Information Channel Capacity Qubit Rs

Channel capacity and recognition overhead

In classical information theory, a qubit is the fundamental unit of quantum information, and its channel capacity is exactly one classical bit: one qubit can transmit one bit of classical information. This is a foundational result of quantum communication, established by Holevo's theorem in 1973. The capacity is a hard upper bound, not an engineering limit; no coding scheme can exceed it.

The Recognition Science framework asks a different question. It models every act of information transfer as a recognition event, a discrete record that reality keeps of a transmission. The framework's central cost function, derived from five plain conditions, is J(x) = (x + 1/x)/2 - 1. This cost is forced, not chosen. The framework proposes that a qubit channel carries not one bit but phi^(-1) bits per qubit, where phi is the golden ratio, approximately 0.618 bits. The number 0.618 is the reciprocal of phi, and it appears because the framework's recognition overhead consumes the remaining capacity.

In Recognition Science, the cost function domainCost takes a sent value m and a received value e and returns J(m/e). The machine-checked library proves three general facts about this function: it vanishes when m equals e, it is nonnegative for positive inputs, and the quantity phi - 3/2 is positive. These are all properties of the cost function J itself, applied to the ratio m/e. The library does not prove that the qubit capacity is 0.618 bits; that statement is a research note, not a theorem.

The distinction matters. The machine-checked library proves facts about the cost function, but it proves nothing specific to qubits, because the definition of domainCost does not reference qubits, channels, or information. The research note records where the idea was meant to go, but the formal content stops at the general cost properties. What would turn this into a theorem about channel capacity is a definition of m and e in terms of qubits and recognition events. That definition does not yet exist in the library.

For a reader, the useful takeaway is precise: the framework has a forced cost function, and that cost function has the properties one would want for a recognition overhead, but the leap from cost to channel capacity is a hypothesis, not a proved result. The number 0.618 bits per qubit is a prediction, not a derivation. The framework's own honesty here is a model for how to separate what is proved from what is proposed.

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

def canonicalThreshold : ℝ := phi - 3 / 2

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

No theorem proves that a qubit channel capacity is 0.618 bits in Recognition Science. No definition of m and e in terms of qubits or recognition events exists in the library. No claim is made that Holevo's theorem is wrong; the framework's proposal is an alternative model, not a refutation.

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)

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