Primitive Isosceles Cyclotomic Nodes in ℤ[φ,i] (Level 20): Exact Enumeration Through Height Six and Finiteness of Refined Norm Fibers

Let φ = (1 + √5)/2, K = ℚ(√5,i), and 𝒪_K = ℤ[φ,i]. For m,n ∈ 𝒪_K, the classical parametrization a = m² − n², b = 2mn, c = m² + n² produces normed triples (A,B,C) = (N(a),N(b),N(c)). This work studies primitive fine isosceles cyclotomic nodes satisfying A > 0 and B = C, with canonical algebraic invariant w = m/n + n/m. The principal theoretical result proves that every fixed refined norm signature Σ([w]) = (A,B,{N(m),N(n)}) has only finitely many canonical classes [w]. The proof reduces a fixed refined fiber to quartic curves over ℚ and to singular branches governed by non-degenerate S-unit equations. The nonsingular cases are treated through the finiteness of S-integral points, while the singular cases are analyzed by explicit factorization and S-unit methods. An exact symmetry-reduced enumeration through coefficient height H = 6 accounts for 815,730,721 ordered pairs before symmetry reduction. It produces 378,400 raw fine hits, 108,288 primitive hits, 772 distinct primitive normed triples, 783 canonical w-classes, and 783 refined signatures. The computation finds no refined-signature collisions. The 783 classes consist of 761 simple normed-triple fibers and 11 double fibers. The associated refined quartic catalogue contains 775 nonsingular quartics and 8 singular quartics, with no exceptional perfect-square quartic. The accompanying reproducibility package contains the SageMath enumeration engine, complete H = 6 catalogues, modular-density data, local prime-parity verification, symbolic proof certificates, SHA-256 checksums, and instructions for reproducing the principal computations. The symbolic certificate verifies the real-quadratic reduction identities, quadratic elimination, quartic model, quartic discriminant factorization, singular branch factorizations, exceptional parameter classification, genus-two squarefreeness, and consistency of the H = 6 catalogue. This corrected and substantially revised version supersedes Version 1, which incorrectly inferred global stabilization, global finiteness, and a complete catalogue of 1,390 nodes from bounded computations. The present work withdraws those claims. It establishes finiteness only for each fixed refined norm fiber and leaves both global finiteness and global injectivity of Σ as open problems.