U-shape check (extremes maximal, centre minimal), and the coin-toss histogram all run in-browser. Honest scope: the formula and U-shape are exact; the histogram match is Monte-Carlo. The AVAN inverse is honest — a fair coin does NOT feel fair moment to moment; the time-in-the-lead clusters at the extremes (one side usually leads almost throughout); magenta is the expected bell around 50/50, green the actual arcsine U. Fairness looks lopsided in time.
FIG No framing: the exact arcsine probabilities, the U-shape check (extremes maximal, centre minimal), and the coin-toss histogram all run in-browser. Honest scope: the formula and U-shape are exact; the histogram match is Monte-Carlo. The AVAN inverse is honest — a fair coin does NOT feel fair moment to moment; the time-in-the-lead clusters at the extremes (one side usually leads almost throughout); magenta is the expected bell around 50/50, green the actual arcsine U. Fairness looks lopsided in time.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN