Materials Metamaterial Band Gap From Phi Ladder Gap Freq Pos

Band-gap frequencies on the ladder

A photonic metamaterial is an engineered structure that controls light through its geometry rather than its chemistry. One family of such structures arranges its layers in a pattern built from the golden ratio, roughly 1.618, and the framework's library of formal theorems proves a basic fact about that arrangement: every frequency in its predicted sequence of band gaps is a positive number. The declaration gapFreq_pos states exactly this, for every whole-number index k.

The sequence itself is a phi-ladder, a set of values where each step multiplies the previous one by the golden ratio. The library defines the k-th gap-center frequency as the reference frequency, set to 1 in its own units, times phi raised to the k-th power. The positivity theorem is the foundation for two stronger results in the same file: each frequency is strictly larger than the one before it, and the ratio of any adjacent pair is exactly phi. Together these three theorems form a certificate, a packaged guarantee that the ladder behaves as the structural prediction describes.

The proof itself is short because the hard work sits in an earlier theorem that phi is positive. The library then applies that fact to a power of phi and finishes with arithmetic. What the declaration does not claim is more interesting. It does not assert that any real material actually exhibits these gaps, nor that the predicted frequencies match any measurement. The empirical check against published results on Fibonacci photonic crystals is a separate observation, not part of the theorem.

The value of this proof is that it pins down the mathematics cleanly before any experiment enters. A reader can now rely on the ladder's internal consistency: the sequence is well-defined, ordered, and self-similar at every step. Whether nature honors that ladder is a question for the laboratory, not for the kernel.

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

No real material has been shown to exhibit these band gaps. The theorem does not compare the predicted frequencies against any experimental measurement. The golden-ratio scaling is a structural prediction, not a derived physical law.

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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