Materials Polymer Blend3 From Jcost

Polymer blend3 from J-cost

Recognition Science (the framework that treats physical law as the forced cost of maintaining a discrete record of events) defines a cost function J(x) = (x + 1/x)/2 - 1. The module Polymer_Blend3_FromJCost applies this cost to the ratio of two real numbers, m and e, by setting domainCost(m, e) = J(m / e). The three proved facts are general properties of J, not polymer-specific results: the cost vanishes when the two inputs are equal, it is nonnegative for positive inputs, and the constant phi - 3/2 is positive. The module proves nothing about polymer blends, because the definitions of m and e never connect to any physical quantity such as polymer chain length or interaction energy.

The name and the research note suggest a goal: polymer blend miscibility, where two polymers mix when a free-energy cost falls below a threshold. The note records an intended criterion, J(chi * N) < J(phi), with the Flory-Huggins critical value chi * N = 2. But the Lean code does not define chi or N, does not prove any inequality involving them, and does not establish miscibility. The three theorems hold for any positive real numbers m and e, so they would apply equally to prices, masses, or temperatures. The module is a template, shared verbatim with 2383 sibling modules, that instantiates the cost function without tying it to the subject named in the file.

The honest summary: the module proves a small, correct, but empty shell. It establishes that the cost function has three desirable properties (zero at equality, nonnegativity, a positive threshold), and it packages those into a certificate structure. What it does not establish is any statement about polymers. The research note is a plan, not a result. A future theorem about miscibility would need a definition of m and e in polymer terms, such as chain length and interaction parameter, before the cost function could say anything about blending.

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

This module does not prove that any polymer blend is miscible or immiscible. The research note's miscibility criterion is not a theorem in the Lean library. The positive threshold phi - 3/2 is not shown to relate to any critical interaction strength.

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/Polymer_Blend3_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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