<jats:title>Abstract</jats:title><jats:p>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-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\beta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math></jats:alternatives></jats:inline-formula>, 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-b’s 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-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\beta $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> 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.</jats:p>