Creative Telescoping via Reduction Quotients

We develop a quotient-theoretic account of creative telescoping for
Laplace-type integrals.  For parameter-dependent phases and amplitudes
admitting reduction, the division-form reduction quotient
\[
\Qred=\mathcal M_\phi/\Ered(\mathcal M_\phi),
\qquad
\Ered(\mathcal M_\phi)=(\Id-\Rphi)(\mathcal M_\phi),
\]
provides the natural algebraic setting for telescoping identities whose
certificates are written in \((\Id-\Rphi)\)-form.  The main result is that
division-form creative telescopers are precisely torsion, or annihilator,
relations in \(\Qred\): a differential operator telescopes an amplitude if
and only if its induced action annihilates the corresponding quotient class.

This perspective converts creative telescoping from an elimination
phenomenon into a statement about finite-dimensional parameter dynamics.
If the cyclic quotient orbit generated by an amplitude is finite-dimensional
and stable under the induced parameter connection, then telescopers exist,
their quotient order is bounded by the dimension of the corresponding cyclic
subspace, and under cyclicity this bound is sharp.  For a chosen scalar
period, the analytic minimal order agrees with the quotient order precisely
when the period functional detects the full cyclic quotient subspace; order
drops reflect loss of observability rather than a smaller intrinsic quotient.

The same mechanism has parallel formulations in the boundary-image quotient
\(\Qbd\), for direct certificates \(C'-(\partial_x\phi)C\), and in the
intrinsic twisted quotient \(\Qtw\), for \(d_\phi\)-exact certificates.  Thus
creative telescoping, Pfaffian closure, holonomicity, order bounds, and
Milnor-number predictions are all organized by the finite-dimensional
quotient module appropriate to the chosen certificate convention.