Geometric Origin of the Primitive-Orbit Normalization log p in the SCT Trace

Research Note 20 in the "Geometry of the Critical Line" programme. Paper 39 established that the twisted heat trace on the SCT critical arithmetic sector factorises, with the transverse torus contributing a Gaussian heat kernel Θₜ (ξ) ∼ (π/t) e^−ξ²/ (4t). Three arithmetic normalization factors are needed to reach the Weil prime-side kernel: the primitive-orbit length log p, the critical-line amplitude p^−r/2, and the dimensional reduction t^−1 → t^−1/2. Paper 39 listed the first of these as "standard orbital" without derivation. This note supplies that derivation within the flat-torus model. The orbital integral over the fundamental domain decomposes via the 1-parameter subgroup generated by the primitive translation Tₚ. The induced length measure on this subgroup assigns arclength log p to one primitive traversal. The iterate T₏⋒ = (Tₚ) ʳ shares the same subgroup, so the orbital factor is log p independent of r — the factor r appears only in the Gaussian displacement exponent. This note does not claim a full arithmetic dictionary or a complete trace-formula theorem; it isolates only the primitive-length factor in the flat torus orbital model. The amplitude factor is treated in RN21, and the dimensional reduction in RN22. 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.