Nuclear Alpha Decay Rs5

A module awaiting its subject

Alpha decay is the process by which an unstable atomic nucleus sheds energy by emitting a helium-4 nucleus, two protons and two neutrons bound together. The empirical rule governing how the half-life depends on the decay energy is the Geiger-Nuttall law, stated in 1911 by Hans Geiger and John Mitchell Nuttall: a nucleus with a higher decay energy generally decays faster. The module called Alpha_Decay_RS5 in the Recognition Science library does not prove or extend this law. Its own docstring says so plainly: the file proves three general facts about the cost function and nothing specific to alpha decay.

The cost function at the center is cost, a measure of the forced expense of a recognition event, defined in the framework as J(x) = (x + 1/x)/2 - 1. The module defines a domain cost as J(m/e), the cost of the ratio of two real numbers m and e. Its three theorems state that this cost is zero when the two numbers are equal, that it is never negative for positive inputs, and that the constant phi - 3/2 is positive, where phi is the golden ratio. These are general properties of the cost function, not statements about nuclear physics.

The file's own documentation marks it as a placeholder. The intended direction was a Recognition Science version of the Geiger-Nuttall law, with a formula for the logarithm of the half-life involving the cost function and the golden ratio. But the definition of domain cost contains no reference to the mass of the alpha particle, the charge of the daughter nucleus, or the decay energy. The research note in the docstring records where the idea was meant to go, not a result. The module is shared verbatim with 2383 sibling files, each named for a different subject but carrying the same three theorems.

What the module does establish, in the machine-checked library of formal theorems, is a certificate. The certificate bundles the three proved facts into a single structure and shows that this structure is inhabited, meaning the facts are consistent. That is a real but modest achievement: it confirms that the cost function behaves well on positive inputs. To make this file a theorem about alpha decay, the library would need a definition of m and e in the subject's own terms, tied to the physics. Until then, the module is a template awaiting its subject.

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)

theorem cert_inhabited : Nonempty AlphaDecay5Cert := ⟨cert⟩

What this page does not claim

This module does not prove the Geiger-Nuttall law. This module does not derive any property of alpha decay from physical quantities. The golden ratio constant phi - 3/2 is not shown to relate to any nuclear threshold.

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/Nuclear/Alpha_Decay_RS5.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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