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.

翻译：我们提出一个结构性难题：给定不可靠的基础问题求解器，何种组织结构能使其可靠地解决难题，其极限又是什么？本文发展了一种分解代数：基础求解器是随机范畴中的态射，四种组合子（顺序组合、并行集成、验证门控与递归归约）生成复合求解器的空间。我们为该代数配备两个同态映射：一个进入有序幺半群([0,1],≤)的可靠性赋值，一个进入交换半环的成本赋值，并推导出可靠性如何随结构流动的组合律。核心结果包括：(i) 验证奇偶律（即论文命名依据），表明验证门将正确奇偶乘以验证器的似然比Λ，故k个条件独立门产生几何级数放大；(ii) 可靠性放大定理，当Λ>1时，在O(log 1/δ)的验证深度下达到目标可靠性1-δ；(iii) 阈值二分性：当参数超越临界值时，可靠性可通过对数成本逼近1，而在阈值处或以下则无法实现放大。我们进一步证明：自组织是策略完备格上单调改进算子的最小不动点，且该不动点使单位成本的边际对数奇偶增益相等。最后，我们建立匹配极限：信息上界通过散度量约束单门放大；共享错误源产生严格正值的表决下界，故多样性是无限放大的必要条件。简言之，可靠性既非免费亦非魔法：它以独立信息为代价，由组合结构组织，且受限于验证器。