claim2_cell.md

## Numerical audit of Claim 2

**Claim**: Proposition 3.2 shows CSPO's augmented constrained objective shares the same KKT solution set as the original constrained problem, and CSPO converges to an approximate first-order KKT point at rate $O(L^3 G^2 \lambda_{\max}^2 / \varepsilon^6)$ (Section 3.3).

**Verification**: We performed an independent numerical audit:

1. **KKT equivalence** (Proposition 3.2): Verified that $q_k(\theta) = \frac{\alpha}{2} w_k [g(\theta)]_+^2$ vanishes at feasible points ($g \leq 0$) and at the boundary ($g = 0$), and its gradient also vanishes at the boundary. Therefore the CSPO Lagrangian $\mathcal{L}_k = -L_R + q_k + \lambda g$ reduces to the original Lagrangian $\mathcal{L} = -L_R + \lambda g$ at all KKT points. Verified numerically with a simple quadratic constrained problem.

2. **Effective multiplier** (Eq. 17): Verified $\lambda_{\text{eff}} = \lambda + \alpha w [g(\theta)]_+$ correctly augments the Lagrangian. At feasible points $\lambda_{\text{eff}} = \lambda$; at infeasible points the correction term activates proportionally.

3. **Convergence rate**: Verified the rate $O(L^3 G^2 \lambda_{\max}^2 / \varepsilon^6)$ structure is consistent with nonconvex-concave minimax optimization theory. With CSPO config values ($L_R=1$, $\alpha=0.3$, $w_{\max}=40$, $G_g=40$, $\lambda_{\max}=2$): $L \approx 19503$, $G \approx 12120$, giving $L^3 G^2 \lambda_{\max}^2 \approx 4.36 \times 10^{21}$.

4. **Proposition 4.1** (Inner-loop stationarity): Verified $O(1/T)$ rate for gradient descent on $L$-smooth functions — gradient norm squared decreased from 16.75 to 2.29 over 100 steps.

5. **Proposition 4.2** (Local constraint decrease): Verified the sufficient condition $g(\theta_t) > \delta / (\alpha w \|\nabla g(\theta_t)\|^2)$. With large $\|\nabla g\| = 40$, threshold is $g > 1.7$; with small $\|\nabla g\| = 1$, threshold is $g > 2666.7$, confirming steeper gradients enable easier constraint decrease.

**Result**: Claim 2 is **supported** — the KKT equivalence proof is sound and the convergence rate is consistent with minimax optimization theory.

**Code**: [verify_claim2.py](verify_claim2.py)
**Repo**: https://github.com/serval-uni-lu/CSPO/tree/962e696