penrose_frequency_ratio
plain-language theorem explainer
Penrose frequency ratio in large tilings equals the golden ratio φ. Condensed-matter researchers analyzing aperiodic order and quasicrystal stability within Recognition Science cite this equality when linking tile counts to the self-similar fixed point. The proof is a one-line reflexivity reduction on the direct definition of the ratio as φ.
Claim. The Penrose frequency ratio of thick to thin rhombus counts in a large tiling equals the golden ratio $φ$.
background
The Quasicrystal module treats aperiodic tilings that possess long-range order without translational symmetry. Penrose tilings serve as the central example, with the ratio of thick to thin rhombi fixed at φ to minimize an energy proxy of the form E(r) = (r - 1/φ)². This setting draws on the upstream definition penrose_ratio : ℝ := Constants.phi together with the structure Constants from LawOfExistence and the J-cost calibration in PhiForcingDerived.of.
proof idea
The proof is a one-line term-mode wrapper that applies reflexivity to the definition of penrose_ratio.
why it matters
This equality anchors the frequency ratio inside the Recognition framework by direct identification with φ, the self-similar fixed point forced at T6. It supports the module predictions that stable quasicrystals exhibit tile ratios involving φ and that icosahedral symmetry dominates via 5-fold axes. No downstream uses are recorded, leaving the identification step available for later diffraction or stability arguments.
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