Projection-Origin Poisson Kernel and Weak-Field Einstein Consistency in a Unified Discrete Framework (Projection-Origin Gravity 2)

⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This work investigates whether a projection-origin spectral model,previously shown to generate a Poisson-type kernel and inverse-squareforce behavior, is geometrically compatible with the weak-fieldstructure of general relativity. Starting from a discrete spectral kernel with zero-mode removal anda self-consistent low-k phase, a potential is generated independentlyof any gravitational field equation. This potential is embedded intoa static weak-field metric ansatz, and geometric quantities areevaluated using a unified lattice-consistent derivative operator. Within this discrete framework three independent geometric relationsare verified simultaneously: tidal consistency between Riemann tensorcomponents and the Hessian of the potential, proportionality betweenthe Einstein tensor component G00 and the lattice Laplacian of thepotential, and correspondence between holonomy density and sectionalcurvature. The analysis is performed without invoking continuum field equationsor mixing discretization schemes, allowing a direct structuralcomparison between projection-origin spectral dynamics andweak-field gravitational geometry. The results provide a numerical consistency test indicating thatprojection-origin dynamics can reproduce core geometric relationscharacteristic of the weak-field limit of general relativity withina unified discrete operator framework. 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.