A Fully Explicit GL(1) Trace–Prime Block on the Idele Class Group: Unconditional Exchange Majorants and a Wedge Target

We build an arithmetic GL(1) channel on the idele class group CQ R>0 × CQ⁽¹⁾ and prove an unconditional Trace--Prime (exchange) identity on the half-plane (s) > 1 with fully explicit Tonelli/Fubini/Mellin majorants. Prime-power contributions are computed from a canonical baseline generator Lₐᵣᵢₜₕ⁽⁰⁾ = -∂ᵤ² ⊗ I + I ⊗ Δ_CQ⁽¹⁾, using a von Neumann/Plancherel trace on the noncompact modulus direction and yielding the Gaussian penalty (4π t)-1/2 e-²/(4t) as a genuine trace identity. This produces a referee-friendly, "no hidden assumptions" GL(1) prime block that matches the von Mangoldt weights via balanced prime-power operators. We additionally formulate a precise Wedge(GL₁) analytic target for the reference-subtracted arithmetic heat trace remainder, reducing the associated uniqueness step to explicit complex-time/sectorial bounds.