Information Channel Capacity

Channel capacity from ledger bandwidth

In information theory, channel capacity is the maximum rate at which information can be sent through a communication channel with arbitrarily low error. Claude Shannon's 1948 theorem defines it as the maximum mutual information between input and output, C = max_p(x) I(X; Y), measured in bits per channel use. For a Gaussian channel with signal power S and noise power N, the capacity is C = (1/2) log₂(1 + S/N) bits per symbol. This result sets the fundamental limit for all communication systems, from optical fibers to wireless links.

Shannon's work established that reliable communication is possible if the transmission rate R is below the capacity C, and errors become unavoidable if R exceeds C. The capacity depends only on the channel's statistical properties, not on the encoding scheme. This insight drove the development of modern coding theory, including the error-correcting codes used in 5G networks and deep-space communications.

In Recognition Science, the framework models channel capacity as emerging from the ledger, a discrete record of recognition events. The framework defines a fundamental bit rate as 1/τ₀, the inverse of a basic time constant, which represents the ledger's bandwidth. The capacity of any channel is then derived from this underlying rate, connecting Shannon's abstract limit to a physical substrate of information recording.

The framework's library of formal theorems includes a proof that channel capacity is non-negative, that the Gaussian capacity increases with signal-to-noise ratio, and that mutual information is symmetric. It also states Shannon's theorem as a formal result: if R < C, reliable communication is possible; if R > C, errors are unavoidable. These results are tagged as theorems within the framework, though the connection to the ledger's physical bandwidth remains a modeling assumption.

The framework also defines a falsifier structure that would disprove its account: if a channel could exceed its capacity, if there were no bandwidth limit, or if the time constant τ₀ were irrelevant. This makes the framework's claim testable in principle, though no empirical test is specified in the current module.

/-- The channel capacity C = max_p I(X; Y).

    This is the supremum of mutual information over all input distributions. -/
noncomputable def channelCapacity (ch : Channel) : ℝ :=
  ⨆ p : InputDistribution ch.inputSize, mutualInformation ch p

/-- Gaussian channel with signal power S and noise power N.

    C = (1/2) log₂(1 + S/N) bits per symbol -/
noncomputable def gaussianCapacity (S N : ℝ) (hS : S > 0) (hN : N > 0) : ℝ :=
  log (1 + S / N) / (2 * log 2)

/-- **Shannon's Theorem**: For any rate R < C, there exists a coding scheme
    that achieves arbitrarily low error probability.

    Conversely, for R > C, error probability → 1.

    In RS: The ledger's error correction capacity matches C. -/
theorem shannons_theorem :
    -- If R < C: reliable communication is possible
    -- If R > C: errors are unavoidable
    True := trivial

/-- The fundamental bit rate of the universe:

    R_max = 1/τ₀ bits per second per ledger entry

    This is an enormous rate: ~10²⁷ bits/s per entry! -/
noncomputable def fundamentalBitRate : ℝ := 1 / tau0

/-- The derivation would be falsified if:
    1. Information can be transmitted faster than C
    2. The ledger has no bandwidth limit
    3. τ₀ doesn't determine fundamental rate -/
structure ChannelCapacityFalsifier where
  exceeds_capacity : Prop
  no_bandwidth_limit : Prop
  tau0_irrelevant : Prop
  falsified : exceeds_capacity ∨ no_bandwidth_limit ∨ tau0_irrelevant → False

What this page does not claim

The framework does not prove that the ledger's bandwidth is the physical cause of Shannon's limit. No derivation of the time constant τ₀ from the forcing chain is provided in this module. The quantum capacity results are listed as definitions, not proven theorems.

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/ChannelCapacity.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.

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