Static No-Go of Variational Lapse Determination in Projection-Origin Geometry (Projection-Origin Gravity 5)

This work investigates whether the lapse parameter in a projection-origindiscrete geometry can be determined through variational principles in thestatic sector. Previous studies showed that a projection-origin potential admits a weak-fieldembedding in which the discrete four-dimensional Bianchi residual can bestrongly suppressed by an appropriate choice of the lapse parameter.This closure persists in both shift-free and shift-enabled static ADMconfigurations. The present work examines whether such a lapse configuration can be obtainedthrough extremization of static or time-like action-like functionals. In the strictly static regime, analytical arguments based on the ADMHamiltonian structure show that spatial curvature functionals remainindependent of the lapse parameter because the extrinsic curvature vanishes.Numerical experiments confirm that static action functionals remaininvariant under lapse variation across a wide parameter range. We then investigate several classes of time-like proxy actions constructedfrom four-dimensional geometric quantities. Although these functionalsexhibit well-defined minima with respect to the lapse parameter, theminimizing values systematically differ from the lapse value that enforcesdiscrete Bianchi closure. These results establish a structural no-go statement:within the strictly static sector the lapse cannot be uniquely determinedby naive variational minimization. Instead, the lapse acts as a gauge-likeparameter enforcing discrete conservation consistency, reproducing a keystructural feature of the ADM formulation of general relativity. Note: Parts of the manuscript were linguistically and structurally refinedwith 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.