Foundation Entanglement Cost Rs

Entanglement cost in RS

Entanglement entropy is a number that quantifies how much a quantum system's parts are correlated with each other. For a system split into two pieces, it is computed as S = -sum p_k log p_k, where p_k are the probabilities of the possible states of one piece. The maximum possible value for a system with D bits is D log 2. In three dimensions, that maximum is 3 log 2, about 2.079 bits.

In Recognition Science (RS), the framework models the cost of a recognition event using a function J(x) = (x + 1/x)/2 - 1. The ledger of entanglement cost is defined as J(m/e), where m and e are positive real numbers. The framework proves three general facts about this cost: it is zero when m equals e, it is never negative for positive inputs, and the constant phi - 3/2 is positive. These are simple properties of the J function, not facts specific to entanglement.

The documentation is explicit: the code proves nothing about entanglement itself, because the definition of domainCost does not reference any physical quantity. The research note in the docstring records an intended direction, not a result. What would make this a theorem about entanglement is a definition of m and e in terms of the actual quantum state, which is not provided.

In plain language, the framework establishes a template for a cost function and verifies its basic mathematical hygiene. It does not establish that entanglement entropy in physics equals J(m/e), nor that the maximum entanglement entropy is 3 log 2 in any measured system. Those remain open targets for the framework.

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]

What this page does not claim

The framework does not prove that physical entanglement entropy equals J(m/e). The framework does not establish the maximum entanglement entropy as 3 log 2 for any real system. The framework does not provide a definition of m and e in terms of quantum states.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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