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

Hidayet Aksu

cs.CRJun 14, 2026

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

Authors: Hidayet Aksu

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.

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Partially Correlated Verifier Cascades in LLM Harnesses: Concave Log-Odds, Polynomial Reliability, and Blind-Spot Ceilings

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if

k

verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in

k

, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable

α\sim G

(de Finetti), the exact cascade posterior is

\ell_k = \ell_0 - \ln m_k

, with

m_k

the

k

-th moment of

G

. Then: (i)

\ell_k

is concave in

k

for every non-degenerate

G

-- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta

(a,b)

latents, failure decays polynomially,

1-r_k \asymp k^{-b}

, with correlation parameter

ρ_v = 1/(a+b+1)

; (iii) a blind-spot atom of mass

1-π

α=1

caps the evidence extractable from any number of gates at

-\ln(1-π)

nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (

β\sim H

) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of

G

and

H

, with closed-form crossover

k^\dagger

. The mechanism is survivorship: errors surviving gates are the high-

α

ones. The theory is measurable:

R

repeated verdicts per instance identify the first

R

moments of

G

, so two verdicts identify

ρ_v

; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at

k=5

and ~3000x at

k=10

; the correlated fit at

R=8

tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

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A single forward pass of a capable model is a fast, fluent, and unreliable problem-solver: it is right often enough to be useful and wrong often enough to be dangerous; in language models, such confident errors are known as hallucinations. We present Maestro Order, a model-agnostic orchestration harness that turns unreliable solvers into reliable problem-solving systems by composing them according to four structural primitives (decompose, ensemble, verify, and recurse) and a budget-aware controller that decides where to spend compute. The harness treats any model as a black-box base solver behind a uniform interface, layers a verifier ensemble whose discrimination is measured online, and allocates verification and voting to the stages with the highest marginal reliability per unit cost. We give the architecture, the message and state schema, the controller algorithm, and the engineering that makes it deterministic, observable, and fault-tolerant. We then specify an evaluation methodology (reliability at fixed cost, coverage, calibration, and ablations) and report results from a faithful Monte Carlo simulation of the harness over a parameterized solver/verifier model. The simulation reproduces the predicted laws quantitatively: verification amplifies reliability geometrically (e.g.

0.55\to0.98

with two gates,

\to0.999

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