Finite Forward-Iterate Schröder Approximation from Geometric Richardson Extrapolation: Reciprocal Filters and Exact Paired Remainders

This theorem note develops a finite forward-iterate realization of geometric Richardson extrapolation through Schröder–Koenigs conjugacy. For an odd analytic germ \(Φ\), with local inverse \(H=Φ⁻¹\), the conjugated multiplication map \(S_m(x)=Φ(mH(x))\) converts geometric Richardson data into finite fixed-weight combinations of forward iterates of \(S_m\). The construction yields reciprocal composition-side and scale-side filters, exact paired remainder identities, explicit first-surviving coefficients and skipped-mode order jumps, an active-mode uniqueness criterion, and finite exact recovery for truncated analytic germs. For the sine/arcsine specialization with odd integer \(m\), the forward maps reduce to classical Chebyshev multiple-angle polynomials. Schröder–Koenigs linearization, composition-operator eigenfunctions, geometric-node Richardson extrapolation, and Richardson order jumps are classical ingredients and are not claimed as new. The emphasis of the present work is the explicit finite forward-iterate realization, its reciprocal pairing with the shrinking-scale filter, and the exact paired remainder formulas in the original variable. Version 0.2.1.