Chemistry Laser Linewidth From Jcost Laser Linewidth Cert

The certificate's scope

The Schawlow-Townes formula gives the minimum possible linewidth of a laser, the narrowest spread of frequencies it can emit, set by quantum noise. In its standard form, Δν = (ν_FSR² × h × ν) / (4π × P_out), it ties the linewidth to the laser's cavity geometry, its output power, and Planck's constant. The Recognition Science framework aims to derive such physical relations from a single cost function, but its library's certificate, a machine-checked bundle of proved facts, currently covers only a small, general part of that ambition.

The declaration LaserLinewidthCert packages three proved statements about the framework's cost function J(x) = (x + 1/x)/2 - 1, applied to a ratio of two real numbers m and e. First, when m equals e, the cost is exactly zero. Second, for positive inputs, the cost is never negative. Third, a constant called the canonical threshold, defined as φ - 3/2 where φ is the golden ratio, is positive. Each of these is a proved statement in the machine-checked library, and the certificate simply bundles them into a single structure.

What the certificate does not do is connect these facts to lasers. The framework's own documentation is explicit: the definition of the cost function contains no reference to cavity decay rates, oscillation frequencies, or any laser parameter. The Schawlow-Townes formula appears only in a research note, as a statement of where the idea was meant to go. To make this certificate a statement about laser linewidth, one would need a definition of m and e in the laser's own terms, a step the library has not taken.

In Recognition Science, this certificate is a template, one of thousands of sibling modules sharing the same three general facts. Its value is as a placeholder: it records the intended target, proves the generic properties that any such derivation would need, and leaves the subject-specific definition open. The reader should take from it a clear picture of what a machine-checked proof can and cannot certify, and where the framework's work on laser physics actually stands.

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]

What this page does not claim

The certificate does not derive the Schawlow-Townes linewidth formula. The certificate does not define m or e in terms of any laser parameter. The certificate does not prove that the framework's cost function applies to laser physics.

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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: