audit_is_O1
The theorem confirms that the audit cost for checking whether an output increases another agent's J-cost equals one in the natural numbers. Researchers modeling ethical constraints inside Recognition Science would cite it to fix the constant-time verification step. The proof is a direct reflexivity application on the definition of auditCost.
claimThe audit cost, defined as the constant-time check determining whether an output increases another agent's J-cost, equals $1$.
background
The Sigma Externalization Audit module defines an O(1) check per output that determines whether the output increases another agent's J-cost. The function auditCost is introduced as the natural number 1, supplying the concrete cost of this verification. The local setting is labeled G-VII-1 and forms part of the ethics component of Recognition Science, where J-cost tracks the functional cost appearing in the Recognition Composition Law.
proof idea
The proof is a one-line term-mode wrapper that applies reflexivity directly to the definition of auditCost.
why it matters in Recognition Science
This result anchors the O(1) property required by G-VII-1 in the Sigma Externalization Audit, confirming constant-time ethical verification inside the Recognition Science framework. It supports efficient checks consistent with the J-uniqueness and phi-ladder constructions but currently has no downstream uses.
scope and limits
- Does not compute or bound the actual J-cost values of any output.
- Does not address multi-agent or multi-output interactions.
- Does not invoke the Recognition Composition Law or the T5-T8 forcing chain.
formal statement (Lean)
28theorem audit_is_O1 : auditCost = 1 := rfl
proof body
Term-mode proof.
29
30end
31
32end IndisputableMonolith.Ethics.SigmaExternalizationAudit