J_phi_ceiling_band
plain-language theorem explainer
J_phi_ceiling_band establishes that the golden-section J-cost ceiling satisfies 0.11 < J(φ) < 0.13. Researchers modeling spin-orbit resonances in the inner Solar System cite this bound to confirm Mercury's 3:2 ratio lies inside the canonical J-cost band. The proof is a term-mode reduction that unfolds the definition and applies linear arithmetic to the supplied numerical bounds on φ.
Claim. The golden-section J-cost ceiling satisfies $0.11 < J(φ) < 0.13$, where $J(φ) := φ - 3/2$.
background
In the Tidal Locking from φ-Resonance module, spin-orbit resonances are minima of the J-cost on the phase manifold. J_phi_ceiling is defined as φ - 3/2 and supplies the canonical ceiling for the deviation of the 3/2 ratio from φ. This sits with sibling results such as phi_cubed_band (φ³ ∈ (4.22, 4.24)) and mercury_deviation_in_J_phi_band (the Mercury deviation lies in the same numerical interval).
proof idea
The proof unfolds J_phi_ceiling to φ - 3/2, introduces the lemmas phi_gt_onePointSixOne and phi_lt_onePointSixTwo, then applies linarith to obtain the two strict inequalities.
why it matters
This supplies clause 8 of the TidalLockingFromPhiResonanceCert structure, completing the master certificate for the φ-resonance account of inner-Solar-System tidal locking. It anchors the J-cost band used across the Recognition Science treatment of resonances and connects to the J-uniqueness step (T5) in the forcing chain. The result closes one of the numerical checks required for the structural theorem on spin-orbit ratios.
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