Nonexistence of consecutive powerful triplets around cubes with mixed prime factorizations

A positive integer is powerful if every prime in its factorization occurs with exponent at least two. The Erdős–Mollin–Walsh conjecture asserts that no three consecutive integers are all powerful. Chan (Integers 25 (2025), #A7) excluded triples of the form x³−1=p³y², x³, x³+1=q³z², and She (Integers 25 (2025), #A103) excluded x³−1=p²a³, x³, x³+1=q²b³. This note closes the remaining mixed configurations: there are no consecutive powerful triples with x³−1=p²a³ and x³+1=q³z², nor with x³−1=p³y² and x³+1=q²b³. The case analysis reduces both families to the single equation t⁶+t³+1=3w², which is solved completely via a rank-0 elliptic quotient of a palindromic genus-2 curve. Complementary results: no finite congruence sieve can decide this equation, so that some global (non-congruence) argument is required, and Beckon's mod-36 constraint on consecutive powerful triples is refined to arbitrary prime-square moduli. The archive includes the LaTeX source and the complete verification scripts (SymPy, SageMath). Version 1.1 (July 2026): editorial revision of v10, no mathematical content changed — the birational correspondence in Section 5 is now written out explicitly, one abstract phrase was made more neutral, spelling was unified to American English, and the bibliography was enriched with verified DOIs. See the archive README for the full version history.Version 1.2 (September 2026): Disclosure revision plus one overview figure; no mathematical content changed. The acknowledgement in v1.0 and v1.1 stated that substantial parts of the work, including the reduction of Section 5, were developed with AI assistance. That was accurate, but too short and not specific enough measured against the broader and more specific disclosure standard the author uses in his other preprints. This version replaces it with a full "Use of AI tools" section. The revision was made on the author's own initiative, not in response to any query or objection. Also new: Figure 1, an overview of the 2x2 grid of shape combinations (Chan, She, and the two mixed cases of this note) and of the reduction of the mixed cases to the rank-0 elliptic quotient; it adds no new mathematics. The verification scripts are now released under the Apache License 2.0 (paper: CC BY 4.0). All theorems, proofs, numbers, and verification scripts are unchanged from version 1.1.Version 1.3 (September 2026): Citation correction; no mathematical content changed. Reference [5] for the Erdős–Mollin–Walsh conjecture previously pointed to Erdős, Publ. Math. Debrecen 23 (1976), 271–282, which discusses consecutive powerful pairs but not the three-consecutive question; it now points to Erdős, Congressus Numerantium XVI (1976), 25–44, where the question is raised (p. 31). The misattribution is common in the literature (Bajpai, Bennett and Chan, Int. J. Number Theory 20 (2024), §7); both Erdős papers were checked in the original for this correction. Nothing else changed from version 1.2.