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

===== You can (and should) modify the proof so that it does not require x^C>0\hat x_C>0x^C>0. The clean fix is: =====
# Define x^\hat xx^ as the restricted minimizer on CCC (with xCˉ=0x_{\bar C}=0xCˉ=0), not by forcing the sign pattern.
# Prove x^C≥0\hat x_C\ge 0x^C≥0 automatically (nonnegativity is true for PageRank-type QQQ with nonpositive off-diagonals).
# Use the restricted KKT subgradient zC∈∂gC(x^C)z_C\in\partial g_C(\hat x_C)zC∈∂gC(x^C) instead of hard-coding zC=λCz_C=\lambda_CzC=λC.
# Then your conductance/connectivity/degree analysis bounds the outside slack and yields a parameterized lower bound on the outside margin

γCˉ:=min⁡i∈Cˉ(λi−∣∇f(x⋆)i∣).\gamma_{\bar C}:=\min_{i\in\bar C}\bigl(\lambda_i-|\nabla f(x^\star)_i|\bigr).γCˉ:=i∈Cˉmin(λi−∣∇f(x⋆)i∣).
If you additionally want the global γ\gammaγ (min over all inactive, including possible zeros inside CCC), you must also control any zeros inside CCC; that is a separate “beta-min / interior slack” assumption and cannot be forced just from ϕ(C)\phi(C)ϕ(C) etc.

Below is drop-in LaTeX that fixes the proof properly.