Constants Hartree Rydberg Score Card Row Hartree Over Rest Bracket

The Hartree ratio bracket

The Hartree energy, often written E_h, is the unit of energy in the atomic system of natural units: roughly the energy an electron feels at one Bohr radius from a proton. Its value in SI units is about 4.3597 × 10^-18 joules, and it appears throughout atomic physics as the natural scale for binding energies. A useful, unit-free way to state it is as a ratio to the electron rest energy m_ec², which is about 8.187 × 10^-14 joules. That ratio is close to 5.33 × 10^-5, a number that comes straight from the fine-structure constant α.

In the Recognition Science framework, this ratio is not an independent measurement but a consequence of the framework's own inverse fine-structure constant, written α^-1. The framework's library of machine-checked formal theorems defines the Hartree-over-rest ratio as α², the square of the fine-structure constant. The key result, named row_hartree_over_rest_bracket, is a theorem: it proves that this ratio lies strictly between 5.32 × 10^-5 and 5.33 × 10^-5. The proof is fully checked, with no unverified assumptions, and it relies on the framework's certified bounds for α^-1, which itself is pinned between 137.030 and 137.039.

This bracket is a consistency check, not a new measurement. The framework does not claim to produce a value for the Hartree energy in joules or in meters; that would require a separate bridge to SI units, using the electron mass, Planck's constant, and the speed of light. What the theorem establishes is the dimensionless ratio, and it does so with a tight interval: the lower bound 5.32 × 10^-5 and the upper bound 5.33 × 10^-5 are both proved, and the conjunction of the two is the bracket theorem. The falsifier for this claim would be a CODATA value of α^-1 falling outside the stated range, or an SI bridge that fails to map the dimensionless α² ratio to the measured Hartree energy.

The practical meaning is that the framework's internal constant, derived from its forcing chain, lands within one part in a thousand of the accepted atomic unit ratio, with a fully audited proof. This is a demonstration that the framework's derived constants are consistent with measured atomic physics, not a derivation of the SI value itself. The bracket theorem is a piece of the framework's score card, a check that its numbers line up with the laboratory, and it does so without fitting any free parameters.

/-- P1-C04 dimensionless Hartree/rest-energy ratio. -/
noncomputable def row_hartree_over_rest : ℝ := alpha ^ 2

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_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⟩

What this page does not claim

The framework does not claim a value for the Hartree energy in SI units (joules or meters). The bracket theorem does not derive the electron mass, Planck's constant, or the speed of light; it only uses the framework's inverse fine-structure constant. The framework does not claim to derive the fine-structure constant α itself; its value is an identification, not a derived coupling.

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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