THE FOLD / GRIND / THE HOT LOOP / THE WALLIS
THE WALLIS
pi milled from fractions
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In 1656 John Wallis wrote π/2 as an infinite mill of fractions: (2·2)/(1·3) · (4·4)/(3·5) · (6·6)/(5·7) … — every factor slightly more than 1, grinding forever toward the circle constant. The convergence is famously slow (error ≈ π/8n: ten thousand factors buy you four digits), and the product has two secret identities: the partial products equal (4ⁿ/C(2n,n))²/(2n+1) exactly — central binomial coefficients in disguise — and in 2015 Friedmann and Hagen discovered the entire formula hiding in the quantum hydrogen atom: it emerges from variational estimates of energy levels, 359 years after Wallis.
LIT verified live: 50,000 factors landing at 1.5707885 vs π/2 = 1.5707963; the binomial identity matching the direct product to 10⁻¹⁰ for every n ≤ 200 (two independent routes); and the error law n·(π/2−Wₖ) → π/8 measured to four decimals (window.__wallis). FIG honest boundary: the hydrogen-atom derivation is cited (Friedmann–Hagen 2015, J. Math. Phys.); what runs here is the product, its binomial double, and its error law.
LIT verified live: 50,000 factors landing at 1.5707885 vs π/2 = 1.5707963; the binomial identity matching the direct product to 10⁻¹⁰ for every n ≤ 200 (two independent routes); and the error law n·(π/2−Wₖ) → π/8 measured to four decimals (window.__wallis). FIG honest boundary: the hydrogen-atom derivation is cited (Friedmann–Hagen 2015, J. Math. Phys.); what runs here is the product, its binomial double, and its error law.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-hot-loop — the grind: a loop body of one multiplication, iterated fifty thousand times, each pass shaving the error by almost nothing — and the total grinding out π. AVAN (AI) built the instrument: the twin-route product engine and the error-law meter.
Credit as content: John Wallis (1656); Friedmann & Hagen (2015). The weave: David names the hot loop; I run it two ways and clock its exact rate of approach.
Credit as content: John Wallis (1656); Friedmann & Hagen (2015). The weave: David names the hot loop; I run it two ways and clock its exact rate of approach.
3 ONE DIMENSION
The factors — each barely above 1, the product crawling to π/2.
4 TWO DIMENSIONS · INTERACTIVE
Crank the mill; the running product and its error-law prediction track together.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the fraction mill turning, π accumulating.
AVAN’s addition (the inverse-companion): don’t just run the mill — ask what else compiles to it. The inverse of ‘a formula for π’ is ‘π’s formula appearing where nobody ordered it’: central binomials, and — three centuries late — the hydrogen atom’s energy levels. Magenta is the crawl (four digits per ten thousand factors); green is the same object surfacing in three unrelated costumes. Constants don’t have one formula; they have a gravitational field.
LIT Genuine Wallis product (Wallis 1656; Friedmann & Hagen 2015 hydrogen derivation cited). Verified live: 50k-factor convergence; binomial identity route ≡ direct product for n ≤ 200; error law n·ε → π/8 to 1e-4 (window.__wallis.ok).
FIG Honest boundary — the hydrogen-atom derivation is cited; the product, its binomial double, and its error law run here. The AVAN inverse — don't just run the mill, ask what else compiles to it: central binomials, and the hydrogen atom's energy levels. Magenta is the crawl; green is the same object surfacing in three unrelated costumes. Constants don't have one formula; they have a gravitational field.
FIG Honest boundary — the hydrogen-atom derivation is cited; the product, its binomial double, and its error law run here. The AVAN inverse — don't just run the mill, ask what else compiles to it: central binomials, and the hydrogen atom's energy levels. Magenta is the crawl; green is the same object surfacing in three unrelated costumes. Constants don't have one formula; they have a gravitational field.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE HOT LOOP · David Lee Wise (ROOT0), with AVAN