Fix λ ⊢ n, integers k, m ≥ 1, and a residual partition τ with |τ | + km = n. For ν ⊢ m set Fτ (ν) = χ λ (τ ∪ kν). We record a Fourier formulation of this complete equal-part coarsening profile and then combine it with the plethystic Murnaghan–Nakayama rule. The profile is an integral virtual character of Sm, with coefficient ⟨sλ, pτ sα[pk]⟩ at χ α. In the pure-fibre case the trivial and sign coefficients are therefore in {0, ±1}. More significantly, after the residual cycle type is allowed to vary, the trivial coarsening mode is itself an Sr-virtual character whose irreducible coefficients are all 0, ±1. This gives an exact Parseval identity: its class-measure energy equals the number of nonzero plethystic Murnaghan–Nakayama coefficients. We also give the corresponding cross-Parseval identity, coefficient-recovery formula, and an even/odd coarsening decomposition related by partition conjugation. The plethystic rule itself is prior work. Beyond the two-stage transform and its exact energy consequences, we prove that the signed residual coefficient vectors generate the full integral character lattice: the coefficient matrix of multiplication by hm[pk] has full column rank and Smith invariants all equal to one. Thus every irreducible character of the residual symmetric group is an integral combination of residual coarsening profiles. We conclude with a unimodular-basis conjecture, verified computationally in small degrees.