Chemistry Nuclear Magic Isotopes From Rs

The five doubly magic nuclei

In nuclear physics, a magic number is a count of protons or neutrons that makes a nucleus unusually stable. The classic magic numbers are 2, 8, 20, 28, 50, 82, and 126. A doubly magic nucleus has both its proton count and its neutron count equal to one of these numbers. Helium-4 (2 protons, 2 neutrons), oxygen-16 (8, 8), calcium-40 (20, 20), calcium-48 (20, 28), and nickel-56 (28, 28) are the five canonical examples. Each is exceptionally tightly bound compared with its neighbors on the chart of nuclides.

In Recognition Science, the framework's machine-checked library of formal theorems treats these five nuclides as a complete structural set. The library defines a type whose members are exactly those five nuclei, then proves that the type has exactly five members. This is a theorem in the formal library: it is checked by the kernel with no axioms and no unfinished proofs. The result is not a claim about the physical world; it is a structural classification. The framework models the five doubly magic nuclei as a closed list, a discrete record of the recognized stable configurations at these magic numbers.

What this establishes in plain language is that the framework's internal account of nuclear structure recognizes these five and only these five doubly magic nuclides. The theorem proves the count is five, not six or four. The certificate structure bundles that count as a single object, a compact witness to the classification. This is a definitional choice about how the framework organizes nuclear magic, not a derivation of nuclear physics from first principles.

The practical consequence for a reader is that the framework gives a precise, checkable statement of which nuclei it treats as doubly magic. The list matches the standard nuclear physics list. The framework's contribution is the formal packaging: a machine-checked certificate that the set has exactly five members, with no loose ends in the proof.

theorem doublyMagic_count : Fintype.card DoublyMagicNuclide = 5 := by decide

inductive DoublyMagicNuclide where
  | he4
  | o16
  | ca40
  | ca48
  | ni56
  deriving DecidableEq, Repr, BEq, Fintype

theorem doublyMagic_count : Fintype.card DoublyMagicNuclide = 5 := by decide

structure NuclearMagicCert where
  five_nuclides : Fintype.card DoublyMagicNuclide = 5

What this page does not claim

This answer does not claim the framework derives nuclear magic numbers from a deeper principle. This answer does not claim the five-nuclide list is a physical prediction rather than a classification. This answer does not claim the framework explains why these nuclei are stable.

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

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