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Abstract

We prove two-sided bounds for the complete inverse-energy Gram of prescribed theta-derivative sources in the original polar-constrained archimedean realization of the Weil form. For every finite source size d + 1, diagonal normalization by explicit powers of U = log(1/ε) gives positive degree-dependent lower and upper bounds whenever U ≥ 64(d + 2)². For each fixed family the normalized Gram is bounded above and below by constants, and source i has exact logarithmic exponent 4i + 13/2. The upper estimate uses exact half-line polar cancellation and a uniformly priced Fourier primitive, retaining the complete exterior polar tail. The lower estimate combines an exact polynomial coefficient map, all-atom theta bounds and a common-weight shell covariance with exact polar shorting. A two-metric residual theorem receives non-square-integrable divided sources, and explicit finite receiving bounds accompany the analytic estimates.