Foundation Entanglement Cost Rs Entanglement Cost Cert

The cost certificate

Entanglement entropy measures how much quantum information is shared between two parts of a system. For a system split into two halves, the entropy S is computed as S = -sum p_k log p_k, where p_k are the probabilities of the outcomes. In the Recognition Science (RS) framework, the maximum entanglement entropy is taken to be S_max = D * log(2) = 3 * log(2) = 2.079 bits, where D is the number of spatial dimensions. This number equals the framework's Count Law bits, a coincidence the framework treats as a pointer toward a deeper link between information and geometry.

The declaration EntanglementCostCert is a formal object in the framework's machine-checked library of formal theorems. It bundles three general facts about a cost function J(x) = (x + 1/x)/2 - 1, which the framework derives from its core axioms. The certificate states that J(m/e) equals zero when m equals e, that J(m/e) is never negative for positive inputs m and e, and that the golden-ratio-based threshold phi - 3/2 is positive. These three facts are proved in the library as theorems, not assumed as postulates.

The certificate's three facts are deliberately general. They hold for any positive real numbers m and e, with no reference to what those numbers mean. The framework's research note suggests the intended application: m and e might stand for the number of measurements and the number of events in a ledger, a discrete record of events. If that identification holds, the certificate would show that the cost of recognition is zero when measurements match events and positive otherwise. But that identification is not part of the formal proof.

In Recognition Science, the ledger is the framework's central model: reality keeps a discrete record of recognition events, and the cost of recognition is forced by the axioms. The certificate is a step toward linking this ledger to entanglement entropy. The framework's library proves the three arithmetic facts once, universally quantified, in a template module; the entanglement module reuses them without adding any entanglement-specific content.

The certificate does not claim that entanglement entropy equals the cost function, nor that the maximum entropy is derived from the cost. It does not define what m and e are in physical terms. The framework's own docstring is explicit: the paragraph about entanglement is a research note recording where the idea was meant to go, not a result. The certificate is a scaffold, not a building.

structure EntanglementCostCert 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

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]

noncomputable def cert : EntanglementCostCert where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The certificate does not define what m and e are in physical terms. The certificate does not prove that entanglement entropy equals the cost function. The certificate does not derive the maximum entropy value from the cost function.

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)

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