Information Channel Capacity Shannons Theorem

The theorem's scope

Claude Shannon's channel capacity, the maximum rate at which information can be sent reliably through a noisy channel, is one of the cornerstones of information theory. The classical result, proved by Shannon in 1948, states that for any channel there exists a rate C such that if you transmit at any rate below C, you can make the error probability arbitrarily small, while if you transmit above C, errors become unavoidable. The capacity is computed as the maximum of the mutual information I(X; Y) over all input distributions p(x), where mutual information measures how much the output reveals about the input. For the common case of a Gaussian channel with signal power S and noise power N, the capacity takes the closed form C = (1/2) log₂(1 + S/N) bits per symbol.

The framework's machine-checked library of formal theorems contains a declaration named shannons_theorem that states exactly this threshold behavior: if the rate R is less than the capacity C, reliable communication is possible, and if R is greater than C, errors are unavoidable. The declaration is a theorem in the sense that it is a statement in the library, but its proof is currently the trivial proof True := trivial, meaning the library does not yet contain a substantive derivation of Shannon's result. The library does, however, prove some supporting facts: mutual information is nonnegative, channel capacity is nonnegative, and mutual information is bounded above by log of the product of input and output sizes. These are real theorems with actual proofs, though they are elementary bounds rather than the full capacity theorem.

In Recognition Science, the library models channel capacity as emerging from the ledger, a discrete record of events, and its bandwidth, the fundamental rate at which the ledger can record and transmit information. The library defines a fundamental bit rate as the reciprocal of a time constant tau0, and it defines capacity formulas for the binary symmetric channel and the Gaussian channel. The declaration capacity_from_ledger is meant to connect the ledger's finite capacity to the channel capacity, but like shannons_theorem, its proof is currently trivial. The library also defines a falsifier structure that would refute the framework's claim if a channel could exceed its capacity, if there were no bandwidth limit, or if the time constant tau0 were irrelevant to capacity.

What the declaration does not claim is as important as what it states. It does not claim that the framework has derived Shannon's theorem from its ledger axioms; the proof is a placeholder. It does not claim that the specific value of tau0 has been derived or measured; the fundamental bit rate is a definition, not a theorem. It does not claim that the framework's channel model matches any particular physical channel; the Gaussian capacity formula is a standard definition, and the theorem that capacity increases with signal-to-noise ratio is proved for that definition, not for a derived physical model. The honest summary is that the library has stated Shannon's theorem as a target and proved some supporting bounds, but the substantive derivation remains open.

/-- **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

/-- **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

/-- **THEOREM**: Mutual information is non-negative. -/
theorem mutual_information_nonneg (ch : Channel) (p : InputDistribution ch.inputSize) :
    mutualInformation ch p ≥ 0 := by
  unfold mutualInformation
  exact le_max_left 0 _

/-- **THEOREM**: Channel capacity is non-negative.
    This follows from mutual information being non-negative for all input distributions. -/
theorem capacity_nonneg (ch : Channel) : channelCapacity ch ≥ 0 := by
  unfold channelCapacity
  -- The supremum of non-negative quantities is non-negative
  apply le_ciSup_of_le
  · -- Bounded above
    use Real.log (ch.inputSize * ch.outputSize)
    intro x ⟨p, hp⟩
    rw [← hp]
    exact mutual_info_bounded ch p
  · -- Use the uniform distribution
    exact mutual_information_nonneg ch (uniformDistribution ch.inputSize ch.input_nonempty)

/-- 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

/-- In Recognition Science, channel capacity comes from the ledger's bandwidth:

    1. **Temporal bandwidth**: τ₀ sets the minimum time per bit
    2. **Spatial bandwidth**: Voxel size sets minimum spatial resolution
    3. **Energy bandwidth**: E_coh sets minimum energy per bit

    C_ledger = (ledger transitions per second) × (bits per transition) -/
theorem capacity_from_ledger :
    -- The ledger has finite capacity
    -- This determines the channel capacity
    True := trivial

What this page does not claim

The framework has not derived Shannon's theorem from its ledger axioms; the proof is a placeholder. The specific value of tau0 has not been derived or measured; the fundamental bit rate is a definition. The framework's channel model has not been shown to match any particular physical channel.

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.

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: