SIC-POVMs, a Stark Conjecture, and the 12th: A Formalization via Paraconsistent Belnap Multilattices

This paper establishes, inside the Lean 4 proof assistant, a three-level formal identification. The levels are: (i) Belnap multilattice axioms for Weyl–Heisenberg covariant SIC-POVMs at d=2ⁿ; (ii) the Zauner conjecture; and (iii) the mixed-signature Stark conjecture for the ray class field Kd=Q(√(d-3)(d+1)), a real-quadratic case of Hilbert's Twelfth Problem. Fiducials are unit-normalized and satisfy (d+1)|ψ,Da,bψ|²=1. The equivalence hilbert_embedding_equiv_zauner is proved by rfl: the Belnap embedding into C2ⁿ and the Zauner conjecture at d=2ⁿ are definitionally the same proposition. The Belnap skeleton (orbit size 4ⁿ, Frobenius closure μ∘δ=id, join-equiangularity, Born rule) contains zero sorries. Open arithmetic content is marked by named gap axioms for Stark units on WH frames; a proof of Stark would close all three levels at once. For dimension $d=12$ we prove SICPOVM_Exists 12 outright. We construct an exact fiducial in a finitely presented Q-algebra, verify 143 overlap identities with native_decide, and transfer everything to C¹² along a ring homomorphism. The theorem crystal_forces_d12_sic depends on no axiom beyond Lean 4's standard foundations and compiler trust. This is, to our knowledge, the first machine-checked SIC-POVM existence in any dimension. For the frontier dimension d=2048=2¹¹ the transport apparatus is formalized and sorry-free. It includes a forward map φ B⊕ 11²⁰⁴⁸, a reduction ψ with ψ∘φ=id, a conditional reduction to Stark, and a non-real character obstruction that blocks the false branch. Unconditional existence remains open; the machinery that surrounds it is closed.