Condensed Matter Hall Resistance Rs

Hall resistance and its RS module

The quantum Hall effect, discovered by Klaus von Klitzing in 1980, produces a resistance that is quantized in exact units. The von Klitzing constant R_K equals h/e^2, where h is the Planck constant and e is the elementary charge. Its measured value is 25812.807 ohms, a number now used as a standard for electrical resistance. In Recognition Science, a module named Hall_Resistance_RS was intended to connect this constant to the framework's cost function, but the machine-checked content proves something far more limited.

The module defines a cost function, cost (a measure of mismatch between two quantities), as J(m/e), where J is the framework's fundamental cost function and m and e are real numbers. The Lean theorems prove three general facts: this cost is zero when m equals e, it is never negative for positive inputs, and a certain threshold involving the golden ratio is positive. These are properties of the cost function itself, not of any physical system. The docstring states plainly that the module proves nothing specific to the Hall effect, because the definition does not reference any physical quantity.

The research note attached to the module records an intended direction: that the Hall resistance could be expressed as (2π/α) times R_K, where α is the fine-structure constant. It notes that α⁻¹ would lie in a narrow range around 137.03 to 137.04. But this paragraph is explicitly labeled as a note about where the idea was meant to go, not a result. The verified theorems establish only the general cost properties, which are shared verbatim across 2383 sibling modules in the library.

What the module does establish, in plain language, is that the framework's cost function behaves sensibly: zero cost when inputs match, nonnegative cost otherwise, and a positive threshold. These are mathematical facts about J, proved in the machine-checked library of formal theorems. They do not derive the Hall resistance, do not compute the fine-structure constant, and do not connect the cost function to any condensed matter system. The module is a template, not a physical derivation.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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

This module derives the quantum Hall resistance from first principles. The fine-structure constant is computed or derived within this module. The cost function is shown to have any physical meaning in condensed matter systems.

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/Hall_Resistance_RS.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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