Legendre's Conjecture: Placeholder, Proof Retracted

This paper, originally uploaded December 2025, claimed a proof of Legendre's conjecture (a prime between every pair of consecutive squares) by reduction to the structural proof of Cramér's conjecture asserted in Keeble (2025; retracted 2026 [4]). On review in May 2026, the proof claim is retracted on four grounds: (a) the foundation has been removed — the Cramér paper has itself been retracted to placeholder status (see Paper 7 [4]), so this paper's conditional claim "Legendre follows from Cramér" has no antecedent to follow from; (b) the original abstract and body contradict each other on the threshold (abstract: "n > 5"; body Section 3.5: n > 40, after the inequality fails at n = 5, 6, 10 and crosses over near n ≈ 30–40); (c) the reduction itself, even granting an effective Cramér bound, conflates asymptotic and effective bounds — the Cramér constant C → 2e^(-γ) is asymptotic, but the body's Section 3.5 argument requires the bound to hold for every prime in [n², (n+1)²] from n = 41 onward, which asymptotic claims do not deliver; (d) the original Section 4 ("The Ternary Structure") is gestural rather than theorem-shaped — assertions that forbidden phase transitions "ensure" prime coverage of residue classes are not derivations and do not earn their place in the proof. What survives is the elementary reduction Cramér ⟹ Legendre (for n past some threshold), which is essentially folklore in the gap-conjecture literature and not original to v1.0. Computational verification of Legendre is in the published literature, well beyond any range the conditional argument actually requires. This v2.1 is a placeholder marking the retraction honestly; the slot is held in case future work in the 2I Universe Programme produces an independent structural argument for Legendre, but this placeholder makes no commitment that it will be filled.