Exact pairwise-insufficiency diagnostic

Four exchangeable binary verifiers; truth prevalence is 1/2. Within either
truth class, let K be the number of verifier errors. Start with
q_k = choose(4,k) (3/20)^k (17/20)^(4-k).
Replace q by q + delta*(1,-3,3,-1,0), for delta=-1/1000, 0, +1/1000.
All probabilities stay nonnegative. Conditional on K, distribute probability
uniformly over all choose(4,K) error patterns. Use the same law in both truth
classes. This defines the complete error law, rather than only a correlation.

The perturbation has zero zeroth, first and second moments. Every verifier
therefore has the same class-conditional error probability 3/20 and every
pair has joint error 9/400. All six off-diagonal error covariances are zero.
The covariance matrices are identical, but the four-way joint laws differ.
Pairwise independence must not be described as full independence.

The variance-equivalent count n^2*p*(1-p)/Var(sum E_i) is 4 in all conditions.
This is a variance summary, not an information count or a new metric.
At equal verifier-query count, majority with random ties and the known-law
Bayes vote classifier both achieve 93.975%, 93.925%, and 93.875% respectively.
The exact gap between the extreme conditions is 1/1000 (0.1 percentage point).
The Bayes calculation uses the specified synthetic class-conditional law;
it is not a practical controller learned without labels. All 16 patterns
were separately enumerated and all four marginal/six pair masses checked.

Consequence: no deterministic function of this covariance matrix alone
uniquely determines voting accuracy over unrestricted joint error laws.
This does NOT rule out useful empirical prediction from covariance, nor
prove that higher-order features improve a finite-data learned controller.
It motivates testing sufficiency, estimation cost and higher-order effects
rather than promoting a variance proxy into an assurance metric.

This is an exact authored binary-classification mechanism diagnostic.
It is not a code-candidate selection experiment, empirical LLM result,
frontier transfer, theorem of universal diversity benefit, new H1 protocol,
external reproduction or established novelty. No fitting, real task text,
model calls or paid compute were used. Original H1 and budgets remain intact.

Relevant prior literature: Kuncheva and Whitaker (2003), Measures of Diversity
in Classifier Ensembles and Their Relationship with the Ensemble Accuracy,
Machine Learning 51(2), 181-207, DOI 10.1023/A:1022859003006.
The university record verifies this citation; this diagnostic does not claim
that its exact construction is original or reproduced from that article.
https://research.bangor.ac.uk/en/publications/measures-of-diversity-in-classifier-ensembles-and-their-relations/

Review roadmap relevance: effective verifier diversity, controlled correlation,
class-conditional errors, oracle ceilings and failure modes of simplified metrics.
A future empirical study must prospectively freeze the diagnostic metrics,
source-only fitting and cost accounting before target outcomes are observed.
