Constants Hartree Rydberg Score Card Hartree Rydberg Score Card Cert Holds

The dimensionless score card

The Hartree energy, the Rydberg constant, and the Bohr radius are the classic scales of the hydrogen atom. The Hartree energy is the unit of atomic energy, about 4.36 x 10^-18 joules. The Rydberg constant sets the scale of hydrogen's spectral lines, and the Bohr radius is the most probable distance between a proton and an electron in the ground state, about 5.29 x 10^-11 meters. These three quantities are usually given in SI units, but their essential content is captured by dimensionless ratios: the Hartree energy divided by the electron's rest energy m_e c^2, the Rydberg constant divided by the same rest energy, and the Bohr radius divided by the reduced Compton wavelength of the electron.

The Recognition Science framework, through its machine-checked library of formal theorems, proves that these ratios take a strikingly simple form. The theorem hartreeRydbergScoreCardCert_holds establishes that the Hartree-to-rest-energy ratio equals alpha^2, the Rydberg-to-rest-energy ratio equals alpha^2 / 2, and the Bohr-to-reduced-Compton ratio equals 1/alpha, where alpha is the fine-structure constant, the dimensionless number near 1/137 that measures the strength of the electromagnetic interaction. The proof also certifies tight numerical intervals for these ratios, using the framework's own certified value of the inverse fine-structure constant, which lies between 137.030 and 137.039.

What the theorem does not claim is any value in joules or meters. The declaration explicitly states that it records only the dimensionless ratios and their interval bounds. Producing a Hartree energy in SI units would require a separate bridge that connects the electron mass, the Planck constant, the speed of light, and the SI display system. The score card is a statement about the structure of the ratios themselves, not about the laboratory units. The falsifier for this claim is a CODATA measurement of the inverse fine-structure constant outside the interval (137.030, 137.039), or an SI bridge that fails to map the dimensionless alpha^2 and alpha^2/2 ratios to the measured Hartree and Rydberg values.

The practical consequence is a clean, unit-free target for the framework's physical derivation plan. The theorem gives a precise, checkable prediction for the hydrogen atom's fundamental scales, independent of the messy business of SI units. It is a score card that says: if the framework's value for the fine-structure constant is right, then these three ratios must hold exactly. The next step, building the SI bridge, remains an open target, but the dimensionless core is now a proved theorem with zero unproved assumptions.

theorem hartreeRydbergScoreCardCert_holds :
    Nonempty HartreeRydbergScoreCardCert :=
  ⟨{ hartree_closed := row_hartree_over_rest_eq
     rydberg_closed := row_rydberg_over_rest_eq
     bohr_closed := row_bohr_over_reduced_compton_eq
     hartree_bracket := row_hartree_over_rest_bracket
     rydberg_bracket := row_rydberg_over_rest_bracket
     bohr_bracket := row_bohr_over_reduced_compton_bracket }⟩

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_rydberg_over_rest_bracket :
    (2.66e-5 : ℝ) < row_rydberg_over_rest ∧
      row_rydberg_over_rest < (2.665e-5 : ℝ) :=
  ⟨row_rydberg_over_rest_lower, row_rydberg_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

This theorem does not provide any value for the Hartree energy, Rydberg constant, or Bohr radius in SI units (joules or meters). It does not prove the value of the fine-structure constant itself; it uses a certified value as a premise. It does not claim that the hydrogen atom's scales are derived from the forcing chain, only that the dimensionless ratios follow from the certified alphaInv.

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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