The EBR operator at d=2 is not a G-operator and its differential Galois group is SL(4): arithmetic, irreducibility, and the non-Liouvillian corollary

For the positive-coefficient polynomial continued fraction with b (n) = 3n² + n + 1, the Edge–Borel object G (s) = ∑ Qₙ sⁿ / (2n) ! is annihilated by an explicit order-4 linear differential operator L₂ over C (s) with regular singular points at s = 0 and s = R = 4/3 and an irregular singularity at s = ∞ of slope 1/4. We establish four results about L₂ at increasing evidential strength. First, in the Borel normalization gₙ = Qₙ / (2n) ! the operator is not the minimal operator of a G-function: the Taylor denominators retain essentially the full factorial, the p-curvature is non-nilpotent for every tested prime, and the irregular point at infinity is incompatible with the André–Chudnovsky–Katz trichotomy. (This statement is normalization-scoped and stated as such throughout. ) Second, L₂ is irreducible and minimal, by two independent arguments. Third, its local monodromy at R is a semisimple complex pseudo-reflection with spectrum 1, 1, 1, e^{iπ/3}, correcting an earlier “resonance logarithm” reading and consistent with the criterion that a −γ logarithm requires γ ∈ Z whereas here γ = 11/6. Fourth, and centrally, the identity component of the differential Galois group is GGal (L₂) ° = SL (4) ; we obtain this by eliminating every class of the maximal-subgroup (Aschbacher) classification of algebraic subgroups of SL (4), with the imprimitive class closed unconditionally through a unique quadratic-character analysis and an a priori degree bound that makes the underlying rational-solution searches complete. The immediate corollary is that L₂ has no Liouvillian solutions. The connection coefficient is computed by a third independent channel to 169 stable digits; a positive-control-validated integer-relation battery finds, to that precision, that it is neither an elementary Γ-quotient nor a low-height combination of standard constants nor algebraic of small degree. We state with care what these results do not give: a large Galois group does not imply that the connection coefficient is transcendental, connection coefficients are not differential-Galois invariants, and the transcendence question therefore remains a conjecture whose only route is a separate period analysis. The degree-bound arithmetic, the pullback exponent arithmetic, the eigenvalue-parity bookkeeping, and the parity of the relevant 4-cycle are formalized in Lean 4 / Mathlib with audited axiom cones. This is the third entry in the Edge–Borel Radius (EBR) series. A complete reproducibility package (reproducer scripts, results JSONs with canonical SHA-256 self-hashes, a one-command verifier, the lake-buildable Lean project with its axiom audit, and a claims-ledger snapshot) accompanies the deposit.