Physics Magnon Dispersion2

Magnon dispersion in the framework

In a ferromagnet, a magnon is a quantized collective excitation of the spin lattice, and its energy depends on the wavevector through a stiffness constant D. For iron, the measured stiffness is about 370 meV·Å². The Recognition Science framework models this stiffness as D = J(φ) · k_B · T_Curie · a² / ħ², where T_Curie is the Curie temperature (1043 K for iron), a is the lattice constant (2.87 Å), and J is the universal cost function derived in the framework. The result is order-of-magnitude consistent with the measured value.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 measures the forced cost of recognition when a ratio deviates from unity. The module defines a domain cost as J(m/e), the cost of a ratio between two quantities m and e. The machine-checked library of formal theorems proves three general facts about this cost: it vanishes when m = e, it is nonnegative for positive inputs, and the threshold φ - 3/2 is positive. These are universal properties of J, not results specific to magnons.

What the module does not do is connect m and e to any physical magnon quantity. The definition of domainCost is just J(m/e) with no reference to spin, wavevector, or stiffness. The docstring states plainly that the physics is a research note, not a theorem. To make this a theorem about magnons, one would need to define m and e in terms of magnon variables. That step is not present.

So the honest summary: the framework provides a candidate formula for the magnon stiffness that matches experiment to order of magnitude, and the formal proof establishes only the general properties of the cost function. The physical identification of m and e remains open.

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

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

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]

What this page does not claim

This module proves a theorem about magnon physics. The stiffness formula is derived from the framework's axioms. The order-of-magnitude agreement is a precise measurement match. The quantities m and e are defined in physical terms.

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/MagnonDispersion2.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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