Information Shannon As Jcost Limit Correction Rs One Band

The finite-N correction

Claude Shannon's 1948 channel capacity formula, C = log₂N, gives the maximum rate at which information can be sent reliably over a noisy channel, where N is the number of possible messages. The formula is exact in the ideal limit of an infinite message set. Real systems always use finitely many messages, and the Recognition Science framework asks what happens to the capacity when N is finite.

The framework's answer is a corrected capacity, C_RS(N) = log₂N - log₂(1 + 1/(φ·N)), where φ is the golden ratio, approximately 1.618. The subtracted term is the finite-N correction. The declaration correction_RS_one_band establishes that for a single message, N = 1, this correction lies strictly between 0 and 1 bit. That is, the penalty for using just one message is a positive amount less than one full bit, a narrow and exactly bounded band.

The correction is not a free parameter. It is forced by the same cost function that drives the framework's other results, and it vanishes as N grows: the classical Shannon formula is recovered in the high-N limit. The machine-checked library of formal theorems establishes the algebraic structure of the correction, including its non-negativity for all positive N and its exact value at N = 1.

In Recognition Science, this is a theorem about the mathematics of the correction, not an empirical claim. The framework predicts that any clean finite-N channel will match the corrected formula to within one percent, but that prediction is a hypothesis with a named falsifier, not a proven result. The theorem establishes the shape and bounds of the correction; the empirical match is a separate, testable claim.

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

/-- 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

The empirical match of the finite-N formula to any real channel is not proven. The correction term is not claimed to be the only possible finite-N correction. The declaration does not establish that the Shannon capacity itself is wrong for finite N.

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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