Constants Hartree Rydberg Score Card Row Rydberg Over Rest Lower

Rydberg energy ratio

The Rydberg constant sets the scale of the energy levels in hydrogen and other simple atoms. In SI units it is about 10,973,731.6 per meter, and it appears in the formula for the wavelengths of light emitted or absorbed by hydrogen. Physicists also express it as an energy, the ionization energy of the hydrogen atom in its ground state, which is about 13.6 electronvolts. A natural way to make this energy comparable across unit systems is to divide it by the electron's rest energy, m_e c^2, the energy locked in the electron's mass itself.

That dimensionless ratio, the Rydberg energy over the electron rest energy, is what the Recognition Science declaration row_rydberg_over_rest_lower pins down. The declaration proves, in the framework's machine-checked library of formal theorems, that this ratio is greater than 2.66e-5. It also proves an upper bound of 2.665e-5, so together the two theorems bracket the ratio in a narrow interval. The proof does not measure the ratio directly; it derives the bound from the framework's certified inverse fine-structure constant, which it takes to lie between 137.030 and 137.039.

The exact value of the ratio, in the framework's account, is alpha^2 / 2, where alpha is the fine-structure constant, the dimensionless number that measures the strength of the electromagnetic interaction. Since alpha is about 1/137, alpha^2 / 2 is about 2.66e-5, matching the bracket. This is not a new measurement. It is a formal derivation: the declaration shows that, given the framework's range for alpha, the ratio must fall in that interval. The declaration itself is a theorem, proved with no gaps and no axioms beyond the standard logical ones.

The declaration does not claim to produce a value in joules or electronvolts. That would require a bridge to SI units, including the electron mass and Planck's constant, which the module explicitly does not provide. It also does not claim that the framework has derived the fine-structure constant from first principles; the value of alpha is an input, certified within a range, not a derived result. The falsifier for this row is a CODATA value of alpha^-1 outside (137.030, 137.039), or an SI bridge that fails to map the dimensionless ratio to the measured Rydberg energy.

What the declaration gives a reader is a clean, machine-checked statement of where the Rydberg energy sits relative to the electron's rest energy, with a tight numerical bound. It is a small piece of a larger score card that ties atomic constants to the framework's dimensionless numbers. For a physicist, the interest is the formal guarantee: the ratio is not just approximately this value, it is proved to lie in this interval, given the certified alpha range.

theorem row_rydberg_over_rest_lower :
    (2.66e-5 : ℝ) < row_rydberg_over_rest := by
  rw [row_rydberg_over_rest_eq]
  have hsqpos : 0 < alphaInv ^ 2 := sq_pos_of_ne_zero (ne_of_gt alphaInv_pos)
  have hpos : 0 < 2 * alphaInv ^ 2 := by nlinarith
  rw [lt_div_iff₀ hpos]
  have hsq : alphaInv ^ 2 < (137.039 : ℝ) ^ 2 := by
    nlinarith [alphaInv_lt, alphaInv_gt]
  have hnum : (2.66e-5 : ℝ) * (2 * (137.039 : ℝ) ^ 2) < 1 := by norm_num
  nlinarith

theorem row_rydberg_over_rest_upper :
    row_rydberg_over_rest < (2.665e-5 : ℝ) := by
  rw [row_rydberg_over_rest_eq]
  have hsqpos : 0 < alphaInv ^ 2 := sq_pos_of_ne_zero (ne_of_gt alphaInv_pos)
  have hpos : 0 < 2 * alphaInv ^ 2 := by nlinarith
  rw [div_lt_iff₀ hpos]
  have hsq : (137.030 : ℝ) ^ 2 < alphaInv ^ 2 := by
    nlinarith [alphaInv_gt, alphaInv_lt]
  have hnum : 1 < (2.665e-5 : ℝ) * (2 * (137.030 : ℝ) ^ 2) := by norm_num
  nlinarith

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]

What this page does not claim

The declaration does not provide a value for the Rydberg energy in SI units. The declaration does not derive the fine-structure constant from first principles. The declaration does not claim a measurement; it is a formal derivation from a certified alpha range.

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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