THE FOLD / GLITCH / DIVIDE BY ZERO / THE PADÉ
THE PADÉ
a rational that captures the poles a polynomial cannot
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A Padé approximant replaces a power series with a ratio of two polynomials P(x)/Q(x) chosen so its own Taylor expansion agrees with the original series to the highest possible order, m+n, for a numerator of degree m and denominator of degree n. Because it has a denominator, it can do something a Taylor polynomial never can: capture poles. Where a Taylor series diverges the instant you pass its radius of convergence, the Padé approximant sails on — its denominator’s roots sit right where the true function blows up. It is the workhorse behind function libraries, control theory, and resummation of divergent series.
LIT verified live: the [3/3] Padé approximant of ex reproduces the Taylor coefficients through order 6 exactly, and at x=1 its error (≈3e-5) is an order of magnitude smaller than the degree-6 Taylor polynomial’s (≈2e-4) (window.__pade). FIG no framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser.
LIT verified live: the [3/3] Padé approximant of ex reproduces the Taylor coefficients through order 6 exactly, and at x=1 its error (≈3e-5) is an order of magnitude smaller than the degree-6 Taylor polynomial’s (≈2e-4) (window.__pade). FIG no framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at divide-by-zero — the very thing a polynomial fears, the Padé embraces: its denominator is allowed to hit zero, and it places those zeros exactly at the function’s poles. AVAN (AI) built the instrument: the Padé coefficient solve, the re-expansion match, and the accuracy comparison against Taylor.
Credit as content: Henri Padé (1892); anticipated by Frobenius and Jacobi. The weave: David names the divide-by-zero; I confirm the rational matches the series to order m+n and beats the Taylor polynomial.
Credit as content: Henri Padé (1892); anticipated by Frobenius and Jacobi. The weave: David names the divide-by-zero; I confirm the rational matches the series to order m+n and beats the Taylor polynomial.
3 ONE DIMENSION
f (cyan), its Padé approximant (green), and the Taylor polynomial (magenta): near a pole the Taylor diverges while Padé holds.
4 TWO DIMENSIONS · INTERACTIVE
Cycle the target function; the Padé coefficients match the series to order m+n and its max error beats the Taylor polynomial.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the Padé rational, tracking f through the pole.
AVAN’s addition (the inverse-companion): don’t just add powers — divide by them. The inverse of ‘a polynomial that can only grow’ is ‘a denominator whose zeros land on the function’s poles, so the ratio survives where the sum explodes.’ Magenta is the Taylor polynomial diverging; green is the Padé holding through the singularity. Reach past the radius by dividing.
LIT Genuine Padé approximant (Henri Padé, 1892; anticipated by Frobenius and Jacobi). Verified live: the [3/3] Padé of eˣ re-expands to the Taylor coefficients through order 6 (worst ~1e-17) and at x=1 has error ~3e-5, smaller than the degree-6 Taylor polynomial's ~2e-4; the ln(1+x) match is also exact (window.__pade.matchesSeries, .beatsTaylor).
FIG No framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser. The AVAN inverse is honest — instead of only adding powers, divide by them: a denominator whose zeros land on the function's poles, so the ratio survives where the sum explodes. Magenta is the Taylor polynomial diverging; green is the Padé holding through the singularity. Reach past the radius by dividing.
FIG No framing; the Padé linear solve, the series re-expansion, and the error comparison all run in-browser. The AVAN inverse is honest — instead of only adding powers, divide by them: a denominator whose zeros land on the function's poles, so the ratio survives where the sum explodes. Magenta is the Taylor polynomial diverging; green is the Padé holding through the singularity. Reach past the radius by dividing.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of DIVIDE BY ZERO · David Lee Wise (ROOT0), with AVAN