representationCost
plain-language theorem explainer
The representationCost definition sums J-costs of phi-power gaps between consecutive indices in a list of occupied phi-ladder rungs. Researchers analyzing Zeckendorf decompositions as J-cost admissible configurations in Recognition Science cite it to quantify stability of non-adjacent Fibonacci sums. The definition is realized directly by foldl accumulation over zipped consecutive pairs, adding Jcost(phi raised to each integer gap).
Claim. For a list of natural numbers $I = [i_1 < i_2 < ... < i_n]$ marking occupied rungs on the phi-ladder, the representation cost equals $0$ when $n < 2$ and otherwise equals the sum over consecutive pairs of $J(phi^{i_{k+1} - i_k})$, where $J$ denotes the J-cost function on positive reals.
background
The ZeckendorfJCost module reinterprets Zeckendorf's theorem as the claim that every positive integer admits a unique J-cost-stable representation on the phi-ladder. Fibonacci numbers occupy the ladder positions via $F_n approx phi^n / sqrt(5)$, and the non-consecutive (gap at least 2) condition is exactly the J-cost admissibility constraint: adjacent occupation collapses under the identity phi^n + phi^{n+1} = phi^{n+2}. J-cost itself is the derived cost of a multiplicative recognizer, given explicitly by $J(x) = (x + x^{-1})/2 - 1$ and non-negative on recognition events.
proof idea
This is a direct definition. It zips the input list with its tail to obtain consecutive index pairs, then folds left from zero while adding Jcost(phi raised to the integer difference) for each pair.
why it matters
The definition supplies the quantitative cost measure that makes the module's central claim precise: Zeckendorf representations are precisely the J-cost-stable ones. It therefore sits inside the Recognition Science chain from T5 J-uniqueness through phi-forcing to the eight-tick octave and D = 3 spatial dimensions. No downstream theorems are recorded, but the definition directly supports the stability analysis that links the greedy algorithm to J-cost descent.
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