The Twisted Trace Assembly and the Final Quotient Target

Research Note 40 in the "Geometry of the Critical Line" programme. This synthesis note assembles the exact provenance of every factor in the twisted heat trace and compares it term by term with the Weil prime-side kernel. The raw geometric trace (Paper 39) gives a product of the 1D sigma-bubble Weyl term and the 2D torus heat kernel, scaling as t⁻³/². Three modifications are needed to reach the Weil kernel: (1) the primitive-orbit factor log p from the centraliser volume (RN20, geometric), (2) the critical-line amplitude p⁻ʳ/² from the KMS state at β = 1/2 (RN21, algebraic), and (3) the torus dimensional collapse t⁻¹ → t⁰ from the Q*-quotient integrating out both transverse torus directions. The first two are supplied at the geometric and algebraic levels. The third — Wall 3 — is the sole remaining open quotient operation. The note does not introduce new mathematical results; it assembles existing proved factors into a single comparison, identifies the sole remaining open operation, and clarifies the programme's wall map.