THE FOLD / BOSS / THE WALL / THE ALTERNATING SIGN
THE ALTERNATING SIGN
the 88-referee formula
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
An alternating sign matrix is a grid of 0s, +1s and −1s where every row and column sums to 1 and the nonzeros alternate in sign. Count them: 1, 2, 7, 42, 429, 7436, … In 1983 Mills, Robbins and Rumsey conjectured the exact formula ∏(3k+1)!/(n+k)! — and the conjecture became a wall. Zeilberger’s eventual proof (1996) ran over eighty pages and was checked by 88 volunteer referees; Kuperberg then felled the same wall in a few pages using the six-vertex model of statistical physics. Bressoud’s Proofs and Confirmations tells the whole siege.
LIT verified live: brute-force enumeration of ALL alternating sign matrices for n = 1…6 (row-by-row DFS with column partial-sum constraints) yields 1, 2, 7, 42, 429, 7436 — exactly matching the Robbins product formula computed in BigInt (window.__asm). FIG the enumeration IS the theorem’s instance for n≤6; the general proof (Zeilberger, Kuperberg) is cited, not re-derived; the 88-referee story is documented in Bressoud.
LIT verified live: brute-force enumeration of ALL alternating sign matrices for n = 1…6 (row-by-row DFS with column partial-sum constraints) yields 1, 2, 7, 42, 429, 7436 — exactly matching the Robbins product formula computed in BigInt (window.__asm). FIG the enumeration IS the theorem’s instance for n≤6; the general proof (Zeilberger, Kuperberg) is cited, not re-derived; the 88-referee story is documented in Bressoud.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-wall — the boss: a formula anyone can state, verified numerically for years, that repelled every proof for thirteen years — the wall wasn’t finding the pattern, it was EARNING it. AVAN (AI) built the instrument: the constrained enumerator and the BigInt formula engine.
Credit as content: Mills, Robbins & Rumsey (1983); Doron Zeilberger (1996, with 88 named referees); Greg Kuperberg (six-vertex proof); David Bressoud (the chronicle). The weave: David names the wall; I count all 7,436 matrices at n=6 and the formula holds the line.
Credit as content: Mills, Robbins & Rumsey (1983); Doron Zeilberger (1996, with 88 named referees); Greg Kuperberg (six-vertex proof); David Bressoud (the chronicle). The weave: David names the wall; I count all 7,436 matrices at n=6 and the formula holds the line.
3 ONE DIMENSION
The seven ASMs of order 3 — and the product formula that counts them.
4 TWO DIMENSIONS · INTERACTIVE
Step n; enumeration and formula march in lockstep.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the tower 1, 2, 7, 42, 429, 7436 rising.
AVAN’s addition (the inverse-companion): don’t confuse verified with proven. The inverse of ‘the formula works for every case we tried’ is ‘thirteen years and 84 pages before anyone knew WHY’: numerical certainty and mathematical understanding are different currencies, exchanged at a brutal rate. Magenta is the mounting numerical evidence that proved nothing; green is the proof that finally paid. I verify instances by the thousand and claim exactly that — instances.
LIT Verified live: brute enumeration of ALL alternating sign matrices for n=1..6 (row DFS under column partial-sum constraints) yields 1,2,7,42,429,7436, matching the Robbins formula computed in BigInt (window.__asm.ok).
FIG Enumeration is the theorem's instance for n≤6; the general proof is cited, not re-derived; the 88-referee story is in Bressoud's Proofs and Confirmations. The AVAN inverse — don't confuse verified with proven: numerical certainty and understanding are different currencies at a brutal exchange rate. Magenta is the mounting evidence that proved nothing; green is the proof that finally paid.
FIG Enumeration is the theorem's instance for n≤6; the general proof is cited, not re-derived; the 88-referee story is in Bressoud's Proofs and Confirmations. The AVAN inverse — don't confuse verified with proven: numerical certainty and understanding are different currencies at a brutal exchange rate. Magenta is the mounting evidence that proved nothing; green is the proof that finally paid.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE WALL · David Lee Wise (ROOT0), with AVAN