Weak-Field Transverse-Traceless Wave Structure in Projection-Origin Geometry (Projection-Origin Gravity 7)

⚠️ 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 examines the existence of a linear dynamical sector withinprojection-origin geometry under fixed spatial kernel and lapse slicing. Previous studies established static Bianchi closure, shift-extendedconsistency within the ADM framework, and the gauge nature of the lapse.The present analysis investigates whether the same unified discreteoperator that governs the static geometric construction also admitspropagating perturbative modes. Small transverse–traceless metric perturbations are evolved using alattice-consistent wave update derived from the projection-origindiscrete operator. The measured dispersion relation agrees with theoperator-predicted form ω = c q, where q denotes the discrete momentumassociated with the unified derivative operator. Throughout the evolution the transverse and trace constraints remainpreserved at numerical precision. A quadratic energy functional isconserved up to machine roundoff and remains stable under latticerefinement. A continuum-limit test with fixed physical wavenumbershows convergence toward the expected linear dispersion relation. A supplementary real-space evolution confirms that the same behaviorpersists beyond spectral mode analysis. These results provide structural evidence that projection-origingeometry admits a dynamically stable weak-field perturbative sectorthat preserves constraints and energy consistency without invokingEinstein field equations. 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.