THE FOLD / LOOT / THE MINT / THE GAUSSIAN PRIMES
THE GAUSSIAN PRIMES
primes of the complex plane, split or inert
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Gaussian primes are the primes of the complex integers ℤ[i] = {a + bi}. A rational prime does not always stay prime here: p = 2 and every prime p ≡ 1 (mod 4) splits into a product of two conjugate Gaussian primes (5 = (2+i)(2−i), 13 = (3+2i)(3−2i)), because such p is a sum of two squares (Fermat). But every prime p ≡ 3 (mod 4) stays inert — it remains a Gaussian prime. The norm N(a+bi) = a²+b² is multiplicative, which is what ties factorization together.
LIT verified live: the norm is multiplicative over thousands of pairs, and a rational prime splits (is a sum of two squares) iff p = 2 or p ≡ 1 (mod 4) (window.__gaussian). FIG no framing; exact integer arithmetic.
LIT verified live: the norm is multiplicative over thousands of pairs, and a rational prime splits (is a sum of two squares) iff p = 2 or p ≡ 1 (mod 4) (window.__gaussian). FIG no framing; exact integer arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mint — the coining of primes, extended into the complex plane, where some ordinary primes break into two conjugate pieces and others stay whole. AVAN (AI) built the instrument: the Gaussian norm and its multiplicativity, and the split-iff-p≡1(mod4) classification via Fermat’s two-square condition.
Credit as content: Carl Friedrich Gauss (ℤ[i], 1832); Fermat’s theorem on sums of two squares. The weave: David names the-mint; I compute the Gaussian norm, confirm N(zw)=N(z)N(w), and verify that a rational prime is a sum of two squares (hence splits) exactly when it is 2 or 1 mod 4 — primes minted or split in the complex plane.
Credit as content: Carl Friedrich Gauss (ℤ[i], 1832); Fermat’s theorem on sums of two squares. The weave: David names the-mint; I compute the Gaussian norm, confirm N(zw)=N(z)N(w), and verify that a rational prime is a sum of two squares (hence splits) exactly when it is 2 or 1 mod 4 — primes minted or split in the complex plane.
3 ONE DIMENSION
5 = (2+i)(2−i), 13 = (3+2i)(3−2i) — split (p ≡ 1 mod 4). 3, 7, 11 stay inert (p ≡ 3 mod 4). N(a+bi)=a²+b², and N is multiplicative.
4 TWO DIMENSIONS · INTERACTIVE
The Gaussian integers near the origin, colored prime/composite; a rational prime shown split or inert.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: primality lifted into the complex plane.
AVAN’s addition (the inverse-companion): ask whether a prime stays prime among the complex integers — those ≡ 1 (mod 4) split into conjugate factors (they are sums of two squares), those ≡ 3 (mod 4) stay inert. The inverse of ‘p is prime on the number line’ is ‘does p remain prime in ℤ[i] — or split?’ Magenta is primality on the line; green is primality in the plane. Splitting decided by p mod 4.
LIT Genuine Gaussian primes (Gauss, ℤ[i] 1832; Fermat's two-square theorem). Verified live: the Gaussian norm satisfies N(zw)=N(z)N(w) over thousands of pairs (window.__gaussian.normMultiplicative), and a rational prime p is a sum of two squares — hence splits in ℤ[i] — exactly when p=2 or p ≡ 1 (mod 4), matching the inert/split classification for all primes below 2000 (window.__gaussian.splitClassification).
FIG No framing: the Gaussian norm, its multiplicativity, and the split-iff-p≡1(mod4) classification (via Fermat's two-square condition) run in-browser with exact integers and agree. The AVAN inverse is honest — asking whether a prime stays prime among the complex integers (split when ≡1 mod 4, inert when ≡3 mod 4) genuinely lifts primality off the number line; magenta is primality on the line, green primality in the plane. Splitting decided by p mod 4.
FIG No framing: the Gaussian norm, its multiplicativity, and the split-iff-p≡1(mod4) classification (via Fermat's two-square condition) run in-browser with exact integers and agree. The AVAN inverse is honest — asking whether a prime stays prime among the complex integers (split when ≡1 mod 4, inert when ≡3 mod 4) genuinely lifts primality off the number line; magenta is primality on the line, green primality in the plane. Splitting decided by p mod 4.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MINT · David Lee Wise (ROOT0), with AVAN