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PreprintCC BY 4.0Zenodo

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

  • Casey Atwell
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Abstract

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.

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Cite

BibTeXRIS
BibTeX
@misc{atwell2026congruenceclass,
  title = {{Congruence-Class Gram Reconstruction for Proper Trees: Multiple Repeated-Weight Classes Without a Common-Divisor Hypothesis}},
  author = {Casey Atwell},
  howpublished = {Zenodo},
  year = {2026},
  month = oct,
  doi = {10.5281/zenodo.23627747},
  url = {https://zenodo.org/records/23627747},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Congruence-Class Gram Reconstruction for Proper Trees: Multiple Repeated-Weight Classes Without a Common-Divisor Hypothesis
AU  - Casey Atwell
PY  - 2026
DA  - 2026/10/24
PB  - Zenodo
DO  - 10.5281/zenodo.23627747
UR  - https://zenodo.org/records/23627747
AB  - 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.
ER  -