Materials Thermoelectric Zt3 From Jcost

Thermoelectric figure of merit

The thermoelectric figure of merit, usually written zT, is a dimensionless number that measures how efficiently a material converts heat into electricity. A good thermoelectric material has a high zT: it produces more voltage for a given temperature difference, and it wastes less heat in the process. The best known materials reach zT values around 1 to 2, and a zT of 3 or higher has long been a target for practical devices like waste-heat generators and solid-state coolers. The figure depends on three competing properties: electrical conductivity, Seebeck coefficient, and thermal conductivity. Raising one usually lowers another, which is why high zT is hard to achieve.

The standard formula is zT = S²σT/κ, where S is the Seebeck coefficient, σ is electrical conductivity, T is absolute temperature, and κ is thermal conductivity. The Seebeck coefficient measures the voltage produced per degree of temperature difference; electrical conductivity measures how easily charge flows; thermal conductivity measures how easily heat flows. A good thermoelectric material must conduct electricity well but heat poorly, a combination that is rare in nature. The temperature T also matters: zT is not constant, it peaks at some optimal temperature T_opt, and the peak value is what engineers usually quote. Finding materials with high peak zT, and at temperatures that match real applications, is an active research area.

In Recognition Science, the framework models this optimization problem with its cost function. The module defines a cost J(m/e), where m and e are two positive real numbers, and it proves three general facts about that cost: it is zero when m equals e, it is never negative when both inputs are positive, and a certain threshold value phi minus 1.5 is positive. The first fact says the cost vanishes when the two quantities are balanced; the second says the cost never rewards imbalance in the negative direction; the third identifies a positive number that could serve as a threshold. These are the only theorems the module proves.

The module does not prove anything specific to thermoelectrics. The cost function is defined as J(m/e) without any reference to Seebeck coefficient, electrical conductivity, or thermal conductivity. The research note attached to the module records an intended direction: peak zT at T_opt = phi times 800 K, or more generally T_opt = phi^k times 300 K. Those are structural guesses, not results. The module itself establishes only the three general facts about the cost function, which hold for any positive m and e. What would make the module a theorem about thermoelectrics is a definition of m and e in terms of the material properties, and that definition does not appear in the file.

The practical consequence is a caution about reading too much into a formal file. The machine-checked library proves exactly what its declarations say, and here they say nothing about zT. The three proved facts are true for any positive inputs, so they are true of the thermoelectric cost only if someone later supplies the missing definitions. Until then, the module is a template: a correct shell around a cost function, waiting for the physical content that would make it a thermoelectric result.

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

No claim that the module proves a peak zT of 3 for any material. No claim that the research note's T_opt formula is a theorem. No claim that the cost function is derived from thermoelectric physics.

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/Thermoelectric_ZT3_FromJCost.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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