ADCI Closure Theorem: Formal Closure of the Value Layer in Decision-Permitting Systems

The ADCI Closure Theorem establishes a minimal and sufficient set of value-layer primitives required for legitimacy in decision-permitting systems. Prior work formalized three governance primitives governing authority, permission, and action selection. This theorem addresses the structural value state of the human subject within such systems. Four irreducible value primitives are defined: A — Agency: The subject retains meaningful control over participation and decision execution.D — Dignity: The subject is treated as a rights-bearing moral subject, not as an object or input.C — Continuity: The subject’s identity and trajectory persist coherently across time.I — Interpretive Authority: The subject retains authority over how their meanings and identity are encoded in system representations. Each primitive is binary: A, D, C, I ∈ {0,1} The value state is defined as: V(h,S) = (A, D, C, I) Value legitimacy holds if and only if: V_valid(h,S) = 1 ⟺ A = 1 ∧ D = 1 ∧ C = 1 ∧ I = 1 The theorem proves: Sufficiency: If all four primitives hold, no structural value-layer harm remains. Necessity: For each primitive, boundary cases exist where its absence produces irreducible structural harm. Irreducibility: No primitive can be expressed as a function of the remaining three. Therefore, the set: V = {A, D, C, I} is minimal and structurally closed. This theorem establishes formal closure of the value layer and provides the second axis of legitimacy within the dual-layer architecture of decision systems.