Physics Topological Insulator3 From Jcost

The certificate module

A topological insulator is a material that conducts electricity on its surface but not through its interior. The surface conduction is carried by special states called Dirac cone states, and in a simple cubic crystal the six faces pair up into three opposite directions. The module Topological_Insulator3_FromJCost records an intended link between that three-direction count and the Recognition Science cost function, but the link is not yet made.

The module defines cost, a measure of recognition effort, as the function J applied to a ratio of two scales m and e. It proves three general facts. First, the cost is zero when the two scales are equal. Second, for positive scales the cost is never negative. Third, the golden ratio minus 3/2 is positive, which gives a threshold value. These facts are proved in the machine-checked library of formal theorems, and the module packages them into a certificate structure named TI_Surface3Cert.

In plain language, the certificate says: if the two scales match, the cost is zero; if both scales are positive, the cost stays nonnegative; and a specific threshold number is positive. The certificate is inhabited, meaning the facts are actually proved, not merely stated. The name suggests a connection to three surface state directions, but the module itself does not define what m and e mean for a topological insulator.

What the module does not do is establish the physical claim. The docstring is explicit: the module proves nothing specific to topological insulators, because the cost is defined without reference to the material. The intended step, defining m and e in the subject's own terms, remains open. The three proved facts are universal properties of the cost function, shared verbatim with 2383 sibling modules.

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]

theorem cert_inhabited : Nonempty TI_Surface3Cert := ⟨cert⟩

What this page does not claim

The module does not prove that a topological insulator has three surface state directions. The module does not define m and e in terms of any material property. The physical recognition-to-linking bridge for surface states is not established.

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/Topological_Insulator3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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