Materials Semiconductor3 Device From Jcost

The semiconductor module

A semiconductor is a material whose electrical conductivity sits between a conductor and an insulator, and the MOSFET, the metal-oxide-semiconductor field-effect transistor, is the switch that nearly every modern chip relies on. Its subthreshold swing measures how sharply the transistor turns off: the voltage change needed to cut the current by a factor of ten. At room temperature, physics sets a floor near 60 millivolts per decade, a limit that follows from the Boltzmann distribution of electrons.

The module named materials semiconductor3 device from jcost in the Recognition Science library does not prove that limit. It defines a cost function, a measure of how far a ratio of two numbers sits from the value one, as Jcost applied to the ratio m over e. The file then proves three general facts about that construction: the cost is zero when the two numbers are equal, the cost is never negative for positive inputs, and the golden-ratio constant phi minus 1.5 is positive. Those three facts are true for any positive real numbers m and e.

In Recognition Science, the framework models recognition events as costs that reality must pay, and the central theorem forces the specific form J(x) = (x + 1/x)/2 - 1 from five plain conditions. The semiconductor file reuses that cost function but never connects it to any physical quantity called m or e. The docstring records a research idea about the 60 mV/decade limit, with guesses like 68 mV/decade, but the Lean code itself contains no definition of m as a mass or e as a charge. The paragraph is a note about where the idea was meant to go, not a result.

What the module does establish, in plain language, is a small certificate: a packaged record showing that the cost function vanishes at equality, stays nonnegative, and has a positive threshold constant. That certificate is inhabited, meaning the three properties hold together. The file is shared verbatim with 2383 sibling modules, each one a template awaiting a subject-specific definition of its own m and e. Until such a definition appears, the module proves nothing about MOSFETs, semiconductors, or any device.

The honest takeaway is that the semiconductor page is a placeholder. The machine-checked library of formal theorems contains a correct but empty shell, and the physical bridge from the cost function to a real transistor remains open. A reader should not mistake the three proved facts for a derivation of the 60 mV/decade limit; that limit is a classical result, and the framework's connection to it is an unproven research note.

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 derive the 60 mV/decade subthreshold swing limit. The module does not define m or e in terms of any physical quantity. The module does not prove any property of a real semiconductor device.

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/Semiconductor3_Device_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.

Derived articles

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