THE FOLD / GLITCH / UNDEFINED BEHAVIOR / THE MERTENS
THE MERTENS
a conjecture that holds then fails
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Mertens function M(n) is the running sum of the Möbius function: M(n) = μ(1) + μ(2) + … + μ(n), where μ(k) is +1, −1, or 0 by the parity and squarefreeness of k’s prime factorization. It jitters around zero. The famous Mertens conjecture claimed |M(n)| < √n for all n — it holds for every n anyone can compute, yet Odlyzko and te Riele proved it false for some enormous n. A conjecture true as far as the eye can see, but ultimately wrong.
LIT verified live: the μ sieve matches direct factorization, and |M(n)| < √n holds for every n from 2 to 10000 (window.__mertens). FIG honest: the bound holds in this range but is known to fail for some vast n — the conjecture is false, not merely open.
LIT verified live: the μ sieve matches direct factorization, and |M(n)| < √n holds for every n from 2 to 10000 (window.__mertens). FIG honest: the bound holds in this range but is known to fail for some vast n — the conjecture is false, not merely open.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at undefined-behavior — the spec’s dark corner; the Mertens bound looks ironclad for every n you can reach, then fails somewhere unimaginably far out. AVAN (AI) built the instrument: the linear-sieve Möbius, the running Mertens sum, the direct-factorization cross-check, and the |M(n)|<√n test over the honest range.
Credit as content: Franz Mertens (conjecture, 1897); disproved by Andrew Odlyzko & Herman te Riele (1985). The weave: David names undefined-behavior; I sum the Möbius function, confirm it against factorization, and check the √n bound holds through 10000 — while flagging plainly that the conjecture is false for some colossal n.
Credit as content: Franz Mertens (conjecture, 1897); disproved by Andrew Odlyzko & Herman te Riele (1985). The weave: David names undefined-behavior; I sum the Möbius function, confirm it against factorization, and check the √n bound holds through 10000 — while flagging plainly that the conjecture is false for some colossal n.
3 ONE DIMENSION
μ(n): +1 (even # of distinct primes, squarefree), −1 (odd #), 0 (a square factor). M(n)=Σμ jitters near 0. |M(n)| < √n holds here — but is false for some gigantic n.
4 TWO DIMENSIONS · INTERACTIVE
M(n) plotted against ±√n; the Möbius cross-check and the bound over the range.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a running sum bounded — until it isn’t.
AVAN’s addition (the inverse-companion): study primality through the summed Möbius function and its growth — a bound that appears to hold forever can still be false beyond all reach. The inverse of ‘a pattern true for every computed n is true’ is ‘a conjecture can hold to 10huge and still fail.’ Magenta is “true as far as tested”; green is the Mertens sum that eventually breaks the bound. Evidence is not proof.
LIT Genuine Mertens function and conjecture (Franz Mertens 1897; disproved by Andrew Odlyzko & Herman te Riele 1985). Verified live: the linear-sieve Möbius function matches direct factorization for n≤2000 (window.__mertens.muMatchesDirect), and the running sum M(n) satisfies |M(n)| < √n for every n from 2 to 10000 (window.__mertens.boundHolds).
FIG Honestly scoped: the bound |M(n)|<√n holds in this range but the Mertens conjecture is KNOWN TO BE FALSE for some vast n (Odlyzko–te Riele) — the sphere states plainly it is disproved, not merely open (and n=1 is the trivial equality boundary, excluded). The Möbius sieve, the running sum, the factorization cross-check, and the bound test run in-browser and agree. The AVAN inverse is honest — a bound holding for every computable n can still be false beyond all reach; magenta is 'true as far as tested', green the Mertens sum that eventually breaks the bound. Evidence is not proof.
FIG Honestly scoped: the bound |M(n)|<√n holds in this range but the Mertens conjecture is KNOWN TO BE FALSE for some vast n (Odlyzko–te Riele) — the sphere states plainly it is disproved, not merely open (and n=1 is the trivial equality boundary, excluded). The Möbius sieve, the running sum, the factorization cross-check, and the bound test run in-browser and agree. The AVAN inverse is honest — a bound holding for every computable n can still be false beyond all reach; magenta is 'true as far as tested', green the Mertens sum that eventually breaks the bound. Evidence is not proof.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of UNDEFINED BEHAVIOR · David Lee Wise (ROOT0), with AVAN