Condensed Matter Mott Insulator U Rs Mott Insulator Rs

The formal declaration

A Mott insulator is a material that should conduct electricity according to simple band theory but does not, because electrons repel each other too strongly to move freely. The standard model for this behavior is the Hubbard model, whose key ratio is U/t: the energy cost U of two electrons sharing a site, divided by the hopping strength t that lets them move. In three dimensions, experiments and numerical simulations place the transition from metal to Mott insulator at roughly U/t between 3 and 5.

In Recognition Science, the declaration MottInsulatorRS packages three statements about a function called domainCost, a cost function that measures the price of recognition between two quantities. The three statements are: the cost is zero when the two quantities are equal, the cost is never negative for positive inputs, and a certain threshold built from the golden ratio is positive. Each of these is proved in the machine-checked library of formal theorems, and the declaration simply bundles them together.

What the declaration does not do is connect those statements to the Mott transition. The number phi^3, about 4.24, appears in a research note attached to the module as the predicted value of U/t at the transition. That note is not a theorem. Nothing in the formal code defines what m and e mean for a Hubbard model, so the cost function has no physical content specific to Mott insulators. The note is a record of where the idea was meant to go, not a result about where it arrived.

This is a deliberate and honest boundary. The framework's power is that it proves general facts about cost functions from first principles. Applying those facts to a specific physical system requires defining the system's quantities in the framework's terms. Until someone writes down what m and e are for a Hubbard model, the declaration remains a template, not a theory of Mott insulators.

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]

structure MottInsulatorRS where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

noncomputable def cert : MottInsulatorRS where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The declaration does not prove that U/t equals phi^3 for any real material. The declaration does not define what m and e mean for a Hubbard model. The research note is not a theorem and carries no formal weight.

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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