Physics Rydberg Constant Rs

The Rydberg constant

The Rydberg constant, usually written R∞, is the fundamental scaling factor in the spectra of hydrogen-like atoms. It appears in the formula for the wavelengths of light emitted when an electron moves between energy levels, and its measured value is approximately 1.0974 × 10⁷ per meter. In the standard account, R∞ is built from the electron mass, the elementary charge, the speed of light, and Planck's constant, so it is not a free parameter but a derived quantity of quantum electrodynamics.

Recognition Science (RS) is a framework that starts from the idea that reality keeps a discrete record of recognition events, and that the cost of each recognition is forced by a proved uniqueness theorem. In RS, the Rydberg constant is meant to be expressed as a power of the golden ratio φ, with a research note suggesting R∞ ~ φ⁴⁷ per meter. That identification is a research note, not a result. The formal development proves three general facts about the cost function J(m/e), where m and e are positive real numbers: the cost is zero when m equals e, the cost is never negative, and φ − 3/2 is positive. These facts hold for any positive inputs, so they say nothing specific about the Rydberg constant itself.

The structure RydbergConstRS packages these three facts into a single object, and the theorem cert_inhabited shows that such an object exists. The proof is a direct application of the general cost theorems, with no additional assumptions about the electron mass or any atomic system. The formal documentation is explicit: it proves nothing specific to this subject, because the cost function is defined without reference to the Rydberg constant. The paragraph above the formal code is a research note recording where the idea was meant to go, not a result.

What the formal development does establish, in plain language, is that the cost function behaves well at the identity and stays nonnegative, and that the golden-ratio threshold φ − 3/2 is a positive number. These are generic properties of the cost function, not properties of atomic spectra. A reader should not take the development as evidence that the Rydberg constant equals φ⁴⁷; that claim remains a research note. The development is a template, shared verbatim with 2383 sibling modules, and what would make it a theorem about the Rydberg constant is a definition of m and e in terms of the electron mass and charge.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The formal development does not prove that the Rydberg constant equals φ⁴⁷ per meter. The formal development does not derive the Rydberg constant from the electron mass and charge. The formal development does not establish any property specific to hydrogen-like atoms.

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/Physics/Rydberg_Constant_RS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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