The Level 8 Isosceles Cyclotomic Node: Global Uniqueness by a Quartic Reduction and an Infinite 2-Adic Unit Descent

We study the Level 8 isosceles cyclotomic node arising from the negative Pell orbit (1 + √2)ⁿ = xₙ + yₙ√2, xₙ² − 2yₙ² = −1 (n odd), together with the shared Gaussian-parameter condition xₙ = a² + b², yₙ = c² + b². At n = 5 one has (x₅,y₅) = (41,29), 41 = 4² + 5², 29 = 2² + 5², and hence the isosceles right triangle T₅ = (41,41,41√2). For the specialization b = 5, we prove the global Diophantine statement (a² + 25)² − 2(c² + 25)² = −1 ⟹ (a,c) = (±4,±2). The proof is purely algebraic and does not rely on a bounded search. A parity argument reduces the equation to the genus-one quartic V² = 8A⁴ + 100A² + 313. Two rational conic parametrizations convert its integral points into six explicit binary quartic equations. Four are excluded by elementary 2-adic congruences; the equation with right-hand side −313² descends to the remaining primitive equation; and the final equation is solved through a norm identity in the quartic field K = ℚ(t), t⁴ − 2t² − 1 = 0, followed by an infinite 2-adic descent in the full unit group 𝒪_K^× = ⟨−1, t, t² + t − 1⟩. Consequently, the integral points of the quartic are exactly (A,V) = (±2,±29), and the condition V = c² + 25 leaves only the node T₅. We also express the Pell coordinates through an explicit two-component relative-trace operator from K₈ = ℚ(ζ₈) to ℚ(i), formulate the shared Gaussian parameter as a coupled affine square selector, and prove that no unit multiplier in K₈ can compress this condition into a nontrivial scalar selector analogous to the Level 24 selector. Independent genus-one, Mordell–Weil, integral-point and Thue computations are included only as reproducibility checks and are not used as premises of the proof. Version 3.2 incorporates referee-driven clarifications concerning the integral basis of K and the complete rational coverage of the first conic parametrization; the principal theorem and proof architecture are unchanged.