Thermodynamics Bosonization From Jcost

The bosonization template

Thermodynamics bosonization is a proposed bridge between two ways of describing the same physical system. In one picture, particles called fermions carry the physics; in the other, fields called bosons do. The classical result, known since the 1970s, is that in one spatial dimension these two descriptions can be exactly equivalent: a theory of interacting fermions can be rewritten as a theory of free bosons. The mapping is called bosonization, and it is a workhorse of condensed matter physics.

In Recognition Science, the framework models this duality through its cost function. The cost function, written J(x), measures the price of a recognition event, a discrete record of something happening. The framework proves that J(x) must equal (x + 1/x)/2 - 1. The proposed thermodynamic version defines a domain cost as J(m/e), where m and e are two energies. The idea is that the ratio of two energies, fed through the cost function, should act as the canonical coupling in the bosonized theory.

What the module actually establishes is far narrower, and the module itself says so. Its docstring states plainly: it proves three general facts about J(m/e), and nothing specific to bosonization. Those facts are that the cost vanishes when the two energies are equal, that it is never negative for positive energies, and that a certain threshold constant, phi minus 3/2, is positive. The machine-checked library of formal theorems proves these three statements, and packages them into a certificate structure.

The gap is the definition of m and e. In the module, m and e are just real numbers. Nothing in the code says what they physically are. To turn the template into a theorem about thermodynamics, someone would need to define m and e in the language of the physical system, for example as a fermion mass and a boson energy scale. Until that definition exists, the module is a scaffold, not a result. The certificate it builds is a proof about the cost function, not about bosonization.

What a reader can take away is a clear picture of how the framework handles a proposed duality. It does not assert the duality; it builds a reusable proof template and marks the missing physical input. The threshold constant, phi minus 3/2, is positive, which is a real fact, but its physical meaning, if any, is not established here.

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

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

Thermodynamics bosonization is a proven result in the framework. The energies m and e have a specific physical meaning assigned by the module. The threshold constant phi minus 3/2 is known to be physically significant.

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

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