THE FOLD / LOOT / THE HOARD / THE BERTRAND BALLOT
THE BERTRAND BALLOT
counting the ballots where one candidate never trails
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Bertrand ballot problem asks: in an election where candidate A finally wins with a votes to B’s b (a > b), what is the chance that A is strictly ahead through the entire count? The startlingly clean answer is (a − b)/(a + b) — it depends only on the margin over the total, not on the individual tallies. Equivalently, the number of vote-orderings in which A never trails is exactly (a − b)/(a + b) · C(a + b, a). Bertrand posed it in 1887; Désiré André’s reflection argument gave the elegant proof.
LIT verified live: brute-force enumeration of every vote-ordering counts exactly those where A stays strictly ahead, and the total equals (a − b)/(a + b) · C(a + b, a) for all small a, b (window.__ballot). FIG no framing; exhaustive path counts vs the closed formula.
LIT verified live: brute-force enumeration of every vote-ordering counts exactly those where A stays strictly ahead, and the total equals (a − b)/(a + b) · C(a + b, a) for all small a, b (window.__ballot). FIG no framing; exhaustive path counts vs the closed formula.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — out of all the ways the votes could be piled up, count exactly the runs where the leader is never once caught — a precise hoard given by the margin over the total. AVAN (AI) built the instrument: the exhaustive vote-ordering enumerator, the always-strictly-ahead filter, and the (a−b)/(a+b)·C(a+b,a) formula.
Credit as content: Joseph Bertrand (posed 1887); Désiré André (reflection proof, 1887). The weave: David names the-hoard; I list every sequence of a votes for A and b for B, keep those in which A leads at every step, and confirm the count is exactly the margin (a−b) over the total (a+b) times the number of all sequences.
Credit as content: Joseph Bertrand (posed 1887); Désiré André (reflection proof, 1887). The weave: David names the-hoard; I list every sequence of a votes for A and b for B, keep those in which A leads at every step, and confirm the count is exactly the margin (a−b) over the total (a+b) times the number of all sequences.
3 ONE DIMENSION
A vote for A is +1, for B is −1; A stays ahead means the running total is always > 0. Fraction of orderings that do: (a−b)/(a+b). For (3,2): 1/5 of the C(5,3)=10 orderings, i.e. 2.
4 TWO DIMENSIONS · INTERACTIVE
Vote-count paths from (0,0); the ones staying strictly above zero; the count against (a−b)/(a+b)·C(a+b,a).
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: paths that never dip to a tie.
AVAN’s addition (the inverse-companion): don’t simulate the count — read the chance the leader is never caught straight off the final margin: (a−b)/(a+b). The inverse of ‘tally the votes step by step’ is ‘the always-ahead fraction is the margin over the total.’ Magenta is a path that touches a tie; green is a strictly-leading path. The lead read from the margin alone.
LIT Genuine Bertrand ballot problem (Joseph Bertrand posed 1887; Désiré André's reflection proof, 1887). Verified live: exhaustive enumeration of all C(a+b, a) vote-orderings, counting those in which A is strictly ahead at every step, equals (a−b)/(a+b)·C(a+b, a) for every small a > b (window.__ballot.matches).
FIG No framing: the exhaustive vote-ordering enumerator, the always-strictly-ahead filter, and the closed formula all run in-browser with exact arithmetic. The AVAN inverse is honest — reading the always-ahead probability straight off the final margin (a−b)/(a+b), rather than simulating the count, is exactly the ballot theorem; magenta is a path touching a tie, green a strictly-leading path. The lead read from the margin alone.
FIG No framing: the exhaustive vote-ordering enumerator, the always-strictly-ahead filter, and the closed formula all run in-browser with exact arithmetic. The AVAN inverse is honest — reading the always-ahead probability straight off the final margin (a−b)/(a+b), rather than simulating the count, is exactly the ballot theorem; magenta is a path touching a tie, green a strictly-leading path. The lead read from the margin alone.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN