Condensed Matter Hall Resistance Rs Hall Resistance Cert

The certificate's actual scope

The quantum Hall effect produces a resistance, the Hall resistance, that is quantized in units of h/e², the von Klitzing constant, measured as 25812.807 ohms. The Recognition Science framework's declaration HallResistanceCert does not prove that this resistance equals that value. It proves three general facts about a function called domainCost, which is defined as Jcost(m/e), where Jcost is the framework's cost function and m and e are positive real numbers.

The three facts are: domainCost is zero when its two inputs are equal; domainCost is never negative for positive inputs; and a threshold constant, phi minus 3/2, is positive. The certificate is a structure that bundles these three theorems together, and the framework's library shows the structure is inhabited, meaning a certificate exists. The proofs rely only on the general properties of Jcost, not on any physics of the Hall effect.

The docstring attached to the file records a research note, an intention to connect this cost to the fine-structure constant and the von Klitzing constant, but the note itself says this is where the idea was meant to go, not a result. The file is one of 2383 siblings that share the same body, each a template applied to a different subject. What would make this file a theorem about the Hall effect is a definition of m and e in terms of the Hall effect's own quantities, which is absent.

In plain terms, the certificate is a machine-checked proof of three harmless inequalities about a cost function. It does not establish the value of the Hall resistance, nor does it connect the cost function to the fine-structure constant. The framework's library proves general facts; this particular file proves nothing specific to condensed matter.

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]

theorem cert_inhabited : Nonempty HallResistanceCert := ⟨cert⟩

What this page does not claim

The certificate does not prove the Hall resistance equals h/e² or any other specific value. The certificate does not connect the cost function to the fine-structure constant or the von Klitzing constant. The certificate does not establish any physical property of the quantum Hall effect.

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)

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.

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