Conditional Bounds on AI Self-Improvement in an Antitone Threshold Model
Description
We establish conditional bounds in a stylized antitone threshold model of recursive AI self-improvement. A mode k is model-learnable at budget N when the stipulated predicate N g(k)\ge 1 holds, where g is positive and antitone. This static threshold model is mathematically separate from an abstract natural-number recurrence and from an auxiliary verifier-yield calculation considered later. In particular, the paper does not prove that values reached by the recurrence are learnable modes. The displayed results are supported by conventional mathematical arguments in the manuscript; no statement-level machine formalization is claimed. (i) Static cutoff. If \(g(k)\to0\), every fixed positive budget has a finite maximal learnable frontier \(F_m(N)\). Summability is one sufficient condition for \(g(k)\to0\). (ii) Finite-target sufficiency. Within the threshold model, every indexed finite frontier has a finite sufficient budget; for \(K>0\), its smallest real-valued budget is \(1/g(K-1)\). (iii) Static frontier bounds. Every admissible count satisfies \(K\le N\sum_{k<K}g(k)\); under summability \(F_m(N)\le N\sum_{k\ge0}g(k)\). This is a consistency bound, not by itself an identified growth rate. (iv) Separate recurrence result. An arbitrary recurrence that preserves a finite invariant set is bounded; if it is also non-regressing, it is eventually constant. No learnability interpretation follows without an additional bridge hypothesis. For stipulated power-law coupling \(g(k)=C(k+1)^{-\beta}\), where \(\beta\) is a positive decay exponent and \(C>0\) is a scale constant, Section 6 gives the exact maximal frontier \(F_m(N)=\lfloor(CN)^{1/\beta}\rfloor\) for \(N>0\). Connecting \(\beta\) to Zipf frequencies, covariance eigenvalues, language-model loss exponents, or real capability scales requires separate validation. The illustrative choice \(\beta_{\mathrm{model}}=3.25\) and its floor-free 24% doubling ratio are scenario arithmetic, not an empirical calibration or prediction. The verifier-yield calculation is an auxiliary motivational model. It is not connected by theorem to \(g\), \(F_m(N)\), or the recurrence, and RLHF or constitutional evaluators are not assumed to be sound correctness oracles. One-sentence summary: A decaying antitone threshold sequence yields static budget frontiers, while bounded recurrence dynamics require separate invariant-set assumptions.
Maturity: Draft. Part of The Latent research program.
Related papers in this program: Universal.
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Subjects
- Machine learning
- http://id.loc.gov/authorities/subjects/sh85079324