The Critical Arithmetic Sector and the Restricted Transverse Trace Conjecture

Paper 38 in the "Geometry of the Critical Line" programme.

The original Transverse Trace Conjecture (TTC) of the SCT programme is formulated on the full coupled spectral space. We show that the combined results of the critical-slice characterisation (Paper 5), the functional-equation isometry (Paper 33), and the winding-confinement analysis (RN7) single out a unique invariant subspace — the critical arithmetic sector — on which the trace identity should be stated. We prove that this sector is an exact reducing subspace for both the geometric spectral operator and the prime translation action, yielding a canonical self-adjoint restricted operator with real discrete spectrum.

The Restricted Transverse Trace Conjecture (RTTC) is stated precisely on this sector. We prove that RTTC implies the identification of the restricted spectrum with the Riemann zero ordinates, and hence implies the Riemann Hypothesis. We decompose RTTC into a ladder of seven sub-conjectures and identify the single hardest sub-piece: the exact extraction of the prime-power coefficients from the noncommutative trace on the critical sector.