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The effective repetition count for detector intervals with shared drift is the number of independent repetitions that would give the same variance for an average indicator as the actual dependent repetitions. In overdispersion of detector-interval counts under shared drift, let $X_i$ indicate an empty interval, with common probability $\overline{p}$ satisfying $0<\overline{p}<1$. If the covariance between distinct indicators is $v=\mathrm{Var}(p(H))$, their correlation coefficient isThe parent's law of total variance calculation therefore givesThis is an observable-specific variance equivalence, not a literal count of independent traces or a guarantee of confidence interval coverage. A different statistic or a different dependence pattern may require a different equivalence.

For an original feasible mixture, take $\overline{p}=0.1$ and $\rho =0.1$. These moments can arise when the block probability is $0$ with probability $9/19$ and $0.19$ with probability $10/19$: its variance is $0.009$. With $N=50$, the variance-inflation multiplier is $5.9$ andDoubling the repetitions to $100$ only raises $N_{\mathrm{eff}}$ to about $9.17$. For a fixed positive $\rho$, its large-$N$ limit is $1/\rho$, here $10$.

The experiment must therefore record which traces share a reference or drift state. Acquiring genuinely independent calibration blocks addresses a different limitation from oversampling one block; effective repetition count across independent calibration blocks depends on that grouping. A cluster bootstrap can preserve the observed within-block dependence when blocks are suitable independent resampling units. NIST's uncertainty law explicitly retains shared-input covariance, supporting the need to model reference reuse rather than discard it.^[ref]

The negative covariance of empty and positive detector-interval counts arises because a fixed number of measurement intervals is divided into mutually exclusive categories. Let $E_i,P_i$ indicate that interval $i$ is respectively physically empty or strictly positive. The same interval cannot satisfy both, so $E_i{P}_i=0$ andFor $N$ independent repetitions with common probabilities, the counts $N_E={\sum }_i{E}_i$ and $N_P={\sum }_i{P}_i$ therefore haveThe joint probability of empty and strictly positive detector-width intervals gives the underlying multinomial model. The sign describes category-count dependence, not physical antagonism between two detectors.

For an original boundary example, take $N=100$ and $p_E=p_P=0.025$. Each count has variance $100(0.025)(0.975)=2.4375$, while their covariance is $-0.0625$. The total tail-category count has variance $2.4375+2.4375-0.125=4.75$. Treating the two counts as independent would give $4.875$ instead. The covariance is needed even though the individual measurement repetitions are independent.

Conditioning on the total is important. If the number of repetitions is random and independent of the category assignments, the law of total covariance givesAn independent total with a Poisson distribution has equal mean and variance, making this covariance zero; a more variable total can make it positive. This is a different sampling protocol, not a contradiction of the fixed-total formula.

Shared calibration drift is another change of model. Repetitions may then have dependent category indicators, and cross-repetition covariance terms must be added. Overdispersion of detector-interval counts under shared drift can make a simple multinomial analysis inadequate.

NIST's measurement guidance retains common influences in uncertainty propagation and requires assumptions for confidence-level interpretation.^[ref] State whether the repetition total is fixed, how the categories are defined and which calibration influences are shared before interpreting their empirical count covariance.