Question Algebra Duality

Abstract

Effective concept refinement is exactly inclusion of answerable Boolean questions.

Theorem 1.1 (Effective refinement is equivalent to question inclusion).

$$\mathrm{Refines}\left(\mathrm{effectiveReadout}\left(coarse\right),\mathrm{effectiveReadout}\left(fine\right)\right)⟺\mathrm{Subset}\left(\mathrm{AnswerableQuestions}\left(\mathrm{effectiveReadout}\left(coarse\right)\right),\mathrm{AnswerableQuestions}\left(\mathrm{effectiveReadout}\left(fine\right)\right)\right).$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/DefinitionEscape/QuestionAlgebraDuality.effective_refinement_iff_question_inclusion (✓ std3). ∎

Source. Repository-derived.

Commentary.

Effective readouts normalize to attained-coordinate subtypes and reuse AnswerableQuestions.

Refinement transports every Boolean question; conversely, a coarse-coordinate question reconstructs the required fiber implication.

References

• Truth anchor: D5/S3/ConceptDynamics/DefinitionEscape/QuestionAlgebraDuality.effective_refinement_iff_question_inclusion
• Dependency: D5/S3/ConceptDynamics/DefinitionEscape/LatentAdequacyCriterion