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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 $\\Lambda$, so that $k$ conditionally independent gates yield geometric amplification; (ii) a $reli","title":"Odds Law: The Decomposition Algebra On How Intelligence Organizes Itself to Solve Difficult Problems Reliably","url":"https://arxiv.org/abs/2606.15712","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.15712v1 Announce Type: cross \nAbstract: 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. 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