J_at_phi_pow
plain-language theorem explainer
The definition supplies the explicit J-cost formula at integer powers of the golden ratio. Derivations of the phi-ladder in recognition cost minimization cite it for energy calculations on the self-similar fixed point. It is realized as a direct substitution of the J functional into φ^n.
Claim. The J-cost evaluated at the nth power of the golden ratio φ is given by J(φ^n) = (φ^n + φ^{-n})/2 - 1 for each natural number n.
background
The φ-Emergence module derives the golden ratio from J-cost minimization under self-similarity constraints. J-cost is the cost of a recognition event, defined as the derived cost induced by a multiplicative recognizer's comparator on positive ratios. Upstream results establish that the cost of any recognition event equals its J-cost and that such cost functions arise from observer forcing on recognition states.
proof idea
The definition is a one-line wrapper that applies the closed-form J expression directly to the argument φ raised to n, using the integer power and its inverse.
why it matters
It provides the concrete values needed to populate the phi-ladder of costs, linking to the self-similar fixed point forced by J-uniqueness. This supports the emergence of φ in the forcing chain from T5 to T6 and feeds computations in the eight-tick octave. No immediate parent theorems appear in the dependency graph.
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