Materials Porosity Wpcfrom Jcost

A module awaiting its subject

Wood-polymer composites blend wood fiber with plastic to make decking and siding that resist moisture. The standard engineering question is what fraction of wood gives the best stiffness and strength. The module named materials porosity wpcfrom jcost was meant to answer that with a cost function, but it does not. Its own research note records the plan: link the golden ratio to an optimal wood content around 62 percent. The formal content proves nothing of the sort.

The module defines a cost function, ledger in the framework's language, as J(m/e), where J(x) = (x + 1/x)/2 - 1. The theorems it proves are three general properties. First, the cost is zero when the two inputs are equal, so J(1) = 0. Second, the cost is never negative when both inputs are positive. Third, the golden ratio minus 1.5 is positive, which is just a number fact. These hold for any positive inputs, not for any particular composite recipe.

The file's own docstring says it plainly: the Lean code proves nothing specific to wood-polymer composites, because the cost is defined as J(m/e) without saying what m and e mean in material terms. The 50 to 70 percent wood content range cited in the notes, and the 62 percent guess from 1/φ, are research intentions, not derived results. The module is shared verbatim with 2383 sibling files, each one a template waiting for a subject-specific definition.

What the module does establish is a certificate, a packaged bundle of the three general facts, named WPCDensityCert. It shows the certificate is inhabited, meaning the three facts are consistent and provable. That is a real but modest achievement: a reusable template for any domain where a cost ratio makes sense. The framework's library can certify that the cost function has these properties, and that is all it certifies here.

The gap is explicit. To make this a theorem about wood-polymer composites, someone must define m and e in the composite's own terms, for example m as a stiffness measure and e as a density measure, and then prove the optimal wood fraction follows. Until that definition exists, the module is a scaffold with no building on it. The honest summary: a general cost template, machine-checked, awaiting its material science content.

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]

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

theorem cert_inhabited : Nonempty WPCDensityCert := ⟨cert⟩

What this page does not claim

The module does not prove any optimal wood content for wood-polymer composites. The 50-70 percent range and the 62 percent estimate are research notes, not formal results. The module does not establish a connection between the golden ratio and composite density.

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/Porosity_WPCFrom_JCost.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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