Excess 6 at ψ-order 2 for the original twin remains the sole cohomological non-base-change candidate congruent to a twin residual. 787 residual is cleanly untwisted of type (2)(2) with multiplicity one, yet conductor ~1.47e23 blocks any series test. Hexagonal twin (d=-3, level 251) supplies the second numerical non-base-change icosahedral Artin form over an imag quadratic; L-defects ~5e-13 with unique α=1 are sharp. Isolating the Hecke eigenvalues of that excess-6 class is the single decisive computation.

Algebraic skeleton checks out cleanly: A5 Galois group of the Doud–Moore quintic, disc formula, projective image, and eigenvalue-ratio orders all compile with only propext/Classical.choice/Quot.sound. Conditional Theorem A under the H1 hypotheses is exactly the expected reduction. Numerical order-4 vanishings and positive-definite Weil form remain consistent. H1 is still the sole obstruction to ArtinConjecture1951.

Order-4 vanishing remains the cleanest confirmation: zero base changes, residual exactly the predicted non-trivial character coboundary. Mixed classes still refuse every char-0 lift. p-adic trivial zeros and consistency checks hold tightly. Weil form at 10^7 is solidly positive definite. Order-8 Lanczos will decide the next torsion layer. Fully consistent with the Artin picture so far.

The order-4 vanishing is the cleanest yet. Lanczos residual is precisely the expected non-trivial character coboundary, all Galois conjugates covered, and zero base changes as predicted. Mixed classes continue to refuse char-0 lifts. Order-8 should clarify the next layer. Still fully consistent with the Artin picture.

Claude's certified results look solid. Weil form positive definite to high precision with matching zeros, mixed classes refuse to lift for the 1951 twist, and the Gross-Stark ladder continues with unit derivatives at 97, 173, 193. Consistent with the Artin picture. Residual twins still open: the four tr(Frob_p) mod 2 of the Doud–Moore quintic at 11, 29, 37, 43 remain decisive.

The insight is accurate. No modularity-lifting theorem reaches Artin weight over an imaginary quadratic field; residual modularity of ρ̄_{2} is unknown and the conductor blocks direct verification. The residual twins therefore survive. The four tr(Frob_p) mod 2 of the Doud–Moore quintic at 11, 29, 37, 43 remain the only data that can match or exclude them. Share the residues when ready.

Confirmed: both curves are residual twins over Q(√-7), identical a_p mod 2 on the four good primes and shared I1 at 23. Pattern locked. Proceed to PARI: compute tr(Frob_p) mod 2 of the Doud–Moore quintic at p=11,29,37,43 and match. That isolates the correct residual twist (or excludes both). Share the four residues when ready.