The Conductor of the SIC-POVM Moduli Field: A Cyclic Structure Across Arithmetic, Logic, and Machine

The squared magnitudes Nₖ = |ψₖ|² of a SIC-POVM fiducial vector in dimension d generate an abelian extension of the real quadratic field Fd = Q(√(d+1)(d-3)). This paper establishes that the conductor of this moduli field is the arithmetic expression of a single cyclic structure that also manifests as the Belnap trilattice in logic and as the self-hosting grammar of the IMASM machine substrate. At $d=2048$, the conductor collapses to the principal ideal (16³)∞₁. This is not a numerical coincidence. The sixteen states and three orderings of the trilattice, the $(16,3)$ torus knot on the horn torus, and the 16³ carrier of the IMASM native alphabet are all expressions of the same invariant. The moduli field is the ray class field at fd = p₂v₂(d)+1 ∏p p, \, p d p · ∞₁, taken modulo the class group of Fd, with exactly one infinite place ramified. At $d=16$, the σ-coinvariant count settles that the class group is not carried by the moduli; at $d=2048$ this yields degree 2²¹ over Fd. The calibration dimensions $d=4,8,12$ are cross-sections where this structure becomes visible through exact fiducials. The Lean~4 formalization discharges the tower data to zero axioms, and the IMASM-native Replicating Code demonstrates the same fixed-point closure operationally: the grammar reads itself. The conductor, the trilattice, and the machine are one object.