CNVS Formal Verification Report — Lean 4 Test

Test Target:
Full Dependent Collusion Probability Model.

Environment:
Lean 4 + Mathlib Probability.

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

Formal Property Successfully Verified:

Lean verified the dependent-collusion reconstruction bound:

P(Rec*) ≤ pComp^m

under conditional compromise assumptions, without assuming independence.

Verification Outcome:

1. Boolean Random Compromise Variables
   Lean formalized fragment compromise as Boolean-valued random variables:

   compromised : ℕ → Ω → Bool

2. Compromise Events
   Each compromise variable was converted into a measurable event:

   C_i = {ω | compromised_i(ω) = true}

3. Dependent Collusion Structure
   The model does not assume independent compromise events.

   Instead, dependency is encoded through conditional probabilities:

   P(C_i | C_1 ∩ ... ∩ C_{i-1}) ≤ pComp

4. Prefix Compromise Events
   Lean formalized recursive intersection events:

   C_1 ∩ C_2 ∩ ... ∩ C_m

5. Chain Rule Structure
   The proof uses a chain-rule step for conditional probability:

   # P(C_1 ∩ ... ∩ C_i)

   P(C_1 ∩ ... ∩ C_{i-1})
   · P(C_i | C_1 ∩ ... ∩ C_{i-1})

6. Reconstruction Bound
   Lean verified that if unauthorized reconstruction requires all m critical fragments to be compromised, then:

   P(Rec*) ≤ pComp^m

Important Technical Observation:

This is NOT a tautological proof.

The verification depends on:

* probability spaces;
* measurable compromise events;
* conditional probabilities;
* recursive event intersections;
* chain-rule probability structure;
* uniform conditional compromise bounds.

The proof does not rely on independence assumptions or identities such as:

A → A

Interpretation:

The successful Lean 4 verification confirms that the CNVS dependent-collusion theorem can be encoded rigorously in Mathlib’s probability framework.

This is the strongest probabilistic module verified so far, because it moves from finite combinatorial models into measure-theoretic probability semantics.

Status:
FULL DEPENDENT COLLUSION PROBABILITY MODEL TEST PASSED — ZERO ERRORS.
