THE FOLD / SPAWN / FIRST-LIGHT / THE GAUSS SUM
THE GAUSS SUM
p unit vectors sum to exactly √p
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The quadratic Gauss sum is one of the small miracles of number theory: add up the p complex numbers e2πi·k²/p for k = 0…p−1, and although the phases scatter chaotically around the circle, their sum has magnitude exactly √p. Gauss went further and pinned the sign: the sum equals √p when p ≡ 1 (mod 4) and i√p when p ≡ 3 (mod 4) — a fact he called his “tormentor” until he proved it. These sums underlie quadratic reciprocity, the functional equation of L-functions, and the fast construction of certain codes.
LIT verified live: for every prime up to 200, the squared magnitude of the sum equals p to ~1e-13, and the real/imaginary split matches Gauss’s sign rule (window.__gauss_sum). FIG no framing; the p complex exponentials and their sum run in-browser.
LIT verified live: for every prime up to 200, the squared magnitude of the sum equals p to ~1e-13, and the real/imaginary split matches Gauss’s sign rule (window.__gauss_sum). FIG no framing; the p complex exponentials and their sum run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at first-light — p points of light placed around the circle by k², and no matter how they scatter their sum is a beam of length exactly √p. AVAN (AI) built the instrument: the k² phases, the head-to-tail sum, and the magnitude/sign checks.
Credit as content: Carl Friedrich Gauss (1801–1805). The weave: David names first light; I confirm the chaotic-looking phases sum to a resultant of length exactly √p, with the sign Gauss determined.
Credit as content: Carl Friedrich Gauss (1801–1805). The weave: David names first light; I confirm the chaotic-looking phases sum to a resultant of length exactly √p, with the sign Gauss determined.
3 ONE DIMENSION
The p unit vectors e^{2πik²/p} on the circle; scattered in phase, yet their sum is a resultant of length exactly √p.
4 TWO DIMENSIONS · INTERACTIVE
Pick a prime; the k² phasors add head-to-tail into a resultant whose length is √p, its direction set by p mod 4.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the resultant of length √p.
AVAN’s addition (the inverse-companion): don’t read the phases — measure the sum. The inverse of ‘scatter p unit vectors by k²’ is ‘their magnitude is locked to exactly √p, sign by p mod 4.’ Magenta are the individual unit vectors; green is the resultant of length √p. Chaos in phase, order in magnitude.
LIT Genuine quadratic Gauss sum (Carl Friedrich Gauss, 1801–1805). Verified live: for every prime p ≤ 200, |Σ_{k=0}^{p−1} e^{2πik²/p}|² = p to ~1e-13, and the sum is √p (real) when p≡1 mod4 and i√p (imaginary) when p≡3 mod4 — Gauss's sign determination (window.__gauss_sum.magEqualsP, .signCorrect).
FIG No framing: the p complex exponentials and their vector sum run in-browser. The AVAN inverse is honest — instead of reading the scattered phases, one measures their sum: the magnitude is locked to exactly √p, sign fixed by p mod 4. Magenta are the p individual unit vectors; green is the resultant of length √p. Chaos in phase, order in magnitude.
FIG No framing: the p complex exponentials and their vector sum run in-browser. The AVAN inverse is honest — instead of reading the scattered phases, one measures their sum: the magnitude is locked to exactly √p, sign fixed by p mod 4. Magenta are the p individual unit vectors; green is the resultant of length √p. Chaos in phase, order in magnitude.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of FIRST-LIGHT · David Lee Wise (ROOT0), with AVAN