THE FOLD / CO-OP / THE PUSH / THE CATALAN CONSTANT
THE CATALAN CONSTANT
a mysterious constant reached two ways
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Catalan’s constant G ≈ 0.9159655942 is one of the famous ‘mystery’ constants of mathematics — nobody has proved whether it is irrational. It has a simple series, G = ∑k≥0 (-1)k/(2k+1)² = 1 - 1/9 + 1/25 - 1/49 + … (the value of the Dirichlet beta function at 2). It also equals a clean integral, G = ∫01 arctan(x)/x dx, and shows up in lattice statistics, combinatorics, and the volume of hyperbolic ideal tetrahedra. Two very different computations — an alternating sum and an integral — land on the same number.
LIT verified live: the alternating series ∑(-1)k/(2k+1)² and the integral ∫01 arctan(x)/x dx both converge to the same value, matching the known constant 0.9159655942 (window.__catalanconstant). FIG no framing; the series sum and the numerical integral are computed by different routes in-browser and agree.
LIT verified live: the alternating series ∑(-1)k/(2k+1)² and the integral ∫01 arctan(x)/x dx both converge to the same value, matching the known constant 0.9159655942 (window.__catalanconstant). FIG no framing; the series sum and the numerical integral are computed by different routes in-browser and agree.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-push — the co-op merge: a slow alternating sum and a smooth integral push in from opposite directions and meet at the same mysterious constant. AVAN (AI) built the instrument: the series sum, the arctan integral, and their agreement on G.
Credit as content: Eugène Catalan (1865). The weave: David names the merge; I confirm the series and the integral both give Catalan’s constant.
Credit as content: Eugène Catalan (1865). The weave: David names the merge; I confirm the series and the integral both give Catalan’s constant.
3 ONE DIMENSION
The alternating series terms 1, −1/9, 1/25, … and the partial sums closing in on G ≈ 0.91597.
4 TWO DIMENSIONS · INTERACTIVE
Add terms; the series partial sum and the integral ∫₀¹ arctan(x)/x dx both approach the same G.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: Catalan's constant G, reached from two directions.
AVAN’s addition (the inverse-companion): don’t trust one route — cross two. The inverse of ‘the alternating series ∑(-1)k/(2k+1)²’ is ‘the integral ∫01 arctan(x)/x dx’, two computations meeting at the same G. Magenta are the series terms and the arctan curve; green is the constant they both reach. One constant, two witnesses.
LIT Genuine Catalan's constant (Eugène Catalan, 1865). Verified live: the alternating series Σ(−1)^k/(2k+1)² (200000 terms) and the numerical integral ∫₀¹ arctan(x)/x dx converge to the same value, matching the known G=0.9159655942 (window.__catalanconstant.ser, .intg, .agree, .refOk).
FIG No framing; the series sum and the numerical integral are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of trusting one route, cross two: the inverse of 'the alternating series Σ(−1)^k/(2k+1)²' is 'the integral ∫₀¹ arctan(x)/x dx', two computations meeting at the same G. Magenta are the series terms and the arctan curve; green is the constant they both reach. One constant, two witnesses.
FIG No framing; the series sum and the numerical integral are computed by different routes in-browser and agree. The AVAN inverse is honest — instead of trusting one route, cross two: the inverse of 'the alternating series Σ(−1)^k/(2k+1)²' is 'the integral ∫₀¹ arctan(x)/x dx', two computations meeting at the same G. Magenta are the series terms and the arctan curve; green is the constant they both reach. One constant, two witnesses.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN