J_phi_sq_identity
plain-language theorem explainer
The identity establishes that the J-cost of φ squared equals the J-cost of φ plus one, following directly from the golden ratio relation. Researchers modeling mass hierarchies on the Recognition Science φ-ladder would cite this when advancing between rungs in the Spectral Emergence framework. The proof reduces the claim via a single rewrite using the defining equation of φ.
Claim. $J(φ^2) = J(φ + 1)$ where $J$ is the J-cost function applied to ratios on the φ-ladder.
background
The Spectral Emergence module derives Standard Model structure and mass hierarchies from the binary cube Q₃ forced by D = 3. The φ-ladder assigns masses via J-cost on successive φ-ratio edges, with each step controlled by the Fibonacci recursion. J-cost quantifies recognition cost or energy for a given ratio, imported from the Cost module. This theorem rests on the upstream phi_sq_eq lemma, which states that φ² = φ + 1 from the quadratic x² - x - 1 = 0.
proof idea
The proof is a one-line wrapper that applies the phi_sq_eq lemma to substitute φ² with φ + 1 inside the Jcost function.
why it matters
This identity supports the φ-ladder mass hierarchy listed among the module's forced consequences from Q₃ symmetries. It connects to T6 where phi is forced as the self-similar fixed point and to the mass formula using yardstick times φ to a rung power. No downstream uses are recorded yet, leaving its role in explicit particle mass calculations open for further closure.
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