Constraint Propagation and Gauge Nature of the Lapse in Projection-Origin Geometry (Projection-Origin Gravity 6)

This study investigates the temporal behavior of the discrete Bianchi structure within a projection-origin geometric framework under controlled lapse reparameterization. Previous works demonstrated that projection-origin geometry admits a weak-field embedding in which the discrete four-dimensional Bianchi residual can be strongly suppressed by an appropriate choice of the lapse parameter. The closure was shown to persist under both shift-free and shift-enabled static ADM configurations. The present work extends this analysis into the time domain. The spatial geometry is fixed to the projection-origin weak-field configuration while time dependence is introduced solely through lapse modulation.Using a unified discrete operator framework, geometric quantities are reconstructed at each time step and a quasi-dynamical constraint diagnostic is evaluated. The results show that lapse modulation produces a coherent, phase-locked response of the constraint residual. A stability phase diagram reveals a sharply defined tolerance boundary in modulation amplitude, while frequency dependence remains comparatively weak within the explored parameter range. Arrival-time diagnostics further demonstrate that lapse reparameterization coherently deforms causal propagation while leaving the spatial curvature kernel unchanged. These findings indicate that within projection-origin geometry the lapse acts as a slicing gauge parameter governing the temporal stability of the conservation structure. The analysis therefore provides a structural bridge between static geometric consistency and future dynamical formulations without assuming Einstein’s equations or introducing propagating gravitational-wave modes. Note: Parts of the manuscript were linguistically and structurally refined with the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own. Note: This work represents Version 1.0 of an ongoing research program on the Order-Projection Principle (OPP). Minor typographical corrections and clarifications may appear in later versions. The core conceptual claims remain unchanged.