Where the twin-prime parity wall stands: the weakest target, multiplicative rigidity, and the limits of expansion on rough support

On the generalised Elliott–Halberstam conjecture the twin prime conjecture is a statement about the signs λ(n)λ(n+2) of the Liouville function on pairs whose entries have no prime factor below xη. This paper locates the obstruction to such statements. The extreme parity configuration (λλ = −1 on all rough pairs) is realised by a sieve-invisible measure, by periodic colourings and by the completely multiplicative characters χ3, χ4; these are one object. The weakest target. Exact multiplicative rigidity for rough pairs (a finite computation shows the only extremal completely multiplicative functions on the core are χ3, χ4) implies asymptotically the existence of a single even rough pair, E(η). A family lemma shows that relations produced by polynomial families, which is what sieve existence arguments supply, always sum to zero. Literature position. E(η) is not known, but each natural relaxation is a theorem: a bounded set of gaps (Pintz), averaged gaps (Matomäki–Radziwiłł–Tao), or fixed roughness (from Tao's two-point theorem). Existence gives no leverage, because the counterfeit has zero even mass. The roughness axis. With ρ = log log z / log log x, the pretentious distance from λ to χ4 on [z, x] is (1−ρ) log log x, bounded exactly at the twin endpoint ρ = 1. Expansion on rough support. The shift-2 dilation identity survives roughness exactly; for the counterfeit the edge form and the mean form of the Helfgott–Radziwiłł decomposition both equal −1, so parity enters only through the averaged-shift mean term. A conditional theorem reaches log z ≤ c√(log x), matching a known upper limit. For the sparsification step: an exact sampling trace identity, a proof that the trace method loses a logarithm and that this loss is real, a pruned template in an independent model (Le–Levina–Vershynin), and deterministic “twin rectangles” through which the real rough graph departs from the model. Nothing here proves or disproves the twin prime conjecture. Every statement is labelled as a theorem, a conditional statement, a heuristic or a numerical observation. Sequel to 10.5281/zenodo.23736249. Stage reports, scripts and data: github.com/maris205/twin_primes/research/parity-wall-route. Web road map: claude.ai/artifact/R4YKS4ABCmSkbRC1ZMFQSJ.