When Can One Obtain Certificates of Optimality Using Positivstellensaetze?

Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric. We formulate this effort as a path-space value problem. Restricted dynamic geometric complexity is the least affine-invariant length of an admissible metric path whose endpoint reaches a Hessian-relative generalized-eigenvalue condition target. Path elimination gives an exact min-plus semigroup and Bellman principle, while fixed-horizon kinetic energy is exactly squared complexity divided by twice the horizon. The main response result is global on a Hadamard state space: geodesic convexity produces a smooth intervention-cube path branch and a uniformly coercive Jacobi form, while one Green inverse generates the value Hessian, two-sided force-to-curvature bounds, exact Möbius effects, and arbitrary prescribed finite-order responses. For the hard condition target, a bordered Jacobi--KKT theorem differentiates the moving projection endpoint and multiplier on every regular active spectral stratum; its indefinite inverse also explains why hard-target interactions need not share the unconstrained sign. The theory specializes to affine-invariant positive-definite geometry. A determinant-one two-dimensional diagonal model has an exact target interval, a closed-form forced path, and a strictly negative-definite interaction matrix. A moving diagonal Hessian gives a closed-form hard-target projection, multiplier, and pair effects of either sign, while a coordinate-sequential three-dimensional protocol yields an exact path metric strictly larger than the ambient projection distance. Thus the global Green and bordered hard-target responses are explicit laws of restricted metric-path elimination built on Bellman composition.

We prove the positive-real $n=9$ case of the Vasc cyclic inequality. The proof was obtained with human-guided assistance from the AI agent MechMath Agent Team: the human-readable part reduces the rational inequality to a homogeneous polynomial inequality, fixes a cyclic maximum, and parametrizes each sorted fixed-maximum cone by cumulative gaps; the finite part is a certificate covering all $8!=40320$ sorted cones. MechMath Agent Team generated the certificate verification workflow through Python tool calls, including the case split, verification programs, and terminal classifications. The published certificate has $36815$ coefficient leaves, $2236$ ordinary Polya multiplier leaves, and $1269$ AM-GM midpoint overlay leaves. Human authors audited the mathematical reductions and verification logic, and a separate artifact contains the certificate, an independent verifier, and a from-source rebuild route.

We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity. This leads to an optimal refutation of the Alon-Saks-Seymour conjecture, as well as an optimal communication lower bound for the Clique versus Independent Set problem, improving the previous results of Balodis, Ben-David, Göös, Jain and Kothari (FOCS 2021, SICOMP 2023) by several doubly logarithmic factors. As further applications of our construction to query complexity and learning theory, we exhibit: (a) a family of Boolean functions that has an optimal quartic separation between certificate complexity and approximate degree, and (b) a sample compression lower bound of $Ω(\sqrt{\log c})$ for multiclass concept classes over $c$ labels.