Information Shannon As Jcost Limit

Shannon capacity as a limit

In information theory, the Shannon channel capacity C = log₂ N gives the maximum number of bits that can be sent reliably per symbol over a noiseless channel with N possible symbols. Claude Shannon proved this in 1948. For N = 2, capacity is 1 bit; for N = 8, it is 3 bits; for N = 1024, it is 10 bits. The logarithm base 2 is what makes the unit a bit, and the formula is exact for the idealized noiseless case.

Recognition Science starts from a different picture. Reality keeps a ledger, a discrete record of recognition events, and the cost of recognizing a message of N symbols is forced by a proved cost function J(x) = (x + 1/x)/2 - 1. From that cost, the framework derives a finite-N channel capacity: C_RS(N) = log₂ N - log₂(1 + 1/(φ·N)), where φ = (1 + √5)/2 is the golden ratio. At large N the correction term log₂(1 + 1/(φ·N)) shrinks to zero, so the classical Shannon capacity is recovered as the high-N limit. At N = 1 the correction is exactly -log₂(1 + 1/φ), a number between 0 and 1.

The module in the machine-checked library of formal theorems proves the algebraic structure of this correction: it is non-negative for positive N, it equals the gap between the classical and RS capacities, and it lies in the band (0, 1) at N = 1. These are theorems, checked with no axioms beyond the standard logical ones. What is not proved is that real finite-N codes actually achieve this capacity; that is a hypothesis with a named falsifier: any clean implementation of a finite-N channel whose capacity violates the formula by more than 1%.

The practical upshot is a correction term that vanishes as messages grow. For a small alphabet, the RS formula predicts a slightly lower capacity than Shannon's classical value, by an amount set by the golden ratio. The framework's claim is that this correction is not an ad hoc patch but follows from the same cost function that forces other constants. The classical Shannon limit remains the large-N truth; the RS contribution is a finite-size refinement with a precise, testable form.

/-- The RS finite-N channel capacity. -/
def C_RS (N : ℝ) : ℝ := C_classical N - correction_RS N

/-- The RS capacity gap equals the correction term: structurally,
    `C_classical(N) - C_RS(N) = correction_RS(N)` for all N. -/
theorem C_classical_minus_C_RS_eq_correction (N : ℝ) :
    C_classical N - C_RS N = correction_RS N := by
  unfold C_RS; ring

/-- For `N = 1`, the correction is `log₂(1 + 1/φ)`. -/
theorem correction_RS_at_one :
    correction_RS 1 = Real.logb 2 (1 + 1 / Constants.phi) := by
  unfold correction_RS
  have : Constants.phi * 1 = Constants.phi := by ring
  rw [this]

/-- The correction at `N = 1` is in the band `(0, 1)`. -/
theorem correction_RS_one_band :
    0 < correction_RS 1 ∧ correction_RS 1 < 1 := by
  rw [correction_RS_at_one]
  have h_phi_pos : (0 : ℝ) < Constants.phi := Constants.phi_pos
  have h_inv_pos : 0 < 1 / Constants.phi := by
    exact div_pos one_pos h_phi_pos
  have h_one_lt : (1 : ℝ) < 1 + 1 / Constants.phi := by linarith
  refine ⟨?_, ?_⟩
  · -- log₂(1 + 1/φ) > 0 since 1 + 1/φ > 1
    apply Real.logb_pos
    · norm_num
    · exact h_one_lt
  · -- log₂(1 + 1/φ) < 1 since 1 + 1/φ < 2 (because 1/φ < 1)
    have h_phi_gt_one : (1 : ℝ) < Constants.phi := Constants.one_lt_phi
    have h_inv_lt_one : 1 / Constants.phi < 1 := by
      rw [div_lt_one h_phi_pos]; exact h_phi_gt_one
    have h_lt_two : 1 + 1 / Constants.phi < 2 := by linarith
    have h_log_lt_log_two : Real.logb 2 (1 + 1 / Constants.phi) < Real.logb 2 2 := by
      apply Real.logb_lt_logb (by norm_num : (1 : ℝ) < 2) (by linarith) h_lt_two
    have h_log_two : Real.logb 2 (2 : ℝ) = 1 := Real.logb_self_eq_one (by norm_num : (1 : ℝ) < 2)
    linarith

/-- The correction is non-negative for `N > 0`. -/
theorem correction_RS_nonneg (N : ℝ) (h : 0 < N) :
    0 ≤ correction_RS N := by
  unfold correction_RS
  have h_phi_pos := Constants.phi_pos
  have h_inv_pos : 0 < 1 / (Constants.phi * N) := by
    apply div_pos one_pos
    exact mul_pos h_phi_pos h
  have h_one_plus : 1 ≤ 1 + 1 / (Constants.phi * N) := by linarith
  have h_log : 0 ≤ Real.logb 2 (1 + 1 / (Constants.phi * N)) := by
    apply Real.logb_nonneg
    · norm_num
    · exact h_one_plus
  exact h_log

/-- **SHANNON-AS-J-COST-LIMIT MASTER CERTIFICATE (Track F7).** -/
structure ShannonAsJCostLimitCert where
  correction_nonneg : ∀ N, 0 < N → 0 ≤ correction_RS N
  correction_at_one : correction_RS 1 = Real.logb 2 (1 + 1 / Constants.phi)
  correction_at_one_band : 0 < correction_RS 1 ∧ correction_RS 1 < 1
  C_RS_decomposition : ∀ N, C_RS N = C_classical N - correction_RS N

What this page does not claim

This does not claim that real finite-N codes achieve the RS capacity; that is a hypothesis, not a theorem. This does not claim that Shannon's classical capacity is wrong; it is the large-N limit of the RS formula. This does not claim that the RS correction applies to noisy channels; the module treats the noiseless case only.

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/ShannonAsJCostLimit.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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