A certified degree-56 totally real realization of SL2(F13)

We develop an explicit construction over the rational numbers starting from a fixed totally real degree-fourteen polynomial with splitting group PSL2(F13). A rational isometry of its full trace form produces a nonzero, totally positive Spin determinant in the splitting field. We use an elementary global norm-separation criterion to turn simultaneous p-adic reconstruction into exact algebraic identities, and apply it to all 1092 rational coordinates of this determinant. An odd-order subgroup then gives a degree-56 irreducible, totally real norm polynomial with splitting group SL2(F13). The starting degree-fourteen polynomial is printed in full, and the complete primitive integral coefficient list of the resulting polynomial, exact construction data, and verification programs accompany the article. An implementation independent of the norm producer verifies exact annihilation in the degree-28 field, all 57 coefficient entries, and primitive normalization. The example is large but fully evaluated: its positive integral leading coefficient has 1,088,453 decimal digits. The construction reported here relies on substantive contributions from GPT-5.6 Sol. This record contains the paper PDF, an arXiv-style LaTeX source archive, and the complete computational verification package.