A declared-data Rutherford check: $\alpha =1$, $\theta =\pi \mathrm{/}2$, $(\mathrm{d}\sigma \mathrm{/}\mathrm{d}\mathrm{\Omega })=4\times {10}^{-24}{\mathrm{m}}^2$, the band values $\pi b({\theta }_1{)}^2$ and the growth of the truncated total as ${\theta }_1\downarrow 0$

Speed and impact parameter: $2E\mathrm{/}\mu =2\times {10}^{-18}\mathrm{/}(2\times {10}^{-26})={10}^8{\mathrm{m}}^2{\mathrm{s}}^{-2}$, so $v_{\mathrm{\infty }}=\sqrt{{10}^8}={10}^4\mathrm{m}{\mathrm{s}}^{-1}$; then $b=J\mathrm{/}(\mu v_{\mathrm{\infty }})=4\times {10}^{-34}\mathrm{/}(2\times {10}^{-26}\cdot {10}^4)=2\times {10}^{-12}\mathrm{m}$.

Special values: at $x=\pi \mathrm{/}4$, the double-angle identity gives $0=\cos (\pi \mathrm{/}2)={\cos }^{2}x-{\sin }^{2}x$; positivity and the Pythagorean identity yield $\sin x=\cos x=1\mathrm{/}\sqrt{2}$, hence $\cot x=1$, ${\csc }^{4}x=4$ and $\arctan 1=\pi \mathrm{/}4$. At $x=\pi \mathrm{/}6$, the addition formulas give $0=\cos (3x)=4{\cos }^{3}x-3\cos x$. Since $\cos x>0$, $\cos x=\sqrt{3}\mathrm{/}2$ and $\sin x=1\mathrm{/}2$, so $\cot x=\sqrt{3}$. Applying the difference formulas to $\pi \mathrm{/}12=\pi \mathrm{/}4-\pi \mathrm{/}6$ now gives $\cot (\pi \mathrm{/}12)=(\sqrt{6}+\sqrt{2})\mathrm{/}(\sqrt{6}-\sqrt{2})=2+\sqrt{3}$. Finally continuity and positivity give $\sin (\pi \mathrm{/}2)=1$; thus $\cot (\theta \mathrm{/}2)\to 0$ as $\theta \uparrow \pi$.

Angle: $\alpha =\kappa \mathrm{/}(2Eb)=4\times {10}^{-30}\mathrm{/}(2\times {10}^{-18}\cdot 2\times {10}^{-12})=1$, so $\arctan \alpha =\pi \mathrm{/}4$ and $\mathrm{\Delta }=\pi \mathrm{/}2-\pi \mathrm{/}4=\pi \mathrm{/}4$, giving $\theta =2\arctan 1=\pi \mathrm{/}2\in (0,\pi )$, consistent with $b(\theta ){\mathrm{\mid }}_{\theta =\pi \mathrm{/}2}={\kappa }_2\cot (\pi \mathrm{/}4)=2\times {10}^{-12}\mathrm{m}=b$.

Band values: using $b(\theta )={\kappa }_2\cot (\theta \mathrm{/}2)$ and the endpoint limit ${\lim }_{\theta \uparrow \pi }b(\theta )=0$, for the band $(\pi \mathrm{/}2,\pi )$ the value is $\sigma =\pi (b(\pi \mathrm{/}2{)}^2-0^2)=\pi (2\times {10}^{-12}{)}^2=4\pi \times {10}^{-24}\approx 1.257\times {10}^{-23}{\mathrm{m}}^2$; for $(\pi \mathrm{/}3,\pi )$, $b(\pi \mathrm{/}3)=2\times {10}^{-12}\sqrt{3}$, so $\sigma =12\pi \times {10}^{-24}\approx 3.770\times {10}^{-23}{\mathrm{m}}^2$; for $(\pi \mathrm{/}6,\pi )$, $b(\pi \mathrm{/}6)=2\times {10}^{-12}(2+\sqrt{3})$, so $\sigma =\pi \cdot 4\times {10}^{-24}(2+\sqrt{3}{)}^2\approx 1.750\times {10}^{-22}{\mathrm{m}}^2$. These are exact expressions in $\pi$ and $\sqrt{3}$ evaluated to four significant digits; all are finite.

Density at $\pi \mathrm{/}2$: ${\kappa }_4=\kappa \mathrm{/}(4E)={10}^{-12}\mathrm{m}$ and ${\csc }^4(\pi \mathrm{/}4)=(\sqrt{2}{)}^4=4$, so $\frac{\mathrm{d}\sigma }{\mathrm{d}\mathrm{\Omega }}(\pi \mathrm{/}2)={\kappa }_4^2\cdot 4=4\times {10}^{-24}{\mathrm{m}}^2$, with dimension ${\mathrm{\ell }}^2$ (Dimensions and SI unit conventions of the non-relativistic classical mechanics framework).

Small-angle growth: the derivative definition and [L3] give $\sin x\mathrm{/}x=(\sin x-\sin 0)\mathrm{/}(x-0)\to {\sin }^{\mathrm{\prime }}(0)=1$ as $x\downarrow 0$, and continuity gives $\cos x\to 1$. Since $\sin x>0$ on $(0,\pi \mathrm{/}2)$, quotient limits yield $x\cot x=\cos x\mathrm{/}(\sin x\mathrm{/}x)\to 1$. With $x={\theta }_1\mathrm{/}2$, this proves $\cot ({\theta }_1\mathrm{/}2)\sim 2\mathrm{/}{\theta }_1$, so $\pi b({\theta }_1{)}^2=\pi {\kappa }_2^2{\cot }^2({\theta }_1\mathrm{/}2)\sim 4\pi {\kappa }_2^2\mathrm{/}{\theta }_1^2=16\pi \times {10}^{-24}\mathrm{/}{\theta }_1^2$ in ${\mathrm{m}}^2$. At ${\theta }_1={10}^{-2}$ rad, $b({\theta }_1)=2\times {10}^{-12}\cot (5\times {10}^{-3})\approx 4.000\times {10}^{-10}\mathrm{m}$ and $\sigma \approx 5.026\times {10}^{-19}{\mathrm{m}}^2$. The positive asymptotic coefficient implies unbounded band values as ${\theta }_1\downarrow 0$, agreeing with Repulsive Coulomb scattering in the model class: global non-colliding motion, the deflection map $\theta =2\arctan (\kappa \mathrm{/}(2Eb))$, the Rutherford density $(\kappa \mathrm{/}4E{)}^2{\csc }^4(\theta \mathrm{/}2)$, finite angular cutoffs and the divergent ideal total clause 4.

Reading: every number above was obtained by substituting the declared model data into the model formulas; no measurement, uncertainty, detector acceptance or sampling model is supplied, so the computation is a check of the model's arithmetic and not a comparison with nature. In particular the divergence of the truncated total at forward angles is a feature of the declared idealisation, and any empirical statement would require a separate experiment item with measured inputs and uncertainties.