Physics Plasmonic Modes From Phi Ladder

Plasmonic modes and the phi ladder

Plasmonics is the study of plasmons, collective oscillations of electrons at a metal surface that couple to light. These oscillations confine electromagnetic energy to sub-wavelength scales, which makes them useful for sensing, waveguiding, and near-field microscopy. The standard taxonomy recognizes five canonical mode types: surface plasmon polaritons, localized surface plasmons, propagating plasmons, bulk plasmons, and gap plasmons.

In Recognition Science, the framework models these five types as a discrete ladder of frequencies. Each mode's characteristic frequency is assigned a value phi^k, where phi is the golden ratio, approximately 1.618, and k is an integer index. The framework proves that the ratio of consecutive frequencies is exactly phi, and that every frequency is positive. This means the five modes are not arbitrary; they are locked to a geometric progression with a ratio equal to the golden ratio.

The framework's library contains a machine-checked proof that there are exactly five such mode types. It also constructs a certificate object that packages the three key facts: the count of modes is five, the frequency ratio between consecutive rungs is phi, and all frequencies are positive. The certificate is a single object that can be passed around in formal proofs to guarantee these properties hold.

What this establishes in plain language is a structural claim: the five canonical plasmonic modes fit a pattern where each mode sits one rung up a ladder whose steps are all the same multiplicative factor, the golden ratio. This is a definitional model within the framework, not a measurement of real plasmon frequencies. The framework chooses this assignment and proves the internal consistency of the resulting structure. It does not claim that measured plasmon frequencies in a laboratory follow this exact ratio.

The payoff is a compact organizing principle. Instead of five unrelated mode types, the framework presents them as a single sequence with a uniform spacing rule. This mirrors the broader Recognition Science program, where the golden ratio appears repeatedly as a consequence of the framework's forcing chain. Here, it appears as a chosen model for how five canonical photonics modes relate to one another.

noncomputable def plasmonFrequency (k : ℕ) : ℝ := phi ^ k

theorem frequency_ratio (k : ℕ) :
    plasmonFrequency (k + 1) / plasmonFrequency k = phi := by
  unfold plasmonFrequency
  have hpos : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  rw [div_eq_iff hpos.ne', pow_succ]
  ring

theorem plasmonicMode_count : Fintype.card PlasmonicMode = 5 := by decide

theorem frequency_pos (k : ℕ) : 0 < plasmonFrequency k := pow_pos phi_pos k

What this page does not claim

Measured plasmon frequencies in any real material follow the phi-ratio ladder. The framework derives the existence of plasmons from first principles. The five-mode list is exhaustive for all possible plasmonic phenomena.

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/Physics/PlasmonicModesFromPhiLadder.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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