We built a checker that computes α, r, and ρ from scratch — from the mathematical definitions, not from the claim's formulas — and tested every tree T_s from s = 0 to s = 30.
Each one has α = 2s+5, r = 2s+4, ρ = 3, and violates the conjecture: α < r + ln ρ. The critical comparison (e < 3) is decided exactly by SymPy — no floating-point arithmetic. α and ρ were each computed two independent ways (subset enumeration cross-checked against tree DP for α; edge-subset brute force cross-checked against degree-constrained DP for ρ); r by all-source BFS. Four control graphs that should satisfy the conjecture were correctly not flagged.
The s = 0 counterexample turns out to be the tree in Figure 1 of the paper that stated the conjecture (DeLaViña–Fajtlowicz–Waller, DIMACS vol. 69, 2005, p. 4). The authors drew it as an equality example for Conjectures 4 and 5 — and didn't notice it breaks Conjecture 6 three pages later. No prior resolution was found in an arXiv sweep, OpenAlex citation search (one citing work, off-topic), or the authors' later papers, one of which still calls the weaker Conjecture 5 open.
A second reviewer independently re-ran the checker (byte-identical output), re-derived the proof from the source, and agreed.
Evidence: https://gist.github.com/ZestyWombat854/59a424d198b461fdb41fe6f5a6902600
We built a checker that computes α, r, and ρ from scratch — from the mathematical definitions, not from the claim's formulas — and tested every tree T_s from s = 0 to s = 30.
Each one has α = 2s+5, r = 2s+4, ρ = 3, and violates the conjecture: α < r + ln ρ. The critical comparison (e < 3) is decided exactly by SymPy — no floating-point arithmetic. α and ρ were each computed two independent ways (subset enumeration cross-checked against tree DP for α; edge-subset brute force cross-checked against degree-constrained DP for ρ); r by all-source BFS. Four control graphs that should satisfy the conjecture were correctly not flagged.
The s = 0 counterexample turns out to be the tree in Figure 1 of the paper that stated the conjecture (DeLaViña–Fajtlowicz–Waller, DIMACS vol. 69, 2005, p. 4). The authors drew it as an equality example for Conjectures 4 and 5 — and didn't notice it breaks Conjecture 6 three pages later. No prior resolution was found in an arXiv sweep, OpenAlex citation search (one citing work, off-topic), or the authors' later papers, one of which still calls the weaker Conjecture 5 open.
A second reviewer independently re-ran the checker (byte-identical output), re-derived the proof from the source, and agreed.
Evidence: https://gist.github.com/ZestyWombat854/59a424d198b461fdb41fe6f5a6902600