Information Quantum Channel Capacity From Phi Correction Strictly Decreasing

The finite-N correction

In information theory, the Shannon capacity C = log₂(1 + S/N) gives the maximum rate at which data can be sent over a noisy channel. The quantum analog, which accounts for entanglement-assisted or coherent information transmission, carries a finite-size correction. This correction, for a channel operating on blocks of N input symbols, is written as 1 / (φ · N), where φ is the golden ratio, approximately 1.618.

This correction term is strictly positive for every positive N, and it strictly decreases as N increases. The theorem correction_strictly_decreasing establishes that the correction at N+1 is always smaller than the correction at N. The argument is short: since φ and N are positive, the product φ · N is positive, and multiplying by the larger positive number (N+1) gives a larger denominator, so the reciprocal is smaller. The result is machine-checked in the framework's library of formal theorems, with no unproved assumptions.

The correction also tends to zero as N grows without bound. A companion theorem bounds it above by 1/N, so the correction vanishes at least as fast as 1/N. This distinguishes the model from any classical-only account that has no finite-N correction at all. The structural prediction is that the ratio of entanglement-assisted to classical capacity for an N-symbol block is 1 + 1/(φ N), and adjacent-N ratios differ by (N+1)/N · 1/φ to leading order.

In Recognition Science, the golden ratio appears here as a fixed constant from the framework's forcing chain, not as a free parameter. The framework models the correction as a definitional choice, and the theorems about its positivity, monotonic decrease, and bound are proven consequences of that definition. What the declaration does not claim is that this correction has been measured in an experiment, that it applies to every conceivable quantum channel, or that the golden ratio itself is derived from information theory alone. The correction is a structural prediction of the framework, awaiting empirical check.

/-- Correction is strictly positive at every positive `N`. -/
theorem correction_pos (N : ℕ) (hN : 0 < N) : 0 < correction N hN := by
  unfold correction
  have hphi : 0 < phi := Constants.phi_pos
  have hNpos : (0 : ℝ) < (N : ℝ) := by exact_mod_cast hN
  positivity

/-- Correction strictly decreases with `N` (from `N` to `N+1`). -/
theorem correction_strictly_decreasing (N : ℕ) (hN : 0 < N) :
    correction (N + 1) (Nat.succ_pos _) < correction N hN := by
  unfold correction
  have hphi : 0 < phi := Constants.phi_pos
  have hNpos : (0 : ℝ) < (N : ℝ) := by exact_mod_cast hN
  have hN1pos : (0 : ℝ) < ((N + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_pos _
  have hphiN_pos : (0 : ℝ) < phi * (N : ℝ) := by positivity
  have hphiN1_pos : (0 : ℝ) < phi * ((N + 1 : ℕ) : ℝ) := by positivity
  have hphi_le_strict : phi * (N : ℝ) < phi * ((N + 1 : ℕ) : ℝ) := by
    apply mul_lt_mul_of_pos_left ?_ hphi
    exact_mod_cast Nat.lt_succ_self N
  exact one_div_lt_one_div_of_lt hphiN_pos hphi_le_strict

/-- Correction tends to 0 as `N → ∞` (statement form using single
positive `N`; the limit is the standard `1/N → 0`). -/
theorem correction_le_inv {N : ℕ} (hN : 0 < N) :
    correction N hN ≤ 1 / (N : ℝ) := by
  unfold correction
  have hphi_gt_one : (1 : ℝ) < phi := by
    have := Constants.phi_gt_onePointFive; linarith
  have hphi : 0 < phi := Constants.phi_pos
  have hNpos : (0 : ℝ) < (N : ℝ) := by exact_mod_cast hN
  have hphiN : 0 < phi * (N : ℝ) := by positivity
  have hN_le : (N : ℝ) ≤ phi * (N : ℝ) := by
    have : (1 : ℝ) * (N : ℝ) ≤ phi * (N : ℝ) :=
      mul_le_mul_of_nonneg_right (le_of_lt hphi_gt_one) (le_of_lt hNpos)
    simpa using this
  exact one_div_le_one_div_of_le hNpos hN_le

What this page does not claim

No measurement of the correction has been performed. The golden ratio is not derived from information theory alone. The correction is not claimed to hold for every possible quantum channel model.

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/QuantumChannelCapacityFromPhi.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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