Information Channel Capacity Capacity From Ledger

The ledger's bandwidth

In information theory, channel capacity is the maximum rate at which data can be sent through a noisy channel with an arbitrarily small error probability. Claude Shannon showed in 1948 that this rate, usually written C, is the maximum mutual information between a channel's input and output: C = max_p(x) I(X; Y) bits per use. For a Gaussian channel with signal power S and noise power N, the formula becomes C = (1/2) log₂(1 + S/N) bits per symbol. The capacity is a hard ceiling: below it, reliable communication is possible; above it, errors become unavoidable.

The framework's machine-checked library of formal statements contains a declaration called capacity_from_ledger. The declaration asserts that the ledger, a discrete record of recognition events, has finite capacity, and that this capacity determines the channel capacity. The library defines a fundamental bit rate as the reciprocal of a base time unit tau0, and it shows that mutual information is non-negative and bounded above by the log of the product of input and output sizes. These are the pieces that make the statement meaningful: a finite ledger cannot record an unbounded amount of information.

In Recognition Science, the claim is that Shannon's capacity is not a free parameter but emerges from the ledger's bandwidth, the fundamental rate at which the ledger can record and transmit information. The library also includes standard results as definitions and statements: the binary symmetric channel capacity formula, the Gaussian capacity increasing with signal-to-noise ratio, and the Holevo bound for quantum channels. A falsifier structure exists that would refute the framework's claim if one could show that capacity could exceed the ledger's bound, that there is no bandwidth limit, or that the base time unit tau0 is irrelevant.

What the declaration does not claim is important. The statement as written is a placeholder: its body is the trivial term True. It does not derive Shannon's formula from first principles, nor does it compute a numerical value for tau0. The library defines the concepts and states the intended relationship, but the actual derivation from ledger structure to the capacity formula remains unfinished. The classical Shannon theory stands on its own; the framework's contribution is a stated target, not a completed derivation.

/-- 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

/-- 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

/-- 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 statement does not derive Shannon's capacity formula from ledger structure. The declaration does not compute a numerical value for the base time unit tau0. The body of capacity_from_ledger is not a completed derivation; it is the trivial term True.

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)

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