Physics Radioactive Decay Types From Config Dim

Five decay modes

Radioactive decay is the process by which an unstable atomic nucleus loses energy by emitting radiation. In the standard physics curriculum, five modes are canonical: alpha decay (emission of a helium-4 nucleus), beta-minus (an electron and an antineutrino), beta-plus (a positron and a neutrino), gamma (a high-energy photon), and spontaneous fission (the nucleus splits into two smaller nuclei plus neutrons). These five are the ones a student meets in every nuclear physics textbook, and they cover the dominant pathways by which naturally occurring and artificial isotopes transform.

The five-mode list has a long history. Alpha decay was identified by Ernest Rutherford in 1899; beta decay was explained by Enrico Fermi in 1934 with his theory of the weak interaction; gamma emission was understood as nuclear de-excitation in the same era; spontaneous fission was discovered by Georgy Flerov and Konstantin Petrzhak in 1940. Together they form the standard taxonomy of radioactive transformation, and the physics of each mode is governed by different forces: the strong force for alpha emission and fission, the weak force for beta decays, and electromagnetism for gamma rays.

In Recognition Science, the framework's machine-checked library of formal theorems contains a module that builds this same list from a single number. The module defines an inductive type with exactly five constructors, one for each decay mode, and proves by direct computation that the count is five. The certificate structure wraps that proof, and a definition supplies the certificate. The plain content is: the five canonical modes form a closed set, and the count is a theorem, not an assumption.

The framework's interest is not in the nuclear physics itself, which it takes as given, but in the pattern that the number five appears where the framework's dimensional structure also produces five. The module is a bridge: it names the five modes in the framework's own language and certifies the count with a kernel-checked proof. What the module does not do is derive the decay modes from first principles, nor does it explain why these five and not others. It establishes a correspondence between a classical taxonomy and a framework-internal count, leaving the physical mechanism to the nuclear physics it imports.

The practical consequence is a clean interface: any later framework theorem that needs the five decay modes can rely on the certified count without re-deriving the taxonomy. For a reader, the module is a small but concrete example of how the framework treats a familiar physical classification: it formalizes the list, proves the count, and then stops, without overclaiming the physics.

inductive DecayMode where
  | alpha
  | betaMinus
  | betaPlus
  | gamma
  | spontaneousFission
  deriving DecidableEq, Repr, BEq, Fintype

theorem decayMode_count : Fintype.card DecayMode = 5 := by decide

structure RadioactiveDecayCert where
  five_modes : Fintype.card DecayMode = 5

What this page does not claim

The module does not derive the decay modes from nuclear forces or quantum mechanics. The module does not explain why these five modes are the canonical ones. The framework does not claim that the five-mode count is a prediction from first principles.

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

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