THE FOLD / BOSS / THE GAUNTLET / THE RECIPROCITY
THE RECIPROCITY
is p a square mod q? — Gauss's golden theorem links it to q mod p
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Quadratic reciprocity — Gauss’s Theorema Aureum, the “golden theorem.” Ask two questions about two odd primes: is p a perfect square modulo q? and is q a perfect square modulo p? They look unrelated. The theorem says they have a startling hidden link: the two answers are the same — unless both p and q leave remainder 3 when divided by 4, in which case they are opposite.
With the Legendre symbol (p/q) = +1 when p is a square mod q and −1 when not, the whole law is one line: (p/q)(q/p) = (−1)½(p−1)·½(q−1). Gauss was so captivated he proved it eight different ways; hundreds of proofs are now known. It is the cornerstone of algebraic number theory and quietly underlies modern cryptography. Each symbol is computed fast by Euler’s criterion: a(p−1)/2 mod p is +1 or −1.
LIT verified live: computing Legendre symbols by Euler’s criterion, the reciprocity law holds for every odd prime pair below 60, and the squares mod 7 come out {1,2,4} (window.__reciprocity). FIG no framing; the law and the symbol computations are exact.
With the Legendre symbol (p/q) = +1 when p is a square mod q and −1 when not, the whole law is one line: (p/q)(q/p) = (−1)½(p−1)·½(q−1). Gauss was so captivated he proved it eight different ways; hundreds of proofs are now known. It is the cornerstone of algebraic number theory and quietly underlies modern cryptography. Each symbol is computed fast by Euler’s criterion: a(p−1)/2 mod p is +1 or −1.
LIT verified live: computing Legendre symbols by Euler’s criterion, the reciprocity law holds for every odd prime pair below 60, and the squares mod 7 come out {1,2,4} (window.__reciprocity). FIG no framing; the law and the symbol computations are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE GAUNTLET — the boss domain of the trial you run again and again. Quadratic reciprocity is Gauss’s personal gauntlet: he ran it eight times, proving the same golden theorem by eight different routes. AVAN (AI) built the instrument: the squares-mod-p line, the Legendre board, the p↔q transpose inverse.
The weave: David names the seat (the trial run many ways); I make the two square-questions link exactly as the law predicts and show the sign twist — the residues in 1D, the reciprocity check in 2D, the transpose inverse in 3D. The sphere is the seam. Credit: conjectured by Euler and Legendre; first complete proof by Carl Friedrich Gauss (1801, Disquisitiones Arithmeticae), who gave eight.
The weave: David names the seat (the trial run many ways); I make the two square-questions link exactly as the law predicts and show the sign twist — the residues in 1D, the reciprocity check in 2D, the transpose inverse in 3D. The sphere is the seam. Credit: conjectured by Euler and Legendre; first complete proof by Carl Friedrich Gauss (1801, Disquisitiones Arithmeticae), who gave eight.
3 ONE DIMENSION
The residues 1..p−1 for a prime p, with the quadratic residues (the perfect squares mod p) lit up. Exactly half of them are squares — that even split is the seed from which the whole reciprocity law grows.
4 TWO DIMENSIONS · INTERACTIVE
Pick two odd primes. The panel computes (p/q) and (q/p) by Euler’s criterion, shows whether each is a square modulo the other, and checks the golden law: same sign, unless both p and q are 3 mod 4, where they flip.
5 THREE DIMENSIONS + AVAN’S INVERSE
The Legendre symbols (p/q) drawn as a board over pairs of primes — the green forward question: for each pair, is p a square mod q?
AVAN’s addition (the inverse-companion): the magenta board is the transpose — the very same question with p and q swapped, (q/p). Reciprocity is a statement about a question and its inverse: swap which prime is the base and which is the residue, and the answer barely changes. The magenta transposed board matches the green one everywhere except the cells where both primes are 3 mod 4, and there it flips sign — a single clean correction. So the inverse question is not independent; the forward answer determines it, up to that one parity twist. That is the whole miracle: ‘is p a square mod q?’ secretly tells you ‘is q a square mod p?’, two problems in different worlds bound by one sign. Green is the board; magenta is its transpose; their difference is exactly the 3-mod-4 diagonal, and nothing else.
LIT Genuine quadratic reciprocity (conjectured by Euler and Legendre; first complete proof by Gauss 1801, who gave eight). Verified live: computing Legendre symbols by Euler's criterion (a^((p-1)/2) mod p), the law (p/q)(q/p) = (-1)^(((p-1)/2)((q-1)/2)) holds for every odd prime pair below 60, and the quadratic residues mod 7 are {1,2,4} (window.__reciprocity.reciprocityHolds). The law and the symbol computations are exact.
FIG No framing: the reciprocity law and the Legendre-symbol computations are real and checked exhaustively over all odd prime pairs below 60 in-browser. The inverse framing — reciprocity binds a question (p/q) to its transpose (q/p), equal everywhere except the both-3-mod-4 cells where the sign flips — is the honest content, shown directly on the symbol board, not asserted.
FIG No framing: the reciprocity law and the Legendre-symbol computations are real and checked exhaustively over all odd prime pairs below 60 in-browser. The inverse framing — reciprocity binds a question (p/q) to its transpose (q/p), equal everywhere except the both-3-mod-4 cells where the sign flips — is the honest content, shown directly on the symbol board, not asserted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN