Jasper Yao

Jasper Yao

We establish fundamental mathematical limits on universal approximation theorem (UAT) system alignment by proving that catastrophic failures are an inescapable feature of any useful computational system. Our central thesis is that for any universal approximator, the expressive power required for useful computation is inextricably linked to a dense set of instabilities that make perfect, reliable control a mathematical impossibility. We prove this through a three-level argument that leaves no escape routes for any class of universal approximator architecture. i) Combinatorial Necessity: For the vast majority of practical universal approximators (e.g., those using ReLU activations), we prove that the density of catastrophic failure points is directly proportional to the network's expressive power. ii) Topological Necessity: For any theoretical universal approximator, we use singularity theory to prove that the ability to approximate generic functions requires the ability to implement the dense, catastrophic singularities that characterize them. iii) Empirical Necessity: We prove that the universal existence of adversarial examples is empirical evidence that real-world tasks are themselves catastrophic, forcing any successful model to learn and replicate these instabilities. These results, combined with a quantitative "Impossibility Sandwich" showing that the minimum complexity for usefulness exceeds the maximum complexity for safety, demonstrate that perfect alignment is not an engineering challenge but a mathematical impossibility. This foundational result reframes UAT safety from a problem of "how to achieve perfect control" to one of "how to operate safely in the presence of irreducible uncontrollability," with profound implications for the future of UAT development and governance.

Jasper Yao

We introduce AlgoSelect, a principled framework for learning optimal algorithm selection from data, centered around the novel Comb Operator. Given a set of algorithms and a feature representation of problems, AlgoSelect learns to interpolate between diverse computational approaches. For pairs of algorithms, a simple sigmoid-gated selector, an instance of the Comb Operator, facilitates this interpolation. We extend this to an N-Path Comb for multiple algorithms. We prove that this framework is universal (can approximate any algorithm selector), information-theoretically optimal in its learnability (thresholds for selection converge almost surely, demonstrated via Borel-Cantelli arguments), computationally efficient, and robust. Key theoretical contributions include: (1) a universal approximation theorem demonstrating that Comb-based selectors can achieve arbitrary accuracy; (2) information-theoretic learnability for selection thresholds; (3) formalization of the Comb Operator within linear operator theory, detailing its boundedness and spectral properties; (4) an N-Path Comb generalization for multi-algorithm selection; and (5) a practical learning framework for the adaptive seeding functions that guide the Comb Operator. Empirical validation on a comprehensive 20$\times$20 problem-algorithm study demonstrates near-perfect selection (99.9\%+ accuracy) with remarkably few samples and rapid convergence, revealing that $H(\text{Algorithm}|\text{Problem}) \approx 0$ in structured domains. AlgoSelect provides a theoretically grounded, practically deployable solution to automated algorithm selection with provable optimality and learnability guarantees, with significant implications for AI and adaptive systems.

Jasper Yao

This paper argues that AI alignment is not merely difficult, but is founded on a fundamental logical contradiction. We first establish The Enumeration Paradox: we use machine learning precisely because we cannot enumerate all necessary safety rules, yet making ML safe requires examples that can only be generated from the very enumeration we admit is impossible. This paradox is then confirmed by a set of five independent mathematical proofs, or "pillars of impossibility." Our main results show that: (1) Geometric Impossibility: The set of safe policies has measure zero, a necessary consequence of projecting infinite-dimensional world-context requirements onto finite-dimensional models. (2) Computational Impossibility: Verifying a policy's safety is coNP-complete, even for non-zero error tolerances. (3) Statistical Impossibility: The training data required for safety (abundant examples of rare disasters) is a logical contradiction and thus unobtainable. (4) Information-Theoretic Impossibility: Safety rules contain more incompressible, arbitrary information than any feasible network can store. (5) Dynamic Impossibility: The optimization process for increasing AI capability is actively hostile to safety, as the gradients for the two objectives are generally anti-aligned. Together, these results demonstrate that the pursuit of safe, highly capable AI is not a matter of overcoming technical hurdles, but of confronting fundamental, interlocking barriers. The paper concludes by presenting a strategic trilemma that these impossibilities force upon the field. A formal verification of the core theorems in Lean4 is currently in progress.