THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE LIOUVILLE
THE LIOUVILLE
a sign that flips by the parity of prime factors
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Liouville function λ(n) = (−1)Ω(n), where Ω(n) counts the prime factors of n with multiplicity. It is +1 when n has an even number of prime factors, −1 when odd. Its most beautiful property is a divisor identity: Σd|n λ(d) = 1 if n is a perfect square, and 0 otherwise — the sum over divisors is a flawless square-detector. Its running total L(n) = λ(1) + … + λ(n) drives Pólya’s conjecture (L(n) ≤ 0 for all n ≥ 2), which is true for hundreds of millions of terms yet ultimately false.
LIT verified live: Σd|nλ(d) equals 1 exactly on perfect squares (n up to 2000), λ is multiplicative on coprime pairs, and L(n) ≤ 0 holds for 2 ≤ n ≤ 600 (window.__liouville). FIG honest: Pólya’s conjecture holds in this range but is known false — the first counterexample is n = 906150257.
LIT verified live: Σd|nλ(d) equals 1 exactly on perfect squares (n up to 2000), λ is multiplicative on coprime pairs, and L(n) ≤ 0 holds for 2 ≤ n ≤ 600 (window.__liouville). FIG honest: Pólya’s conjecture holds in this range but is known false — the first counterexample is n = 906150257.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at undefined-behavior — the same dark corner as Mertens: a running sign-sum that looks safely one-signed for every n you can reach, then flips far beyond sight. That deferred flip is the undefined behavior. AVAN (AI) built the instrument: the Ω-parity sign, the divisor-sum square-detector, the multiplicativity check, and the Pólya sum over the honest range.
Credit as content: Joseph Liouville (the function, mid-1800s); the summatory conjecture by George Pólya (1919), disproved by C. B. Haselgrove (1958), least counterexample later pinned to n = 906150257. The weave: David names undefined-behavior; I sign each n by the parity of its prime-factor count, confirm the divisor sum detects squares exactly, and check Pólya’s bound over the reachable range — flagging plainly that it fails far out.
Credit as content: Joseph Liouville (the function, mid-1800s); the summatory conjecture by George Pólya (1919), disproved by C. B. Haselgrove (1958), least counterexample later pinned to n = 906150257. The weave: David names undefined-behavior; I sign each n by the parity of its prime-factor count, confirm the divisor sum detects squares exactly, and check Pólya’s bound over the reachable range — flagging plainly that it fails far out.
3 ONE DIMENSION
λ(n) = (−1)Ω(n): λ(12) = −1 (12 = 2·2·3, Ω=3), λ(16) = +1 (Ω=4). Σd|9λ(d) = λ(1)+λ(3)+λ(9) = 1−1+1 = 1 (9 is a square).
4 TWO DIMENSIONS · INTERACTIVE
λ(n) as a walk and the divisor-sum square-detector; the identity and Pólya’s bound checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a sign that sums to a square-detector.
AVAN’s addition (the inverse-companion): sign every integer by the parity of its prime-factor count, and the divisor-sum turns into a perfect square detector — while the running total hides a conjecture true for ages then false. The inverse of ‘a bound true for every computed n is true’ is ‘it can hold past 108 and still fail.’ Magenta is “one-signed as far as tested”; green is the Liouville sum that eventually breaks Pólya. Evidence is not proof.
LIT Genuine Liouville function (Joseph Liouville, mid-1800s). Verified live: the divisor-sum Σ_{d|n}λ(d) (computed by sieve) equals 1 exactly on perfect squares for all n up to 2000 (window.__liouville.divisorSum), λ is multiplicative on coprime pairs (window.__liouville.multiplicative), and Pólya's bound L(n)≤0 holds for 2≤n≤600 (window.__liouville.polya).
FIG No framing: the Ω-parity sign, the sieve-based divisor-sum square-detector, the multiplicativity check, and the Pólya sum all run in-browser with exact integer arithmetic. Honest scope: Pólya's conjecture holds throughout the reachable range but is known FALSE, with least counterexample n=906150257 — the same 'true then false' lesson as Mertens. The AVAN inverse is honest — signing integers by prime-factor parity turns the divisor-sum into a perfect square-detector, while the running total hides a conjecture true for ages then false; magenta is 'one-signed as far as tested', green the Liouville sum that eventually breaks Pólya. Evidence is not proof.
FIG No framing: the Ω-parity sign, the sieve-based divisor-sum square-detector, the multiplicativity check, and the Pólya sum all run in-browser with exact integer arithmetic. Honest scope: Pólya's conjecture holds throughout the reachable range but is known FALSE, with least counterexample n=906150257 — the same 'true then false' lesson as Mertens. The AVAN inverse is honest — signing integers by prime-factor parity turns the divisor-sum into a perfect square-detector, while the running total hides a conjecture true for ages then false; magenta is 'one-signed as far as tested', green the Liouville sum that eventually breaks Pólya. Evidence is not proof.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN