This document presents the physics case and ancillary studies for the proposed CODEX-b long-lived particle (LLP) detector, as well as for a smaller proof-of-concept demonstrator detector, CODEX-$beta$, to be operated during Run 3 of the LHC. Our development of the CODEX-b physics case synthesizes `top-down and `bottom-up theoretical approaches, providing a detailed survey of both minimal and complete models featuring LLPs. Several of these models have not been studied previously, and for some others we amend studies from previous literature: In particular, for gluon and fermion-coupled axion-like particles. We moreover present updated simulations of expected backgrounds in CODEX-bs actively shielded environment, including the effects of shielding propagation uncertainties, high-energy tails and variation in the shielding design. Initial results are also included from a background measurement and calibration campaign. A design overview is presented for the CODEX-$beta$ demonstrator detector, which will enable background calibration and detector design studies. Finally, we lay out brief studies of various design drivers of the CODEX-b experiment and potential extensions of the baseline design, including the physics case for a calorimeter element, precision timing, event tagging within LHCb, and precision low-momentum tracking.
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