Information Holevo Bound Rs Holevo Bound Rs

What the declaration proves

The Holevo bound is a theorem in quantum information theory: it limits how much classical information can be reliably extracted from a quantum system, measured in bits per use of the channel. The Recognition Science declaration named HolevoBoundRS sounds like it should extend or sharpen that bound. It does not. The machine-checked library of formal theorems proves three general facts about a function called cost, defined as J(m/e) for two real numbers m and e, where J is a fixed cost function from the framework's core. The three facts are: the cost is zero when m equals e, the cost is never negative for positive inputs, and the number phi minus 3/2 is positive.

Those three facts are true for any positive real numbers m and e. They say nothing about quantum channels, ensembles, or information transmission. The declaration's own documentation states this plainly: the Lean code proves nothing specific to the subject of the Holevo bound, because the cost function is defined without any reference to a quantum system. The paragraph in the source that mentions a recognition overhead of 0.618 times the Shannon capacity is a research note recording where the idea was meant to go, not a result. The formal content is a template shared verbatim with 2383 sibling modules, each one a placeholder awaiting a definition of m and e in that subject's own terms.

What the declaration does establish, as a theorem, is a small piece of the framework's own mathematics. The cost function J, which the framework derives from five plain conditions, has the properties that J(1) = 0 and J(x) is nonnegative for positive x. The declaration packages those properties, plus the positivity of phi - 3/2 (which is about 0.118), into a structure named HolevoBoundRS and proves that this structure exists. A reader who wants to know whether the framework has anything to say about the actual Holevo bound will find no answer here. The page is a placeholder, honestly labeled as one.

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 HolevoBoundRS := ⟨cert⟩

What this page does not claim

No new bound on quantum information transmission is proved. No connection between the cost function and any specific quantum channel or ensemble is established. The research note about 0.618 times Shannon capacity is not a theorem.

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/Information/Holevo_Bound_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

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: