CNVS Formal Verification Report — Lean 4 Test

Test Target:
Binomial Tail Security Model — Final Version Without Temporary Axioms.

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 binomial-tail security model without relying on the previous temporary numerical axiom.

Verified Formula:

Pr(Rec*) ≤ Σᵢ₌ᵣᵏ (k choose i) p_eff^i (1 - p_eff)^(k-i)

Verification Outcome:

1. Binomial Tail Definition
   Lean formalized the upper binomial tail using:

   * finite sums;
   * binomial coefficients;
   * real-valued probabilities;
   * threshold filtering.

2. Non-Negativity
   Lean verified that binomial terms and the full binomial tail are nonnegative when:

   0 ≤ p_eff ≤ 1

3. Security Bound
   Lean verified that reconstruction probability is bounded by the binomial tail.

4. Numerical Example Without Axiom
   The previous temporary axiom for the value:

   binomialTail 3 2 (1/2) = 1/2

   was removed and replaced by direct Lean verification.

Important Technical Observation:

This closes the only previously identified shortcut in the Binomial Tail module.

The final version no longer depends on an added numerical axiom.

Status:
BINOMIAL TAIL FINAL TEST PASSED — ZERO ERRORS.
