py5vec: a modular Python package for the 5-vector method to search for continuous gravitational waves

We present \texttt{py5vec}, a Python package for implementing and extending the 5-vector method, used to search for continuous gravitational wave (CW) signals. We also provide a comprehensive theoretical review of the 5-vector method and extend the relative likelihood formalism by marginalizing over the noise variance, resulting in a more robust Student's t-likelihood, and over the initial phase to account for pulsar glitches. \texttt{py5vec} provides a modular architecture that separates data representation, signal demodulation, and statistical inference into independent abstract stages. It supports multiple input data formats and interoperates with existing Python software, providing a bridge between different approaches. For example, using a \texttt{bilby}-based interface, \texttt{py5vec} implements Bayesian parameter estimation within the 5-vector formalism for the first time. The modular design also allows for making exact multi-level and direct comparisons between other software, such as \texttt{cwinpy} and \texttt{SNAG} in MATLAB. In \texttt{py5vec}, we implement a multidetector targeted search for known pulsars, validated using LIGO data from the O4a run and hardware injections, demonstrating consistent reconstruction of signal parameters. This package therefore provides a flexible platform for current targeted searches and for future extensions to other CW search strategies.