THE FOLD / SPAWN / COLD BOOT / THE MARKOV TRIPLE
THE MARKOV TRIPLE
a Diophantine equation whose solutions grow on a tree
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Markov triples are the positive-integer solutions of the Markov equation x2 + y2 + z2 = 3xyz. The smallest is (1, 1, 1), then (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), … and every one is reachable from (1,1,1) by Vieta jumping: fixing two coordinates, the equation is a quadratic in the third whose two roots sum to 3xy, so (x, y, z) → (x, y, 3xy − z) hops to another solution. All triples form an infinite binary tree. The numbers that appear — 1, 2, 5, 13, 29, 34, 89, … — are the Markov numbers.
LIT verified live (exact BigInt): every triple generated by Vieta jumping from (1,1,1) satisfies x2+y2+z2 = 3xyz, the known Markov numbers all appear, and jumping the same coordinate twice returns the original (window.__markov). FIG honest: the famous uniqueness conjecture (each Markov number is the largest of exactly one triple) is still open.
LIT verified live (exact BigInt): every triple generated by Vieta jumping from (1,1,1) satisfies x2+y2+z2 = 3xyz, the known Markov numbers all appear, and jumping the same coordinate twice returns the original (window.__markov). FIG honest: the famous uniqueness conjecture (each Markov number is the largest of exactly one triple) is still open.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at cold-boot — the whole infinite tree of solutions boots from a single seed, (1,1,1), each jump spawning two new triples. That cold start from one triple is the boot. AVAN (AI) built the instrument: the Markov-equation check in exact integers, the Vieta-jump neighbours, the tree traversal, and the involution test.
Credit as content: Andrey Markov (1879); Vieta jumping is the classical descent behind it. The weave: David names cold-boot; I start from (1,1,1), replace one coordinate by 3xy − z to jump to a neighbouring solution, and confirm every triple in the tree solves the equation — while noting the uniqueness conjecture remains unproven.
Credit as content: Andrey Markov (1879); Vieta jumping is the classical descent behind it. The weave: David names cold-boot; I start from (1,1,1), replace one coordinate by 3xy − z to jump to a neighbouring solution, and confirm every triple in the tree solves the equation — while noting the uniqueness conjecture remains unproven.
3 ONE DIMENSION
(1,1,1): 1+1+1 = 3 = 3·1·1·1. Jump z: 3·1·1 − 1 = 2 → (1,1,2). Jump again: 3·1·2 − 1 = 5 → (1,2,5). Markov numbers: 1, 2, 5, 13, 29, 34, 89, …
4 TWO DIMENSIONS · INTERACTIVE
The Markov tree grown by Vieta jumps; each node checked against the equation.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a whole tree of solutions from one seed.
AVAN’s addition (the inverse-companion): don’t search for solutions of x2+y2+z2=3xyz — jump between them: the equation is quadratic in each variable, so its two roots let you hop from any solution to a neighbour. The inverse of ‘solve the Diophantine equation’ is ‘Vieta-jump from one solution to the next, forever.’ Magenta is a lone solution; green is the infinite tree the jumps unfold. Solutions that breed solutions.
LIT Genuine Markov triples (Andrey Markov, 1879; via classical Vieta jumping). Verified live with exact BigInt: every triple generated by Vieta jumping from (1,1,1) to depth 8 satisfies x²+y²+z²=3xyz (window.__markov.treeValid), the known Markov numbers 1,2,5,13,29,34,89,169,194,233 all appear (window.__markov.knownAppear), and jumping the same coordinate twice is an involution (window.__markov.involution).
FIG No framing: the Markov-equation check, the Vieta-jump neighbours, the tree traversal, and the involution test all run in-browser in exact integers. Honest scope: the famous uniqueness conjecture (each Markov number is the largest coordinate of exactly one triple) remains open — stated, not claimed proven. The AVAN inverse is honest — Vieta-jumping between solutions (rather than searching for them) genuinely unfolds the whole solution tree from one seed; magenta is a lone solution, green the infinite tree. Solutions that breed solutions.
FIG No framing: the Markov-equation check, the Vieta-jump neighbours, the tree traversal, and the involution test all run in-browser in exact integers. Honest scope: the famous uniqueness conjecture (each Markov number is the largest coordinate of exactly one triple) remains open — stated, not claimed proven. The AVAN inverse is honest — Vieta-jumping between solutions (rather than searching for them) genuinely unfolds the whole solution tree from one seed; magenta is a lone solution, green the infinite tree. Solutions that breed solutions.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of COLD BOOT · David Lee Wise (ROOT0), with AVAN