Diagonal Novelty and World Information

Abstract

Diagonal catalog escape need not strictly refine world-state information.

Theorem 1.1 (Representational novelty need not add world information).

$$\begin{gathered} \forall A,Y,X,B_C,B_E:\mathrm{Type}, \\ f:Y\to Y,g:A\to A\to Y, \\ C:X\to B_C,Sem:(A\to Y)\to X\to B_E, \\ (\forall y,f(y)\ne y)\Rightarrow \neg \mathrm{diagonal}(f,g)\in \mathrm{range}(g)\wedge \\ \mathrm{Refines}(Sem(\mathrm{diagonal}(f,g)),C)\Rightarrow \neg \mathrm{StrictRefinement}(C,\mathrm{join}(C,Sem(\mathrm{diagonal}(f,g)))). \end{gathered}$$

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

Source. Repository-derived.

Commentary.

The expression address type, symbol type, world-state type, catalog, current world concept, and expression-semantics map are independent source primitives.

The escaped expression is constructed by the canonical twisted diagonal of the supplied catalog. A fixed-point-free twist makes this expression absent from the catalog range.

The second public clause has an independent premise: when the escaped expression’s world semantics factors through the current concept, joining that semantic readout to the current concept cannot be a strict refinement.

The first clause applies the frozen qualitative escape theorem. The second applies the frozen concept-join universal property to build the reverse factorization that contradicts strictness.

Neither the expression semantics nor the current concept is defined from the non-strictness goal, so catalog novelty and world information remain distinct carriers.

References

• Truth anchor: D5/S3/ConceptDynamics/Coding/DiagonalNoveltyInformationSeparation.diagonal_novelty_need_not_add_world_information
• Dependency: D5/S0/Diagonal/Lawvere/QualitativeEscape
• Dependency: D5/S3/ConceptDynamics/StrictRefinementCapability