Chemistry Laser Linewidth From Jcost

Laser linewidth and the cost function

In laser physics, the Schawlow-Townes linewidth is the fundamental lower limit on the spectral width of a laser's output, set by spontaneous emission. It is usually written Δν = (ν_FSR² × h × ν) / (4π × P_out), where ν_FSR is the free spectral range, h is Planck's constant, ν is the oscillation frequency, and P_out is the output power. The formula, derived by Arthur Schawlow and Charles Townes in 1958, tells an engineer how narrow a laser can get before quantum noise blurs the line.

Recognition Science proposes a different route to the same idea. Its framework models a ledger, a discrete record of recognition events, and defines a cost for each event: the forced overhead of maintaining the record. The proposed linewidth relation is Δν/ν = J(φ) × κ/ω, where κ is the cavity decay rate and ω is the oscillation frequency. Here J is the framework's cost function, and φ is the golden ratio. The module LaserLinewidthFromJCost records this as a research note, not a proved result.

What the module actually proves is three general facts about the cost function J(m/e). First, the cost vanishes when the two inputs are equal: J(r/r) = 0 for any nonzero r. Second, the cost is never negative for positive inputs: J(m/e) ≥ 0 when m and e are both positive. Third, the constant φ − 3/2 is positive, which is a small numerical fact about the golden ratio. These are proved in the machine-checked library of formal theorems, but they hold for any positive inputs, not for any specific laser.

The gap is the definition of m and e. The module defines the cost as J(m/e) without saying what m and e mean for a laser cavity. Until a definition ties those variables to cavity decay, output power, or photon number, the module is a template, not a chemistry theorem. The research note records where the idea was meant to go; the formal proof stays at the level of general cost properties.

In plain language: the framework offers a candidate formula for laser linewidth, but its verified content is about the cost function's abstract behavior, not about lasers. The chemistry-specific step, identifying m and e in physical terms, remains open. A reader should not take the module as evidence that the Schawlow-Townes limit has been rederived; it is a placeholder with a clean general lemma attached.

structure LaserLinewidthCert 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

theorem domainCost_at_eq (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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The module does not prove the Schawlow-Townes formula or any chemistry-specific linewidth result. The research note's proposed linewidth relation is not a theorem; it is an unverified identification. The golden ratio φ does not appear in the proved theorems; it only appears in the unproved research note.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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