Congruence-Class Gram Reconstruction for Proper Trees: Multiple Repeated-Weight Classes Without a Common-Divisor Hypothesis

We prove a restricted reconstruction theorem for proper trees: if their weighted skeleton has all vertex weights in one congruence class modulo some integer at least two, at least two uniquely occurring weights, and an independent set of all repeated-weight vertices, equality of the first power-sum derivative of the chromatic symmetric function for proper-tree competitors implies isomorphism.

An explicit two-edge Gram decoder and acyclic neighborhood reconstruction yield the result. The theorem depends on cited results in Saad A. Awan, arXiv:2609.13464. It does not resolve Stanley’s unrestricted conjecture. Independent human peer review and historical priority are not certified.