Physics Spontaneous Emission From Jcost

The recognition-limited rate

Spontaneous emission is the process by which an excited quantum system, such as an atom, drops to a lower energy state and emits a photon without any external stimulus. The standard quantum mechanical treatment, developed by Albert Einstein in 1917 and refined by Paul Dirac in 1927, gives the transition rate through the Einstein A coefficient. That coefficient, which has units of inverse time, is proportional to the cube of the emitted photon's angular frequency, the square of the transition dipole moment between the initial and final states, and a factor of the fine-structure constant divided by the square of the reduced Planck constant. The rate is a central element of laser physics, astrophysics, and quantum optics.

In Recognition Science (RS), the framework models physical processes as constrained by a discrete record of recognition events, and it assigns a forced cost to each such event. The cost function J(x) = (x + 1/x)/2 - 1, proved unique in the framework's machine-checked library, measures the price of a recognition event when the ratio of two involved quantities is x. The framework's library proves three general facts about this cost: it is zero when the two quantities are equal, it is nonnegative for all positive inputs, and the golden ratio phi minus 3/2 is a positive threshold. These facts hold for any positive real inputs, because the cost function is defined without reference to any specific physical system.

In the module SpontaneousEmission_FromJCost, the framework's library applies this cost to the ratio of a mass m to a charge e, defining the domain cost as J(m/e). The module proves, in machine-checked form, that this domain cost vanishes when m equals e, that it is nonnegative for all positive masses and charges, and that the canonical threshold phi - 3/2 is positive. These three results are the entire content of the module. They establish that the cost function has the basic properties one would want from a measure of mismatch between two physical quantities, but they do not yet connect that cost to the actual rate of spontaneous emission.

The research note attached to the module records the intended next step: the Einstein A coefficient for spontaneous emission should be multiplied by the cost factor J(phi), evaluated at the golden ratio, yielding a recognition-limited radiation rate. In this account, the cost factor would modify the standard rate by a constant of order phi + 1/phi - 2, which is approximately 0.236. That correction would represent the price the framework assigns to the mismatch between the mass and charge scales in the transition. This is a research note, not a theorem. The module proves nothing about the actual emission rate, because the definition of domain cost does not reference the dipole moment, the frequency, or any other physical quantity that appears in the Einstein A coefficient.

What the module does establish is a template. It shows how the framework's cost function can be attached to a ratio of physical quantities, and it proves the minimal sanity conditions that any such attachment must satisfy. The step from this template to a physical prediction requires a definition of m and e in the subject's own terms, a derivation of the frequency dependence, and a comparison against measured emission rates. Until that work is done, the module remains a formal scaffold, not a physical law.

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]

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 that spontaneous emission rates are actually modified by the cost factor. The module does not derive the Einstein A coefficient from the cost function. The research note's proposed rate is a hypothesis, not a theorem.

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

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