Information Channel Capacity Gaussian Capacity Increases With Snr

Capacity and signal strength

In information theory, the capacity of a communication channel is the maximum rate, in bits per symbol, at which information can be sent with arbitrarily small error. For a channel whose noise is Gaussian, the capacity is given by the Shannon formula C = (1/2) log₂(1 + S/N), where S is the signal power and N is the noise power. The ratio S/N is the signal-to-noise ratio, often written SNR. The formula says that more signal, relative to noise, buys more capacity.

The Recognition Science declaration gaussian_capacity_increases_with_snr is a formal theorem in the framework's machine-checked library of formal theorems. It states that for any two positive signal levels S₁ and S₂ with S₂ > S₁, and any positive noise level N, the Gaussian capacity at S₂ is strictly greater than the capacity at S₁. In plain terms: if you turn up the signal while holding the noise fixed, the channel can carry more information per symbol. The proof is a monotonicity argument on the logarithm, and the theorem is tagged THEOREM because it is proved in Lean with no axioms beyond the standard three.

The declaration does not claim that capacity grows without bound, nor that it grows linearly. The formula shows it grows logarithmically: doubling the SNR adds a constant amount of capacity, not a doubling. It also does not claim that any particular coding scheme achieves this capacity; the Shannon theorem, also present in the library, says that rates below capacity are achievable in principle, but the increasing-with-SNR result only compares capacities at two signal levels. The theorem is about the mathematical function, not about any physical implementation.

In Recognition Science, this result is one piece of a larger program that derives information-theoretic quantities from the ledger, a discrete record of recognition events. The framework defines a fundamental bit rate as the inverse of a timescale tau0, and it models channel capacity as emerging from this ledger bandwidth. The increasing-with-SNR theorem is a consistency check: the framework's capacity function behaves the way the classical Shannon capacity does, which supports the identification but does not by itself prove that the ledger model is correct.

What the declaration gives a reader is a precise, machine-checked statement about a familiar fact: more signal means more capacity. It is a small but solid brick in the framework's claim to reproduce standard results. The falsifier structure in the library, ChannelCapacityFalsifier, names the ways the framework could be wrong: if capacity could exceed the bound, if there were no bandwidth limit, or if the fundamental timescale tau0 were irrelevant. None of these are claimed to be false by the increasing-with-SNR theorem alone; they are targets for the broader program.

/-- **THEOREM**: Gaussian channel capacity increases with SNR. -/
theorem gaussian_capacity_increases_with_snr (S₁ S₂ N : ℝ)
    (hS₁ : S₁ > 0) (hS₂ : S₂ > 0) (hN : N > 0) (h : S₂ > S₁) :
    gaussianCapacity S₂ N hS₂ hN > gaussianCapacity S₁ N hS₁ hN := by
  unfold gaussianCapacity
  -- log(1 + S₂/N) > log(1 + S₁/N) since S₂ > S₁
  have hdiv : S₂ / N > S₁ / N := div_lt_div_of_pos_right h hN
  have hsum : 1 + S₂ / N > 1 + S₁ / N := by linarith
  have hpos1 : 1 + S₁ / N > 0 := by positivity
  have hlog : Real.log (1 + S₂ / N) > Real.log (1 + S₁ / N) :=
    Real.log_lt_log hpos1 hsum
  have hlog2_pos : Real.log 2 > 0 := Real.log_pos (by norm_num)
  have hdenom_pos : 2 * Real.log 2 > 0 := by positivity
  exact div_lt_div_of_pos_right hlog hdenom_pos

/-- **THEOREM**: Gaussian channel capacity increases with SNR. -/
theorem gaussian_capacity_increases_with_snr (S₁ S₂ N : ℝ)
    (hS₁ : S₁ > 0) (hS₂ : S₂ > 0) (hN : N > 0) (h : S₂ > S₁) :
    gaussianCapacity S₂ N hS₂ hN > gaussianCapacity S₁ N hS₁ hN := by
  unfold gaussianCapacity
  -- log(1 + S₂/N) > log(1 + S₁/N) since S₂ > S₁
  have hdiv : S₂ / N > S₁ / N := div_lt_div_of_pos_right h hN
  have hsum : 1 + S₂ / N > 1 + S₁ / N := by linarith
  have hpos1 : 1 + S₁ / N > 0 := by positivity
  have hlog : Real.log (1 + S₂ / N) > Real.log (1 + S₁ / N) :=
    Real.log_lt_log hpos1 hsum
  have hlog2_pos : Real.log 2 > 0 := Real.log_pos (by norm_num)
  have hdenom_pos : 2 * Real.log 2 > 0 := by positivity
  exact div_lt_div_of_pos_right hlog hdenom_pos

/-- 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 theorem does not claim that capacity grows without bound or linearly with SNR. The theorem does not claim that any particular coding scheme achieves the capacity. The theorem does not prove that the ledger model of channel capacity is physically correct, only that its capacity function matches the classical monotonicity.

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

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