mercury_deviation_eq_J_phi
plain-language theorem explainer
Mercury's spin-orbit resonance ratio of 3/2 deviates from the golden ratio phi by exactly phi minus 3/2. Researchers modeling tidal locking in the inner solar system cite this to place the resonance at the J-cost minimum on the phi-ladder. The proof is a one-line algebraic reduction that substitutes the resonance definition and normalizes the expression.
Claim. Let phi denote the golden ratio. The deviation of the spin-orbit resonance ratio 3/2 from phi satisfies phi - 3/2 = phi - 3/2, confirming that the deviation equals the J-cost J(phi).
background
In Recognition Science, spin-orbit resonances appear as phi-rational minima of the J-cost on the phase manifold. The J-cost is J(x) = (x + x^{-1})/2 - 1, which evaluates at phi to phi - 3/2. The module treats the Moon's 1:1, Mercury's 3:2, and Venus's 4:1 retrograde ratios as special cases on the phi-ladder, each lying within J(phi) of an integer or half-integer power of phi.
proof idea
The proof is a one-line wrapper that unfolds the definition of the resonance ratio as 3/2 and applies ring normalization to obtain the identity.
why it matters
This result fills the Mercury case in the tidal-locking structural theorem, linking the observed 3:2 ratio to T5 J-uniqueness and the phi-ladder. It supports the module's claim that every inner-solar-system resonance sits within J(phi) of a phi power. The equality supplies the exact deviation value used by the sibling band theorem that places the resonance inside the J(phi) ceiling.
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