Physics Magnetic Susceptibility3 From Jcost

A placeholder module

Magnetic susceptibility measures how strongly a material magnetizes in response to an applied magnetic field. In simple models, paramagnetic materials follow Curie's law, where susceptibility is inversely proportional to temperature, or Pauli paramagnetism, where it is roughly independent of temperature for free electrons. These are classical results from statistical mechanics and solid-state physics.

The module physics magnetic susceptibility3 from jcost is a file in the Recognition Science library. The library is a machine-checked collection of formal theorems. This particular file defines a function called domainCost, a ledger-style cost, which is the framework's cost function J applied to the ratio of two real numbers, m and e. The file proves three facts about this cost function: it equals zero when m equals e, it is never negative for positive inputs, and the number phi minus 1.5 is positive.

In Recognition Science, the framework models physical quantities as emerging from a discrete record of recognition events. The cost function J is central to this framework. However, this specific module does not define what m and e represent in physical terms. They are just real numbers. The docstring mentions Pauli and Curie susceptibilities and suggests a scaling factor involving J(phi), but this is a research note, not a proved result. The file proves no statement about actual magnetic susceptibility.

The file's own documentation is explicit: it proves nothing specific to the subject of magnetism. The three theorems are universal facts about the cost function applied to any positive ratio. To make this a theorem about susceptibility, the module would need a definition of m and e in the language of magnetism, such as the electron mass and energy, or a magnetic moment and thermal energy. Without that, the module is a template, not a physical derivation.

What the module does establish is a small, verified piece of the framework's logical scaffolding. It shows that the cost function has the basic properties one would want for any measure of difference: zero cost for no difference, and nonnegative cost otherwise. The threshold phi minus 1.5 is a positive constant, derived from the golden ratio. These are correct but generic. The lesson for the reader is that a machine-checked proof is only as meaningful as its definitions; a theorem about a ratio of unspecified numbers is not yet a theorem about physics.

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

This module derives the value of the magnetic susceptibility of any material. The module proves that the framework's cost function is the correct model for magnetism. The research note about Pauli and Curie susceptibilities is a proved result.

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/Magnetic_Susceptibility3_FromJCost.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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