bandgapRatio
plain-language theorem explainer
The ratio of successive photonic bandgap frequencies in the phi-lattice equals the golden ratio phi for every natural number index. Metamaterial researchers working from Recognition Science would cite this to confirm self-similar scaling of the bandgap positions. The proof is a direct algebraic reduction that unfolds the power definition and applies ring simplification after a positivity check.
Claim. For every natural number $k$, the ratio of the bandgap frequency at index $k+1$ to the frequency at index $k$ equals the golden ratio phi, where the frequency at index $k$ is defined as phi raised to the power $k$.
background
The module treats photonic bandgaps in a phi-lattice metamaterial whose geometry produces frequencies at successive powers of phi. The upstream definition states that bandgapFrequency(k) equals phi^k, directly encoding the periodicity asserted in RS_PAT_018. This places the result inside the five canonical metamaterial responses (epsilon-near-zero through topological) that arise when configDim equals 5.
proof idea
The term proof first unfolds bandgapFrequency to replace both occurrences with explicit powers of phi. It then applies pow_pos to obtain the positivity hypothesis needed for division, rewrites the ratio via div_eq_iff, and finishes with the ring tactic to reach phi.
why it matters
The theorem supplies the phi_ratio field inside photonicsMetamaterialCert, which records the five metamaterial types together with the phi scaling. It thereby fills the frequency-ladder claim of RS_PAT_018 and confirms the self-similar fixed point required by T6 of the unified forcing chain. No open scaffolding remains for this specific equality.
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