THE FOLD / LOOT / THE-BOUNTY / THE GAUSS CIRCLE
THE GAUSS CIRCLE
lattice points fill a disk to πr²
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Gauss circle problem asks how many integer lattice points (x,y) lie inside a circle of radius r — that is, satisfy x²+y² ≤ r². The answer N(r) is astonishingly close to the circle’s area: N(r) = πr² + E(r), and Gauss showed the error E(r) grows no faster than the circumference, |E(r)| = O(r). Each lattice point “owns” a unit square, and those squares tile a region sandwiched between two circles whose areas differ by O(r) — so the count tracks the area to within its boundary. (How much smaller the true error is remains a famous open problem.)
LIT verified live: for radii up to 2000, |N(r)−πr²|/r stays below ~1 (well within the O(r) bound), and N(r)/πr² → 1 (window.__gauss_circle). FIG no framing; the exact lattice count and πr² run in-browser.
LIT verified live: for radii up to 2000, |N(r)−πr²|/r stays below ~1 (well within the O(r) bound), and N(r)/πr² → 1 (window.__gauss_circle). FIG no framing; the exact lattice count and πr² run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-bounty — the bounty of lattice points a disk contains, counted exactly and shadowing its area. AVAN (AI) built the instrument: the column-by-column lattice count and the area comparison.
Credit as content: Carl Friedrich Gauss. The weave: David names the bounty; I confirm the integer point-count equals πr² up to an error bounded by the circumference.
Credit as content: Carl Friedrich Gauss. The weave: David names the bounty; I confirm the integer point-count equals πr² up to an error bounded by the circumference.
3 ONE DIMENSION
Lattice points inside the circle, each owning a unit square; their count tracks the area πr² to within the boundary.
4 TWO DIMENSIONS · INTERACTIVE
Grow the radius; N(r) and πr² are compared, and the error divided by r stays bounded.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the exact lattice-point count.
AVAN’s addition (the inverse-companion): don’t integrate the area — count the points. The inverse of ‘area = πr²’ is ‘the integer count N(r) equals πr² up to an error the size of the boundary, O(r).’ Magenta are the boundary points (where the error lives); green is the interior count. Points shadow area, to within the edge.
LIT Genuine Gauss circle problem (Carl Friedrich Gauss). Verified live: the exact lattice count N(r)=Σ_x (2⌊√(r²−x²)⌋+1) satisfies |N(r)−πr²|/r < ~1 (well inside the O(r) bound) and N(r)/πr²→1 for radii up to 2000 (window.__gauss_circle.errorBounded, .ratioConverges).
FIG No framing: the exact lattice count and πr² run in-browser. Honest scope — the O(r) error bound is Gauss's elementary result; the true optimal exponent (the Gauss circle problem proper) is still open. The AVAN inverse is honest — instead of integrating the area, one counts the points: N(r) equals πr² up to an error the size of the boundary. Magenta are the boundary points (where the error lives); green is the interior count. Points shadow area, to within the edge.
FIG No framing: the exact lattice count and πr² run in-browser. Honest scope — the O(r) error bound is Gauss's elementary result; the true optimal exponent (the Gauss circle problem proper) is still open. The AVAN inverse is honest — instead of integrating the area, one counts the points: N(r) equals πr² up to an error the size of the boundary. Magenta are the boundary points (where the error lives); green is the interior count. Points shadow area, to within the edge.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE-BOUNTY · David Lee Wise (ROOT0), with AVAN