Blind Kernel Obstruction

Abstract

A nonempty blind residual obstructs every finite or pointwise language extension.

Theorem 1.1 (Blind residuals obstruct every package extension).

$$\begin{gathered} (\mathrm{blindResidual}\left(Gamma,q,T\right)=\varnothing \Rightarrow (\exists r,T=r\circ \mathrm{languageExtension}\left(q,Gamma\right))\wedge (\mathrm{finiteSelectionSufficient}\left(Gamma,q,T\right)\vee \mathrm{compactificationRequired}\left(Gamma,q,T\right)))\wedge \\ (\mathrm{Nonempty}\left(\mathrm{blindResidual}\left(Gamma,q,T\right)\right)\Rightarrow (\forall n,D:\mathrm{Fin}\left(n\right)\to Gamma,\neg (\exists r,T=r\circ \mathrm{languageExtension}\left(q,D\right)))\wedge \\ (\forall Delta\subseteq Gamma,\neg (\exists r,T=r\circ \mathrm{languageExtension}\left(q,Delta\right)))\wedge \neg \mathrm{finiteSelectionSufficient}\left(Gamma,q,T\right)). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The package is viewed as a family indexed by its subtype. The imported jointKernel and jointReadout are reused directly. The blind residual is only the named intersection of jointKernel with the canonical defectRelation; no second kernel, joint readout, or target-defect relation is introduced.

If the residual is empty, adjoining the full pointwise language to the baseline admits a target recovery factor. This uses the accepted target recovery criterion and the required inhabited-state hypothesis. The remaining exhaustive alternative is either a sufficient finite selection or the compactification condition: full pointwise factorization with no finite sufficient selection.

If the residual contains a pair, the baseline and every package definition agree on that pair while the target differs. Hence no finite indexed selection and no arbitrary subpackage pointwise union admits a target factor map. Repeated indices add no readout information, so arbitrary indexed unions are represented by their subpackage of values.

The proof applies the accepted target recovery criterion to each persisting canonical defect. Thus the obstruction is inherited from the repository factorization theorem rather than reproved.

References

• Truth anchor: D5/S3/ConceptDynamics/DefinitionEscape/BlindKernelObstruction.blind_kernel_obstruction
• Dependency: D5/S3/ConceptDynamics/Faithfulness/JointFaithfulnessLeibnizCriterion
• Dependency: D5/S3/ConceptDynamics/Restoration/TargetRecoveryCriterion