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

arXiv: https://arxiv.org/abs/2606.15712 · HTML: https://mlsanity.com/paper/2606.15712/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 1 of its 17 references that are in the map.
- First posted on arXiv 14 Jun 2026.

## Builds on (most cited first, 1 of 1)

- [A synthetic approach to Markov kernels, conditional independence and theorems on sufficient statistics](https://mlsanity.com/paper/1908.07021/) (2019)

## Built on by (most cited first, 2 of 2)

- [Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings](https://mlsanity.com/paper/2607.13918/) (2026)
- [Maestro Order: A Model-Agnostic Orchestration Harness](https://mlsanity.com/paper/2606.23983/) (2026)

## Abstract (arXiv)

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],≤) 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 Lambda, 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 Lambda>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.