= Solution
Start with a fast massive projectile following an approximately straight path. Its time-dependent <electric field> perturbs bound absorber <electrons>. At large impact parameter, treating the perturbation to first order and summing excitation and <ionization> transition probabilities gives an <energy transfer> weighted by the absorber's oscillator strengths. The oscillator-strength sum rule reduces the leading coefficient to the number of target <electrons>, while their logarithmically weighted excitation energies define $I$.
The classical impulse picture explains the logarithm: a transverse impulse scales as $1/(bv)$, so transferred <energy> scales as $1/(b^2v^2)$; multiplying by the number of encountered <electrons> in $2\pi b\,db\,dx$ gives $db/b$. Quantum <scattering> replaces ad hoc short- and long-distance cutoffs by recoil kinematics and bound-state excitation. Relativistic projectile kinematics supplies $T_{\max}$ and the $-\beta^2$ term. Summation over many independent encounters gives the mean <stopping power>, rather than the <energy-loss straggling> of individual tracks.
The leading approximation assumes $M\gg m_e$, a sufficiently fast projectile for perturbation theory, a largely fixed projectile charge, and weak correlations between collisions. At low <velocity>, electron binding, charge exchange and <nuclear stopping> become important; at high <velocity>, polarization screens distant interactions and gives the density correction. The projectile's <bremsstrahlung> is suppressed by its large <mass> and is excluded from the <collision stopping power>.
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