Defect Relation Minimum Coloring

Abstract

Canonical defect-relation coloring computes the exact finite repair-label count.

Theorem 1.1 (Minimum repair labels equal chromatic number and fiber diversity).

$$\begin{array}{c}\forall X,B,Y:\mathrm{Type},\\ [\mathrm{Fintype}(X)],\\ C:X\to B,T:X\to Y,\\ (\forall m\in \mathbb{N},\mathrm{RepairLabelFeasible}\left(C,T,m\right)⟺\mathrm{Colorable}\left(\mathrm{defectGraph}\left(C,T\right),m\right))\wedge \\ (\mathrm{minimumRepairLabels}\left(C,T\right)=\mathrm{chromaticNumber}\left(\mathrm{defectGraph}\left(C,T\right)\right))\wedge \\ (\mathrm{chromaticNumber}\left(\mathrm{defectGraph}\left(C,T\right)\right)=\mathrm{effectiveWorstFiberDiversity}\left(C,T\right)).\end{array}$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/GraphColoring/DefectRelationMinimumColoring.minimum_repair_labels_eq_chromatic_eq_fiber_diversity (✓ std3). ∎

Source. Repository-derived.

Commentary.

The state carrier is finite. The concept and target codomains need not be finite because the fiber maximum is taken over the canonical effective image of the concept readout.

The graph adapter reads the family’s canonical target-defect relation: two states are adjacent exactly when the concept identifies them and the target distinguishes them.

A finite repair label is feasible exactly when it is a proper coloring. The least feasible count is therefore both the graph’s chromatic number and the largest target diversity in one concept fiber.

References

• Truth anchor: D5/S3/ConceptDynamics/GraphColoring/DefectRelationMinimumColoring.minimum_repair_labels_eq_chromatic_eq_fiber_diversity
• Dependency: D5/S3/ConceptDynamics/Coding/DefectGraphMinimumColoring
• Dependency: D5/S3/ConceptDynamics/TargetRisk/RefinementRiskCostTradeoff