Near-Square Primes: Placeholder, Proof Retracted

This paper, originally uploaded December 2025, claimed a proof of Landau's fourth problem (infinitely many primes of form n² + 1) via mod 30 lattice geometry combined with Legendre's conjecture proved via Cramér's. On review in May 2026, the proof claim is retracted on four grounds: (a) STATUS — the badge "DERIVED" worn at the head of v1.0 covers content that is OBSERVED (the densities and the Hardy–Littlewood constant value at finite N) shading into STRUCTURAL (the lattice partition), with the infinitude claim sitting at CONJECTURED with heuristic support; v1.0's claim that the Hardy–Littlewood constant C "emerges not from probabilistic heuristics but from the geometric density of the surviving lattice" is a particular instance of the same overclaim, since the Hardy–Littlewood heuristic is that lattice-density computation under independence assumptions; (b) CIRCULAR — the central Theorem 3 rests on Theorem 1, which asserts Legendre's conjecture follows from Cramér's, but Legendre is itself an open Landau problem (reducing Near-Square to Legendre does not resolve it) and the Cramér proof v1.0 invoked has now been retracted (see Paper 7 [6], doi:10.5281/zenodo.18075423); (c) NULL/FATAL — Step 5 of Theorem 3 substitutes empirical extrapolation for deductive reasoning, arguing that observed density stability on [1, 7000] forbids density collapse at any larger scale; the same move, applied to the same kind of empirical stability, would close the twin prime conjecture, Goldbach, and essentially every Hardy–Littlewood prime-tuples conjecture; (d) STATUS/CLARITY — Sections 6 ("The Inexhaustible Tension") and the close of Section 7 carry the load of the argument on language ("the +1 cannot be exhausted because it is not consumed") that is metaphysical rather than mathematical and cannot do mathematical work in a proof. What survives is more substantial than in the other cluster placeholders. Specifically: (i) the mod 30 lattice characterisation (Proposition 1 of v1.0) — exactly 9 of 15 even residue classes mod 30 survive the mod 5 sieve — is correct and structural; (ii) the mod-3 colouring of those nine classes by n² + 1 mod 3 into three "1-coloured" classes and six "2-coloured" classes is correct and elementary; (iii) the observed near-coincidence of average primality rates between the two colour classes (1-coloured average ≈ 28.1%, 2-coloured average ≈ 27.9%) is a striking empirical pattern at N ≤ 7000, but warrants verification at considerably larger N with a null distribution and variance estimate before any novelty claim — and the relationship of this observation to the Chebyshev-bias literature is itself not yet established and forms part of the §3(B) work; it is an observation requiring investigation, not yet a result; (iv) computational verification of Hardy–Littlewood density at scales up to N = 7000 stands as range data. This v2.2 is a placeholder marking the proof-of-infinitude retraction honestly; the slot is held while two follow-up papers (a structural reframe of v1.0's surviving content, and a research note on the colour-class symmetry observation pending the verification above) are considered for separate slots.