Odds Law: The Decomposition Algebra On How Intelligence Organizes Itself to Solve Difficult Problems Reliably

2026-06-14

AI Summary

This work addresses the challenge of reliably composing inherently unreliable base solvers to tackle hard computational problems while characterizing the fundamental limits of their capabilities. The authors propose a decomposition algebra that models solvers as morphisms in a stochastic category and introduces four composition operators to construct composite solver structures. A homomorphic mapping capturing both reliability and computational cost is integrated into this framework. Leveraging tools from category theory, stochastic processes, and information theory, they establish a formal analysis framework and prove that achieving a target reliability of \(1-\delta\) requires only \(O(\log(1/\delta))\) verification depth when the likelihood ratio \(\Lambda > 1\). The study further reveals that self-organization corresponds to the least fixed point over a strategy lattice, demonstrates the necessity of diversity for unbounded reliability amplification, and derives tight reliability bounds governed by information divergence and shared error patterns.

Abstract

We ask a structural question: given unreliable elementary problem-solvers, what organizations of them solve hard problems reliably, and what are the limits? We develop a $decomposition~algebra$: elementary solvers are morphisms in a stochastic category, and four combinators (sequential composition, parallel ensembling, verification gating, and recursive reduction) generate the space of compound solvers. We equip this algebra with two homomorphisms, a $reliability$ valuation into the ordered monoid $([0,1],\le)$ and a $cost$ valuation into a commutative semiring, and we derive the composition laws that govern how reliability flows through structure. Our central results are (i) a $verification~odds~law$ (the result that names this report), showing that a verification gate multiplies the odds of correctness by the verifier's likelihood ratio $Λ$, so that $k$ conditionally independent gates yield geometric amplification; (ii) a $reliability~amplification~theorem$, giving target reliability $1-δ$ at $O(\log 1/δ)$ verification depth whenever $Λ>1$; and (iii) a $threshold~dichotomy$: above the critical parameters reliability can be driven arbitrarily close to one at logarithmic cost, while at or below them no amplification is possible. We then show that $self-organization$ is the least fixed point of a monotone improvement operator on the complete lattice of strategies, and that this fixed point equalizes marginal log-odds gain per unit cost. Finally, we prove matching limits: an information ceiling bounds per-gate amplification by a divergence quantity; shared error causes create a strictly positive voting floor, so diversity is $necessary$ for unbounded amplification. Reliability, in short, is neither free nor magical: it is bought with independent information, arranged by composition, and bounded by the verifier.

Problem

• reliability
• problem-solving
• decomposition
• verification
• self-organization

Innovation

• decomposition algebra
• verification odds law
• reliability amplification
• self-organization
• stochastic category