Condensed Matter Mott Insulator U Rs

What the module actually proves

A Mott insulator is a material that should conduct electricity by the usual band theory but does not, because electrons repel each other too strongly to move freely. The key quantity is U/t: the energy cost U of putting two electrons on the same site divided by the hopping energy t that lets them move. When U/t is small, electrons hop and the material conducts; when it is large, they stay pinned and the material insulates. The transition between the two is the Mott transition.

In the framework called Recognition Science, the transition is identified with the golden ratio cubed, phi^3 = 4.24. The empirical range for three dimensions is U/t roughly 3 to 5, so 4.24 sits inside that range. That identification is a research note, not a theorem: the machine-checked library proves no statement specific to Mott insulators.

What the module does prove is three general facts about a cost function J(x) = (x + 1/x)/2 - 1. First, J(1) = 0, so the cost vanishes when the two arguments are equal. Second, J(x) is nonnegative for positive x. Third, phi - 3/2 > 0, a small positivity fact. These are universal statements about the cost function, stated once in a shared template and reused here; they apply to any subject whose domain cost is defined as J(m/e).

The module defines domainCost(m, e) = J(m/e) and proves the three facts for that definition, but it never defines what m and e mean for a Mott insulator. To turn this into a theorem about the Mott transition, one would need a definition of m and e in the subject's own terms, say m as U and e as t. Without that, the module is a template instantiated, not a physical result.

The honest takeaway: the framework provides a general cost function with clean formal properties, and the golden-ratio threshold is a plausible identification awaiting a subject-specific definition. The machine-checked part is the cost function's behavior, not the 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

The module does not prove that the Mott transition occurs at U/t = phi^3. The module does not define U or t in terms of recognition cost. The golden-ratio threshold is a research note, not a machine-checked 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/CondensedMatter/Mott_Insulator_U_RS.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.

Derived articles

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