KNOTT-5.2 — Zeckendorf’s Theorem (Canonical Fibonacci Base Representation)

Every positive integer N ≥ 1 decomposes uniquely as a sum of non-consecutive Fibonacci numbers F_{k₁} + F_{k₂} + … + F_{kᵣ} (kᵢ ≥ kᵢ₊₁ + 2, kᵣ ≥ 2), producing a unique 11-free binary numeral in Fibonacci place values (…, 13, 8, 5, 3, 2, 1).

Curated Source & Catalog Provenance

Catalog ID	KNOTT-5.2
Curated Source	Dr. Ron Knott, Fibonacci Representations of Numbers (#zeck)
Source Attribution	© 1996–2025 Dr. Ron Knott, University of Surrey — Explore Dr. Knott’s original exposition
Exact SymPy Validator	geometor.knott.golden_string.construct_zeckendorf_staircase_model() (0 sp.Float atoms)
GEOMETOR Concordance	geometor/divine, geometor/trees

Constructive Synthesis & Exact Proof

On fibrep.html (KnottFibRep), Dr. Ron Knott pairs phigit expansions with Édouard Zeckendorf’s theorem (Zeckendorf1972): every positive integer \(N \ge 1\) decomposes uniquely as a sum of non-consecutive Fibonacci numbers \(F_{k_1} + \cdots + F_{k_r}\) (\(k_i \ge k_{i+1} + 2,\; k_r \ge 2\)).

• Greedy Non-Consecutive Fibonacci Staircase: Plotting Fibonacci place-value bars A2B2C2D2 (\(F_2 = 1\)), A3B3C3D3 (\(F_3 = 2\)), A4B4C4D4 (\(F_4 = 3\)), A5B5C5D5 (\(F_5 = 5\)), and A6B6C6D6 (\(F_6 = 8\)), any two adjacent bars \(F_{k-1} + F_k\) immediately coalesce into the next single bar \(F_{k+1}\), eliminating every 11 pair in Fibonacci base.

• Canonical Decomposition of N = 12: Below the staircase, integer \(12\) decomposes uniquely along S0 \(\to\) S8 \(\to\) S11 \(\to\) S12 into non-adjacent summands S0S8 (\(F_6 = 8\), matching A6D6), S8S11 (\(F_4 = 3\), matching A4D4), and S11S12 (\(F_2 = 1\), matching A2D2), encoding \(12 = (10101)_F\) in Zeckendorf representation.

Machine-Verified Exact Symbolic Invariants

Invariant	Exact Symbolic Value
zeckendorf 12 sum	\(12\)
zeckendorf 100 term count	\(3\)
F2	\(1\)
F4	\(3\)
F6	\(8\)