We propose a formal economic framework for mathematical problem-solving, treating discovery as a production process within a computational factory. By shifting the role of Artificial Intelligence from a direct “solver” to a “heuristic bound-finder,” we demonstrate that even marginal gains in search-space reduction yield superlinear returns as time approaches infinity \((T \to \infty)\).

We define the “Compounding Search Engine,” where identifying patterns in mathematical structures leads to tighter operational bounds, which in turn accelerates subsequent discovery. The model suggests that the human role in this system transitions from manual calculation to high-level “stall detection” and system redirection.

The Mechanics of Bounded Search#

Rather than treating math solutions as absolute axioms, we treat them as operational strategies. For any given problem (e.g., an Ordinary Differential Equation or a 3D search problem), known constraints can reduce the domain or range.

  • Case Example (3D Search): If the search direction is known via a vector, the search space is immediately reduced to a single octant (1/8th of the original space), achieving an 87.5% efficiency gain.

Machine Learning (ML) acts as a “Pattern Scout.” Its role is to identify where boundaries likely exist within the computational landscape before the heavy compute of a full search is triggered.

The Asymptotic Efficiency Model
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The core of our efficiency model relies on the Asymptotic Efficiency Theorem (see: Original Theorem Post). We define \(C\) as the fixed cost of discovering a boundary and \(G\) as the efficiency gain (the factor by which the search space is reduced).

  • Key Formula: The discovery of a boundary is economically viable if the cumulative savings over \(T\) iterations exceed the cost \(C\).
  • The Multiplier: Because finding a boundary is a fixed upfront cost while its application provides a recurring benefit, the “Efficiency Multiplier” approaches infinity as \(T \to \infty\). Even a marginal 1% gain (\(G = 1.01\)) eventually offsets any finite discovery cost.

We define the Compounding Search Engine as a closed feedback loop where each discovered bound (output) becomes high-fidelity training data that lowers the cost \(C_{t+1}\) for the next discovery.

The Economic Factory of Discovery
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We propose a formal Ontology of Math Goods, where a solved theorem or a validated search result is treated as a manufactured commodity. In this model, computational biology or space-search problems are not “riddles” but “production units” moving through a factory line.

The Singularity Factory is the ultimate expression of this ontology: a system where the AI discovers a new bound, uses it to solve a problem, extracts a new pattern from that solution, and generates a tighter bound–all in a millisecond loop. This “200 MPH highway” represents the transition from linear human-led research to exponential automated discovery.

Conclusion: The Human as Roomba-Kicker
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In this high-speed “Discovery Factory,” the human role is refined to system-level maintenance. We call this “Stall Detection.” Just as a user might kick a stuck Roomba, the human mathematician intervenes when the AI’s pattern recognition hits a logic wall or when search-space reduction falls below a critical threshold. The human provides the “out-of-domain” kick that redirects the factory back onto a productive path.

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QED

References
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  1. Tao, T. (2025). AI-Assisted Mathematics and the Future of Formal Proofs. Available Online.
  2. David. (2026). Asymptotic Efficiency Theorem. Original Thesis.