Materials Metamaterial Band Gap From Phi Ladder Gap Freq Adjacent Ratio

The adjacent-rung ratio

A photonic metamaterial is an engineered structure that controls light the way a crystal controls electrons: certain frequencies cannot pass through, forming band gaps. In a one-dimensional Fibonacci arrangement, theory and experiment (Macià-Barber 2009, Wang et al. 2017) show that these gaps appear in a self-similar cascade. The declaration gapFreq_adjacent_ratio states the clean version of that observation: for any rung k on the ladder, the ratio of the next gap-center frequency to the current one is exactly the golden ratio φ ≈ 1.618.

Formally, the framework defines a reference frequency referenceFreq as 1, then sets the gap-center frequency at rung k to referenceFreq * φ^k. The theorem proves that gapFreq (k+1) / gapFreq k = φ for every natural number k. It also proves that these frequencies are all positive and strictly increasing. These are not empirical claims; they are consequences of the definition, checked in the machine-checked library of formal theorems with no unproved assumptions.

What the declaration does not claim: it does not say that any real material actually exhibits this ratio. The theorem only describes the mathematical model. The empirical match to Fibonacci photonic crystals is a separate observation, cited from the literature, not a theorem. Nor does it claim that the golden ratio emerges from deeper physics here; the φ-ladder is a structural choice, not a derived law. The theorem is a precise statement about the model's internal consistency, not a prediction about a specific fabricated sample.

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']

theorem gapFreq_pos (k : ℕ) : 0 < gapFreq k := by
  unfold gapFreq referenceFreq
  have : 0 < phi ^ k := pow_pos Constants.phi_pos k
  linarith [this]

theorem gapFreq_strictly_increasing (k : ℕ) :
    gapFreq k < gapFreq (k + 1) := by
  rw [gapFreq_succ_ratio]
  have hk : 0 < gapFreq k := gapFreq_pos k
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have : gapFreq k * 1 < gapFreq k * phi :=
    mul_lt_mul_of_pos_left hphi_gt_one hk
  simpa using this

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 claim that any real material exhibits the φ ratio. The theorem does not claim that the φ-ladder is derived from first principles; it is a structural definition. The theorem does not claim the empirical cascade is exact; that is a literature observation, not a formal result.

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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