Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

==== - This kind of theorem is plausible and standard (it’s exactly the strict dual feasibility logic), but it does require assumptions that are stronger than just “small conductance”. In particular, without something like “no low-degree boundary whiskers” (or at least “inactive nodes have large degree”), you cannot hope for γ≳ρ\gamma\gtrsim \sqrt{\rho}γ≳ρ, because γ≤min⁡i∈I⋆λi=ραmin⁡i∈I⋆di\gamma \le \min_{i\in I^\star}\lambda_i = \rho\alpha\min_{i\in I^\star}\sqrt{d_i}γ≤mini∈I⋆λi=ραmini∈I⋆di. ====
* If you’re thinking in the usual sparse-graph local clustering regime (many degree-1/2 nodes), the best “natural” order you can hope for is more like γ=Θ(ρ)\gamma=\Theta(\rho)γ=Θ(ρ) (up to constants), not Θ(ρ)\Theta(\sqrt{\rho})Θ(ρ).

If you want, I can convert the theorem + corollary into a compact LaTeX section and align notation precisely with your earlier appendices (including a short proof: it’s a one-page KKT/primal–dual witness argument).