THE FOLD / GLITCH / STACK OVERFLOW / THE BRUCK-RYSER
THE BRUCK-RYSER
orders of projective planes ruled out by two squares
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Bruck–Ryser theorem forbids certain finite projective planes using a fact about sums of two squares. A projective plane of order n is a highly symmetric geometry with n²+n+1 points and the same number of lines. Bruck and Ryser proved a necessary condition: if n ≡ 1 or 2 (mod 4), then a projective plane of order n can exist only if n is a sum of two integer squares. This single arithmetic test rules out infinitely many orders — the first being order 6 (6 ≡ 2 mod 4, and 6 is not a sum of two squares), which is why no 6×6 pair of orthogonal Latin squares (Euler’s 36 officers) exists. It is a necessary, not sufficient, condition.
LIT verified live: among orders n ≤ 50 with n ≡ 1 or 2 (mod 4), the ones that are not sums of two squares — and so ruled out by Bruck–Ryser — are exactly 6, 14, 21, 22, 30, 33, 38, 42, 46; the small orders with known planes (2,3,4,5,7,8,9) are never excluded (window.__bruckryser). FIG no framing; the mod-4 test and the sum-of-two-squares check run in-browser. Honest: order 10 passes Bruck–Ryser yet has no plane — that was proved only later by massive computation.
LIT verified live: among orders n ≤ 50 with n ≡ 1 or 2 (mod 4), the ones that are not sums of two squares — and so ruled out by Bruck–Ryser — are exactly 6, 14, 21, 22, 30, 33, 38, 42, 46; the small orders with known planes (2,3,4,5,7,8,9) are never excluded (window.__bruckryser). FIG no framing; the mod-4 test and the sum-of-two-squares check run in-browser. Honest: order 10 passes Bruck–Ryser yet has no plane — that was proved only later by massive computation.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at stack-overflow — the glitch where whole orders of geometry overflow into impossibility, ruled out by a two-squares test. AVAN (AI) built the instrument: the mod-4 condition, the sum-of-two-squares check, and the list of excluded orders — with an honest note that the condition is necessary, not sufficient.
Credit as content: R. H. Bruck & H. J. Ryser (1949). The weave: David names the overflow; I confirm which orders Bruck–Ryser rules out, and flag order 10 as passing yet impossible.
Credit as content: R. H. Bruck & H. J. Ryser (1949). The weave: David names the overflow; I confirm which orders Bruck–Ryser rules out, and flag order 10 as passing yet impossible.
3 ONE DIMENSION
Orders 2..50: those ≡1,2 (mod 4) and not a sum of two squares (magenta) are ruled out by Bruck–Ryser.
4 TWO DIMENSIONS · INTERACTIVE
Cycle orders; the mod-4 class and the sum-of-two-squares test decide whether Bruck–Ryser excludes it.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the orders Bruck–Ryser permits (and the magenta ones it forbids).
AVAN’s addition (the inverse-companion): don’t search for a plane — test the arithmetic. The inverse of ‘does a projective plane of order n exist?’ is (for n≡1,2 mod 4) ‘is n a sum of two squares?’ — if not, no plane can exist. Magenta are the forbidden orders; green are the orders that survive the test. Geometry gated by two squares.
LIT Genuine Bruck–Ryser theorem (R. H. Bruck & H. J. Ryser, 1949). Verified live: among orders n≤50 with n≡1,2 (mod 4), those NOT sums of two squares — excluded by Bruck–Ryser — are exactly 6,14,21,22,30,33,38,42,46; known-plane orders 2,3,4,5,7,8,9 are never excluded, and order 10 passes BR yet has no plane (window.__bruckryser.ok, .knownOk, .ten).
FIG No framing; the mod-4 test and the sum-of-two-squares check run in-browser. HONEST: this is a necessary, not sufficient, condition — order 10 passes Bruck–Ryser yet has no projective plane (proved only later, by massive computation, 1989). The AVAN inverse is honest — instead of searching for a plane, test the arithmetic: the inverse of 'does a projective plane of order n exist?' is (for n≡1,2 mod 4) 'is n a sum of two squares?'. Magenta are the forbidden orders; green are the orders that survive the test. Geometry gated by two squares.
FIG No framing; the mod-4 test and the sum-of-two-squares check run in-browser. HONEST: this is a necessary, not sufficient, condition — order 10 passes Bruck–Ryser yet has no projective plane (proved only later, by massive computation, 1989). The AVAN inverse is honest — instead of searching for a plane, test the arithmetic: the inverse of 'does a projective plane of order n exist?' is (for n≡1,2 mod 4) 'is n a sum of two squares?'. Magenta are the forbidden orders; green are the orders that survive the test. Geometry gated by two squares.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of STACK OVERFLOW · David Lee Wise (ROOT0), with AVAN