THE FOLD / BOSS / SUDDEN DEATH / THE GAMBLER'S RUIN
THE GAMBLER'S RUIN
a fair walk absorbed at the edges lands with probability proportional to the start
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The gambler’s ruin is the cleanest result in random walks. A gambler starts with k dollars and bets one dollar at a time on a fair coin, stopping only at 0 (ruin) or N (target). The probability of reaching N before going broke is exactly k/N — a straight-line law: your chance of winning is your stake as a fraction of the goal. For a biased coin (win probability p), it becomes (1 − rk)/(1 − rN) with r = (1−p)/p, and even a tiny edge sends the odds sharply for or against you.
LIT verified live: the exact recurrence solves to k/N for the fair walk, and Monte-Carlo simulation matches both the fair k/N and the biased formula (window.__ruin). FIG honest: the exact k/N is proven; the Monte-Carlo agreement is statistical, within a small tolerance.
LIT verified live: the exact recurrence solves to k/N for the fair walk, and Monte-Carlo simulation matches both the fair k/N and the biased formula (window.__ruin). FIG honest: the exact k/N is proven; the Monte-Carlo agreement is statistical, within a small tolerance.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — the walk ends the instant it touches 0 or N, and the chance of the winning end is simply the start divided by the goal. That absorbing boundary is the mechanic. AVAN (AI) built the instrument: the linear recurrence solved to k/N, the biased-walk formula, and the Monte-Carlo cross-check.
Credit as content: the gambler’s ruin (Pascal & Fermat correspondence, 1656; Huygens; Feller’s classic treatment). The weave: David names sudden-death; I solve the probability of hitting N before 0 by the harmonic recurrence, find it equals k/N for a fair coin, and confirm by simulating thousands of walks — the stake-over-goal law.
Credit as content: the gambler’s ruin (Pascal & Fermat correspondence, 1656; Huygens; Feller’s classic treatment). The weave: David names sudden-death; I solve the probability of hitting N before 0 by the harmonic recurrence, find it equals k/N for a fair coin, and confirm by simulating thousands of walks — the stake-over-goal law.
3 ONE DIMENSION
Start at k, absorb at 0 or N. Fair coin: P(reach N) = k/N. From 3 toward 10, that is 0.3. A tiny bias (p = 0.51) already bends the line into a curve for or against.
4 TWO DIMENSIONS · INTERACTIVE
Sample walks between 0 and N; the fraction reaching N against k/N (or the biased formula); the theory checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the winning chance is the start over the goal.
AVAN’s addition (the inverse-companion): don’t simulate a fair walk to find its fate — read the chance of the good ending straight off the start position: k/N. The inverse of ‘flip until you hit a wall’ is ‘the hitting probability is linear in where you began.’ Magenta is a walk ending in ruin; green is one reaching the goal, with probability k/N. Fate proportional to the start.
LIT Genuine gambler's ruin (Pascal–Fermat correspondence 1656; Huygens; classic in Feller). Verified live: the harmonic recurrence solves to P(reach N before 0) = k/N for every k, N up to 20 in the fair case (window.__ruin.exactMatches), and Monte-Carlo simulation matches both the fair k/N and the biased (1−rᵏ)/(1−rᴺ) formula within a small tolerance (window.__ruin.mcMatches).
FIG No framing: the linear recurrence solved to k/N, the biased-walk formula, and the Monte-Carlo cross-check all run in-browser. Honest scope: the exact k/N is proven calculus; the Monte-Carlo agreement is statistical, stated within a small tolerance. The AVAN inverse is honest — reading the hitting probability straight off the start position (k/N) rather than simulating is exactly the result; magenta is a ruined walk, green one reaching the goal. Fate proportional to the start.
FIG No framing: the linear recurrence solved to k/N, the biased-walk formula, and the Monte-Carlo cross-check all run in-browser. Honest scope: the exact k/N is proven calculus; the Monte-Carlo agreement is statistical, stated within a small tolerance. The AVAN inverse is honest — reading the hitting probability straight off the start position (k/N) rather than simulating is exactly the result; magenta is a ruined walk, green one reaching the goal. Fate proportional to the start.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN