The Villaber Formula: Convergence to One — Perfect Closure, Core Formula V_n = Sum(1/2)^k + 2*(1/2)^n = 1 EXACT

**The Villaber Formula: Convergence to One — Perfect Closure**

This work introduces the Villaber Formula, a novel exact decomposition of unity that achieves perfect closure from a geometric halving sequence.

**Core Formula:**

For all integers n ≥ 2,

V_n = Σ_{k=1}^{n-1} (1/2)^k + 2 * (1/2)^n = 1 EXACT

Unlike the classical infinite series Σ_{k=1}^{∞} (1/2)^k = 1 which only converges at infinity, the Villaber Formula achieves EXACT equality to 1 in finite steps by duplicating the final term.

**Special Case (n=3) - The Square Partition Model:**

1/2 + 1/4 + 1/8 + 1/8 = 1 EXACT

This is visualized as a perfect square partition, where a unit square is divided into 1/2, 1/4, and two 1/8 blocks, demonstrating complete coverage with no gap and no overlap.

**Significance and Applications:**

1. **Information Theory:** The formula provides a finite, exact solution to the Kraft Equality: Σ 2^{-l_i} = 1. The sequence [1, 2, 3, 3] satisfies Kraft with equality, proving it defines an optimal and complete prefix code.

2. **Optimal Prefix Codes:** It constructs an optimal binary prefix code (e.g., Huffman coding) that is 100% efficient with zero redundancy.

3. **Mathematical Pedagogy:** A tangible, visual proof for convergence, fractions, and geometric series.

4. **Philosophy of Mathematics:** From Infinite Halving to Perfect Closure — showing how infinity can be closed perfectly in finite form.

**Keywords:** Villaber Formula, Kraft Equality, Geometric Series, Prefix Code

**Author: Ruel Villaber

**Affiliation:** Independent Researcher, Philippines

**License:** CC BY 4.0

This record contains the core formula poster and abstract.