Sampling and finite-bootstrap errors for a Bernoulli mean

Given: The stated IID two-point observation law, observed sample $(0,1)$, finite unit space and map $T_2$, bootstrap procedure, and four realized resamples. Prove the conditional bootstrap law and error separation.

The error ledger uses a declared scalar target and explicit successive reference values, with each stage error equal to the difference of adjacent references. (Successive statistical and computational error targets)

The total realized error along the finite reference chain is the sum of its signed stage errors. (Pathwise error decomposition along a reference chain)

Identical distribution and IID families: IID means a common law and mutual independence.

Size-n bootstrap from the empirical law: for a fixed tuple of length $n\ge 1$, draw $n$ indices independently and uniformly from $[n]$, with replacement, and apply the declared measurable statistic to the resampled tuple.

Empirical-law bootstrap draws are conditionally IID: conditional on the fixed observed tuple, the resampled coordinates are IID with common law equal to its empirical measure.

For $B\ge 2$ IID finite-variance replicate outputs, $[B(B-1){]}^{-1}{\sum }_b(Z_b-{\bar{Z}}_B{)}^2$ is unbiased for the variance of their average. (Unbiased variance estimate from independent computation replicates)

The observations are IID with the stated two-point law; the target is their mean $\theta =3\mathrm{/}4$.

The observed ordered tuple is $(0,1)$ in the finite measurable unit space $(\{0,1\},2^{\{0,1\}})$; each entry is one resampling unit and $T_2(x_1,x_2)=(x_1+x_2)\mathrm{/}2$ is measurable on the full power-set sample space.

Four independent repetitions of the stated bootstrap algorithm produce the listed ordered index pairs.