Why the Q*-Quotient Cannot Live on the SCT Torus Alone

Research Note 41 in the "Geometry of the Critical Line" programme. RN40 identified the sole remaining discrepancy between the KMS-weighted twisted trace and the Weil prime kernel: a factor of t⁻¹ from the 2D transverse torus, to be removed by the Q*-quotient (Wall 3). This note shows that the quotient cannot be implemented on the SCT torus alone. Three torus-only quotient attempts fail: radial scaling on R² does not respect the torus lattice; continuous scaling on the compact torus is incompatible with the irrational flow parameter α = 2ln2/π; and the archimedean-only quotient R/Q*₊ collapses the flow line to a single point. The quotient requires non-archimedean data: the profinite completion Ẑ = ∏ₚ Zₚ that provides discrete structure preventing total collapse. The programme already contains algebraic candidates in the Hecke correspondence operators μₚ from RN9–10. Walls 3 and 5 converge to the same prerequisite: identifying the abstract Hecke algebra with a canonical completion of the SCT translation group.