The Universal Smoothing Bridge

We construct a formal bridge between the space of distributions \(\mathcal{D}'\) and the Latent framework by proving that mollification provides a universal smoothing adapter. For any distribution \(T\), mollifier \(\varphi\), smoothing parameter \(\varepsilon > 0\), and truncation order \(N\), the Latent representation \(\Lambda_N(T * \varphi_\varepsilon)\) satisfies

\[d(T, \Lambda_N(T \varphi_\varepsilon)) \leq \varepsilon + \frac{1}{\rho(T \varphi_\varepsilon)^N},\]

where \(\rho\) is the Latent Number of the smoothed function. We prove that derivatives commute with the full pipeline: \(\partial(\Lambda_N(T \varphi_\varepsilon)) = \Lambda_N((\partial T) \varphi_\varepsilon)\). As a concrete application, we show that the derivative of the Latent-compressed smoothed Heaviside step function equals the Latent-compressed mollifier: \(\partial(\Lambda_N(H * \varphi_\varepsilon)) = \Lambda_N(\varphi_\varepsilon)\). The theory encompasses higher-order derivative commutation (by induction on \(\mathbb{N}\)), mollifier composition, quantitative double-smoothing error bounds via the triangle inequality, Fourier characterization as low-pass filtering, heat kernel smoothing, Colombeau distribution products with Leibniz rule, and PDE existence via Friedrichs mollifiers. All 35 theorems are formally verified in the proof kernel (exportable to Lean 4), with 106 declarations and 0 errors.