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Absolute error on a logarithmic scale from 0.1 ppb to 10,000 ppm. Ramanujan I reaches 1400 ppm while the SCOR champion stays below 0.01 ppb.
Top band: lowest error among all methods. Bottom band: best excluding R6 (shows the competition). At low eccentricity all errors are below 10−12 ppm, so the “winner” is just floating-point noise.
Peak (max): guaranteed worst-case bound. RMS: typical performance. Mean: average error. Optimized for peak error (minimax), so not optimal under other norms.
Positive = overestimate, negative = underestimate. R2/3exp swings to ±83 ppb. Sub-ppb methods are invisible at this scale.
Y clamped to ±1.5 ppb. Error equioscillates at N+1 Chebyshev points. SCOR champion (gold) has the tightest band at ±0.007 ppb.
Each method is tested at 998 eccentricities from 0.001 to 0.998. The table shows the single worst error encountered anywhere (the L∞ norm). This is the guaranteed accuracy bound. See the Error Metrics table above for RMS and mean.
The key idea: Ramanujan II’s formula evaluates to 22/7 when the ellipse degenerates (b→0). We correct this by adding Padé layers that pin the degenerate value to successively deeper rational approximations of 4/π from its continued fraction [3; 7, 15, 1, 292, 1, 1, ...]. Each layer matches 5 more Taylor coefficients of the exact perimeter series. The remaining error (the “budget” Tk) shrinks by orders of magnitude at each level, and is exhausted by a sum of exponentials.
Each tower level adds a [3/2] Padé correction at h5k, matching 5 more Taylor coefficients and pinning the endpoint R(1) = P(a,0)/(π·2a) to a deeper convergent of 4/π from the continued fraction of π = [3; 7, 15, 1, 292, ...].
The exponential correction exhausts the tower remainder budget Tk = 1 − qkπ/pk across N basis functions.
| Level | Pinned to | Tk | Best nexp | Best Error |
|---|---|---|---|---|
| R2 | 22/7 | 4.0 × 10−4 | 3 | 0.083 ppm |
| R3 | 355/113 | 8.5 × 10−8 | 15 | 1.43 ppb |
| R4 | 103993/33102 | 1.8 × 10−10 | 20 | 0.492 ppb |
| R5 | 104348/33215 | 1.1 × 10−10 | 20 | 0.508 ppb |
| R6 | 208341/66317 | 3.9 × 10−11 | 16 | 0.018 ppb ☆ |
| R6 SCOR | 208341/66317 | 3.9 × 10−11 | 16+3log | 0.007 ppb ☆☆ |
For any fixed set of rates, the (N+2)-dimensional optimization over N coefficient ratios + λ + q separates into:
1. An inner linear Chebyshev problem in the N coefficients (solved exactly for each (λ, q) via LP)
2. An outer 2D search over (λ, q)
This reduces the optimization from 17D to 2D, making exhaustive rate search over C(50,4) ≈ 230K combinations feasible.
Every ellipse has two foci. From any point on the curve, draw lines to both foci: their lengths r1 and r2 sum to 2a, but their RATIO varies around the ellipse. The “certainty” R = c/ℓ measures how sharply the foci resolve each other, while the “uncertainty” U averages the angular disagreement. Their ratio is ALWAYS π/2, regardless of eccentricity. A universal geometric constant.
For ANY ellipse with eccentricity e in (0, 1), foci at distance c = ae from center, and semi-latus rectum l = b²/a:
where R = c/l = e/(1-e²) is the focal resolution and U = ⟨|M·D|⟩ = R·κ is the sweep uncertainty, with κ = 2/π.
The ellipse's two foci pin each other's location to EXACTLY π/2 natural units per unit of angular sweep disagreement.
The ratio R/U = π/2 = 1.5708... is CONSTANT for all eccentricities. The red dot shows the current eccentricity.
Maximum at α=0 (periapsis). Zero at α=90° (latus rectum). Mean = R·κ = R·(2/π). Ceiling = c/l = R.
An ellipse can be thought of relativistically: the Doppler factor Δ(θ) = r2/r1 at each point acts like a redshift/blueshift. The rapidity η = atanh(e) is the hyperbolic “speed” of the ellipse. The information surplus If − Hu = log2(π/2) ≈ 0.6514 bits is the irreducible geometric advantage of having two foci rather than one center.
The focal information ALWAYS exceeds the uncertainty entropy by exactly log2(π/2) bits, the irreducible geometric advantage of having two foci.
The constant κ = 2/π appears in five independent contexts related to the ellipse. This is not a coincidence; all five are consequences of the R/U = π/2 duality.
| Context | Identity | κ acts as |
|---|---|---|
| Focal duality | U·(l/c) = κ | coupling |
| Gauss connection | Γ(1)²/(Γ(3/2)Γ(1/2)) | amplitude |
| Flat-limit deficiency | P(a,0)/(2πa) = κ | limit ratio |
| MEPB coefficient | 7π/704 = 7/(352κ) | scale factor |
| Angular average | ⟨|cosα|⟩ = κ | mean value |
Unit closure: κ × (R/U) = (2/π)(π/2) = 1.