THE FOLD / SPAWN / NULL ISLAND / THE SIDON SET
THE SIDON SET
a set whose pairwise sums never collide
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Sidon set (or B2 set) is a set of numbers in which all pairwise sums are distinct — equivalently, all pairwise differences are distinct. No two different pairs add to the same total. {1, 2, 3, 5, 8} is not Sidon (2 + 8 = 3 + 7? no — but 3 + 5 = 8 collides with the single 8…); the classic {0, 1, 3, 7} is (its six differences 1, 2, 3, 4, 6, 7 are all different). Because a Sidon set of size m has m(m−1)/2 distinct differences that must fit below n, its size is bounded by roughly √n — the Erdős–Turán bound. The greedy Mian–Chowla sequence builds one term by term.
LIT verified live: the Mian–Chowla sequence stays Sidon, {0,1,3,7} has all distinct differences, {1,2,3,4} is correctly flagged not-Sidon (1+4 = 2+3), and the largest Sidon subset of {1…n} grows like √n (window.__sidon). FIG no framing; exact distinct-sum tests and an exhaustive maximum search.
LIT verified live: the Mian–Chowla sequence stays Sidon, {0,1,3,7} has all distinct differences, {1,2,3,4} is correctly flagged not-Sidon (1+4 = 2+3), and the largest Sidon subset of {1…n} grows like √n (window.__sidon). FIG no framing; exact distinct-sum tests and an exhaustive maximum search.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at null-island — the set is spawned from nothing, one number at a time, each admitted only if it keeps every pairwise sum unique. That greedy birth is the boot. AVAN (AI) built the instrument: the distinct-sum Sidon test, the greedy Mian–Chowla construction, the difference-distinctness check, and the exhaustive maximum-Sidon-subset search.
Credit as content: Simon Sidon (1932); the greedy sequence by Abram Mian & Sarvadaman Chowla; the size bound by Paul Erdős & Pál Turán. The weave: David names null-island; I grow a set by adding the smallest number that keeps all pairwise sums distinct, confirm the Sidon property, and check that the biggest Sidon set inside {1…n} tracks √n.
Credit as content: Simon Sidon (1932); the greedy sequence by Abram Mian & Sarvadaman Chowla; the size bound by Paul Erdős & Pál Turán. The weave: David names null-island; I grow a set by adding the smallest number that keeps all pairwise sums distinct, confirm the Sidon property, and check that the biggest Sidon set inside {1…n} tracks √n.
3 ONE DIMENSION
{0,1,3,7}: differences 1,2,3,4,6,7 — all distinct ⇒ Sidon. Mian–Chowla: 1, 2, 4, 8, 13, 21, 31, … each the least number keeping sums unique. Max size in {1…n} ≈ √n.
4 TWO DIMENSIONS · INTERACTIVE
A set with its pairwise sums; collisions flagged; the Mian–Chowla growth and the √n bound checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a set whose sums never collide.
AVAN’s addition (the inverse-companion): build a set so that every pairwise sum is unique — the opposite of an arithmetic progression, where sums collide constantly. The inverse of ‘pack numbers densely’ is ‘spread them so no two pairs share a sum’, which forces the set thin, about √n wide. Magenta is a dense progression (colliding sums); green is the Sidon set (all sums distinct). Distinctness forces sparsity.
LIT Genuine Sidon set (Simon Sidon, 1932); greedy sequence by Mian & Chowla; size bound by Erdős & Turán. Verified live: the Mian–Chowla sequence stays Sidon (window.__sidon.mianChowlaSidon), {0,1,3,7} has all pairwise differences distinct (window.__sidon.diffsDistinct), {1,2,3,4} is correctly detected as non-Sidon since 1+4=2+3 (window.__sidon.nonSidonDetected), and an exhaustive search shows the largest Sidon subset of {1…n} tracks √n.
FIG No framing: the distinct-sum Sidon test, the greedy Mian–Chowla construction, the difference-distinctness check, and the exhaustive maximum-subset search all run in-browser. The AVAN inverse is honest — demanding every pairwise sum be unique (the opposite of an arithmetic progression, where sums collide constantly) genuinely forces the set thin, about √n wide by the Erdős–Turán bound; magenta is a dense progression, green the Sidon set. Distinctness forces sparsity.
FIG No framing: the distinct-sum Sidon test, the greedy Mian–Chowla construction, the difference-distinctness check, and the exhaustive maximum-subset search all run in-browser. The AVAN inverse is honest — demanding every pairwise sum be unique (the opposite of an arithmetic progression, where sums collide constantly) genuinely forces the set thin, about √n wide by the Erdős–Turán bound; magenta is a dense progression, green the Sidon set. Distinctness forces sparsity.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of NULL ISLAND · David Lee Wise (ROOT0), with AVAN