Fresh Independent Checkpoint Guarantee

Abstract

Fresh checkpoints governed by the deployment product law certify a frozen implementation.

Theorem 1.1 (Fresh deployment checkpoints give an exponential all-pass guarantee).

$$\begin{array}{c}\forall Input,Output:\mathrm{Type},\\ [\mathrm{MeasurableSpace}(Input)],[\mathrm{MeasurableSingletonClass}(Input)],[\mathrm{Countable}(Input)],\\ \forall \mathcal{D}:\mathrm{PMF}\left(Input\right),P,xStar:Input\to Output,\\ \forall m:\mathbb{N},epsilon:\mathbb{R},\\ 0\le epsilon\wedge epsilon\le 1\wedge epsilon\le \mathrm{real}\left(\mathrm{toMeasure}\left(\mathcal{D}\right),\left\{P(x)\ne xStar(x)\mid x\in Input\right\}\right)\Rightarrow \\ \mathrm{let}(m{u}_{suite}:\mathrm{Measure}\left(\mathrm{Fin}\left(m\right)\to Input\right)=\mathrm{pi}\left(j\mapsto \mathrm{toMeasure}\left(\mathcal{D}\right)\right),\\ Apass:\mathrm{Set}\left(\mathrm{Fin}\left(m\right)\to Input\right)=\left\{\forall j:\mathrm{Fin}\left(m\right),P(suite(j))=xStar(suite(j))\mid suite\in \mathrm{Fin}\left(m\right)\to Input\right\});\\ \mathrm{real}\left(m{u}_{suite},Apass\right)=\mathrm{real}{\left(\mathrm{toMeasure}\left(\mathcal{D}\right),\left\{P(x)=xStar(x)\mid x\in Input\right\}\right)}^m\wedge \\ \mathrm{real}\left(m{u}_{suite},Apass\right)\le \exp \left(-(epsilon\times (m:\mathbb{R}))\right).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Interpretation/FreshIndependentCheckpointGuarantee.fresh_independent_checkpoint_deployment_guarantee (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let deployment be an arbitrary probability mass function on a countable measurable input carrier. The implementation and expected behavior are fixed before the suite law is constructed.

The checkpoint tuple is governed by the finite product measure of copies of deployment. This joint law is the independence premise; it is not represented by a family of matching marginal assertions.

The exact all-pass mass is the single-check pass mass raised to the suite budget. If deployment loss is at least epsilon, that mass is at most (1 - epsilon)^m and hence at most exp(-epsilon m).

Pinned Mathlib supplies Measure.pi_pi, ENNReal.toReal_prod, and the real probability-complement identity. The frozen repository theorem independent_sampling_exponential_bound supplies the final step directly. The existing interpretation witnesses are Boolean special cases and do not state this arbitrary frozen-implementation guarantee.

References

• Truth anchor: D5/S3/ConceptDynamics/Interpretation/FreshIndependentCheckpointGuarantee.fresh_independent_checkpoint_deployment_guarantee
• Dependency: D5/S3/TotalVariation/IndependentSamplingExponentialBound