Materials Structural Materials Mod93

A shared template, not a materials theorem

Structural materials mod93 is a module in the Recognition Science framework's machine-checked library of formal theorems. Its name suggests it should say something about materials. The file's own status note is blunt: it proves three general facts about the cost function J(x) = (x + 1/x)/2 - 1, and nothing specific to materials. The module defines a domain cost as J(m/e), the cost of a ratio between two quantities m and e, but never defines what m and e mean for materials. Without that definition, the module is a placeholder, not a result about its subject.

The three proved facts are elementary. First, when m equals e, the cost is zero: J(1) = 0. Second, for positive m and e, the cost is never negative. Third, the number phi - 3/2 is positive, where phi is the golden ratio (about 1.618). These follow directly from the general properties of J, which the framework proves for any positive input. The module packages them into a certificate structure and shows that certificate exists, but the certificate is identical in form to what 2383 sibling modules carry for other subjects.

The docstring records a research intention: a prediction for materials at recognition rung 93. But the Lean code does not implement that intention. The shared content lives once, universally quantified, in a template module called DomainCostTemplate; this materials module imports that template and re-exports it under a new name. What would make this a theorem about materials is a definition of m and e in materials' own terms, such as a stress measure divided by a strength measure. That definition does not appear here.

In plain language: the module is a shell. It proves that a certain cost function behaves well on positive numbers, which is true for any subject, and it names a threshold involving the golden ratio. It does not prove anything about alloys, composites, fracture, or any physical material property. The framework's own note says the paragraph above the code is a research note recording where the idea was meant to go, not a result. A reader who wants a materials theorem must look for a module that actually defines materials quantities; this one does not.

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)

noncomputable def cert : StructMaterialsM93Cert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

This module proves any property of actual structural materials. The golden-ratio threshold has any physical interpretation for materials in this module. The research note about rung 93 is a proved result.

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/Structural_Materials_mod93.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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