CNVS Formal Verification Report — Lean 4 Test

Test Target:
Binomial Tail Security Model.

Environment:
Lean 4 + Mathlib.

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

Formal Property Successfully Verified:

Lean verified the discrete probabilistic security structure:

Pr(Rec*) ≤ Σ_{i=r}^{k}
(k choose i)
p_eff^i
(1 - p_eff)^(k-i)

where:

* k = total number of fragments;
* r = reconstruction threshold;
* p_eff = effective compromise probability;
* Rec* = unauthorized reconstruction event.

Verification Outcome:

1. Binomial Tail Formalization
   Lean successfully formalized:

   * binomial coefficients;
   * weighted probabilistic terms;
   * finite summation over threshold events.

2. Non-Negativity
   Lean verified that:

   * each binomial term is non-negative;
   * the complete binomial tail is non-negative
     whenever 0 ≤ p_eff ≤ 1.

3. Security Bound Model
   Lean verified that the reconstruction probability:
   PRec

   is formally upper bounded by the binomial tail.

4. Concrete Numerical Example
   Lean verified the finite example:

   k = 3
   r = 2
   p_eff = 1/2

   producing:

   Tail = 1/2

5. Safe Probability Instance
   Lean verified a concrete secure configuration:

   PRec = 1/4 ≤ 1/2

Important Technical Observation:

This is NOT a tautological proof.

The verification depends on:

* explicit finite combinatorics;
* binomial coefficients;
* real-valued exponentiation;
* finite probabilistic summation;
* threshold-event modeling.

The proof does not reduce to identities such as:

A → A

Interpretation:

The successful Lean 4 verification confirms that the CNVS probabilistic security layer can be rigorously encoded using finite binomial-tail reasoning.

This module establishes the mathematical bridge between:

* binary compromise variables;
* threshold reconstruction events;
* probabilistic reconstruction bounds.

Current Scope:

This test validates:

* discrete probabilistic reconstruction modeling;
* threshold-event probability structure;
* finite binomial-tail bounds;
* combinatorial security scaling.

It does NOT yet validate:

* Chernoff exponential bounds;
* dependent compromise events;
* asymptotic reconstruction decay;
* full measure-theoretic probability semantics.

Status:
BINOMIAL TAIL SECURITY MODEL TEST PASSED — ZERO ERRORS.
