Long-wavelength density response and momentum resolution in strange-metal charge spectroscopy

Momentum-resolved electron energy-loss spectroscopy (EELS) on strange-metal Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$ (Bi-2212) observes a broad charge continuum that is nearly momentum independent over an extended finite-momentum regime, whereas optical spectroscopy determines a metallic local conductivity. We analyze the long-wavelength response function that connects these regimes: the proper homogeneous bulk charge-density polarization. For any homogeneous $U(1)$-conserving metal whose longitudinal matter conductivity remains finite in the $q\to 0$ limit at fixed nonzero frequency, the Ward identity gives ${\chi }_{\rho \rho }^{\mathrm{\prime }\mathrm{\prime }}(\mathbf{q},\omega )=q^2\mathrm{Re}{\sigma }_L(\mathbf{q},\omega )\mathrm{/}\omega$. We then prove that any nonnegative normalized momentum-resolution kernel with a self-similar $q$-scaled profile, uniformly bounded second moment, and tight second-moment tails preserves this $q^2$ asymptotic behavior, changing only the prefactor. Consequently, a nearly local finite-$q$ continuum in Bi-2212, if continuously connected to a regular metallic optical limit of the same proper bulk response, is naturally described by a crossover surface $q_{\ast }(\omega ,T)$. The theorem is a constraint on the proper bulk density response, or on a positive $q$-scaled convolution of it; screened loss functions and surface EELS observables require the corresponding electrodynamic conversion before the constraint is applied.