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.

Frozen crystalline snapshot: branch crystalline/manuscripts3-2026-07-07, tag crystalline-manuscripts3-v1; p4rakernel commit eea2c0c. Manuscript trio frozen in ig-docs on the same branch and tag. Acknowledgements: Thanks to Harry T. Larson for imparting the importance of catching rising problems and never letting them go~. The conductor was found by refusing to accept a tool's refusal: when bnrstark declined the coordinate field for nonzero $r_2$, the question became which field the moduli actually generate, and that question had an answer. %────────────────────────────────────────────────────────────────────────────── Publisher: umpolungfish. License: LUNLICENSE (other-open on Zenodo).