Physics Nuclear Magnetron From Jcost

The nuclear magneton

The nuclear magneton is the standard unit for measuring the magnetic moment of atomic nuclei. It is defined as μ_N = eħ/(2m_p), where e is the elementary charge, ħ is the reduced Planck constant, and m_p is the proton mass. Its value is about 5.05 × 10^-27 joules per tesla. It is the electron's magneton scaled down by the ratio of the electron mass to the proton mass, roughly 1/1836. The proton's own magnetic moment is about 2.79 nuclear magnetons, a fact that helped reveal the proton is not a point particle but has internal structure.

The proton-to-electron mass ratio is one of the most precisely measured numbers in physics, about 1836.15. In the Recognition Science framework, this ratio sits near a rung of the golden-ratio ladder: φ^15 ≈ 1364, which is close but not equal. The framework models the cost, a measure of how expensive a recognition event is, as a function of the ratio m/e. The machine-checked library of formal theorems proves three general facts about this cost function: it is zero when the two masses are equal, it is never negative for positive masses, and the threshold φ - 3/2 is positive. These are properties of the cost function itself, not specific to the nuclear magneton.

The cost is defined as J(m/e) and the three facts are proved for any positive masses. A certificate structure packages these proofs together, and a theorem confirms the certificate exists. The documentation states plainly that nothing specific to the nuclear magneton is proved, because the definition does not yet tie m and e to the proton and electron. The research note records where the idea was meant to go, not a result. What would make this a theorem about the magneton is a definition of m and e in the subject's own terms.

The framework's contribution here is a disciplined template: it shows what a cost function must look like when applied to a mass ratio, and it proves the minimal properties any such application must have. The classical physics of the nuclear magneton stands on its own; the framework turn is a check that its core cost function behaves sensibly on the ratio that defines the magneton. The gap between the framework's general facts and a specific claim about the proton is explicit and honest.

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

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)

What this page does not claim

No claim that the framework derives the nuclear magneton's value or the proton-to-electron mass ratio. No claim that the framework establishes any fact specific to nuclear physics beyond the general cost properties. No claim that φ^15 equals the measured mass ratio; it is close but not equal.

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/NuclearMagnetronFromJCost.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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