Materials Metamaterial Band Gap From Phi Ladder Gap Freq Strictly Increasing

The ladder's monotone guarantee

A photonic metamaterial is an engineered structure that controls light through its geometry rather than its chemistry. In one design family, the geometry follows the golden ratio φ ≈ 1.618, the same proportion found in pentagons and Fibonacci spirals. The framework's library, a machine-checked collection of formal theorems, defines a sequence of band-gap center frequencies: recognition, the framework's term for a discrete record of events, assigns each integer rung k a frequency gapFreq(k) = φ^k in dimensionless units. The declaration gapFreq_strictly_increasing proves that this sequence is strictly increasing: each rung's frequency is larger than the one before it.

The proof is short and rests on two facts. First, every gapFreq(k) is positive, since φ is positive and powers of positive numbers stay positive. Second, the ratio between adjacent rungs is exactly φ, so gapFreq(k+1) = gapFreq(k) × φ. Because φ is greater than 1, multiplying a positive number by φ makes it strictly larger. The theorem formalizes this chain of reasoning in the machine-checked library, with no unproved assumptions and no axioms beyond the standard logical ones. The certificate structure MetamaterialBandGapCert bundles this monotonicity together with positivity, the one-step ratio, and the adjacent-ratio-equals-φ property into a single object, showing the four claims are mutually consistent.

What the theorem does not claim is just as important. It does not assert that any real physical metamaterial exhibits these gaps; that is an empirical question, supported separately by published observations of Fibonacci photonic crystals (Macià-Barber 2009, Wang et al. 2017), not by the formal proof. It does not derive the golden ratio from first principles; φ is taken as given from the framework's constants. It does not say anything about the width, depth, or location of the gaps in physical units, only about the dimensionless center frequencies. The theorem is a structural guarantee about a defined sequence, not a prediction about any specific material sample.

The practical consequence is a clean ordering principle: if the ladder model holds for a real structure, the band gaps must appear in ascending frequency order, each centered at φ times the previous one. That ordering is not an approximation or a heuristic; it is a proved consequence of the definitions. An engineer designing a multi-band filter can rely on the monotonic cascade as a design constraint, knowing the formal library guarantees the sequence never folds back on itself.

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_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

structure MetamaterialBandGapCert where
  freq_pos : ∀ k, 0 < gapFreq k
  one_step_ratio : ∀ k, gapFreq (k + 1) = gapFreq k * phi
  strictly_increasing : ∀ k, gapFreq k < gapFreq (k + 1)
  adjacent_ratio_eq_phi : ∀ k, gapFreq (k + 1) / gapFreq k = phi

What this page does not claim

The theorem does not assert that any real physical metamaterial exhibits these gaps. It does not derive the golden ratio from first principles; φ is taken as given. It says nothing about gap widths, depths, or locations in physical units.

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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