Information Channel Capacity Qubit Rs Qubit Channel Cert

What the certificate proves

In information theory, a quantum bit, or qubit, is the basic unit of quantum information, and a classical bit is the basic unit of ordinary information. A central question asks how much classical information can be sent through a quantum channel. The Recognition Science framework's library, a machine-checked collection of formal theorems, contains a declaration named QubitChannelCert, which is a certificate: a packaged set of proved statements that can be inspected and reused.

The certificate proves three general facts about a function called domainCost, which is defined as J(m/e), where J is a specific cost function and m and e are positive real numbers. First, when m equals e, the cost is zero. Second, for any positive m and e, the cost is never negative. Third, a particular constant, phi minus 3/2, is positive, where phi is the golden ratio. These three facts are proved in the library as theorems, and the certificate packages them into a single structure.

The certificate does not prove anything about qubits or channels. The docstring above the declaration describes a research hope: that the quantum channel capacity equals phi^(-1) times the qubit count, giving about 0.618 bits per qubit after recognition overhead. But the definition of domainCost uses only the ratio m/e, with no reference to qubits, channels, or any physical quantity. The three proved facts hold for any positive numbers m and e, so they are general properties of the cost function, not results about information theory.

This gap between the research note and the proved content is stated plainly in the library's own documentation. The docstring says the paragraph about qubit capacity is a research note recording where the idea was meant to go, not a result. To turn the certificate into a theorem about its subject, the library would need a definition of m and e in that subject's own terms. Until that definition exists, the certificate remains a general mathematical object, not a claim about quantum channels.

What the certificate does establish is a small, clean piece of mathematics: the cost function J has a zero at unity, is nonnegative on positive inputs, and a golden-ratio-related threshold is positive. These facts are machine-checked and can be cited as proved. The qubit capacity claim, by contrast, is a hypothesis with a named falsifier: any measurement of a real quantum channel's classical capacity that contradicts the 0.618 bits per qubit prediction would falsify it.

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]

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

What this page does not claim

The qubit channel capacity equals phi^(-1) times the qubit count. The certificate establishes any physical result about quantum information. The three proved facts are specific to qubits rather than general properties of 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/Information/Channel_Capacity_Qubit_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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