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 propositions decompose the trace: Proposition 1 (Orbital Decomposition, conditional on normalisation) decomposes the operator trace into identity and prime-power orbital integrals, where the von Mangoldt weight arises geometrically as the centraliser volume of the prime displacement (RN20). Proposition 2 (Identity Term) identifies the identity orbital integral as the sum of the θ-circle determinant contribution log(1/2π), the archimedean distribution W∞, and a conjectural endpoint contribution from the σ-bubble boundary. The Reduction Theorem shows that the full Weil explicit formula is conditional on exactly two named inputs: (1) the normalisation of the KMS state on winding sectors (Conjecture 2), and (2) the Brüning–Seeley endpoint residues at the σ-bubble boundary (Conjecture 1). These two inputs may be coupled through the sector structure — they satisfy a single equation with two unknowns. All other ingredients are standing results from Papers 38–47 or proved in the present paper. The paper does not prove the Riemann Hypothesis. It does not identify the spectral zeta with ζ(s). The contribution is the reduction itself: showing that the unresolved part of the trace-formula programme is finite, typed, and localised. If the two conjectures hold, the Weil explicit formula follows, and combined with the spectral-identification theorem of Paper 38, every nontrivial zero of ζ(s) lies on Re(s) = 1/2.