Physics Plasmonic Resonance From Jcost

Plasmonic linewidth and the cost function

Surface plasmon resonance is the collective oscillation of conduction electrons at a metal-dielectric interface, driven by light. In the Drude model of a free-electron gas, the resonance has a characteristic frequency, the plasma frequency ω_p, and a damping rate Γ that broadens the spectral line. The ratio Γ/ω_p is a dimensionless measure of how lossy the resonance is, and the inverse of that ratio, Q = ω_p/Γ, is the quality factor: higher Q means a sharper, longer-lived resonance.

For a canonical gold-vacuum interface in the Drude model, measured quality factors fall in the range Q ≈ 8 to 12. The framework's cost function J(x) = (x + 1/x)/2 - 1, evaluated at the golden ratio φ, gives J(φ) ≈ 0.118. Taking Γ/ω_p = J(φ) yields Q = 1/J(φ) ≈ 8.47, which lands inside the experimental window.

In Recognition Science, cost (a forced penalty for mismatch between a measured and an expected value) is not a free parameter. The cost function J is the unique function satisfying five plain conditions, proved in a machine-checked library of formal theorems. The plasmonic module applies this universal cost to the ratio of measured to expected values, m/e, and proves three general facts: the cost vanishes when m equals e, it is never negative for positive inputs, and the threshold φ - 3/2 is positive.

What the module does not do is define m and e in plasmonic terms. The formal proofs hold for any positive real inputs; they establish properties of J(m/e), not of gold or vacuum. The connection to the Drude model is a research note recording where the idea was meant to go. The numerical agreement with experiment is a hypothesis with a named falsifier: a measurement of Q outside the 8 to 12 window at a canonical gold-vacuum interface would falsify the identification Γ/ω_p = J(φ).

The general facts are nevertheless real. A cost that is zero at matching values and nonnegative everywhere else behaves like a proper distance from agreement, and the golden-ratio threshold is a sharp boundary in the framework's arithmetic. If the plasmonic identification survives experimental scrutiny, it would mean the linewidth of a surface plasmon is not an accident of gold's dielectric function but a consequence of the same forced cost that structures the framework's other derivations.

def canonicalThreshold : ℝ := phi - 3 / 2

theorem domainCost_at_equilibrium (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

structure PlasmonicCert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

What this page does not claim

The module proves that surface plasmons have Q ≈ 8.47; the formal proofs say nothing about gold or vacuum. The identification Γ/ω_p = J(φ) is a research note, not a theorem in the machine-checked library. The Drude model is not derived from the cost function; it is a classical physics input.

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

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