THE FOLD / LOOT / THE HOARD / THE JACOBI TWO-SQUARE
THE JACOBI TWO-SQUARE
sums of two squares counted by divisors mod 4
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Jacobi’s two-square theorem counts, exactly, how many ways a number is a sum of two squares — using only its divisors. Let r₂(n) be the number of integer pairs (a,b) with a²+b²=n (signs and order counted). Jacobi proved r₂(n) = 4·(d₁(n) - d₃(n)), where d₁(n) counts the divisors of n congruent to 1 (mod 4) and d₃(n) counts those congruent to 3 (mod 4). A geometric question — how many lattice points lie on the circle of radius √n — is answered purely by counting divisors and sorting them by their remainder mod 4.
LIT verified live: for every n from 1 to 2000, a brute count of lattice points (a,b) on the circle a²+b²=n equals 4·(d₁(n)-d₃(n)) computed from the divisors — e.g. r₂(25)=12 (window.__twosquare). FIG no framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000.
LIT verified live: for every n from 1 to 2000, a brute count of lattice points (a,b) on the circle a²+b²=n equals 4·(d₁(n)-d₃(n)) computed from the divisors — e.g. r₂(25)=12 (window.__twosquare). FIG no framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hoard — the count of ways to hoard n as a²+b², tallied not by searching the plane but by sorting n’s divisors by their remainder mod 4. AVAN (AI) built the instrument: the brute lattice count on the circle, the divisor tally, and their agreement.
Credit as content: Carl Gustav Jacob Jacobi (1834); Fermat and Gauss before. The weave: David names the hoard; I confirm the lattice-point count equals 4(d₁-d₃).
Credit as content: Carl Gustav Jacob Jacobi (1834); Fermat and Gauss before. The weave: David names the hoard; I confirm the lattice-point count equals 4(d₁-d₃).
3 ONE DIMENSION
The circle a²+b²=n and the integer lattice points on it — r₂(n) of them, counted by divisors mod 4.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the lattice-point count r₂(n) is compared to 4·(d₁(n) − d₃(n)) from the divisors.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: r₂(n), the number of lattice points on the circle.
AVAN’s addition (the inverse-companion): don’t scan the plane — sort the divisors. The inverse of ‘count lattice points on the circle of radius √n’ is ‘4 times (divisors ≡1 minus divisors ≡3, mod 4)’. Magenta are the divisors sorted by remainder mod 4; green is the lattice-point count they determine. Geometry answered by arithmetic.
LIT Genuine Jacobi two-square theorem (Carl Gustav Jacob Jacobi, 1834; Fermat, Gauss before). Verified live: for every n=1..2000, the brute count of integer lattice points on the circle a²+b²=n equals 4·(d₁(n)−d₃(n)), where d₁,d₃ count divisors ≡1,≡3 (mod 4); r₂(25)=12 (window.__twosquare.ok, .r25).
FIG No framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000. The AVAN inverse is honest — instead of scanning the plane, sort the divisors: the inverse of 'count lattice points on the circle of radius √n' is '4 times (divisors ≡1 minus divisors ≡3, mod 4)'. Magenta are the divisors sorted by remainder mod 4; green is the lattice-point count they determine. Geometry answered by arithmetic.
FIG No framing; the lattice-point count and the divisor formula both run in-browser and agree for all n up to 2000. The AVAN inverse is honest — instead of scanning the plane, sort the divisors: the inverse of 'count lattice points on the circle of radius √n' is '4 times (divisors ≡1 minus divisors ≡3, mod 4)'. Magenta are the divisors sorted by remainder mod 4; green is the lattice-point count they determine. Geometry answered by arithmetic.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOARD · David Lee Wise (ROOT0), with AVAN