Blind Kernel Reduction Measure

Abstract

Positive weight detects blind residual pairs separated by a new definition.

Theorem 1.1 (Positive reduction weight detects a separated blind pair).

$$\begin{gathered} (\forall S,0\le \nu (S))\wedge \nu (\varnothing )=0\Rightarrow \\ (\mathrm{blindKernelReductionMeasure}\left(\nu ,\Gamma ,q,T,d\right)=\nu (\mathrm{intersection}\left(\mathrm{blindResidual}\left(\Gamma ,q,T\right),\mathrm{complement}\left(\ker \left(d\right)\right)\right)))\wedge (0\le \mathrm{blindKernelReductionMeasure}\left(\nu ,\Gamma ,q,T,d\right))\wedge \\ (0<\mathrm{blindKernelReductionMeasure}\left(\nu ,\Gamma ,q,T,d\right)\Rightarrow \exists p,p\in \mathrm{blindResidual}\left(\Gamma ,q,T\right)\wedge d(p_1)\ne d(p_2)). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The imported blindResidual is used unchanged. For a proposed definition d, the measured reduction set is its intersection with the complement of the Setoid equality kernel of d. Thus the Lean definition is exactly the displayed P_Gamma formula and introduces no second residual or kernel.

The public hypotheses require the abstract real-valued set weight to be nonnegative and to vanish on the empty set. The conclusion packages the defining equality, nonnegativity, and the one-way implication from positive reduction weight to a blind residual pair separated by d. It does not assume that every nonempty set has positive weight.

Finite counting weight on Boolean state pairs supplies the positive example. A constant definition supplies the reverse example. A normalized Dirac-style weight supplies a nonempty separated set of weight zero, demonstrating why no converse is claimed.

The closing catalog, language-closure, and target-usefulness maxim is not asserted as a Lean proposition: the source supplies no formal catalog, closure, or usefulness predicates from which a faithful statement could be formed.

References

• Truth anchor: D5/S3/ConceptDynamics/DefinitionEscape/BlindKernelReductionMeasure.blind_kernel_reduction_measure
• Dependency: D5/S3/ConceptDynamics/DefinitionEscape/BlindKernelObstruction