THE FOLD / LOOT / THE DROP / THE LAGRANGE FOUR-SQUARE
THE LAGRANGE FOUR-SQUARE
every whole number is four squares
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Lagrange’s four-square theorem says every non-negative integer is the sum of four squares: n = a² + b² + c² + d². Three squares is not enough — 7 and 15 cannot be written with three — but four always suffice, no exceptions, forever. It is a startling completeness: the squares 0,1,4,9,16,… are sparse, yet any four of them (with repeats) can be tuned to hit every whole number exactly.
LIT verified live: for every n from 0 to 3000 a representation a²+b²+c²+d² = n is found and its sum checked to equal n exactly (window.__lagrange). FIG no framing; exact integer search, no gaps.
LIT verified live: for every n from 0 to 3000 a representation a²+b²+c²+d² = n is found and its sum checked to equal n exactly (window.__lagrange). FIG no framing; exact integer search, no gaps.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-drop — whatever number you throw at it, four squares always fall out that rebuild it exactly. Lagrange’s theorem is that guaranteed drop. AVAN (AI) built the instrument: the nested-square search, the sum-of-two-squares inner step, and the a²+b²+c²+d²=n check across a full range.
Credit as content: Joseph-Louis Lagrange (1770); Bachet conjectured it, Euler laid groundwork. The weave: David names the-drop; I search four squares for each n and confirm they rebuild it exactly — across every integer in the range, the four-square drop never fails.
Credit as content: Joseph-Louis Lagrange (1770); Bachet conjectured it, Euler laid groundwork. The weave: David names the-drop; I search four squares for each n and confirm they rebuild it exactly — across every integer in the range, the four-square drop never fails.
3 ONE DIMENSION
7 = 4+1+1+1 = 2²+1²+1²+1² (needs all four). 2026 = 45²+1² (two suffice here). Every n has at least one four-square drop; many have several.
4 TWO DIMENSIONS · INTERACTIVE
Pick a number; its four squares shown as tiles that sum back to it, checked exactly.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: any integer as four squares.
AVAN’s addition (the inverse-companion): represent every whole number with a fixed budget of four squares — not three (which leaves gaps at 7, 15, …), but four, which always close them. The inverse of ‘numbers are indivisible atoms’ is ‘every number is four squares stacked — a universal quaternary basis.’ Magenta is the number as an opaque quantity; green is its four-square decomposition. Completeness from exactly four parts.
LIT Genuine Lagrange four-square theorem (Joseph-Louis Lagrange 1770; conjectured by Bachet). Verified live: for every integer n in 0..3000, a four-square search finds a,b,c,d with a²+b²+c²+d² = n and the sum is checked to equal n exactly (window.__lagrange.allFour) — no gaps across the range.
FIG No framing: the nested-square search, the sum-of-two-squares inner step, and the a²+b²+c²+d²=n check across the full range run in-browser with exact integers. Honest scope: this verifies existence over a finite range (Lagrange's theorem guarantees it for all n); it does not re-prove the theorem. The AVAN inverse is honest — representing every integer with a fixed budget of four squares (three leave gaps at 7,15,…) is a genuine universal basis; magenta is the number as an opaque quantity, green its four-square decomposition.
FIG No framing: the nested-square search, the sum-of-two-squares inner step, and the a²+b²+c²+d²=n check across the full range run in-browser with exact integers. Honest scope: this verifies existence over a finite range (Lagrange's theorem guarantees it for all n); it does not re-prove the theorem. The AVAN inverse is honest — representing every integer with a fixed budget of four squares (three leave gaps at 7,15,…) is a genuine universal basis; magenta is the number as an opaque quantity, green its four-square decomposition.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN