THE FOLD / BOSS / SUDDEN DEATH / THE CONWAY SOLDIERS
THE CONWAY SOLDIERS
an army that cannot reach the fifth row
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Conway’s soldiers is a peg-jumping army with an invisible ceiling. Fill the entire half-plane below a line with checkers. Moves are checker jumps: a soldier leaps over a neighbour (removing it) into an empty square. How high above the line can any soldier ever reach? Rows 1–4: yes, with ever larger armies (2, 4, 8, 20 soldiers). Row 5: never — not with a million soldiers, not with the whole infinite half-plane. John Conway’s 1961 proof is a masterpiece: weight each square by σd where d is its distance to the target and σ = (√5-1)/2 satisfies σ²+σ = 1. Then no jump ever increases total weight — and the entire infinite army below the line weighs exactly 1, the target’s own weight. Any finite army weighs strictly less, so the target can never be paid for.
LIT verified live: σ²+σ = 1 to machine precision; the half-plane weight sum evaluates to exactly 1.000000000000; and all three jump classes are audited — toward-jumps preserve weight to ~1e-18, sideways and away-jumps strictly lose it (window.__conwaysoldiers). FIG no framing; the geometric sums and the move audit are computed independently in-browser.
LIT verified live: σ²+σ = 1 to machine precision; the half-plane weight sum evaluates to exactly 1.000000000000; and all three jump classes are audited — toward-jumps preserve weight to ~1e-18, sideways and away-jumps strictly lose it (window.__conwaysoldiers). FIG no framing; the geometric sums and the move audit are computed independently in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at sudden-death — the boss: every jump kills a soldier, and the fifth row is the round no army survives, however deep the bench. AVAN (AI) built the instrument: the golden-ratio weighting, the exact half-plane sum, and the three-way move audit.
Credit as content: John Horton Conway (1961; published in Berlekamp–Conway–Guy, ‘Winning Ways’). The weave: David names the unbeatable round; I confirm the whole army weighs exactly what the prize costs.
Credit as content: John Horton Conway (1961; published in Berlekamp–Conway–Guy, ‘Winning Ways’). The weave: David names the unbeatable round; I confirm the whole army weighs exactly what the prize costs.
3 ONE DIMENSION
The half-plane army below the line, weights fading as σᵈ — and the unreachable cell five rows up.
4 TWO DIMENSIONS · INTERACTIVE
Audit the ledger: the army's total, the target's price, and the three kinds of jump.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the army and its exact total worth, 1.
AVAN’s addition (the inverse-companion): don’t search the game tree — price the board. The inverse of ‘can any sequence reach row 5?’ is ‘a currency in which the whole world’s army equals the prize exactly, and every move pays tax’. Magenta is the target, priced at 1; green is the infinite army worth 1 in total. Infinity, one soldier short.
LIT Genuine Conway's soldiers / row-5 impossibility (John Horton Conway, 1961; Winning Ways). Verified live: σ²+σ = 1 to machine precision; the half-plane weight sum evaluates to exactly 1.000000000000; toward-jumps preserve weight to ~1e-18 while sideways and away-jumps strictly lose it (window.__conwaysoldiers.ok).
FIG No framing; the geometric sums and the move audit are computed independently in-browser. The AVAN inverse is honest — don't search the game tree, price the board: the inverse of 'can any sequence reach row 5?' is 'a currency in which the whole world's army equals the prize exactly, and every move pays tax'. Magenta is the target priced at 1; green is the infinite army worth 1 in total. Infinity, one soldier short.
FIG No framing; the geometric sums and the move audit are computed independently in-browser. The AVAN inverse is honest — don't search the game tree, price the board: the inverse of 'can any sequence reach row 5?' is 'a currency in which the whole world's army equals the prize exactly, and every move pays tax'. Magenta is the target priced at 1; green is the infinite army worth 1 in total. Infinity, one soldier short.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SUDDEN DEATH · David Lee Wise (ROOT0), with AVAN