Two-channel duality, fold memory, and arithmetic blindness in polymer contact networks

The contact network of a collapsed polymer is a finite graph, and graph invariants of the kind used in algorithmic class field theory can be read off it exactly: the Laplacian, its sandpile group, the discrete Hodge decomposition, the Albanese metric of a periodic cover, and the Perron branch of the adjacency. This paper applies them to 82 conformations of the prime-driven topological polymer, whose bending stiffness follows the prime gaps, against surrogates that shuffle the gaps or keep only their smooth prime-number-theorem envelope. Unweighted invariants, including the exact sandpile group, do not distinguish the arms. The sequence enters through a divisor, the rank of the stiffness on the vertices, and the two channels then separate. In the bulk (Laplacian) channel the divisor energy separates the true chain from its shuffles at Z_seq = +11.4 in both partially collapsed and fully relaxed globules, while the boundary (Perron) channel separates no pair of arms. The bulk channel reads the envelope: the PNT-only surrogate matches the true chain (p = 0.76), and once the smooth positional modes are removed every arm contrast vanishes (Holm-adjusted p = 1 for all 16 tests). A marginal difference seen in an eight-globule pilot (p = 0.043) does not replicate on the full library. What survives in every channel and every arm is a strong memory of the globule's own sequence, z of about 6 to 24 against gap shuffles on the same conformation, equal for primes, shuffles and PNT-only surrogates. The folded network remembers its sequence, and the arithmetic it can tell apart is the envelope's. Two identities hold exactly on all 82 graphs: the backbone winding has Albanese energy N, and the spanning-tree count times the closure-flux energy is an integer. This paper closes a four-part series. The record contains the manuscript (LaTeX source and PDF), the contact-graph, Hodge, Legendre-filter and Albanese libraries, the scripts behind every table and figure, the data, and an automated test suite. The conformations are those of the PDTP v4.0 repository. Source repository: https://github.com/Ruqing1963/contact-network-two-channels