Expert review brief

Prepared 17 September 2026. This brief has not been sent to anyone.

The attached manuscript attempts a complete planar proof for , all , with one positive-central decaying solution for every prescribed node count. It reconstructs CGY11's comparison proof and explicitly supplies two planar/infinite-endpoint arguments. It is not represented as a novel result.

Most important questions

Update: Zhang–Zhang, arXiv:2606.17840v2, Section 6, explicitly claims this same planar extension. Its Section 6 is unchanged from v1, although the abstract's planar sentence was removed. Please include this exact argument in the literature/gap assessment. See the assessment for the version comparison.

1. Literature status. CGY11 Theorem 1.1 and its example explicitly include the full planar power case. Is a mathematical gap or correction known? If so, which exact lemma or inference fails? Tang's later conjecture concerns both whole-space and finite-ball statements; could the discrepancy be a difference in scope rather than a failure of the planar whole-space result?
2. Transfer lemma. Does manuscript Lemma 5.1 correctly propagate the common-level strict comparison through every extremum? Please check the first-intersection cases, endpoint passage, and the uniform neighborhood for every pair of nearby parameters.
3. Tail. Does the preserved gap in Proposition 6.1 correctly rule out all alternatives except a new node on the upper side and negative-energy trapping on the lower side?

Exact locations

• CGY11, published p. 614, Proposition 4.3: vanishes at , so the printed strict inequality between and is unavailable.
• Manuscript equation (5.4): compare at instead, obtaining .
• CGY11, published p. 614, Proposition 4.4: the endpoint is infinite for a bound state.
• Manuscript equations (6.2)-(6.3): retain a fixed positive gap while both decaying endpoints would have .

The manuscript includes the variation intersection lemma, elementary ODE preliminaries, interval topology, and verification of the prescribed-node existence theorem's hypotheses. The exact identities were symbolically checked; the logical comparison still needs independent scrutiny.

Publication decision

A correct reconstruction is not automatically a new solution. If the older theorem is accepted, this could at most be an explanatory or corrective note, subject to editorial interest. If a substantive known gap remains, its precise identification is needed before evaluating whether this argument closes it. No priority or new-paper-count claim should be made merely because the argument has been typeset.