fibonacci_lattice_is_complete
plain-language theorem explainer
Every positive integer admits a representation as a sum of non-consecutive Fibonacci numbers. Number theorists and Recognition Science researchers mapping classical partitions onto the phi-ladder cite this completeness statement. The proof is a term-mode construction that applies the greedy Zeckendorf function, verifies the non-consecutive property, and confirms the weighted sum via the Fibonacci recurrence identity.
Claim. For every positive integer $n$, there exists a finite list $l$ of natural numbers such that $l$ forms a Zeckendorf representation (no two consecutive indices) and the sum of the Fibonacci numbers $F_k$ for $k$ in $l$ equals $n$, where the Fibonacci sequence begins $F_0=1$, $F_1=1$, $F_{k+2}=F_k+F_{k+1}$.
background
Zeckendorf representations encode integer partitions under the non-consecutive constraint, which the module identifies with J-cost stability on the phi-ladder. The Fibonacci sequence is defined by fib(0)=1, fib(1)=1, and fib(n+2)=fib(n)+fib(n+1). In Recognition Science this constraint matches the admissibility condition derived from J(x)=(x+x^{-1})/2-1, since adjacent rungs collapse by the recurrence while gaps of at least 2 remain stable under the Recognition Composition Law.
proof idea
The proof introduces n and refines the existential witness to the standard Zeckendorf list. It applies the lemma that this list satisfies the Zeckendorf representation predicate and invokes the sum identity for Fibonacci weights. The entire argument is a one-line term wrapper relying on the fib recurrence and the known greedy decomposition property.
why it matters
This result supplies the completeness half of Zeckendorf's theorem inside the RamanujanBridge module, equating classical representations with J-cost stable states on the phi-ladder. It anchors the RS decipherment that links the greedy algorithm to J-cost descent and feeds the broader forcing chain (T5 J-uniqueness, T6 phi fixed point). No immediate downstream theorems are recorded, yet the statement closes the classical-to-RS translation for subsequent complexity and spectral structures.
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