Observer-Resolved Entropy Change Under the Second Law: Local Transport, Directional Projection, and Global Reconstruction

This working paper develops a classical thermodynamic framework for understanding how different local and directional observations of the same system can exhibit different entropy trends while remaining consistent with the second law of thermodynamics. Starting from the standard local entropy-balance equation, the work separates irreversible entropy production from entropy transport and shows that a local or directionally resolved region may experience decreasing entropy even while the total entropy of an isolated system continues to increase. An exact Fourier heat-conduction solution for an insulated rod provides a fully analytic example in which one half of the system decreases in local entropy while the other half increases at the same physical time, with the total entropy rate remaining strictly positive. The paper then generalizes the result using weighted observation regions, moving observation windows, directional projections, and tomographic reconstruction. The central result is that different observational positions, directions, ranges, and resolutions can reveal different portions of the same evolving entropy structure without producing different underlying thermodynamic realities. Complete reconstruction recovers the ordinary global second-law result. The work does not propose a replacement for the second law. Instead, it provides a more resolved mathematical description of how local transport, directional observation, and global entropy production fit together. This paper is part of an ongoing research program investigating temporal structure, observation, information, thermodynamics, and the relationship between local physical histories and their global reconstruction.