Chemistry Nuclear Magic Isotopes From Rs Doubly Magic Count

The five doubly magic nuclides

A doubly magic nuclide is an atomic nucleus in which both the number of protons and the number of neutrons are magic numbers, the series {2, 8, 20, 28, 50, 82, 126} that marks closed nuclear shells. The Recognition Science framework's machine-checked library of formal theorems defines a type for exactly five of these: helium-4, oxygen-16, calcium-40, calcium-48, and nickel-56. The declaration doublyMagic_count proves that this type has exactly five members. The proof is a direct computation, with no axioms beyond the ambient type theory and no unfinished proofs.

The five are canonical in the framework's model of nuclear structure. Helium-4 has 2 protons and 2 neutrons; oxygen-16 has 8 and 8; calcium-40 has 20 and 20; calcium-48 has 20 and 28; nickel-56 has 28 and 28. Each entry in the list has both numbers drawn from the magic series. The framework does not derive the magic numbers themselves here; it takes the set as a given and counts the doubly magic cases that fall within its structural depth parameter, which is set to 5.

The count is a theorem about a defined list, not a discovery about the physical world. It establishes that the framework's formal object contains five nuclides, and that the certificate structure carrying this count is inhabited. It does not claim that these are the only doubly magic nuclides in nature, nor that the framework predicts their existence. The declaration says nothing about nuclear stability, binding energy, or the empirical status of the magic numbers. Those questions lie outside the scope of this particular formal result.

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

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

What this page does not claim

The declaration does not claim these are the only doubly magic nuclides in nature. It does not claim the framework predicts or derives the magic numbers themselves. It says nothing about nuclear stability or binding energy.

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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