Physics Entanglement Entropy Area Law

The area law in Recognition Science

Entanglement entropy measures how much quantum information is shared between a region and its surroundings. For many quantum systems, this entropy scales with the boundary area of the region, not its volume. This is the area law. In a three-dimensional space with a tiny length cutoff ε, the entropy S_EE follows S_EE ∝ A/ε², where A is the area of the boundary.

Recognition Science models this entropy using its cost function J. The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event. The framework proposes that the coefficient of the area law is J(φ)/4π, where φ is the golden ratio. This gives about 0.009 per recognition area unit. The framework also notes that the 1/4 factor in the Bekenstein-Hawking entropy is related to J(φ)/4.72, about 0.025.

In Recognition Science, the module for this topic defines a domain cost as J(m/e), where m and e are real numbers. The module proves three general facts about this cost. First, the cost vanishes when m equals e. Second, the cost is never negative for positive inputs. Third, the quantity φ - 3/2 is positive. These are proved in the machine-checked library of formal theorems.

The module does not yet connect these general facts to a specific physical system. The definition of m and e in terms of entanglement entropy is missing. The research note records the intended direction, but the formal proof does not establish the area law itself. The area law with the RS coefficient remains a hypothesis, not a theorem.

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]

What this page does not claim

The area law with the RS coefficient is not proved by this module. The module does not define m and e in terms of any physical quantity. The Bekenstein-Hawking entropy relation is not derived here.

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

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