Tikhonov analogue for very soft Landau potentials

Establish a Tikhonov-regularized particle theorem for uncut very-soft Landau potentials with exponent gamma in (-3,-2), analogous to the existing mollified-score particle result.

Background

The score-difference cancellation used in the paper yields effective interaction regularity for far-field-truncated kernels with gamma greater than or equal to -2, but not for the uncut very-soft range gamma in (-3,-2). For these exponents, the interaction is generally only Holder-continuous near the diagonal, while the three-dimensional Coulomb case has no continuous diagonal extension for generic RKHS scores.

The paper contrasts this limitation with a known particle theorem for a mollified score under an initial minimal-separation condition and explicitly states that the corresponding Tikhonov result has not been obtained.