Condensed Matter Mott Transition From Jcost Mott Transition Cert

What the certificate proves

The Mott transition is a classic problem in condensed matter physics: certain materials switch from conducting to insulating when the repulsion between electrons becomes strong compared with their ability to hop between atoms. The standard criterion, named after Nevill Mott, compares the Hubbard on-site repulsion U with the bandwidth W, and predicts a transition when U/W exceeds about 1. In the Recognition Science framework, the declaration MottTransitionCert packages three machine-checked facts about a cost function, not about electrons in a crystal.

The cost function here is domainCost, defined as J(m/e), where J is the framework's recognition cost and m and e are two real numbers standing for 'measured' and 'expected'. The certificate records three properties. First, when the measured value equals the expected value, the cost is exactly zero. Second, for positive inputs, the cost is never negative. Third, the number phi minus 3/2, called the canonical threshold, is positive. These three facts are proved in the machine-checked library of formal theorems, and the certificate simply bundles them into one structure.

What the certificate does not do is more important than what it does. The framework's research note imagines a Mott transition at U/W = phi, the golden ratio, where J(phi) is about 0.118. But the formal definition of domainCost never mentions U, W, electrons, or any material property. The m and e in the definition are just real numbers. The note itself says the module proves nothing specific to this subject, because the cost is defined without reference to one. The certificate is a template, shared verbatim with thousands of sibling modules, waiting for a definition of m and e in a real physical context.

In plain terms, the certificate establishes a small, general fact about a mathematical function: a particular cost measure vanishes at equality, stays nonnegative, and has a positive threshold constant. It does not establish that any real material undergoes a Mott transition at the golden ratio, nor that the framework's J-cost describes Hubbard physics. Those remain research targets, not proved results.

theorem cert_inhabited : Nonempty MottTransitionCert := ⟨cert⟩

theorem domainCost_at_equilibrium (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 certificate does not prove that any real material undergoes a Mott transition at U/W = phi. The certificate does not define m and e in terms of Hubbard U or bandwidth W. The certificate does not establish that J-cost describes any condensed matter system.

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/MottTransitionFromJCost.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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