Productive Diagonal Escape Criterion

Abstract

A diagonal catalog escape is productive iff it creates a new question.

Theorem 1.1 (Productive diagonal escape creates a newly answerable question).

$$\mathrm{Nonempty}\left(World\right)\wedge \mathrm{FixedPointFree}\left(twist\right)\Rightarrow (\mathrm{ProductiveCatalogEscape}\left(catalog,current,expressionSemantics,\mathrm{diagonal}\left(twist,catalog\right)\right)⟺\exists question:World\to Bool,(\exists !answer:\mathrm{EffectiveCoordinate}\left(\mathrm{conceptJoin}\left(current,\mathrm{expressionSemantics}\left(\mathrm{diagonal}\left(twist,catalog\right)\right)\right)\right)\to Bool,question=\mathrm{compose}\left(answer,\mathrm{effectiveReadout}\left(\mathrm{conceptJoin}\left(current,\mathrm{expressionSemantics}\left(\mathrm{diagonal}\left(twist,catalog\right)\right)\right)\right)\right))\wedge \neg (\exists answer:\mathrm{EffectiveCoordinate}\left(current\right)\to Bool,question=\mathrm{compose}\left(answer,\mathrm{effectiveReadout}\left(current\right)\right))).$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Coding/ProductiveDiagonalEscapeCriterion.productive_diagonal_escape_iff_new_question (✓ std3). ∎

Source. Repository-derived.

Commentary.

ProductiveCatalogEscape combines catalog novelty with strict refinement by the diagonal target.

Under the fixed-point-free and nonempty hypotheses, strict effective refinement is equivalent to a new Boolean question.

References

• Truth anchor: D5/S3/ConceptDynamics/Coding/ProductiveDiagonalEscapeCriterion.productive_diagonal_escape_iff_new_question
• Dependency: D5/S0/Diagonal/Lawvere/QualitativeEscape
• Dependency: D5/S3/ConceptDynamics/DefinitionEscape/QuestionAlgebraDuality