phi_cubed
plain-language theorem explainer
The definition sets the real number phi cubed to the cube of the golden ratio. Researchers modeling inner solar system spin-orbit resonances cite it to position Venus's retrograde 4:1 ratio near 4.236 on the phi ladder. It is a direct abbreviation that reuses the phi constant from the upstream Fibonacci universality module.
Claim. Define the real number equal to the cube of the golden ratio by $x := phi^3$, where $phi$ satisfies $phi^2 = phi + 1$.
background
The Tidal Locking from Phi Resonance module treats observed spin-orbit ratios as phi-rational minima of the J-cost on the phase manifold. Moon-Earth sits at exact 1:1 with J-cost zero. Mercury-Sun 3:2 lies near phi. Venus-Sun retrograde 4 lies near phi cubed. The upstream FibonacciPhiUniversality module states: Already proved: φ³ = 2φ + 1 = F(3)·φ + F(2).
proof idea
This is a direct definition that unfolds to phi raised to the third power. It inherits algebraic closure under the phi recurrence from the upstream phi cubed theorem.
why it matters
The definition supplies the phi cubed term required by the Venus clause inside the TidalLockingFromPhiResonanceCert structure and by the one-statement theorem. It completes the AS6 track prediction that every inner solar system resonance ratio sits within J(phi) of a phi-ladder position, consistent with phi as the self-similar fixed point. The module closes with zero axioms.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.