Relative Semantic Diagonal

Abstract

Complete decoder catalogs yield diagonal targets outside the latent closure.

Theorem 1.1 (A complete decoder catalog leaves a nonempty blind residual).

$$\forall decoderCatalog:X\to (Current\times (Gamma\to InputOutput))\to Output,\mathrm{Nonempty}\left(X\right)\wedge \mathrm{FixedPointFree}\left(twist\right)\wedge \mathrm{Surjective}\left(decoderCatalog\right)\Rightarrow \mathrm{Nonempty}\left(\mathrm{blindResidual}\left(Gamma,current,\mathrm{relativeSemanticDiagonal}\left(twist,\mathrm{languageExtension}\left(current,\mathrm{familyReadout}\left(Gamma\right)\right),decoderCatalog\right)\right)\right).$$

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

Source. Repository-derived.

Commentary.

The diagonal target uses languageExtension, a decoder catalog, and a fixed-point-free twist; blindResidual remains canonical.

Surjectivity puts every decoder at an address where the diagonal differs, and the recovery criterion turns inadequacy into a nonempty blind residual.

References

• Truth anchor: D5/S3/ConceptDynamics/DefinitionEscape/RelativeSemanticDiagonal.complete_catalog_diagonal_blindResidual_nonempty
• Dependency: D5/S3/ConceptDynamics/DefinitionEscape/DefinitionKernelGalois
• Dependency: D5/S3/ConceptDynamics/DefinitionEscape/QuestionAlgebraDuality