Hybrid Hierarchical Representation of Numbers
Based on the Zeckendorf System
A Synthesis of Sparsity and Efficiency
Vladimir Veselov | Moscow Institute of Electronic Technology (MIET), Moscow | February 26, 2026
CID: bafybeied4oksujc4ipublvkqvsfnocqeytuoxwdzh2otveve2dd7blk774
Abstract
This paper proposes a novel approach to the representation and processing of non-negative integers, combining the hierarchical sparse weight structure introduced by V. Veselov with the canonical Zeckendorf representation based on Fibonacci numbers. The hybrid model is designed for compact storage and accelerated arithmetic processing of giant numbers with extremely low density of non-zero “units” (). Empirical estimates indicate a potential speedup of 100–500 times compared to standard arbitrary-precision libraries for specific problem classes in cryptography and computer algebra.
Modules
11
6 base + 5 extension
Theorems
47
Machine-verified
Lines
1,329
Proof code
Sorry
0
Zero gaps
Audit Rounds
7
Hostile audits passed
Formal Verification Certificate
Every theorem in this project has been machine-checked by the Lean 4 kernel. No axiom is assumed without proof. No gap exists in the verification chain.
Hybrid Zeckendorf • Lean 4 + Mathlib • Apoth3osis Labs
Paper ↔ Proof Correspondence
Every section of the paper that defines a structure, algorithm, or theorem has a corresponding Lean 4 module with a machine-checked proof. Sections without a Lean counterpart are expository (introduction, applications, conclusion).
| Paper Section | Lean Module | Status |
|---|---|---|
| 1 | expository | N/A |
| 2.1 | WeightSystem.lean | PROVED |
| 2.2 | FibIdentities.lean | PROVED |
| 2.3 | HybridNumber.lean | PROVED |
| 2.4 | Normalization.lean | PROVED |
| 3.1 | Addition.lean | PROVED |
| 3.2 | Multiplication.lean | PROVED |
| 3.3 | Normalization.lean | PROVED |
| 4 | Multiplication.lean (density) | PROVED |
| 5 | expository | N/A |
| 6 | expository | N/A |
| 7 | expository | N/A |
| 8 | expository | N/A |
Proof Modules
Each module below shows two views. Mathematics renders the definitions and theorems in standard mathematical notation. Lean 4 shows the corresponding formal proof code.
Sheaf-Glue Closure Extension
Five additional modules extending the base formalization with quantitative bounds, representation injectivity, closure-operator semantics, and SGI26 sheaf-glue bridge layers. Together they upgrade the original eval-level correctness proofs to full representational uniqueness and unconditional idempotent closure.
Key result: — normalization is an unconditional idempotent closure operator (nucleus).
Sharp bound: Representational (pointwise) monotonicity of carry is impossible — machine-checked counterexample via Euclidean division discontinuity at weight thresholds. Eval-level monotonicity is the strongest available form.
Module Dependencies
Hover over a module to trace its imports. Solid lines show direct dependencies; dashed lines show transitive imports used for specific operations.
Verified C Artifacts
Generated via CAB pipeline: Lean 4 → LambdaIR → MiniC → C. Each file carries a correctness certificate verifying semantic preservation.
hybrid_zeckendorf_weight.c
Weight sequence computation | from WeightSystem
hybrid_zeckendorf_fib.c
Fibonacci identity helpers | from FibIdentities
hybrid_zeckendorf_eval.c
Evaluation and constructors | from HybridNumber
hybrid_zeckendorf_add.c
Lazy addition | from Addition
hybrid_zeckendorf_normalize.c
Two-stage normalization | from Normalization
hybrid_zeckendorf_multiply.c
Structural multiplication | from Multiplication
Extension Artifacts
Safe Rust Artifacts
Transpiled from verified C to idiomatic safe Rust. Raw pointers replaced with owned types, slices, and iterators. Correctness inherited from the C verification chain. Download the full crate (includes Cargo.toml and lib.rs) and run cargo test.
Cargo.toml
Crate manifest (cargo build) | from Crate
lib.rs
Crate root — declares all modules | from Crate
hybrid_zeckendorf_weight.rs
Weight sequence (safe Rust) | from WeightSystem
hybrid_zeckendorf_fib.rs
Fibonacci helpers (safe Rust) | from FibIdentities
hybrid_zeckendorf_eval.rs
Eval and constructors (safe Rust) | from HybridNumber
hybrid_zeckendorf_add.rs
Lazy addition (safe Rust) | from Addition
hybrid_zeckendorf_normalize.rs
Normalization (safe Rust) | from Normalization
hybrid_zeckendorf_multiply.rs
Multiplication (safe Rust) | from Multiplication
Extension Artifacts
Provenance Chain
Research Paper
Veselov, V. “Hybrid Hierarchical Representation of Numbers Based on the Zeckendorf System.” February 2026. Defines Def 2.1, Algorithms 1-3, complexity analysis.
Lean 4 Formalization
11 modules (6 base + 5 extension), 47 theorems, 1,329 lines. Kernel-verified by Lean 4 + Mathlib. 7 rounds of hostile adversarial audit passed. Zero sorry/admit.
CAB-Certified C
Lean 4 → LambdaIR → MiniC → C. Each transformation step carries a certificate hash verifying semantic preservation.
Safe Rust
C → Rust transpilation. Ownership semantics, no unsafe blocks. Correctness inherited from verified C source.
IPFS Pinned
All artifacts content-addressed and pinned to IPFS. CID: bafybeied4oksujc4ipublvkqvsfnocqeytuoxwdzh2otveve2dd7blk774. Immutable, globally retrievable via any IPFS gateway.
Hybrid Zeckendorf | Paper → Proof → Code | Apoth3osis
All proofs machine-verified using Lean 4 and Mathlib