QST φn and −1/12: Golden Scale Sums, Finite Parts, and the VSS Refinement Question

This memo records a concrete mathematical bridge between a Golden scale hierarchy and the finite part associated with 1+2+3+⋯. For φ=(1+√5)/2 it introduces the weighted family Sp(η)=∑_{n=1}^∞ n φ^{-p η n} (p>0, η>0), proves S1(1)=φ³ and S2(1)=1, and shows the finite part lim_{η→0+}[Sp(η)−1/(p η ln φ)²]=−1/12. It develops this as a candidate VSS module and identifies the calculation needed to turn its weights, subtraction, and readouts into physical results. Memo v0.1, 13 October 2026.