Physics Hall Conductance From Jcost

Hall conductance and the cost ledger

The integer quantum Hall effect is a precise phenomenon: a two-dimensional electron gas in a strong magnetic field conducts electricity in exact integer multiples of e²/h, where e is the electron charge and h is Planck's constant. The integer n is the filling factor, and each plateau in the conductance corresponds to a quantized step. This is one of the most accurate measurements in physics, and it is completely classical in its description: σ_xy = n × e²/h.

The framework of Recognition Science starts from a different question. It asks what happens when a physical quantity is compared to its own unit. The framework defines a cost function J(x) = (x + 1/x)/2 - 1, which measures the mismatch between a value and its reference. For the Hall conductance, the module defines the cost of the ratio σ_xy / (e²/h). The central result is that this cost vanishes exactly when the ratio equals one, which is the n = 1 plateau. At that point, the conductance is exactly e²/h, and the cost function returns zero.

The machine-checked library of formal theorems proves three general facts about this cost. First, the cost is zero when the measured value equals the reference value: J(1) = 0. Second, the cost is nonnegative for all positive inputs, so it never goes below zero. Third, a threshold constant φ - 3/2 is positive, where φ is the golden ratio. These three facts are assembled into a certificate structure that is inhabited, meaning the proof exists. The module proves these facts in full generality, but it does not prove that any particular physical system realizes them.

In Recognition Science, the framework models each plateau as one φ-rung. The idea is that the recognition threshold n = 1 is the fundamental step, and each additional plateau corresponds to another rung on a ladder. The framework's library shows that the cost function forces the golden ratio as a natural scale, and the Hall conductance module connects that scale to the quantized steps of the quantum Hall effect. The module itself establishes only the general cost facts, not the physical identification of m and e in the Hall system's own terms.

What this means in practice is a bridge between two languages. The classical language says conductance is quantized. The framework's language says the cost of recognition is zero at the threshold. The module proves the cost facts; the physical mapping remains a research note. A reader who wants the full derivation must wait for a definition of m and e in the Hall system's own terms, which the module does not provide.

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 QHConductanceCert := ⟨cert⟩

What this page does not claim

The module does not prove that any physical system realizes the cost facts. The module does not derive the integer quantum Hall effect from first principles. The module does not identify m and e in the Hall system's own terms.

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/Physics/HallConductanceFromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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