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

Computer Science > Cryptography and Security [Submitted on 14 Jun 2026] Odds Law: The Decomposition Algebra On How Intelligence Organizes Itself to Solve Difficult Problems Reliably Hidayet Aksu 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 \Lambda, so that k conditionally independent gates yield geometric amplification; (ii) a reliability~amplification~theorem, giving target reliability 1-\delta at O(\log 1/\delta) 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. Comments: 10 pages, 2 figures Subjects: Cryptography and Security (cs.CR); Artificial Intelligence (cs.AI); Multiagent Systems (cs.MA) ACM classes: I.2.7; I.2.6; I.2.11; F.1.2; G.3 Cite as: arXiv:2606.15712 [cs.CR]   (or arXiv:2606.15712v1 [cs.CR] for this version)   https://doi.org/10.48550/arXiv.2606.15712 Focus to learn more Submission history From: Hidayet Aksu [view email] [v1] Sun, 14 Jun 2026 09:53:11 UTC (25 KB) Access Paper: HTML (experimental) view license Current browse context: cs.CR < prev   |   next > new | recent | 2026-06 Change to browse by: cs cs.AI cs.MA References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)