Why expansion methods for Chowla do not reach twin primes: a method-limitation theorem and an explicit parity compiler

The logarithmically averaged two-point Chowla conjecture was proved by Tao and quantitatively improved by Helfgott–Radziwiłł and Pilatte, through a dilation identity followed by a spectral bound for a divisibility operator. On the generalised Elliott–Halberstam conjecture (GEH), Bombieri's asymptotic sieve turns the twin prime conjecture into a Chowla-type statement on almost primes, and Tao remarked that this restriction destroys the multiplicativity the Chowla arguments rely on. We make this remark a theorem. We define a class of dilation–spectral certificates containing the Helfgott–Radziwiłł and Pilatte certificates adapted to the shift 2, with arbitrary smooth, rough or mixed dilators up to xκ, and prove that every such certificate has resolution >> η min(1, η/κ) log x on twin-type parity (Theorem A). The loss is the factor 1/log xη that two-sided roughness costs on every edge. To state what such a certificate would have to resolve, we write the Bombieri–Tao reformulation as an exact, numerically testable identity for ∑n≤X Λ(n)Λ(n+2)/n in terms of a logarithmic parity defect ρ(X) of the Liouville function on almost-prime pairs (Theorem B), and show that on GEH the joint parity of rough pairs is governed by the twin parameter alone (Theorem C). A road map (Figure 1) summarises the structure. Nothing here proves or disproves the twin prime conjecture. AI assistance. This work was produced with substantial assistance from an AI system (Claude, Anthropic). Appendix D records what was explored numerically, which claims were corrected along the way, and how each result was verified. The proofs have not yet been independently checked by a human specialist; corrections are welcome. Files: the paper (PDF), the LaTeX source, and the Python scripts with their JSON outputs that produce every numerical claim.