Locally finite p-groups in Golod groups and an AT section

We give an affirmative answer to Kourovka Notebook Problem 21.131 of A. V. Timofeenko. For every prime p, we construct a countable infinite locally finite AT p-group satisfying Rozhkov's original infinite-support definition and an epimorphism onto it from a subgroup of a classical homogeneous, residually finite Golod p-group. More generally, every countably infinite locally finite p-group embeds in a Golod group in the precise canonical nil-adjoint sense of Problem 9.76, and is a quotient of a subgroup of such a classical homogeneous Golod group. The argument uses the established stable-nil embedding construction of Alahmadi, Alsulami, Jain and Zelmanov, first published in 2017. Relative elementary matrices place the needed unit derived subgroup in a finitely generated canonical adjoint subgroup. A direct scalar-extension argument gives absolute nilness for the selected realization, allowing Hammoudi's homogeneous Golod cover to be applied. The AT target is infinitely generated; no answer to Problem 13.55 or finitely generated AT-target strengthening is claimed. The seven-page note is accompanied by LaTeX source, standard-library exact verifiers, expected outputs, source and priority audits, and independent AI adversarial reviews. Finite computations supplement the infinite proofs and do not certify their universal quantifiers. The extensive bounded literature search through 24 October 2026 found no earlier complete same-scope result; this is not an exhaustive certification of first priority. Original 1986 AT text was inspected through primary indexed full text; its PDF bytes were not acquired. AI tools were used extensively in discovery, development, writing and verification. This preprint has not undergone independent human peer review, and no formal proof-assistant verification is claimed. Paper, prose and code: CC BY 4.0.