Chemistry Crosslink Density3 From Jcost

Cross-link density and its formal scaffold

Cross-link density is a standard quantity in polymer science: the number of covalent bridges that tie polymer chains into a network, per unit volume. It is what turns a viscous melt into a solid rubber, and it is measured in bridges per cubic meter, typically in the range 10^20 to 10^22 m^-3. A rubber with more cross-links is stiffer and less able to flow; a rubber with too few is a weak gel. The quantity is defined by chemistry, by the reaction that forms the bridges, not by any single formula.

In Recognition Science, the framework's library of machine-checked formal theorems contains a module named for this subject. The module defines a cost-like quantity domainCost, a measure of mismatch between two real numbers, as the framework's cost function applied to their ratio. It proves three general facts about that quantity: it is zero when the two numbers are equal, it is never negative when both inputs are positive, and a certain threshold built from the golden ratio is positive. These are true statements about real arithmetic, and the proofs are checked by the machine.

The module proves nothing about actual cross-link density. Its definition of domainCost never mentions polymer molar volume, Avogadro's number, or any chemical measurement. The research note inside the module records an intended direction, a formula relating cross-link density to the golden ratio and molar volume, but that formula is not stated as a theorem and no chemical meaning is attached to it. The same formal skeleton appears verbatim in 2383 sibling modules, each named for a different scientific subject, each proving the same three arithmetic facts.

What the module does establish, in plain language, is a reusable formal pattern: a cost function that vanishes at equality, stays nonnegative, and has a positive golden-ratio threshold. That pattern is a candidate scaffold for a future chemical theory, but the bridge from the scaffold to the chemistry is open. A theorem about cross-link density would require defining the two numbers in chemical terms, molar volume and something like a count of bridges, and then proving the relationship. Until that definition exists, the module is a placeholder, not a result.

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)

theorem cert_inhabited : Nonempty CrosslinkDens3Cert := ⟨cert⟩

What this page does not claim

The module does not derive or predict any measured cross-link density value. The golden-ratio threshold phi - 3/2 has no established chemical meaning in this module. The intended formula rho_X = phi^k * (N_A / V_mol) is a research note, not a proved statement.

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/Chemistry/Crosslink_Density3_FromJCost.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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