Materials Electrical Conductance From Jcost

Conductance and its cost

Electrical conductance, usually written G, measures how easily electric current passes through a material. The standard quantum picture, the Landauer-Büttiker formalism, treats a conductor as a channel with a transmission probability T. The conductance is then G = (2e²/h) × T, where e is the electron charge and h is Planck's constant. The factor 2e²/h is the conductance quantum, about 77.5 microsiemens, and T ranges from 0 (no transmission) to 1 (perfect transmission). A perfect single channel therefore conducts exactly one quantum of conductance.

The Landauer-Büttiker formula emerged from work by Rolf Landauer in 1957 and Markus Büttiker in the 1980s. It connects the macroscopic, measurable property of conductance to the microscopic, probabilistic idea of transmission. The formula is central to mesoscopic physics, the study of devices small enough that quantum effects dominate, such as quantum point contacts and nanowires. In those systems, conductance is quantized in integer multiples of 2e²/h when transmission is perfect.

In Recognition Science, the framework models the cost of a recognition event with a specific function J(x) = (x + 1/x)/2 - 1. The framework's library proves that any cost function meeting five plain conditions must equal this J. The framework then applies J to the ratio of two quantities, m and e, in the module for electrical conductance. The definition is simple: domainCost(m, e) = J(m / e). The framework proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the number phi - 3/2 is positive. The number phi is the golden ratio, about 1.618, so phi - 3/2 is about 0.118.

In Recognition Science, the framework models the cost of a recognition event with a specific function J(x) = (x + 1/x)/2 - 1. The framework's library proves that any cost function meeting five plain conditions must equal this J. The framework then applies J to the ratio of two quantities, m and e, in the module for electrical conductance. The definition is simple: domainCost(m, e) = J(m / e). The framework proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the number phi - 3/2 is positive. The number phi is the golden ratio, about 1.618, so phi - 3/2 is about 0.118.

The framework's research note suggests a specific identification: at a threshold, the transmission T equals J(phi), which is approximately 0.118. The conductance would then be J(phi) × (2e²/h), about 0.118 times the conductance quantum. This is a hypothesis about what the threshold transmission should be, not a derived result. The module itself proves only the three general facts about J, and its own docstring states that it proves nothing specific to electrical conductance because the quantities m and e are not defined in the subject's own terms.

What the module does establish, in plain language, is a small set of formal facts about the cost function. It proves that the cost of a perfect match is zero, that cost is never negative, and that the golden-ratio threshold is a positive number. These are the building blocks the framework would need if a future definition linked m and e to physical quantities. The research note records where the idea was meant to go, but the proof stops at the general cost properties. The step from cost to conductance remains a target, not a theorem.

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

The module does not prove that electrical conductance equals J(phi) times the conductance quantum. The module does not define what m and e physically represent. The framework does not derive the Landauer-Büttiker formula from its cost function.

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/Materials/ElectricalConductanceFromJCost.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: