Foundation Entanglement Monogamy3 From Jcost

The module's actual scope

Entanglement monogamy is a quantum information principle: if two particles A and B are strongly entangled, then A's entanglement with a third particle C is limited. In the standard form, the mutual information satisfies S(A:B) + S(A:C) ≤ S(A:BC), meaning entanglement cannot be freely shared. The Recognition Science framework aimed to derive an analogous inequality for its cost function J, which measures the forced cost of recognition events.

The module EntanglementMonogamy3FromJCost defines a domain cost as J(m/e), the cost function applied to a ratio of two real numbers. It proves three general facts: the cost vanishes when the two inputs are equal, it is nonnegative for positive inputs, and a threshold constant phi - 3/2 is positive. These are true for any positive real numbers m and e, with no reference to entanglement, particles, or any physical subject.

The file's docstring is explicit: the Lean code proves nothing specific to entanglement monogamy, because the domain cost is defined without linking m and e to any physical quantities. The paragraph describing the intended monogamy inequality is a research note, not a result. The content is shared verbatim across 2383 sibling modules, stated once in a common template. What would make this a theorem about monogamy is a definition of m and e in that subject's own terms, which the module does not provide.

In Recognition Science, the framework's library of machine-checked formal theorems shows the cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions. The three proved facts here are consequences of J's general properties, not of any entanglement structure. The intended monogamy inequality J(A,BC) ≤ J(A,B) + J(A,C) remains a research target, not a proved statement.

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]

theorem cert_inhabited : Nonempty EntMonogamy3Cert := ⟨cert⟩

What this page does not claim

The module proves the entanglement monogamy inequality for any physical system. The module establishes a connection between the cost function and quantum mutual information. The three proved facts are specific to entanglement rather than general properties of J.

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/EntanglementMonogamy3FromJCost.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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Derived articles

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