Cell Relabeling and Apparent Fluxes in Causal Spatial Subdivision

This technical note provides a clarification of cell updating in the framework of causal spatial subdivision used to construct the Friedmann equation from Newtonian energy conservation. In this framework, space is partitioned into causally defined regions (“cells”) based on light-cone accessibility, without assuming a discrete spacetime structure. We show that the updating of the subdivision should be understood as a relabeling of causally defined spatial regions rather than as physical transport or the creation and annihilation of spatial elements. When this relabeling is made explicit, apparent flux terms naturally arise from the time-dependent reassignment of spatial points to cell labels. These flux terms do not represent physical flows and do not modify the underlying dynamics. We further demonstrate that the standard matter conservation law $\rho a^3=\mathrm{const}$ is recovered as a consistency condition once the relabeling nature of cell updating is properly accounted for, rather than as an independent dynamical assumption. The cosmological constant $\Lambda$ does not participate in cell updating and therefore appears as a non-updating reference term, consistent with its constant contribution in the Friedmann equation. This note serves as a technical supplement to the main construction of the Friedmann equation in the causal spatial subdivision framework and provides a clean foundation for subsequent applications, while remaining fully compatible with continuous spacetime descriptions and standard cosmological dynamics.