MetamaterialType
plain-language theorem explainer
Researchers modeling photonic bandgaps in Recognition Science cite this inductive definition of five metamaterial response types. It enumerates epsilon-near-zero, mu-near-zero, double-negative, hyperbolic, and topological cases under phi-lattice periodicity. The definition enables the direct cardinality result of five and supports certification of phi-scaled bandgaps. It is a straightforward enumeration that derives equality and finiteness instances.
Claim. The inductive type enumerating metamaterial responses consists of five constructors: $epsilon$-near-zero, $mu$-near-zero, double-negative, hyperbolic, and topological.
background
This module develops photonic metamaterials from the phi-lattice geometry under RS_PAT_018. The geometry produces photonic bandgap positions at phi-ladder frequencies, with periodicity setting bandgaps at $phi^k$ times the fundamental frequency.
proof idea
This is an inductive definition that directly lists the five constructors. The deriving clause automatically supplies instances for decidable equality, representation, boolean equality, and finite type structure.
why it matters
The definition supplies the enumeration used to prove there are five types and to certify the phi ratio for consecutive bandgaps. It implements the RS_PAT_018 claim equating five canonical responses to configDim D equals 5. This aligns with Recognition Science use of phi for self-similar structures and ladder scalings in physical classifications.
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