Diagnostic Obstructions for Energy-Weighted Mean-Field Closure in Softened Coulomb Phase Dynamics

This preprint develops a diagnostic obstruction framework for energy-weighted mean-field closure in softened Coulomb and logarithmic phase dynamics. The manuscript does not claim a universal no-go theorem for singular mean-field limits. Instead, it audits two natural macroscopic energy-weighted closure architectures: a global fractional stabilizer and a localized relative-gradient stabilizer. For the global fractional route, the analysis identifies a summability obstruction in the generic correlated qq sector and a direct Schatten-index bottleneck under the physical regularity cap. For the localized relative-gradient route, the analysis isolates the third-cutoff-current obstruction: after ring-volume recovery, the leading cutoff contribution can generate an inadmissible active scale coefficient in the Gronwall estimate unless an additional parity or microscopic cancellation mechanism is available. The resulting conditional obstruction theorem clarifies why these two audited macroscopic strategies do not close the generic softened-Coulomb route without additional structure. The paper converts these failures into a roadmap of missing mechanisms, including relative-coordinate weak Schatten multiplier estimates, odd-current cancellation, Jastrow/Bogoliubov-type pair-correlation correctors, BBGKY-renormalized defect variables, and hybrid correlation-corrected energy functionals.