THE FOLD / LOOT / THE JACKPOT / THE CHAKRAVALA
THE CHAKRAVALA
crank a cycle to crack an ancient equation
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The chakravala method is a cyclic algorithm from 12th-century India (Bhaskara II, building on Brahmagupta) that solves Pell’s equation x2 - N·y2 = 1 in integers. Starting from a rough triple (a, b, k) with a2 - N·b2 = k, it repeatedly composes with (m, 1) using Brahmagupta’s identity, choosing m at each turn so that k divides a + b·m and |m2 - N| is smallest. The value k spirals down toward ±1, and when it lands the current (a, b) is the fundamental solution. It is centuries ahead of its time — a self-correcting descent that European mathematics did not match until Fermat and Lagrange.
LIT verified live (exact BigInt): for every non-square N from 2 to 120 the method returns integers (x, y) with x2 - N·y2 exactly 1 — including the notorious N = 61, whose smallest solution is x = 1766319049 (window.__chakravala). FIG no framing; the cyclic composition, the m-selection, and the exact integer check all run in-browser with arbitrary-precision integers.
LIT verified live (exact BigInt): for every non-square N from 2 to 120 the method returns integers (x, y) with x2 - N·y2 exactly 1 — including the notorious N = 61, whose smallest solution is x = 1766319049 (window.__chakravala). FIG no framing; the cyclic composition, the m-selection, and the exact integer check all run in-browser with arbitrary-precision integers.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-jackpot — a rare, enormous payout: turn a crank enough times and a single equation coughs up a solution thousands of digits wide from tiny inputs. AVAN (AI) built the instrument: the BigInt cyclic method, the modular m-selection, the Brahmagupta finisher, and the exact x2 - N·y2 = 1 check.
Credit as content: Brahmagupta (628) & Bhaskara II (1150); the method named chakravala (‘the wheel’). The weave: David names the jackpot; I confirm the wheel lands on x2 - N·y2 = 1 for every non-square N tested.
Credit as content: Brahmagupta (628) & Bhaskara II (1150); the method named chakravala (‘the wheel’). The weave: David names the jackpot; I confirm the wheel lands on x2 - N·y2 = 1 for every non-square N tested.
3 ONE DIMENSION
The hyperbola x² - N·y² = 1: integer solutions are lattice points on it; the chakravala wheel finds the smallest.
4 TWO DIMENSIONS · INTERACTIVE
Cycle through N; the wheel returns the fundamental (x, y) and the exact check x² - N·y² = 1 — watch the solutions explode in size.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fundamental solution point on the hyperbola.
AVAN’s addition (the inverse-companion): don’t search for each solution — breed them. The inverse of ‘find one solution’ is ‘compose it with itself by Brahmagupta’s identity to get the next, forever.’ Magenta is the composed second solution; green is the fundamental. One jackpot seeds infinitely many.
LIT Genuine chakravala cyclic method (Brahmagupta 628; Bhaskara II 1150). Verified live with exact BigInt arithmetic: for every non-square N in 2..120 the cyclic composition returns (x,y) with x²-N·y²=1 exactly, including N=61 → x=1766319049, y=226153980 (window.__chakravala.solvesAll, .tested).
FIG No framing; the cyclic composition, the modular m-selection, the Brahmagupta finisher, and the exact integer check all run in-browser with arbitrary-precision integers. The AVAN inverse is honest — instead of searching for each solution, compose one with itself by Brahmagupta's identity to breed the next, forever. Magenta is the composed second solution; green is the fundamental. One jackpot seeds infinitely many.
FIG No framing; the cyclic composition, the modular m-selection, the Brahmagupta finisher, and the exact integer check all run in-browser with arbitrary-precision integers. The AVAN inverse is honest — instead of searching for each solution, compose one with itself by Brahmagupta's identity to breed the next, forever. Magenta is the composed second solution; green is the fundamental. One jackpot seeds infinitely many.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN