Claimed extension to nonnegative weighted graphs

PDF p. 10: Section 2 opening paragraph

The article states that “all results in the paper can be extended to graphs with weighted edges, provided that all the weights are non-negative” (Section 2, PDF p. 10). Retain this as an attributed extension assertion, with network terminology from Finite symmetric networks and loop removal. Its energy and rotating-frame differential expressions extend to every finite symmetric nonnegative matrix, including arbitrary loops. Common positive rescaling of weights and the reference degree preserves the normalized centered spectral inequalities. However, an unchanged unit-loop degree bound is not valid for arbitrary loop weights: the correct general envelope contains the actual maximal loop weight. The five local results Loop removal in the weighted Kuramoto calculus, Covariance under a common positive rescaling of weights, Weighted spectral bounds control the degree without loops, Loop-dependent weighted degree envelope and A complete weighted network distinguishes a loop cap from spectral control establish these precise assertions. They do not specify or prove every weighted probabilistic or construction analogue covered by the article's informal sentence.

The quoted extension sentence occurs in the opening of Section 2, PDF p. 10. Under the original binary convention in Section 1.2, Lemma 3.1 on PDF p. 18 is valid: its proof explicitly uses $A_{xx}\in \{0,1\}$. Footnote 7 states that self-loops do not affect Kuramoto dynamics. The local example is therefore a counterexample only to retaining that displayed degree conclusion unchanged after allowing arbitrary nonnegative loop weights. It does not refute the original binary lemma, the existence of suitably modified weighted extensions, or any synchronization conclusion. The corrected envelope supplies such a weighted extension for this lemma. The full scope of unspecified weighted random-graph models and construction results remains outside the established local assertions; no universal proof is claimed.