Constants Hartree Rydberg Score Card

The score card

The Hartree energy, the Rydberg constant, and the Bohr radius are the natural measuring sticks of atomic physics. The Hartree energy is the potential energy of two electrons separated by one Bohr radius. The Rydberg constant sets the scale of hydrogen's spectral lines. The Bohr radius is the characteristic size of a hydrogen atom in its ground state. These three constants are usually quoted in joules or meters, but their physical content survives when you divide out the electron's rest energy and its reduced Compton wavelength.

In those dimensionless terms, the relationships are exact and simple. The Hartree energy over the electron rest energy equals the square of the fine-structure constant, α². The Rydberg energy over the rest energy equals α²/2. The Bohr radius over the reduced Compton wavelength equals 1/α. The fine-structure constant α is the dimensionless number, about 1/137, that measures the strength of the electromagnetic force. These ratios are not new physics; they are the standard textbook definitions rewritten without units.

In Recognition Science, the module called the Hartree Rydberg Score Card records these three ratios as certified statements. A machine-checked library of formal theorems proves, with no gaps and no extra axioms, that the Hartree ratio lies between 5.32e-5 and 5.33e-5 and that the Rydberg ratio lies between 2.66e-5 and 2.665e-5. It also proves the Bohr ratio falls between 137.030 and 137.039. These brackets come from the framework's own certified inverse fine-structure constant, not from any fitted parameter.

The module deliberately stops short of claiming a value in joules or meters. Producing those SI numbers would require a separate bridge that connects the framework's dimensionless constants to the measured electron mass, Planck's constant, and the speed of light. That bridge is not part of this score card. What the module does establish is that the dimensionless skeleton of atomic physics, the ratios that survive any choice of units, is forced by the single number α and is provably correct within tight bounds.

theorem row_hartree_over_rest_eq : row_hartree_over_rest = 1 / alphaInv ^ 2 := by
  unfold row_hartree_over_rest alpha
  field_simp [ne_of_gt alphaInv_pos]

theorem row_rydberg_over_rest_eq :
    row_rydberg_over_rest = 1 / (2 * alphaInv ^ 2) := by
  unfold row_rydberg_over_rest alpha
  field_simp [ne_of_gt alphaInv_pos]

theorem row_bohr_over_reduced_compton_eq :
    row_bohr_over_reduced_compton = alphaInv := rfl

theorem row_hartree_over_rest_bracket :
    (5.32e-5 : ℝ) < row_hartree_over_rest ∧
      row_hartree_over_rest < (5.33e-5 : ℝ) :=
  ⟨row_hartree_over_rest_lower, row_hartree_over_rest_upper⟩

theorem row_bohr_over_reduced_compton_bracket :
    (137.030 : ℝ) < row_bohr_over_reduced_compton ∧
      row_bohr_over_reduced_compton < (137.039 : ℝ) :=
  ⟨alphaInv_gt, alphaInv_lt⟩

What this page does not claim

No Joule or meter values for the Hartree energy, Rydberg constant, or Bohr radius are claimed. No claim that the fine-structure constant itself is derived within this module. No claim that the score card proves the physical interpretation of the constants, only the dimensionless ratios.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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