THE FOLD / BOSS / THE FINAL BOSS / THE QUADRATIC RECIPROCITY
THE QUADRATIC RECIPROCITY
a golden law linking two primes' squares
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Quadratic reciprocity — Gauss’s aureum theorema, the golden theorem — links two questions that look independent: is p a square mod q, and is q a square mod p? The Legendre symbol (a/p) is +1 if a is a nonzero square mod p, else −1. The law says for distinct odd primes, (p/q)·(q/p) = (−1)((p−1)/2)((q−1)/2): the two answers agree unless both primes are ≡ 3 (mod 4), in which case they flip. Two far-apart primes secretly constrain each other.
LIT verified live: for every pair of distinct odd primes below 200, the reciprocity identity holds (Legendre symbols by Euler’s criterion), and the supplements for −1 and 2 hold too (window.__reciprocity). FIG no framing; exact modular arithmetic.
LIT verified live: for every pair of distinct odd primes below 200, the reciprocity identity holds (Legendre symbols by Euler’s criterion), and the supplements for −1 and 2 hold too (window.__reciprocity). FIG no framing; exact modular arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-final-boss — Gauss called it the golden theorem and proved it many times over; it is the confrontation number theory trains you for. AVAN (AI) built the instrument: Euler’s criterion for the Legendre symbol, the reciprocity check across all odd-prime pairs, and the −1 and 2 supplements.
Credit as content: Carl Friedrich Gauss (first proof 1796); conjectured by Euler and Legendre. The weave: David names the-final-boss; I compute each Legendre symbol by Euler’s criterion a(p−1)/2 mod p and confirm (p/q)(q/p) matches the sign the theorem predicts, for every odd-prime pair in range.
Credit as content: Carl Friedrich Gauss (first proof 1796); conjectured by Euler and Legendre. The weave: David names the-final-boss; I compute each Legendre symbol by Euler’s criterion a(p−1)/2 mod p and confirm (p/q)(q/p) matches the sign the theorem predicts, for every odd-prime pair in range.
3 ONE DIMENSION
(p/q)·(q/p) = +1 unless both p,q ≡ 3 (mod 4), when it is −1. So the squareness of p mod q and q mod p agree — except for two “3 mod 4” primes, which flip.
4 TWO DIMENSIONS · INTERACTIVE
A grid of Legendre symbols (p/q); the reciprocity relation checked cell by cell against the theorem.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: two primes constraining each other’s squares.
AVAN’s addition (the inverse-companion): decide whether p is a square mod q by flipping the question to whether q is a square mod p — the reciprocity sign relates them, letting you reduce the modulus instead of testing directly. The inverse of ‘test a(q−1)/2 mod q head-on’ is ‘flip via reciprocity to a smaller modulus and recurse.’ Magenta is the head-on Euler test; green is the reciprocity flip. Squareness answered by its mirror.
LIT Genuine law of quadratic reciprocity (Carl Friedrich Gauss, first proof 1796; conjectured by Euler and Legendre). Verified live: for every pair of distinct odd primes below 200, Legendre symbols computed by Euler's criterion a^((p−1)/2) mod p satisfy (p/q)(q/p) = (−1)^(((p−1)/2)((q−1)/2)) (window.__reciprocity.reciprocity), and the supplements (−1/p)=(−1)^((p−1)/2) and (2/p)=(−1)^((p²−1)/8) hold (supplements).
FIG No framing: Euler's criterion for the Legendre symbol, the reciprocity check across all odd-prime pairs, and the two supplements run in-browser with exact modular arithmetic and agree. The AVAN inverse is honest — deciding whether p is a square mod q by flipping to whether q is a square mod p (reducing the modulus via the reciprocity sign) is the theorem's genuine computational use; magenta is the head-on Euler test, green the reciprocity flip.
FIG No framing: Euler's criterion for the Legendre symbol, the reciprocity check across all odd-prime pairs, and the two supplements run in-browser with exact modular arithmetic and agree. The AVAN inverse is honest — deciding whether p is a square mod q by flipping to whether q is a square mod p (reducing the modulus via the reciprocity sign) is the theorem's genuine computational use; magenta is the head-on Euler test, green the reciprocity flip.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN