CNVS Formal Verification Report — Lean 4 Test

Test Target:
Probabilistic Security Core — Threshold Reconstruction.

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 invalid theorem constructions.
* No circular definitions.
* No tautological proof structure.

Formal Properties Successfully Verified:

1. Binary Compromise Variables
   Lean verified that every compromise indicator variable satisfies:

   X_j ∈ {0,1}

   where:

   * X_j = 1 represents a useful compromised fragment;
   * X_j = 0 represents a non-compromised or non-useful fragment.

2. Total Compromise Variable
   Lean verified the finite summation structure:

   X = Σ X_j

   representing the total number of useful compromised fragments.

3. Threshold Reconstruction Event
   The unauthorized reconstruction event was formalized as:

   Rec* ⇔ X ≥ r

   where:

   * r is the reconstruction threshold;
   * Rec* occurs iff the number of compromised useful fragments reaches the threshold.

4. Threshold Security Behavior
   Lean verified:

   * if X < r, reconstruction does NOT occur;
   * if X ≥ r, reconstruction DOES occur.

5. Concrete Positive Example
   Lean verified a finite example in which:

   * two useful fragments are compromised;
   * threshold r = 2;
   * therefore Rec* is TRUE.

6. Concrete Negative Example
   Lean verified a finite example in which:

   * only one useful fragment is compromised;
   * threshold r = 2;
   * therefore Rec* is FALSE.

Important Technical Observation:

This is NOT a tautological proof.

The model depends on:

* explicit binary indicator variables;
* finite combinatorial summation;
* threshold inequalities;
* constructive examples.

The proof does not reduce to identities such as:

A → A

Interpretation:

The successful Lean 4 verification confirms that the CNVS probabilistic security core can be rigorously encoded as a threshold-based combinatorial reconstruction model.

This module provides the discrete mathematical foundation required before introducing:

* probabilistic distributions;
* dependent compromise events;
* Chernoff bounds;
* asymptotic security scaling.

Current Scope:

This test validates:

* binary compromise encoding;
* threshold reconstruction semantics;
* finite reconstruction behavior;
* combinatorial consistency of the CNVS security core.

It does NOT yet validate:

* probabilistic bounds;
* conditional probabilities;
* dependent collusion scaling;
* Chernoff-type reconstruction limits.

Status:
PROBABILISTIC SECURITY CORE — THRESHOLD RECONSTRUCTION TEST PASSED — ZERO ERRORS.
