Constants Hartree Rydberg Score Card Row Rydberg Over Rest Bracket

The Rydberg ratio bracket

The Rydberg constant is a fundamental physical constant that sets the scale of atomic spectral lines. In the hydrogen atom, it determines the energy required to ionize the atom from its ground state, a quantity known as the Rydberg energy. When this energy is divided by the electron's rest energy (m_e c^2), the result is a pure number, a dimensionless ratio. This ratio is central to the Recognition Science framework's score card for atomic constants.

In Recognition Science, the framework's machine-checked library of formal theorems defines this ratio as recognition, a discrete record of events, cost function. The specific declaration, row_rydberg_over_rest, defines the dimensionless Rydberg-over-rest-energy ratio as alpha^2 / 2, where alpha is the framework's certified inverse fine-structure constant. The theorem row_rydberg_over_rest_bracket then proves, with no unproven axioms, that this ratio lies strictly between 2.66e-5 and 2.665e-5. This is a tight numerical bracket, a certified interval bound on a pure number.

The bracket is not a measurement itself. It is a derived consequence of the framework's definition of the Rydberg ratio and its independently certified value for the inverse fine-structure constant. The framework's library proves the inequality from the definition and the certified bounds on alphaInv, using only the standard axioms of the ambient type theory. The proof is a formal, machine-checked derivation, not an empirical fit.

What this declaration does not claim is a value for the Rydberg constant in Joules or meters. It explicitly does not provide an SI-unit value, because doing so would require a separate bridge connecting the dimensionless ratio to the electron mass, Planck's constant, and the speed of light. The declaration also does not claim to have derived the fine-structure constant itself; the value of alphaInv is an input, not an output. The bracket is a statement about a ratio, not about the absolute scale of atomic physics.

The practical consequence is a certified, unit-free anchor. Anyone working within the framework can rely on the fact that, if the framework's inverse fine-structure constant is correct, then the Rydberg energy is between 2.66 and 2.665 hundred-thousandths of the electron's rest energy. This provides a clean, checkable prediction that connects the framework's internal constants to a well-known physical quantity, without overstepping into claims about SI units or the origin of alpha itself.

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⟩

/-- P1-C02 dimensionless Rydberg/rest-energy ratio. -/
noncomputable def row_rydberg_over_rest : ℝ := alpha ^ 2 / 2

What this page does not claim

A value for the Rydberg constant in SI units such as Joules or inverse meters. A derivation of the fine-structure constant itself; its value is an input. A measurement of the Rydberg ratio; the bracket is a derived theorem.

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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