  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.
