Thermodynamics Kibble Zurek From Jcost

The cost of a domain

The Kibble-Zurek mechanism describes how defects form when a system is driven through a phase transition quickly. As a system cools through its critical point, regions that should order in the same way cannot communicate fast enough, so they freeze into different states. The density of the resulting defects, like cracks in a crystal or vortices in a superfluid, follows a power law in the quench rate. The framework of Recognition Science attempts to derive this defect density from its core cost function.

In Recognition Science, the cost of recognition, a measure of the forced expense of matching a new event to an existing pattern, is fixed by a proved theorem to be J(x) = (x + 1/x)/2 - 1. The module under question defines a domain cost as the cost evaluated on the ratio of two quantities, m and e, which are meant to represent competing scales during a transition. The machine-checked library of formal theorems proves three facts about this domain cost: it is zero when the two scales are equal, it is never negative for positive scales, and a specific threshold value, phi minus 3/2, is positive. These are the only results the file establishes.

The library file itself is explicit that it proves nothing specific to thermodynamics. The definition of domain cost does not reference any physical quantity, so the theorems hold for any positive real numbers m and e. The intended connection to the Kibble-Zurek mechanism, where m and e would be defined in terms of quench time and relaxation time, is recorded in a research note. The note states a hope that the correlation length exponent nu equals 1/phi at the RS critical point, and that defect density scales as phi cubed per unit volume, but these are not derived results.

What the module does establish, in plain language, is a minimal consistency check. It shows that the cost function, when applied to a ratio of scales, behaves sensibly: equal scales cost nothing, and unequal scales cost a positive amount. The positive threshold phi minus 3/2, approximately 0.118, is a number that emerges from the golden ratio, but the file does not connect it to any measurable transition property. The gap between the proved facts and the intended physics is the entire subject of the research note.

The consequence for a reader is a clear picture of where the framework stands. The core cost function is proved, but its application to phase transitions is not. The module is a template, not a theorem about Kibble-Zurek. To become a result, someone must define m and e in terms of the transition's own parameters, such as quench rate and relaxation time, and then prove the defect density law from the cost. Until that definition exists, the physics remains an open target, not a derivation.

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]

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

What this page does not claim

The Kibble-Zurek mechanism is derived from the cost function. The defect density phi cubed per unit volume is a proved result. The module establishes any specific thermodynamic prediction.

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

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