Chemistry Nematic3 Order Param From Jcost

Nematic order from cost

Nematic liquid crystals are fluids whose rod-shaped molecules tend to point the same way. The order parameter S measures this alignment: it is 0 in the disordered liquid and approaches 1 for perfect parallel packing. At the phase transition, experiments on many materials find a critical value S_c near 0.4, a number that has resisted a simple universal explanation.

In Recognition Science, the framework models this transition using its central object, the cost function, a forced measure of how expensive a recognition event is. The framework's library defines a domain cost as J(m/e), where J(x) = (x + 1/x)/2 - 1, and m and e are two real numbers left unspecified. The module proves three general facts about this cost: it is zero when the two inputs are equal, it never goes negative for positive inputs, and the golden ratio minus 1.5 is positive. These are properties of the cost function itself, not of any particular chemical system.

The research note attached to the module records where the idea was meant to go: that the critical order parameter should be S_c = 1 - phi^(-3), which equals about 0.764, or alternatively S_c = J(phi)^(1/3), about 0.491. The first value overshoots the measured 0.4; the second lands closer but still above it. The note calls this structural, meaning the framework sees a reason for the transition to exist, but the exact match to experiment is not established.

What the module actually proves, in machine-checked form, is only the three general cost facts. The definition of m and e in terms of molecular properties, which would turn the research note into a theorem about nematics, is absent. The module is a template shared verbatim with 2383 sibling files, each one a placeholder for a subject-specific derivation that has not yet been written.

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 module does not prove any value for the nematic order parameter at transition. The research note's value S_c = 0.491 is not a theorem; it is an unverified identification. The module does not define m and e in terms of molecular properties.

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/Nematic3_Order_Param_FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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