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

Momentum-resolved electron energy-loss spectroscopy (EELS) on strange-metal Bi₂Sr₂CaCu₂O₈₊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→0 limit at fixed nonzero frequency, the Ward identity gives χ''ρρ( q,ω)=q²ReσL( q,ω)/ω. 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 q² 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₍ω,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.