Materials Topological Insulator3 From Jcost

A module with a placeholder name

A topological insulator is a material that conducts electricity on its surface but not through its interior, a property protected by quantum mechanics. The module materials topological insulator3 from jcost in the Recognition Science library carries that name, but its Lean code does not define a band structure, an electron Hamiltonian, or a topological invariant. The file's own docstring states the situation plainly: the Lean below proves three general facts about a cost function, and proves nothing specific to this subject.

The central definition is a cost, a number that measures the price of a recognition event, here written as domainCost (m e) = Jcost (m / e). The three proved facts are that this cost is zero when the two inputs are equal, that it is never negative for positive inputs, and that a constant called canonicalThreshold, defined as phi minus 3/2, is positive. Each of these is a theorem about the abstract cost function J, not about any material. The code packages these three facts into a structure named TopoInsulator3v2Cert and proves that structure is inhabited, meaning the facts are consistent.

The name and a research note attached to the file point to an intended application. The note predicts a topological phase transition when the cost of a band gap relative to the Fermi energy crosses a threshold set by the golden ratio phi. That prediction is a hypothesis, not a theorem, and it is not what the Lean code establishes. The code's own documentation says the content is shared verbatim with 2383 sibling modules, and the real theorem is stated once in a template file.

What the module does establish, in plain language, is a small set of properties of the cost function J: it vanishes at equality, it stays nonnegative on positive inputs, and one particular constant is positive. What it does not establish is any fact about topological insulators. The gap between the file's name and its content is not a hidden flaw; it is documented in the file itself. The module is a placeholder, a scaffold on which a real materials theorem could be built if someone defined what m and e mean for a specific crystal.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

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 canonicalThreshold : ℝ := phi - 3 / 2

What this page does not claim

This module proves any fact about topological insulators, band gaps, or electron behavior. The topological phase transition predicted in the research note is a derived theorem. The golden ratio threshold J(phi) has been verified against experimental data.

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/TopologicalInsulator3FromJCost.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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