Explicit Closed Elementary Nets That Cannot Be Completed, Including Fields of Transcendence Degree One

An elementary net (carpet) of order n over a field K is closed if its elementary net group contains no new elementary transvections, and it is completable if its diagonal can be supplemented to a full net. Completable nets are closed. Koibaev gave closed nets that are not completable over fields of characteristic 0 and 2 and asked for such nets in odd characteristic (Kourovka Notebook, Problem 21.76). Nuzhin (2026) has answered this question, with examples in every characteristic, using closures of finitely generated subgroups over a rational function field in two variables. We give explicit examples with short proofs. An elementary lifting lemma turns a closed pair of additive subgroups of a quotient ring R/J into a closed elementary net of every order n ≥ 3. With F[x,y] → F[x] and a degree argument it gives, for every field F, the net with Fx + yF[x,y] in positions (1,2) and (2,1) and yF[x,y] elsewhere. Lifting the two exceptional pairs of Levchuk's refinement of Dickson's theorem along F_3[t] → F_9 and F_2[t] → F_4 gives closed nets over F_3(t) and F_2(t) that are not completable. These one-variable nets also satisfy the hypotheses of Kourovka Problem 19.48. Together with a theorem of Koibaev and Nuzhin on algebraic extensions, it follows that in characteristics 2 and 3 a field carries such nets if and only if it is not algebraic over its prime field. For p ≥ 5 we do not know whether examples exist over F_p(t); by Levchuk's theorem, lifting from a finite field cannot produce them. This is an unrefereed note. Scope and priority: to our knowledge Problem 21.76 was first answered by Ya. N. Nuzhin (Sib. Math. J. 67 (2026) 840–845, doi:10.1134/S0037446626040099); no priority is claimed for that answer. The new points are the explicit nets with elementary proofs, the one-variable examples in characteristic 3, and the characterisation for p = 3. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: KOU-21.76. Paper page: https://eulersolve.org/papers/kou-21-76/