Physics Bohr Radius Exact Rs

The Bohr radius in RS

The Bohr radius is the average distance between the proton and the electron in a hydrogen atom in its lowest energy state. Its standard value is about 0.529 angstroms, where one angstrom is one ten-billionth of a meter. In conventional physics, the radius is built from the electron mass, the elementary charge, the speed of light, and Planck's constant: a_0 = hbar / (m_e * alpha * c). It is a fundamental scale in atomic physics.

In Recognition Science, the framework derives physical constants from a single forced cost function, J(x) = (x + 1/x)/2 - 1, which is proved to be the only cost function satisfying five plain conditions. The Bohr radius appears in this framework as a_0 = phi^(-D-1) in RS natural units, where phi is the golden ratio and D is the number of spatial dimensions, which the framework forces to be three. This gives a_0 = phi^(-4) in RS units, a purely geometric value with no fitted constants.

The module Bohr_Radius_Exact_RS in the machine-checked library of formal theorems does not prove the Bohr radius formula. Its docstring states that the formula is a research note, not a result. What the module proves are three general facts about the cost function applied to a ratio m/e: the cost vanishes when the two quantities are equal, it is nonnegative for positive inputs, and the threshold phi - 3/2 is positive. These are universal properties of the cost function, not specific to the Bohr radius.

To turn these general facts into a theorem about the Bohr radius, one would need a definition of the mass m and charge e in the framework's own terms. That definition is absent. The module is therefore a template, shared verbatim with 2383 sibling modules, that becomes a real statement about its subject only when such a definition is supplied.

What the module does establish, in plain language, is that the cost function behaves sensibly: it is zero at equality, never negative for positive inputs, and its canonical threshold is positive. These are the building blocks for any future derivation of the Bohr radius, but they are not that derivation.

structure BohrRadiusExactRS where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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

structure BohrRadiusExactRS where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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]

What this page does not claim

The module proves the Bohr radius formula. The golden ratio value for the Bohr radius is a measured result. The framework derives the fine-structure constant.

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

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