Imagine a drum with seven straight sides. Keep its area fixed, and move its corners slightly. Does the perfectly regular shape have the lowest fundamental vibration frequency among its nearby competitors? This research candidate answers yes for both seven and eight sides. The word nearby matters. The result does not compare the regular polygon with every possible polygon, and it does not say how large a change is allowed. What makes the result more than a numerical picture is its treatment of error. A computer first supplies approximate functions on a triangular mesh. A separate checker then asks whether those functions are good enough to support the mathematical argument. It checks geometry, continuity, residual errors and a small matrix that measures curvature across every genuine shape direction. There are ten such directions for the heptagon and twelve for the octagon. Translation, rotation and scaling are removed because they do not represent the shape changes under study. At unit circumradius, the certified curvature bounds are three fifths and seven twentieths. They are conservative guarantees, not claims to the best possible constants. One useful identity makes the error from an auxiliary response equation enter quadratically. Other errors remain, and in these examples they dominate the remaining uncertainty. Earlier work established validated local results for five and six sides; related torsion problems use a different physical quantity. The package includes the paper, exact stored trials, executable checks and tests designed to reject corrupted evidence. This remains an unrefereed candidate, not a formally verified theorem. This is Evidence Press, Certified local minima for two regular polygons, dated the thirtieth of September twenty twenty-six. The paper and evidence package are linked on the release page. This synthetic AI voice explains the research; it is not additional scientific evidence.