Optical theorem; Rayleigh limit; lossless absorbs nothing

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Reference: optical theorem; Rayleigh limit; lossless absorbs nothing
Kind: analytic: a closed form, limit or conservation law: independent of any code
Calculated: Q_ext, Q_abs
Agreement: 1e-12 / 1e-7

A 50 nm silica core with a 5 nm gold shell (n = 0.27 + 2.93i) in air: extinction and scattering two ways The extinction from the coefficients against 4π/k² Re S(0), the forward amplitude; the scattering from the coefficients against the far-field pattern integrated over 4001 angles (the trapezoid rule limits that pair to about 1e-7). The fixed gold index keeps the comparison about the method, not a material.

Q_ext, sum over orders (PyStratify) Q_sca, sum over orders (PyStratify) Q_ext, optical theorem (forward amplitude) Q_sca, integrated pattern

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largest relative difference: 9.55×10⁻⁸
PyStratify the lines, the reference the circles

A small sphere of m = 1.5 + 0.2i: the electric dipole coefficient against Rayleigh's Rayleigh's a₁ = −(2i/3) x³ (m² − 1)/(m² + 2) is the first term in x, so divided by x³ it is a constant: PyStratify meets it as x → 0 and leaves it as the sphere grows (11% at x = 1); the difference falls as x² (1e-7 at x = 1e-4 in the test).

|a₁| / x³, PyStratify Rayleigh: (2/3) |(m² − 1)/(m² + 2)|

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relative difference for x ≤ 0.01: 2.26×10⁻⁶
PyStratify the lines, the reference the circles

Random lossless eight-shell spheres in water: what they absorb Eight shells of random real indices 1.2–3.5: absorption is the difference of extinction and scattering, so anything above rounding would be energy the recursion invents; it stays below 1e-12 (the test's bound) to x = 300, growing slowly with the number of orders summed; floored at 1e-17 for the logarithmic axis.

|Q_abs| / Q_ext, PyStratify

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largest |Q_abs| / Q_ext: 8.51×10⁻¹³

Run it yourself: sphere_checks.py · test_physics.py · figures.py · SPHERE_CHECKS.md on GitHub, PyStratify 97e52a9