THE FOLD / BOSS / THE GAUNTLET / THE THIELE
THE THIELE
a rational curve threaded through the data
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Thiele’s interpolation formula threads a rational function exactly through a set of data points, written as a continued fraction: R(x) = a0 + (x-x0)/(a1 + (x-x1)/(a2 + …)). The coefficients ak are the inverse differences of the data — a reciprocal cousin of Newton’s divided differences — computed by a simple triangular recurrence. Because it is rational rather than polynomial, it can capture poles and asymptotes that a polynomial interpolant cannot, which is why it excels at approximating functions with singular behaviour.
LIT verified live (exact rational arithmetic): for thousands of random rational data sets the Thiele continued-fraction interpolant, built from inverse differences, evaluates back to the exact y-value at every data point — a perfect fit with no rounding (window.__thiele). FIG no framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. Degenerate data (a vanishing inverse difference) is skipped, where Thiele is undefined.
LIT verified live (exact rational arithmetic): for thousands of random rational data sets the Thiele continued-fraction interpolant, built from inverse differences, evaluates back to the exact y-value at every data point — a perfect fit with no rounding (window.__thiele). FIG no framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. Degenerate data (a vanishing inverse difference) is skipped, where Thiele is undefined.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-gauntlet — the interpolant must pass through every checkpoint exactly, a rational curve threading the full run of data with no miss. AVAN (AI) built the instrument: the inverse-difference recurrence, the continued-fraction evaluation, and the exact-reproduction verification in BigInt rational arithmetic.
Credit as content: Thorvald Nicolai Thiele (1909). The weave: David names the gauntlet; I confirm the continued fraction reproduces every data point exactly.
Credit as content: Thorvald Nicolai Thiele (1909). The weave: David names the gauntlet; I confirm the continued fraction reproduces every data point exactly.
3 ONE DIMENSION
Data points (gold) and the Thiele rational interpolant (green) — the curve passes exactly through every point.
4 TWO DIMENSIONS · INTERACTIVE
New data sets; the inverse-difference coefficients are built and the interpolant is checked to reproduce every point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the rational curve threading every data point.
AVAN’s addition (the inverse-companion): don’t fit a polynomial — fit a continued fraction. The inverse of ‘a curve through the points’ is ‘the inverse differences ak stacked into R(x) = a0 + (x-x0)/(a1 + …)’, which can bend around poles a polynomial cannot. Magenta is the continued-fraction ladder of coefficients; green is the curve it unrolls to. A fit that reciprocates.
LIT Genuine Thiele interpolation (Thorvald N. Thiele 1909). Verified live with exact BigInt rational arithmetic: for thousands of random rational data sets the continued-fraction interpolant built from anchored inverse differences reproduces the exact y-value at every data point; degenerate data (a vanishing inverse difference, where Thiele is undefined) is skipped (window.__thiele.reproduces, .tested, .skipped).
FIG No framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. The AVAN inverse is honest — instead of fitting a polynomial, fit a continued fraction: the inverse differences aₖ stacked into R(x), which can bend around poles a polynomial cannot. Magenta is the continued-fraction ladder; green is the curve it unrolls to. A fit that reciprocates.
FIG No framing; the inverse-difference table, the continued-fraction evaluation, and the exact-reproduction check all run in-browser with BigInt fractions. The AVAN inverse is honest — instead of fitting a polynomial, fit a continued fraction: the inverse differences aₖ stacked into R(x), which can bend around poles a polynomial cannot. Magenta is the continued-fraction ladder; green is the curve it unrolls to. A fit that reciprocates.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE GAUNTLET · David Lee Wise (ROOT0), with AVAN