# Three binary verifiers: pairwise independence does not fix decision risk

This is an elementary authored finite classification counterexample, not a novelty claim, code-selection result, verifier measurement, learned-controller result or H1 amendment. Pairwise independence without mutual independence is standard probability. It supports only the narrow claim that covariance/equicorrelation effective-count summaries do not generally determine decision utility for unrestricted higher-order laws.

Let hidden truth Y be Bernoulli(1/2). Conditional on either truth class, draw an exchangeable error vector E=(E1,E2,E3) independent of Y. Define K=sum(E), and for rational0<=t<=1/100 let

    q0=73/100-t
    q1=24/100+3t
    q2=3/100-3t
    q3=t.

Assign probability qk / binomial(3,k) to each vector with k errors. All masses are nonnegative and sum1. Reported labels are Xi=Y xor Ei. Each class-conditional error marginal is

    E[K]/3=(q1+2q2+3q3)/3=1/10,

so each verifier has90% conditional accuracy. Each pair joint error is

    E[binomial(K,2)]/3=(q2+3q3)/3=1/100.

For two binary errors these marginals/joint determine the full pair distribution: P11=.01, P10=P01=.09, P00=.81. Thus each error pair is independent in each truth class. Error covariance0 and rho0 give the variance-based equicorrelation count3/[1+2rho]=3 in both laws. This rho is for errors, not unconditional reported labels Xi: shared truth makes reported labels correlated even with independent errors.

For observed label vector x containing m ones, likelihood under Y=0 is qm/binomial(3,m); under Y=1 it is q(3-m)/binomial(3,m). Balanced prior gives equal weights to these likelihoods. For m0 versus3, q0>q3 throughout the interval. For m1 versus2, the per-pattern masses are q1/3=2/25+t and q2/3=1/100-t, so q1>q2 throughout. Therefore the posterior-optimal action is exactly majority for every one of the eight possible observation vectors. No tie or zero-observation cell occurs under the balanced prior. Its full-coverage Bayes risk is

    P(K>=2)=q2+q3=3/100-2t.

At t=0 the K masses are(.73,.24,.03,0): Bayes risk3%, success97%. At t=.01 they are(.72,.27,0,.01): Bayes risk1%, success99%. Both query all three verifiers at the same cost3 abstract units (one per verifier, not dollars or generation cost), with equal marginal conditional accuracy and complete pairwise error laws. The2-percentage-point success difference is a comparison of different specified known laws, not a causal improvement obtainable from a measured verifier portfolio. No sampling interval is appropriate for exact authored values.

The interior t=.001 reproduces the mutually independent Bernoulli(.1) error law: K masses(.729,.243,.027,.001), risk2.8%. The endpoints are pairwise independent but not mutually independent because P(E1=E2=E3=1)=t differs from(.1)^3=.001. Higher-order error structure is the missing degree of freedom.

Balanced truth prior is essential to the majority-optimality proof. Under arbitrary prior pi, posterior likelihoods multiply by1-pi and pi; optimal actions can differ from majority. At pi0/1, Bayes always chooses the known truth on supported cells and risk0; null cells retain undefined posterior/action. Reproduction tests verify these cases instead of forcing majority universally.

This THREE-verifier example does not show insufficiency of the full class-conditional pair law for an original TWO-judge binary setting: for two errors, that law already specifies the entire conditional error vector. It neither invalidates the frozen pair-interaction H1 nor establishes the effectiveness of a pair interaction model. Known full-law Bayes classification, real code candidate selection, source-fitted continuation policies, cost-sensitive adaptive acquisition and empirical transfer are different claims. No general effective-count metric is proposed or refuted as a possible empirical predictor; only unrestricted decision-utility sufficiency is disproved.