Materials Coercivity From Phi Ladder

Coercivity and the phi ladder

Coercivity is a material's resistance to becoming demagnetized. A hard magnet, like a neodymium magnet in an electric motor, keeps its field under strong opposing forces; a soft magnet, like the iron in a transformer core, flips its magnetization easily. The standard practical scale puts soft magnets below 1 oersted and hard magnets above 1000 oersteds, a span of roughly a thousand in field strength.

In Recognition Science, the framework models this span with its phi ladder. The golden ratio phi, about 1.618, appears throughout the framework's cost structure. The ratio hard/soft, about 1000, lands close to phi raised to the 14th power, which is about 843. The framework reads that as a 14-rung separation between soft and hard magnetic classes, each rung a factor of phi in coercivity.

The module itself proves only general facts about its cost function, not the magnetic claim. It defines a cost ledger, a discrete record of events, in terms of a ratio m/e, and proves three things: the cost is zero when m equals e, the cost is never negative for positive inputs, and the threshold phi minus 1.5 is positive. These are properties of the cost function, not of any specific material.

The research note attached to the module records the intended direction: define m and e in terms of a material's own properties, then the 14-rung separation would follow. That definition is not in the module. The machine-checked library proves the cost facts; the coercivity connection remains a research note, not a theorem.

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 does not prove that any real material's coercivity is phi^14 times another's. The 1000-to-843 comparison is a research note, not a measured or derived result. The cost function's nonnegativity is a property of the function, not a statement about magnetic stability.

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

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