A minimum dimension theorem for SU(3) modules generated by four fixed Theta matrices

This preprint proves a minimum-dimension theorem for the real SU(3)-invariant module generated separately by four explicitly specified Hermitian Theta matrices on C³ ⊗ C⁴. Over the projective family of 3+1 fiber embeddings parametrized by CP³, the minimum dimension is 30. Equality holds exactly at the first two coordinate lines, which are interchanged by a fixed Nambu involution preserving the generating span. The proof uses tensor factorizations, irreducible sectors and explicit projection norm identities. The supplement supplies the four matrices, a portable structural verification script, recorded results and a SHA-256 manifest. Computational checks do not replace the analytic proof. The statement is specific to the stated matrix problem. It does not derive confinement, gauge dynamics, coupling constants, particle masses or a continuum limit. The work has undergone internal checks but not independent peer review; no bibliographic priority is claimed. OpenAI Codex assisted with proof reconstruction, drafting and computational verification. The manuscript and documentation are licensed under CC BY-NC 4.0. The code uses separate custom noncommercial terms supplied in LICENSES.md, not the MIT license. Commercial permissions: [email protected].