Probability of zero reversions under independent checks

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If each of N checks independently has the same reversion probability p, the chance of observing no reversion is (1−p)ᴺ. A finite record without a return therefore remains compatible with small positive reversion probabilities. The formula quantifies a sampling model; correlated or unequal checks require another model.

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Original context

9.4 Finite persistence cannot establish “never again” empirically

Suppose no return to baseline is observed in N independent checks. Under a constant independent reversion probability p,

Pzero reversions=1−pN.

The one-sided 95% upper bound is 1−0.051/N. For N=1000, it is about 0.002991 per check. It is not zero. Dependence between checks makes this simple bound less informative, not a proof of eternity.

A proof of permanent divergence is possible for a fully specified system when a forward-invariant set excludes the reference, as in the affine example. But that proves a property of those transition rules. A finite sequence of neural-model transcripts does not supply such a proof unless the required model and invariant are established independently.

10. A prospective study that would meaningfully test the proposal

P\left(zero reversions\right)={\left(1-p\right)}^{N}.