Foundation Entanglement Monogamy3 From Jcost Ent Monogamy3 Cert

A certificate and its limits

In mathematics, a certificate is a compact bundle of verified facts. The Recognition Science library contains a machine-checked collection of formal theorems, and within it the declaration EntMonogamy3Cert is a certificate: a structure that packages three proven statements about a function called domainCost, which is defined as J(m/e), the J-cost of the ratio of two real numbers m and e. The three facts are that this cost is zero when m equals e, that it is never negative when both inputs are positive, and that the number phi minus 3/2 is positive. The library proves each of these and then shows the certificate exists, meaning the three facts hold together consistently.

The name suggests a connection to entanglement monogamy, a quantum information principle where two systems cannot both be maximally entangled with a third. The certificate itself, however, proves nothing about entanglement. The function domainCost is defined purely as J(m/e), with no definition of m or e in terms of quantum states. The three proven facts are general properties of the J-cost function, not results about physics. The library's own documentation states this plainly: the module proves three general facts and nothing specific to the subject, because the cost function is defined without reference to one.

What the certificate does establish is a small, solid foundation. It shows that J-cost behaves sensibly as a cost: zero at equality, nonnegative for positive inputs. The positivity of phi minus 3/2 is a concrete numerical fact, roughly 0.118, that could serve as a threshold in future work. The certificate format itself is useful: it bundles these facts into a single object that other theorems can refer to, so any later result about entanglement monogamy can build on a verified base rather than re-proving these basics.

The gap between the name and the content is the honest story. The certificate is a placeholder, a scaffold. What would turn it into a theorem about entanglement is a definition of m and e in quantum terms, for example as measures of entanglement between subsystems. Until such a definition exists, the certificate remains a general-purpose tool waiting for its subject. The library records this openly as a research note, not as a result.

structure EntMonogamy3Cert where
  cost_at_eq : ∀ r : ℝ, r ≠ 0 → domainCost r r = 0
  cost_nonneg : ∀ m e : ℝ, 0 < m → 0 < e → 0 ≤ domainCost m e
  threshold_pos : 0 < canonicalThreshold

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⟩

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

What this page does not claim

Entanglement monogamy is proven for any physical system. The quantities m and e are defined in terms of quantum states. The certificate provides a numerical bound on entanglement. The name EntMonogamy3Cert reflects a proven result about monogamy.

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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