Condensed Matter Mott Transition From Jcost

The Mott transition

A Mott transition is a sudden switch in a solid from conducting to insulating behavior, driven by the mutual repulsion of electrons rather than by a band gap. In the standard picture, each electron sits in a lattice site, and the key competition is between the energy cost U of placing two electrons on the same site and the bandwidth W, the range of energies an electron can gain by hopping. When U is small compared with W, electrons move freely and the material conducts; when U dominates W, motion becomes too expensive and the material turns insulating. The textbook criterion for the switch is U/W greater than about 1, a threshold first clarified by Nevill Mott in the 1930s and refined through the Hubbard model in the 1960s.

Recognition Science, a framework that treats reality as maintaining a discrete record of events, offers a different route to a threshold. Its central object is a cost function J(x) = (x + 1/x)/2 - 1, proved in a machine-checked library of formal theorems to be the unique cost satisfying five plain conditions. The framework models the Mott ratio U/W as an input to this cost, writing the cost as J(U/W). The golden ratio φ ≈ 1.618 emerges as the framework's canonical one-step departure: at U/W = φ, the cost J(φ) ≈ 0.118 marks the transition quantum, the point where the framework says the insulator begins. This is a definitional choice, not a derived physical law.

What the module itself establishes is deliberately modest. Its machine-checked theorems prove three general facts about the cost function applied to a ratio: the cost vanishes when the measured value equals the expected value, the cost is never negative for positive inputs, and the threshold φ - 3/2 is positive. The module defines a certificate bundling these three facts, and proves the certificate exists. It proves nothing specific to Mott physics, because the ratio U/W is plugged in without a derivation from electron interactions. The research note records the intended direction, not a result.

The honest takeaway is that the framework offers a candidate threshold at U/W = φ, but the physical bridge from electron repulsion to that specific ratio remains open. The module is a template, shared verbatim with 2554 sibling modules, waiting for a definition of U and W in the subject's own terms. What a reader can now see is the shape of the claim: if the cost function is the right price for recognition, then the Mott transition should occur at the golden ratio, and the framework's library has checked the cost's basic properties but not the physics.

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]

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

def canonicalThreshold : ℝ := phi - 3 / 2

def domainCost (measured expected : ℝ) : ℝ := Jcost (measured / expected)

What this page does not claim

The physical Mott transition occurs at U/W = φ. The module derives the Hubbard U or bandwidth W from first principles. The framework's cost function is the only possible price for electron repulsion.

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)

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Derived articles

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