Constants Hartree Rydberg Score Card Row Hartree Over Rest Lower

The Hartree bound

The Hartree energy is the atomic unit of energy, roughly the strength of the electric pull between an electron and a proton at the Bohr radius. In the Recognition Science (RS) framework, this quantity appears not as a measured number in joules but as a pure ratio: the Hartree energy divided by the electron's rest energy, m_ec². The declaration row_hartree_over_rest_lower is a theorem in the framework's machine-checked library of formal theorems. It proves that this dimensionless ratio is greater than 5.32 × 10^-5.

The theorem works from a single certified input: the RS inverse fine-structure constant, written alphaInv, which the framework brackets between 137.030 and 137.039. The Hartree ratio is defined as the square of the fine-structure constant, α², which equals 1 divided by alphaInv². Since alphaInv is known to lie in that interval, the framework derives the lower bound by arithmetic: if alphaInv is at most 137.039, then α² is at least 1/(137.039)², which is about 5.325 × 10^-5. The companion theorem row_hartree_over_rest_upper proves the ratio stays below 5.33 × 10^-5, so together they bracket the Hartree ratio between 5.32 and 5.33 × 10^-5.

This is a statement about a dimensionless number, not about any particular unit system. The framework explicitly does not claim a value for the Hartree energy in joules or electronvolts. Converting the ratio to SI units would require a separate bridge that supplies the electron mass, the Planck constant, the speed of light, and a display convention. The theorem's falsifier is named: if CODATA's inverse fine-structure constant ever fell outside (137.030, 137.039), or if an SI bridge failed to map the dimensionless α² to the measured Hartree energy, the claim would fail.

The practical payoff is that the framework pins the hydrogen ground-state binding scale to about five significant figures using only the fine-structure constant. It is a consistency check: the dimensionless ratio must sit in that narrow window, and the measured Hartree energy, once converted, must agree. The theorem does not derive the fine-structure constant itself; it takes the certified interval as input and derives the consequence for the Hartree ratio.

theorem row_hartree_over_rest_lower :
    (5.32e-5 : ℝ) < row_hartree_over_rest := by
  rw [row_hartree_over_rest_eq]
  have hpos : 0 < alphaInv ^ 2 := sq_pos_of_ne_zero (ne_of_gt alphaInv_pos)
  rw [lt_div_iff₀ hpos]
  have hsq : alphaInv ^ 2 < (137.039 : ℝ) ^ 2 := by
    nlinarith [alphaInv_lt, alphaInv_gt]
  have hnum : (5.32e-5 : ℝ) * (137.039 : ℝ) ^ 2 < 1 := by norm_num
  nlinarith

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_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⟩

theorem row_hartree_over_rest_upper :
    row_hartree_over_rest < (5.33e-5 : ℝ) := by
  rw [row_hartree_over_rest_eq]
  have hpos : 0 < alphaInv ^ 2 := sq_pos_of_ne_zero (ne_of_gt alphaInv_pos)
  rw [div_lt_iff₀ hpos]
  have hsq : (137.030 : ℝ) ^ 2 < alphaInv ^ 2 := by
    nlinarith [alphaInv_gt, alphaInv_lt]
  have hnum : 1 < (5.33e-5 : ℝ) * (137.030 : ℝ) ^ 2 := by norm_num
  nlinarith

What this page does not claim

This theorem does not provide a value for the Hartree energy in joules or electronvolts. This theorem does not derive the fine-structure constant itself; it takes the certified interval as input. This theorem does not claim that the Hartree energy is measured; it only establishes a dimensionless ratio bound.

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.

Derived articles

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