Pólya Bounds from Rectangular Geometry: Sharp interval deficits, retained boundary conditions, and perforated domains

Pólya’s planar Dirichlet conjecture asserts that the kth eigenvalue of a bounded planar domain is at least 4πk divided by its area. These two papers develop complementary geometric and operator approaches to this inequality.The geometry paper proves sharp interval-deficit constants and uses them to obtain all-energy sufficient criteria for rectangular assemblies, perforated domains, and short openings in Dirichlet screens. Half-turn pairing gives an explicit positive counting margin for offset pairs of equal-width rectangles, uniformly over window positions and finite splittings. Hole subtraction transfers host margins to perforations. The interval constants are optimal for their stated inequalities; the resulting geometric thresholds are sufficient conditions and are not asserted necessary.The operator paper expresses the coupled spectral count through exact interface inertia. Finite comparisons retain explicit tail bounds, while saturated, clipped-power, and concentrated polynomial filters preserve coupling between boundary traces. These constructions complement the geometric criteria by identifying the operator quantity that an unrestricted proof must control. The arguments combine min–max comparison, zero extension, exact Schur complements, positive operator inequalities, Fourier analysis, and scalar quadrature. An exact signed Fourier-leakage identity isolates a missing geometric sign; separate-face estimates can lose essential cancellation near resonance. Shape variation gives necessary boundary-flux balance conditions for smooth extremizers. Sharp quadrature constructions show that abstract spectra can satisfy the heat bound and every Riesz bound above a fixed positive order while violating the indexed area inequality. These are limitations of scalar data, not planar-domain counterexamples. The unrestricted conjecture remains unproved. The remaining interface step is a universal area bound for exact inertia in arbitrary assemblies. Editable LaTeX, verification scripts, and supporting analyses accompany both papers.