The Conditional Trace Formula on the SCT Spectral Triple: A Reduction Theorem

Paper 48 in the Geometry of the Critical Line programme. This paper reduces the remaining trace-formula problem on the SCT spectral triple to two well-posed analytic inputs, after Paper 47 assembled all five factors of the Weil prime-side kernel. Two theorems are proved. Theorem 1 (Conditional GNS Equivalence) establishes that the enlarged algebra — generated by the Hecke operators and the spectral projections of the Dirac operator — admits a faithful cyclic representation on the geometric Hilbert space; GNS equivalence follows conditional on normalisation. Theorem 2 (Tracial Property) proves that the KMS state at inverse temperature β = 1/2, restricted to the modular fixed-point algebra, is a trace. Two…