  This is the first of a series of detailed papers on string amplitudes with
highly excited strings (HES). In the present paper we construct a generating
function for string amplitudes with generic HES vertex operators using a
fixed-loop momentum formalism. We generalise the proof of the chiral splitting
theorem of D'Hoker and Phong to string amplitudes with arbitrary HES vertex
operators (with generic KK and winding charges, polarisation tensors and
oscillators) in general toroidal compactifications
$\mathcal{E}=\mathbb{R}^{D-1,1}\times \mathbb{T}^{D_{\rm cr}-D}$ (with generic
constant K\"ahler and complex structure target space moduli, background
Kaluza-Klein (KK) gauge fields and torsion). We adopt a novel approach that
does not rely on a "reverse engineering" method to make explicit the loop
momenta, thus avoiding a certain ambiguity pointed out in a recent paper by
Sen, while also keeping the genus of the worldsheet generic. This approach will
also be useful in discussions of quantum gravity and in particular in relation
to black holes in string theory, non-locality and breakdown of local effective
field theory, as well as in discussions of cosmic superstrings and their
phenomenological relevance. We also discuss the manifestation of wave/particle
(or rather wave/string) duality in string theory.
