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Definition 4 (Definition: Fundamental Group) Let \(X\) be a topological space and \(x_0 \in X\) a basepoint. A loop based at \(x_0\) is a continuous map \(\gamma: [0,1] \to X\) with \(\gamma(0) = \gamma(1) = x_0\). Two loops \(\gamma, \delta\) are homotopic relative to \(x_0\) (written \(\gamma \simeq \delta\)) if there exists a continuous map \(H: [0,1] \times [0,1] \to X\) such that \[H(s,0) = \gamma(s), \quad H(s,1) = \delta(s), \quad H(0,t) = H(1,t) = x_0\] for all \(s,t \in [0,1]\). This is an equivalence relation; denote the equivalence class of \(\gamma\) by \([\gamma]\).
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A small, trusted kernel: a few thousand lines of code that check every step of every proof mechanically. Everything else (the AI, the automation, the human guidance) is outside the trust boundary. Independent reimplementations of that kernel, in different languages (Lean, Rust), serve as cross-checks. You do not need to trust a complex AI or solver; you verify the proof independently with a kernel small enough to audit completely. The verification layer must be separate from the AI that generates the code. In a world where AI writes critical software, the verifier is the last line of defense. If the same vendor provides both the AI and the verification, there is a conflict of interest. Independent verification is not a philosophical preference. It is a security architecture requirement. The platform must be open source and controlled by no single vendor.
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