Adelic Quotient Reduction and the Missing t^−1/2 Factor on the Weil Prime Side

Research Note 22 in the "Geometry of the Critical Line" programme. RN20 and RN21 supplied the primitive-orbit factor log p and the thermal KMS amplitude p^−r/2. The remaining discrepancy between the raw SCT geometric trace and the Weil explicit formula is the power of t: the arithmetic prime side requires t^−1/2, whereas the 2-dimensional transverse carrier yields t^−1. This note assembles the final formal ingredient of the Weil prime-side triad. The adelic quotient by Q* (Kill #67, Paper 40) integrates out the 1-dimensional global scaling action of R*_+, converting the 2D prefactor (π/t) into the 1D prefactor 1/√ (4πt). The quotient trace prescription is modeled by integrating the heat kernel over the scaling orbits with the Haar measure dx/x on R*_+. Assembled with the previous results, the formal twisted arithmetic trace becomes: Trₐrith (T₏⋒ e^−tD²) ∼ (log p / p^r/2) · (1/√ (4πt) ) · e^− (r log p) ²/ (4t) which formally reproduces the prime side of the Weil explicit formula, conditional on the quotient trace prescription and up to standard global sign and measure conventions. This note does not claim a rigorous implementation of the quotient trace on AQ/Q*; the assembly remains formal and conditional. The rigorous implementation, with explicit normalisation and domain control, remains one of the missing structural ingredients required before any arithmetic upgrade. Part of a 46-paper open-access programme on the geometry of the Riemann zeta function's critical line, anchored by the SCT 5-Manifold and the cover equation Φ + e^iπ − 1/Φ = 0.