From Measure to Flux: Energy-Conjugate Bookkeeping and Exact Semidiscrete Scattering in Finite Radial Wave Models

We study a finite spatial window on the radial wave equation---the setting in which a bounded observer accounts for wave energy on a Schwarzschild background---and develop its discrete bookkeeping to the point of exact solvability. For the standard second-order semidiscretization with trapezoid endpoint weights, we construct the boundary flux conjugate to the discrete energy, extend it to a one-parameter characteristic reflection family with an exact energy identity, and derive the exact per-mode reflection coefficient of the family: $r(k) = (\cos(kh/2)-\alpha)/(\cos(kh/2)+\alpha)$. The discrete boundary is therefore the continuum characteristic boundary with a secant-dressed impedance, $\alpha_{eff}(k)=\alpha\sec(kh/2)$, equivalently a Möbius composition of the target reflection coefficient with one additional factor $β_k=\tan^2(kh/4)$. From the exact symbol we obtain, for Gaussian packets: an exact odd reflected-power law whose first two defect orders share the cubic kernel $R(1-R^2)$, with extremum at $1/\sqrt3$; the packet impedance coefficient $\kappa=\tfrac{3}{16}(h/\sigma)^2[1-\tfrac{5}{48}(h/\sigma)^2]$; the continuum-transparent target member's reflected-power floor $\rho_0=\tfrac{15}{1024}(h/\sigma)^4$; for the leading continuum Gaussian energy weight, a spectral mean--variance decomposition that is exactly $3/5$--$2/5$; and a $1/|R|$ reconstruction law near the transparent point. The leading packet coefficients $\kappa$ and $\rho_0$ close against frozen solver receipts at relative discrepancies $4.71\times10^{-9}$ and $1.41\times10^{-8}$; higher-order and reconstruction-law checks are reported at their own independently tabulated accuracies. Finally, we close the gap between the semidiscrete theory and the fully discrete solver: a boundary-parameter-dependent timestep policy converts the classical dissipation of fourth-order Runge--Kutta integration into a signed, odd-in-$R$ reflected-power bias, whose scaling law $h^5\sigma^{-6}$ we derive, and which a pre-registered timestep-halving falsifier removes to $96.407\%$ of the observed residual, meeting a locked target to $0.41\%$. The complete attribution chain---policy $\to$ branch-dependent dissipation $\to$ odd bias $\to$ matched-pair closure $\to$ locked falsifier---was not found in the sources examined.