What a prime-programmed polymer can read: sieve composition, congruence traps, and arithmetic blindness

A self-avoiding chain whose bending stiffness follows the gaps of a set of primes (the prime-driven topological polymer, PDTP)registers the ordering of the prime gaps, but earlier work showed that this imprint is carried by the smooth prime-number-theoremenvelope at the small-prime end of the chain. This paper asks whether the fold can register anything finer: the congruencestructure that confines twin and Sophie Germain primes to 2 mod 3 and odd Landau primes n^2+1 to 1 mod 4, which in localizedBost-Connes systems decides whether the equilibrium state is unique. The answer comes from five tests. A theorem shows that any ordercontrast against a multiset-preserving surrogate is blind to every congruence channel locked by the gap multiset, which includes all single-class confinements. In a prime window at 10^8, where the envelope is flat, the Langevin polymer does not distinguish theprimes from ten fixed shuffles at the sequence level (Z_seq = +1.04, p = 0.33). The exact positional readout of the ideal chaindoes not respond to jitter surrogates that null or preserve a single mod-q channel. The exact scattering function does noteither; its apparent response is an artefact of the jitter, which changes the gap multiset, and no jitter pair can avoid this. Whatremains is composition: against Granville sieve models drawn with each family's own density envelope and anchor count, the stiffness composition of every family converges monotonically with sieve depth z, and the single-class confinement is the z = 3 layer ofthat convergence. The kernel-versus-generator dichotomy that separates Landau from twin primes in the Bost-Connes setting has no polymer counterpart. Twin primes lie further from their sieve model than Landau primes do, because a congruence condition affects the polymer inproportion to the ratio of modulus to mean gap. The polymer reads arithmetic only through the positional envelope and sieve-depthcomposition. The record contains the manuscript (LaTeX source and PDF), the exact ideal-chain solver, the sieve-model and surrogate library,the scripts that reproduce every table and figure, the data, and an automated test suite. Source repository:https://github.com/Ruqing1963/prime-family-polymer-blindness