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We address the two issues raised by Bayle, Vallisneri, Babak, and Petiteau(in their gr-qc document arXiv:2106.03976) about our matrix formulation ofTime-Delay Interferometry (TDI) (arXiv:2105.02054) \cite{TDJ21}. In so doing weexplain and quantify our concerns about the results derived by Vallisneri,Bayle, Babak and Petiteau \cite{Vallisneri2020} by applying their dataprocessing technique (named TDI-$\infty$) to the two heterodyne measurementsmade by a two-arm space-based GW interferometer. First we show that thesolutions identified by the TDI-$\infty$ algorithm derived by Vallisneri,Bayle, Babak and Petiteau \cite{Vallisneri2020} {\underbar {do}} depend on theboundary-conditions selected for the two-way Doppler data. We prove this byadopting the (non-physical) boundary conditions used by Vallisneri {\it et al.}and deriving the corresponding analytic expression for a laser-noise-cancelingcombination. We show it to be characterized by a number of Doppler measurementterms that grows with the observation time and works for any time-dependenttime delays. We then prove that, for a constant-arm-length interferometer whosetwo-way light times are equal to twice and three-times the sampling time, thesolutions identified by TDI-$\infty$ are linear combinations of the TDIvariable $X$. In the second part of this document we address the concernexpressed by Bayle {\it et al.} regarding our matrix formulation of TDI whenthe two-way light-times are constant but not equal to integer multiples of thesampling time. We mathematically prove the homomorphism between the delayoperators and their matrix representation \cite{TDJ21} holds in general. Bysequentially applying two order-$m$ Fractional-Delay (FD) Lagrange filters ofdelays $l_1$, $l_2$ we find its result to be equal to applying an order-$m$ FDLagrange filter of delay $l_1 + l_2$.

Reply to the Bayle {\it et al.} gr-qc document dated June 7, 2021}