Three Conditions of Entity Strength

Abstract

Process stability, target fidelity, and nontrivial resolution are independent yet jointly realizable.

Theorem 1.1 (The three entity-strength conditions are independent).

$$\left(\exists c1\in Bool\to Unit,t1\in Bool\to Unit,P1\in \mathrm{Set}\left(Bool\to Bool\right),\mathrm{ProcessStable}\left(P1,c1\right)\wedge \left(\mathrm{TargetFaithful}\left(t1,c1\right)\wedge \left(\neg \mathrm{NontrivialResolution}\left(c1\right)\right)\right)\right)\wedge \left(\left(\exists c2\in Bool\times Bool\to Bool,t2\in Bool\times Bool\to Bool,P2\in \mathrm{Set}\left(Bool\times Bool\to Bool\times Bool\right),\mathrm{ProcessStable}\left(P2,c2\right)\wedge \left(\mathrm{NontrivialResolution}\left(c2\right)\wedge \left(\neg \mathrm{TargetFaithful}\left(t2,c2\right)\right)\right)\right)\wedge \left(\exists c3\in Bool\to Bool,t3\in Bool\to Bool,P3\in \mathrm{Set}\left(Bool\to Bool\right),\mathrm{TargetFaithful}\left(t3,c3\right)\wedge \left(\mathrm{NontrivialResolution}\left(c3\right)\wedge \left(\neg \mathrm{ProcessStable}\left(P3,c3\right)\right)\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Identity/EntityStrengthThreeConditions.three_conditions_are_independent (✓ std3). ∎

Source. Repository-derived.

Commentary.

None of process stability, target fidelity, and nontrivial resolution follows from the other two. Three explicit concepts witness the three missing implications.

A constant Boolean-to-Unit concept is stable under every process and faithful to a constant target, but it distinguishes no states. On a Boolean pair, the first-coordinate concept is stable under all processes that preserve that coordinate and has nontrivial resolution, but it cannot recover the second-coordinate target.

Finally, the identity concept on Bool is faithful to the identity target and distinguishes false from true, while Boolean negation is an allowed process that violates stability.

Lemma 1.2 (The three entity-strength conditions are jointly realizable).

$$\exists c\in Bool\times Bool\to Bool,t\in Bool\times Bool\to Bool,P\in \mathrm{Set}\left(Bool\times Bool\to Bool\times Bool\right),\mathrm{ProcessStable}\left(P,c\right)\wedge \left(\mathrm{TargetFaithful}\left(t,c\right)\wedge \mathrm{NontrivialResolution}\left(c\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ConceptDynamics/Identity/EntityStrengthThreeConditions.three_conditions_jointly_realizable (✓ std3). ∎

Source. Repository-derived.

Commentary.

Reading the first coordinate of a Boolean pair satisfies all three conditions when the designated processes are exactly those that preserve that coordinate.

Preservation gives process stability directly, using the same first-coordinate readout as the target gives fidelity through the identity decoder, and pairs with different first coordinates supply nontrivial resolution.

References

• Truth anchor: D5/S3/ConceptDynamics/Identity/EntityStrengthThreeConditions.three_conditions_are_independent
• Truth anchor: D5/S3/ConceptDynamics/Identity/EntityStrengthThreeConditions.three_conditions_jointly_realizable
• Dependency: D5/S3/ConceptDynamics/ConceptJoinUniversal