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

The squared magnitudes $N_k = |\psi_k|^2$ of a SIC-POVM fiducial vector in dimension $d$ generate an abelian extension of the real quadratic field $F_d = \mathbb{Q}(\sqrt{(d+1)(d-3)})$. 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 $(16^3)\infty_1$. 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 $16^3$ carrier of the IMASM native alphabet are all expressions of the same invariant. The moduli field is the ray class field at $\mathfrak{f}_d = \mathfrak{p}_2^{v_2(d)+1} \prod_{\mathfrak{p}\mid p, \, p\mid d} \mathfrak{p} \cdot \infty_1$, taken modulo the class group of $F_d$, with exactly one infinite place ramified. At $d=16$, the $\sigma$-coinvariant count settles that the class group is not carried by the moduli; at $d=2048$ this yields degree $2^{21}$ over $F_d$. 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.