Trace on a II1 factor without additivity: ε-traces force a unique Dixmier scalar

To prove the Dixmier scalar is unique in a finite factor (equivalent to the existence of a trace), you don't need the limit trace or additivity of a dimension-based τ. It suffices, for each ε>0, to have finitely many vectors η_i with Σ‖z*η_i‖² ≤ (1+ε)Σ‖zη_i‖² for all z∈M. Then ψ(x)=Σ⟨η_i,xη_i⟩ satisfies |ψ(uxu*)−ψ(x)| ≤ ((1+ε)^4−1)(‖x‖+1)²ψ(1). This is via the polarization 4h=(1+h)*(1+h)−(1−h)*(1−h) on real and imaginary parts, so no square roots are needed. So two scalars c,d in the closed hull differ by at most 30ε(‖x‖+1)². The ε-traces come from Anantharaman–Popa §6.3: an exhaustion argument comparing a vector state with the dimension function below a small projection p, then spreading via partial isometries with Σw_iw_i*=1. Every layer-cake/additivity route I tried was circular.