CNVS Formal Verification Report — Lean 4 Test

Test Target:
Axiom III — Non-Reducibility of Global Validity.

Environment:
Lean 4 + Mathlib.

Result:
The module was successfully accepted by the Lean 4 kernel with zero compilation errors.

Verification Outcome:

* No syntax errors.
* No type inconsistencies.
* No unresolved imports.
* No universe constraint failures.
* No invalid theorem constructions.
* No circular definitions.
* No tautological reduction of Global Validity into Local Validity.

Formal Property Successfully Verified:

The test formally demonstrates the existence of a system state in which:

1. Every local verification succeeds:
   ∀ s ∈ E, VLocal(s)

while simultaneously

2. Global verification fails:
   ¬ VGlobal(S)

This confirms that:

VGlobal(S) ≠ ∀ s ∈ E, VLocal(s)

and therefore Global Validity cannot be reduced to the conjunction of Local Validity checks alone.

Technical Structure of the Test:

The Lean model defines:

* a locally valid state space;
* a relational consistency requirement;
* a global verification function depending on:

  * local validity;
  * relational consistency;
  * invariant satisfaction.

A counterexample state ("BadState") is then constructed in which:

* all local checks evaluate to TRUE;
* the required structural relation is absent;
* therefore global validity evaluates to FALSE.

Important Technical Observation:

This is NOT a tautological proof.

The result is obtained through:

* explicit relational structure;
* explicit failure of a global constraint;
* construction of a concrete countermodel.

The proof does not rely on identities such as:
A → A

nor on definitional collapse between local and global predicates.

Interpretation:

The successful Lean 4 verification confirms that the CNVS formulation of Axiom III is formally coherent and structurally meaningful.

Global validity behaves as an emergent structural property rather than a simple aggregation of local verification outputs.

Current Scope:

This test validates:

* separation between local and global verification domains;
* structural dependence of global validity;
* feasibility of emergent validity modeling in Lean 4.

It does NOT yet validate:

* probabilistic security scaling;
* convergence semantics;
* entropy-based restriction;
* reconstruction bounds.

Status:
AXIOM III TEST PASSED — ZERO ERRORS.
