Materials Metamaterial Band Gap From Phi Ladder Gap Freq Succ Ratio

The ladder ratio

The golden ratio φ, approximately 1.618, is the number that solves r² = r + 1. A photonic metamaterial is an engineered structure that controls light through its periodic geometry rather than its chemistry. In a one-dimensional Fibonacci photonic crystal, alternating layers of two materials arranged in a Fibonacci sequence, measured studies by Macià-Barber (2009) and Wang et al. (2017) report a self-similar cascade of band gaps: the frequency ranges where light cannot propagate. The cascade's fundamental scaling constant is the golden ratio.

Within the Recognition Science framework, the declaration gapFreq_succ_ratio proves a precise statement about that cascade. The framework defines a reference band-gap center frequency, set to 1 in its native dimensionless units, and places band-gap centers on a φ-ladder: the gap at rung k has frequency φ^k times the reference. The theorem states that for every rung k, the next rung's frequency equals the current one multiplied by φ, written gapFreq (k+1) = gapFreq k * φ. A companion theorem proves the adjacent-rung ratio equals exactly φ. This is a proved result in the framework's machine-checked library of formal theorems, with no unproved assumptions and no axioms beyond the ambient type theory's standard three.

The theorem itself is a statement about a definitional choice. The framework models the band-gap frequencies as lying on this ladder; the proof verifies that the model's arithmetic is consistent. It does not derive the ladder from physical first principles, and it does not claim that every photonic metamaterial exhibits this exact ratio. The empirical observations from the cited papers are measurements, not consequences of the theorem. The framework's contribution is a certified calculation: given the ladder placement, the ratio property follows rigorously.

What the declaration changes is the epistemic status of one step in a larger account. A reader can now distinguish sharply between the proved arithmetic fact, the definitional choice of the ladder, and the measured physical behavior. The theorem guarantees that if the model is right, the ratio is exactly φ, not approximately φ. The gap between model and measurement remains a question for experiment, not for the proof.

theorem gapFreq_succ_ratio (k : ℕ) :
    gapFreq (k + 1) = gapFreq k * phi := by
  unfold gapFreq
  rw [pow_succ]; ring

theorem gapFreq_adjacent_ratio (k : ℕ) :
    gapFreq (k + 1) / gapFreq k = phi := by
  rw [gapFreq_succ_ratio]
  have hpos : 0 < gapFreq k := gapFreq_pos k
  field_simp [hpos.ne']

What this page does not claim

The theorem does not prove that any physical photonic crystal exhibits the φ ratio. The theorem does not derive the φ-ladder from physical laws; the ladder is a definitional choice. The theorem says nothing about the width or depth of any band gap, only about the center frequencies.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Materials/MetamaterialBandGapFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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