Condensed Matter Cuprate Tc From Phi Ladder Cuprate Tc Cert

What the certificate proves

In the Recognition Science framework, a ledger is a discrete record of events, and the cost of recognition is a number that measures how far a measured value sits from an expected one. The declaration CuprateTcCert packages three such facts together. First, when the measured value equals the expected value, the cost is exactly zero. Second, for any two positive numbers, the cost is never negative. Third, the constant φ − 3/2 is positive, where φ is the golden ratio. These are the only three claims the certificate makes.

The certificate is a structure, a formal container, and it is inhabited: a machine-checked library of formal theorems provides a proof that the structure exists. The proofs rely on the general cost function J(x) = (x + 1/x)/2 − 1, which is defined for any positive ratio. The certificate itself is not a theorem about cuprates. It is a template, shared verbatim with 2554 sibling modules, that states facts about the cost function in the abstract.

What the certificate does not claim is the interesting part. It does not say that the optimal doping of a high-temperature superconductor is 0.16, nor that the maximum critical temperature follows from the golden ratio. Those ideas appear in a research note above the code, but the formal proof never defines the measured mass or the expected mass in terms of copper-oxide planes, phonons, or any physical quantity. The note records where the research was meant to go, not a result that has been established.

In plain terms, the certificate proves a small piece of mathematics about a cost function, and it leaves the physics entirely untouched. A reader who wants a theorem about cuprate superconductors must look elsewhere, or must supply the missing definitions that connect the cost function to a real material.

theorem domainCost_at_equilibrium (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]

What this page does not claim

The certificate does not prove that the optimal doping of a cuprate superconductor is 0.16. The certificate does not establish any relation between the golden ratio and the maximum critical temperature of a cuprate. The certificate does not define the measured or expected mass in terms of any physical quantity.

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/CondensedMatter/CuprateTcFromPhiLadder.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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