bandgapFrequency
plain-language theorem explainer
The declaration assigns the k-th photonic bandgap frequency in a phi-lattice metamaterial to phi raised to the power k. Researchers deriving Recognition Science predictions for photonic structures cite it when establishing ratio properties or certifying metamaterial configurations. The definition is a direct one-line assignment with no lemmas or reductions.
Claim. The frequency of the photonic bandgap at rung $k$ is given by $f_k = phi^k$, where $phi$ is the golden-ratio self-similar fixed point.
background
The module treats phi-lattice metamaterials under RS_PAT_018, where lattice periodicity places bandgaps at phi-ladder frequencies. Five canonical responses (epsilon-near-zero, mu-near-zero, double-negative, hyperbolic, topological) correspond to configDim D = 5. The phi-ladder itself traces to the self-similar fixed point forced at T6 of the UnifiedForcingChain.
proof idea
Direct definition that sets bandgapFrequency k equal to phi raised to k; no tactics or lemmas are applied.
why it matters
This supplies the explicit frequency values required by the downstream bandgapRatio theorem, which proves consecutive ratios equal phi, and by the PhotonicsMetamaterialCert structure that records five metamaterial types together with the phi-ratio property. It fills the RS_PAT_018 claim that phi-lattice periodicity sets bandgaps at phi^k times the fundamental, linking to the eight-tick octave and D = 3 spatial dimensions.
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