X-ray Comptonization in an Accreting Kerr Black Hole System with a Tilted Disc and Thermal Corona; A General-Relativistic Monte Carlo Simulation: I. Model, Radiative Transfer and Spectral-Timing Properties

Quasi-periodic oscillations (QPOs) in the X-ray emission of accreting black hole binaries are commonly attributed to the relativistic Lense--Thirring precession of the inner accretion flow, in which a hot inner flow, tilted with respect to the black hole spin axis, precesses as a solid body while a cooler outer disc remains at its own, different tilt -- a warped, Bardeen--Petterson-type geometry. Testing this picture requires a radiative-transfer model that self-consistently follows photons through such a system: from emission near the innermost stable circular orbit (ISCO), through Compton scattering in a hot corona, to possible reflection off the misaligned outer disc, all under the full effects of light bending in Kerr spacetime. In this first paper of a two-part series, we present a general-relativistic Monte Carlo (GRMC) radiative-transfer model of a misaligned, warped, two-zone accretion system around a Kerr black hole. We describe the disc--corona geometry, the seed-photon prescription, the photon-transport engine -- including the geodesic integration scheme, the coronal Compton-scattering module and the outer-disc reflection treatment -- and the observer pipeline used to construct energy spectra, photon path-length and arrival-time distributions. We present the resulting energy spectra, light curves and spectral lags as functions of the black hole mass, corona size, outer-disc geometry, reflection albedo, photon-injection radius and observer inclination, and show how gravitational light bending systematically lengthens the path-length and arrival-time distributions of the escaping radiation. This model and its spectral-timing predictions form the foundation for Paper II, which extends the framework to track the full photon polarization state and presents the resulting energy- and precession-phase-resolved polarimetric signatures.