GP: A Geometric Polynomial Complexity Class Uniform Definition and Complete Proofs

This groundbreaking article introduces GP (Geometric Polynomial), a novel complexity class that revolutionizes the intersection of computational complexity and continuous optimization. By formalizing a uniform, bit-precise definition, it establishes a new paradigm for solving decision problems in polynomial time through geometric representations: a differentiable energy function $E_x$, a Riemannian metric \(g_x\), and controlled navigation procedures (e.g., adaptive gradient descent guided by spectral quantities). This approach avoids abusive encodings of non-polynomial power, imposing strict polynomial constraints on arithmetic circuits, evaluability, and precision $\epsilon = 1/poly(n)$. Key innovations include rigorous proofs in the bit-complexity model: Theorem 3.1 demonstrates P ⊆ GP by encoding Turing machine executions into polynomial energies with local strong convexity, ensuring polynomial-bit convergence. Theorem 3.2 conditionally places GP ⊆ NP, leveraging quantified gaps $(\Delta \geq 1/$exp(poly(n)$)$) for verifiable certificates. Technical appendices provide complete, effective bounds on gradient descent, Riemannian reductions, spectral approximations (Lanczos/LOBPCG), floating-point errors, and regularization—making GP the first complexity class to operationalize "geometric oracles" with quantitative numeric guarantees. This paradigm shift bridges discrete Turing computation with continuous geometric flows, opening avenues for theoretical advances (e.g., intermediate classes between P and NP) and practical applications in optimization, AI, and beyond. As a pioneering framework, it invites experimental validation and extensions to capture "natural" problems unsolvable by traditional methods.