Changing the language does not move the parity wall: a dynamical anatomy of the twin-prime obstruction and the function-field bypass

It is sometimes suggested that the parity barrier for twin primes might disappear once the problem is translated into the language of dynamical systems or physics. We test this suggestion channel by channel: symbolic dynamics of sieve encodings, spectral statistics of zeta zeros, Furstenberg systems and disjointness, the ax+b semigroup, the function-field proof of Sawin and Shusterman, hyperbolic prime geodesics, and the primon gas and the Bost–Connes system. None of them moves the wall, and the reasons organise into layers. The additive layer degenerates: along rough pairs the Furstenberg system collapses and its Palm system keeps one parameter; pairwise-random counterfeits satisfy every Sarnak-type disjointness. The multiplicative layer (dilation consistency) excludes every non-multiplicative counterfeit but not the characters χ3, χ4. Averaged statistics separate λ from χ3, χ4 but are functions of one-point (pretentious) data. The missing link is robust rigidity, which implies twin non-extremality. The function-field proof avoids this link by an algebraic identity: Pellet's formula together with a derivative that commutes with translation turns the Möbius function, on each translation subspace, into a Dirichlet character of large varying modulus. The natural substitutes over Z produce only characters of fixed modulus, i.e. the counterfeits themselves: uniform permutation signs are Legendre symbols (via Fried's theorem), Buium's p-derivation does not commute with translation and its Pellet analogue collapses to quadratic reciprocity, the primon gas has the shift outside its solvable algebra, and the Bost–Connes system quotients out integer translations. The missing object over Z is a derivation in a variable common to all primes. Nothing here proves or disproves the twin prime conjecture. The work was carried out with substantial AI assistance; every statement is labelled as a theorem, a heuristic, an analysis or a numerical observation.