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A Linear-Size Block-Partition Fibonacci Encoding for Gödel Numbering

Zoltán Sóstai

Abstract

We construct an encoding of finite strings over a fixed finite alphabet as natural numbers, based on a block partition of the Fibonacci sequence. Each position in the string selects one Fibonacci number from a dedicated block, with unused indices between blocks guaranteeing non-adjacency. The encoded number is the sum of the selected Fibonacci numbers, and Zeckendorf's theorem guarantees that this sum uniquely determines the selection. The encoding is injective, the string length is recoverable from the code, and the worst-case digit count of the encoded number grows as $Θ(m)$ for strings of length $m$, matching the information-theoretic lower bound up to a constant factor. We also prove that the natural right-nested use of Rosko's (2025) binary carryless pairing for sequence encoding has worst-case $Θ(2^m)$ digit growth, an exponential blowup that the block-partition construction avoids entirely.

A Linear-Size Block-Partition Fibonacci Encoding for Gödel Numbering

Abstract

We construct an encoding of finite strings over a fixed finite alphabet as natural numbers, based on a block partition of the Fibonacci sequence. Each position in the string selects one Fibonacci number from a dedicated block, with unused indices between blocks guaranteeing non-adjacency. The encoded number is the sum of the selected Fibonacci numbers, and Zeckendorf's theorem guarantees that this sum uniquely determines the selection. The encoding is injective, the string length is recoverable from the code, and the worst-case digit count of the encoded number grows as for strings of length , matching the information-theoretic lower bound up to a constant factor. We also prove that the natural right-nested use of Rosko's (2025) binary carryless pairing for sequence encoding has worst-case digit growth, an exponential blowup that the block-partition construction avoids entirely.

Paper Structure

This paper contains 19 sections, 8 theorems, 13 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fibonacci sequence
  4. Zeckendorf's theorem
  5. The block-partition encoding
  6. The alphabet
  7. Block partition
  8. Encoding
  9. Correctness
  10. Non-adjacency
  11. Injectivity
  12. Valid codes and length recovery
  13. Decoding
  14. Complexity analysis
  15. Code length
  16. ...and 4 more sections

Key Result

Theorem 2.1

Every positive integer $n$ has a unique representation as a sum of non-consecutive Fibonacci numbers: where $e_1 > e_2 > \cdots > e_s \ge 2$ and $e_i - e_{i+1} \ge 2$ for all $i$.

Theorems & Definitions (21)

  • Remark 1.1
  • Theorem 2.1: Zeckendorf Zeckendorf1972, Lekkerkerker Lekkerkerker1952
  • Definition 3.1: Block partition
  • Definition 3.2: Block-partition encoding
  • Example 3.3
  • Example 3.4
  • Lemma 4.1: Non-adjacency
  • proof
  • Theorem 4.2: Injectivity
  • proof
  • ...and 11 more