Twin Prime Conjecture: Placeholder, Proofs Retracted

This paper, originally uploaded December 2025 (titled The Crystallographic Proof of Twin Prime Infinitude), claimed two complementary proofs of the Twin Prime Conjecture — algebraic (Section 4) and geometric (Section 5) — via Eisenstein partition (inert/split mod 3) and shadow sub-lattice analysis. On review in May 2026, both proof claims are retracted on five grounds of distinct kinds: two direct mathematical errors that refute the proofs (a, b); one status mislabelling whose correction itself collapses the algebraic proof's independence step (c); and two further structural reasons (d, e) to be confident the retraction is correct. Specifically: (a) MATHEMATICAL ERROR in the algebraic proof — Section 4 Step 3 asserts ∏_r≥5(r−2)/(r−1) converges to C₂/2 > 0, but this product diverges to zero, since by Euler [1] the sum ∑1/(r−1) over primes diverges; the correct Hardy–Littlewood [3] form is ∏(1 − 1/(p−1)²), which is a different product and the one that converges. (b) STRUCTURAL ERROR in the geometric proof — Section 5 Step 4 invokes "finite union of incommensurate sub-lattices cannot cover an infinite lattice," but the argument requires an infinite union (one sub-lattice for every prime r ≤ √(p+2) as p → ∞), and infinite unions of pairwise-coprime sub-lattices can cover an infinite lattice with density approaching one (the Eratosthenes sieve, where ∏_p≤x(1−1/p) → 0 by Mertens [2]). (c) STATUS — Step 2 cites CRT as justification for treating blocking events as independent, but CRT gives equidistribution modulo ∏r_i, not independence of "p+2 divisible by r" events; the Hardy–Littlewood heuristic [3] uses approximate independence to conjecture C₂, but this is not derived. (d) PARITY BARRIER — Selberg's obstruction (see Iwaniec & Kowalski [4] for the modern treatment; subsequent breakthroughs on bounded gaps between primes [5, 6] do not engage parity either) is not engaged at any point. For v1.0's specific algebraic proof — which sieves the inert primes by blocking residues mod r and lower-bounds escape density — the parity barrier is close to a direct refutation in the same sense (a) is: it rules out exactly this class of argument. (e) META-TEST — the argument structure, if accepted, would establish infinitude for primes of form n²+1 (Landau's fourth, see Paper 10 [8]) and for primes in every admissible k-tuple (Hardy–Littlewood prime-tuples conjecture [3]). Both are open; an argument that proves too much proves nothing. What survives, with appropriate framing as motivation rather than derivation: (i) the Eisenstein inert/split partition (p ≡ 2 mod 3 vs p ≡ 1 mod 3) with the +2 step carrying a candidate across the boundary is pretty and pedagogically valuable; (ii) the shadow-lattice visualisation in v1.0 Section 6 is accurate at small scale with correct sub-lattice periods (3×5, 3×7, 3×11, 3×13, ...); (iii) the framing of twin primes as "bridges between structurally independent classes" is a useful intuition pump for why the conjecture ought to be true — distinct from the claim that it is true. This v2.2 is a placeholder marking the retraction of both proof claims; the slot is held for possible future structural-motivation work in the directions described in §3.