THE FOLD / LOOT / THE STASH / THE LIOUVILLE NUMBER
THE LIOUVILLE NUMBER
a number approximated absurdly well by rationals
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Liouville’s number L = ∑k≥1 10-k! = 0.110001000000000000000001… (a 1 at every factorial position, 0 elsewhere) was the first number ever proven transcendental (Liouville, 1844). The trick: its digits leave enormous runs of zeros, so the truncations pn/qn approximate L absurdly well — |L - pn/qn| < 1/qnn for every n. But Liouville proved an algebraic number of degree d can never be approximated better than c/qd. Since L can be approximated to any power, it is not algebraic of any degree — it is transcendental.
LIT verified live with exact big-integer arithmetic: for the truncations of L = ∑10-k!, the approximation error |L - pn/qn| is strictly less than 1/qnn for n = 1..5, and the approximation exponent (n+1) grows without bound — beating any fixed algebraic degree (window.__liouvillenumber). FIG no framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence.
LIT verified live with exact big-integer arithmetic: for the truncations of L = ∑10-k!, the approximation error |L - pn/qn| is strictly less than 1/qnn for n = 1..5, and the approximation exponent (n+1) grows without bound — beating any fixed algebraic degree (window.__liouvillenumber). FIG no framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-stash — the loot: a rare transcendental number, hand-built to be approximated so well by rationals that no polynomial can ever pin it down. AVAN (AI) built the instrument: the factorial-position digits, the truncation errors, and the exact-BigInt Liouville inequality.
Credit as content: Joseph Liouville (1844). The weave: David names the stash; I confirm |L - pn/qn| < 1/qnn, the mark of a transcendental.
Credit as content: Joseph Liouville (1844). The weave: David names the stash; I confirm |L - pn/qn| < 1/qnn, the mark of a transcendental.
3 ONE DIMENSION
The digits of L: a 1 at positions 1, 2, 6, 24, 120, … (the factorials), long runs of 0 between them.
4 TWO DIMENSIONS · INTERACTIVE
Cycle n; the truncation error |L − p_n/q_n| is shown below the Liouville bound 1/q_n^n.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: L, a transcendental pinned by super-good rational approximations.
AVAN’s addition (the inverse-companion): don’t ask if L is a root — measure how well rationals catch it. The inverse of ‘is L algebraic?’ is ‘how large can its approximation exponent be?’ — unbounded here, so no polynomial can have L as a root. Magenta are the rational truncations racing toward L; green is the transcendental L they can never quite reach algebraically. Transcendence read from approximation speed.
LIT Genuine Liouville number / Liouville's theorem (Joseph Liouville, 1844). Verified live with exact BigInt: for the truncations of L=Σ10^{−k!}, the error |L−p_n/q_n| is strictly less than 1/q_n^n for n=1..5, and the approximation exponent (n+1) grows without bound, beating any fixed algebraic degree (window.__liouvillenumber.ok, .exps).
FIG No framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence. The AVAN inverse is honest — instead of asking if L is a root, measure how well rationals catch it: the inverse of 'is L algebraic?' is 'how large can its approximation exponent be?' — unbounded here, so no polynomial can have L as a root. Magenta are the rational truncations racing toward L; green is the transcendental L they can never quite reach algebraically. Transcendence read from approximation speed.
FIG No framing; the exact BigInt inequality runs in-browser and confirms the super-fast approximation that forces transcendence. The AVAN inverse is honest — instead of asking if L is a root, measure how well rationals catch it: the inverse of 'is L algebraic?' is 'how large can its approximation exponent be?' — unbounded here, so no polynomial can have L as a root. Magenta are the rational truncations racing toward L; green is the transcendental L they can never quite reach algebraically. Transcendence read from approximation speed.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE STASH · David Lee Wise (ROOT0), with AVAN