Escape-Zero Completion Point

Abstract

Faithful escape zero is equivalent to determination by the joined readout, and a unique audited parameter supplies the regularized completion point.

Theorem 1.1 (Escape zero characterizes the audited completion point).

$$\begin{array}{c}\forall A,X,Q,Y:Type,\\ D:A\to Type,\\ q:X\to Q,d:\forall a:A,X\to D(a),\\ T:X\to Y,w:\mathrm{EscapeWeight}\left(\mathrm{Prod}\left(X,X\right)\right),\\ Cost:\forall a:A,\left(X\to D(a)\right)\to \mathbb{R},\lambda :\mathbb{R},a:A,\\ (\forall S:\mathrm{Set}\left(\mathrm{Prod}\left(X,X\right)\right),\mathrm{mass}\left(w,S\right)=0\iff S=\varnothing )\Rightarrow \begin{array}{c}(\mathrm{parameterEscapeDefect}\left(q,d,T,w,a\right)=0\Rightarrow \mathrm{FactorsThrough}\left(T,\mathrm{conceptJoin}\left(q,d(a)\right)\right))\wedge \\ (\mathrm{FactorsThrough}\left(T,\mathrm{conceptJoin}\left(q,d(a)\right)\right)\Rightarrow \mathrm{parameterEscapeDefect}\left(q,d,T,w,a\right)=0)\wedge \\ ((uniqueCompletion:\exists !\kappa :A,\mathrm{IsAuditedCompletionParameter}\left(q,d,T,w,Cost,\lambda ,\kappa \right))\Rightarrow \\ \begin{array}{c}\mathrm{IsMinOn}\left(\mathrm{regularizedCompletionObjective}\left(q,d,T,w,Cost,\lambda \right),\mathrm{SetUniv}\left(A\right),\mathrm{choose}\left(uniqueCompletion\right)\right)\wedge \\ \mathrm{parameterEscapeDefect}\left(q,d,T,w,\mathrm{choose}\left(uniqueCompletion\right)\right)=0\wedge \\ \mathrm{FactorsThrough}\left(T,\mathrm{conceptJoin}\left(q,d(\mathrm{choose}\left(uniqueCompletion\right))\right)\right)\wedge \\ \forall candidate:A,\mathrm{IsAuditedCompletionParameter}\left(q,d,T,w,Cost,\lambda ,candidate\right)\Rightarrow candidate=\mathrm{choose}\left(uniqueCompletion\right)\end{array})\end{array}\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Completion/EscapeZeroCompletionPoint.escape_zero_iff_determined_with_audited_minimizer (✓ std3). ∎

Source. Repository-derived.

Commentary.

A baseline readout q is joined with the parameter-dependent definition readout d(a). The escape defect is the supplied weight of target pairs that the joint readout still identifies.

Faithfulness says that a set has zero weight exactly when it is empty. The repository’s sufficiency-escape equivalence then gives both directions between zero defect and target factorization through the joined readout.

An audited parameter has exactly three properties: it globally minimizes Delta(a) + lambda Cost(d(a)), its joint readout determines the target, and its escape defect is zero. Under the source’s unique-existence condition, the selected witness kappa has each property and every other audited parameter equals it.

References

• Truth anchor: D5/S3/ConceptDynamics/Completion/EscapeZeroCompletionPoint.escape_zero_iff_determined_with_audited_minimizer
• Dependency: D5/S3/AnalyticClosure/Budget/BudgetedEscapeRateAntitone
• Dependency: D5/S3/ConceptDynamics/RefinementFactorization/SufficiencyEscapeEquivalence