It is shown that the fundamental standard hadron cascade diffusion equation in the Mellin transform space is not rigorously correct because of the inconsistent double energy integral evaluation which generates the function $<\eta^{s}>$ with its associated parametizations. To ensure an exact basic working equation, the Faltung integral representation is introduced which has the elasticity distribution function $u(\eta)$ as the only fundamental input function and $<\eta^{s}>$ is just the Mellin transform of $u(\eta)$. This Faltung representation eliminates standard phenomenological parameters which serve only to mislead the physics of cascade. The exact flux transform equation is solved by the method of characteristics, and the hadron flux in real space is obtained by the inverse transform in terms of the simple and essential residues. Since the essential residues are given by the singularities in the elasticity distribution and particle production transforms that appear in the exponentials, these functions should not be parametized arbitrarily to avoid introducing non-physical essential residues. This Faltung formulation is applied to the charged hadron integrated energy spectra of halo events detected at Mt. Chacaltaya by the Brazil-Japan Collaboration, where the incoming flux at the atmospheric boundary is a single energetic nucleon.
