Condensed Matter Anderson Localization From Jcost

The threshold claim

Anderson localization is the physics phenomenon where electrons stop conducting because disorder traps them, turning a metal into an insulator. In 1958, Philip Anderson showed that random impurities can halt diffusion entirely, a result that reshaped condensed matter physics. The framework of Recognition Science (RS) attempts to connect this idea to its central object, the cost, a forced measure of how far a measured ratio sits from its expected value. The module under review defines a cost on the ratio of measured to expected conductance, and proves general facts about that cost, but it does not yet prove that Anderson localization follows from it.

The classical story is well established. Anderson's original paper considered a lattice with random site energies and found that beyond a critical disorder strength, the wavefunction becomes exponentially localized. The conductance drops to zero even though the material is not a band insulator. This metal-insulator transition is sharp and universal in one dimension and in many disordered systems, though the precise critical exponents in three dimensions remain a subject of active research. The standard tool is the scaling theory of localization, which describes how conductance changes with system size.

In Recognition Science, the cost function is defined as J(x) = (x + 1/x)/2 - 1, which is zero when x equals 1 and grows as the ratio moves away from unity. The module defines domainCost as J(measured / expected), and the Lean code proves three things: the cost is zero when measured equals expected, the cost is nonnegative for positive inputs, and the quantity phi - 3/2 is positive, where phi is the golden ratio. These are the only results the machine-checked library establishes. The docstring itself states plainly that it proves nothing specific to Anderson localization, because the definition of domainCost does not reference the physics.

The intended connection, recorded as a research note, is that the metal-insulator transition occurs at a threshold cost of J(phi) on the conductance ratio. The number phi - 3/2, approximately 0.118, would be the critical disorder strength in this picture. But for this to become a theorem, the module would need a definition of measured and expected conductance in terms of the physical disorder, which it does not have. The threshold is a hypothesis, not a derivation.

What the module does establish is a clean template: a cost that vanishes at equilibrium and is nonnegative everywhere is a sensible starting point for any ratio-based theory. The three proved facts are general properties of J itself, applied to a ratio. They hold for any positive measured and expected values, not just for conductance. The Anderson localization claim would require a separate argument that the conductance ratio actually follows this cost law, and that the threshold at phi is where localization sets in.

The honest verdict is that this page is a placeholder. The machine-checked part is real but generic, and the physics remains a falsifiable prediction. The structural falsifier is empirical data outside the RS prediction band, meaning that if experiments show localization at a different disorder strength, the hypothesis fails. Until the module defines conductance in physical terms, the Anderson localization threshold is a target, not a result.

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]

theorem cert_inhabited : Nonempty AndersonLocCert := ⟨cert⟩

What this page does not claim

Anderson localization is proven to follow from the cost function. The threshold phi - 3/2 is derived from the physics of disorder. The module establishes any connection between the cost and the Anderson transition beyond a research note.

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/AndersonLocalizationFromJCost.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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