Materials Magnetization Saturation From Jcost

Saturation from the cost function

Magnetization saturation is the point where a ferromagnetic material, such as iron, nickel, or cobalt, can absorb no further magnetic field strength; all its atomic magnetic moments align. The measured saturation flux densities are 1.71 tesla for iron, 0.61 tesla for nickel, and 1.44 tesla for cobalt. The ratio of iron to nickel saturation, about 2.8, is close to the square of the golden ratio φ, approximately 2.618.

In Recognition Science, the framework models this saturation through a cost function. The cost, a measure of recognition effort, is defined as J(m/e), where m is a magnetic moment and e is a reference electron moment. The framework's library, a machine-checked collection of formal theorems, proves three facts about this cost: it equals zero when m equals e, it is never negative for positive inputs, and the threshold φ minus 1.5 is positive. These are general properties of the cost function, not specific to any material.

The framework defines a certificate structure that bundles these three proved facts. The certificate exists, meaning the structure is inhabited. However, the framework proves nothing about actual saturation values for iron, nickel, or cobalt, because the definition of domainCost does not reference any specific material. The paragraph about measured saturation values is a research note recording where the idea was meant to go, not a result.

What the framework establishes in plain language is a template. It shows that any ratio of a magnetic moment to an electron moment, when plugged into the cost function, yields a nonnegative cost that vanishes at equality. The golden-ratio threshold is a positive constant, ready to be used as a boundary. But the step from these general facts to a prediction about iron's 1.71 tesla remains open; the definition of m and e in terms of the material's own properties is missing.

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]

theorem cert_inhabited : Nonempty SatMagnetizCert := ⟨cert⟩

What this page does not claim

The framework does not prove that iron saturates at 1.71 tesla or that any material's saturation follows the cost function. The measured saturation values and the φ ratio are empirical observations, not results derived from the cost function. The cost function's general properties do not by themselves establish a physical connection to ferromagnetism.

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

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