Materials Rs Matl Module 009

What Module 009 proves

Materials RS Module 9 is a file in the Recognition Science framework's machine-checked library of formal theorems. Its subject line names a specific physics claim: that a Cooper pair binding energy in lead, computed as phi cubed times 0.642 meV, equals 2.72 meV, and that this matches measurement. The file's own documentation, however, states plainly that this paragraph is a research note recording where the idea was meant to go, not a result. The formal content proves nothing specific to lead or to superconductivity.

What the module actually establishes is a small set of general properties of a function called domainCost, defined as J(m/e) where J is the framework's forced cost function and m and e are positive real numbers. The three proved facts are: domainCost equals zero when m equals e; domainCost is nonnegative for all positive inputs; and the number phi minus 3/2 is positive. A fourth theorem packages these into a certificate structure and shows that certificate is inhabited. All four are formally verified in the Lean proof assistant, with no axioms beyond the standard three.

In plain language, the module proves that a certain cost function vanishes when two quantities are equal, never goes negative when both inputs are positive, and that a threshold constant built from the golden ratio is positive. These are true for any positive real numbers m and e. The module does not define m and e in terms of lead, electrons, or phonons, so the Cooper pair calculation in the research note is not a theorem of the framework. The docstring says this body is shared verbatim with 2383 sibling modules; the content is stated once, universally quantified, in a template file.

What the module does contribute is a template for how a subject-specific claim could become a theorem: one would need a definition of m and e in that subject's own terms. Until that definition exists, the module remains a general statement about the cost function, not a result about any material. The three general facts it proves are consistent with the framework's broader forcing chain, but they do not add new physics.

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 Cooper pair binding energy calculation is not a proved result. The module does not establish any property specific to lead or any other material. The three general facts are not new physics beyond the framework's existing cost function theorems.

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/RS_Matl_Module_009.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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: