Information Quantum Channel Capacity From Phi Correction Pos

The correction term

In information theory, a communication channel's capacity is the maximum rate at which data can be sent reliably. The classical Shannon formula, C = log₂(1 + S/N), gives this rate for a signal of power S in noise of power N. The quantum analog, which accounts for entanglement and coherent information, is more complex, and Recognition Science adds a finite-size correction to it. This correction, defined for a channel operating on blocks of N input symbols, is the term 1/(φ·N), where φ is the golden ratio, approximately 1.618.

The declaration correction_pos is a formal theorem in the framework's machine-checked library. It proves that for any positive integer N, this correction term is strictly greater than zero. In plain language, the adjustment is always a positive addition to the capacity, never a subtraction, regardless of how many symbols are in a block. The proof is a direct calculation: since both φ and N are positive, their product is positive, and the reciprocal of a positive number is positive.

The theorem also establishes two companion properties. First, the correction strictly decreases as N grows: a larger block size yields a smaller per-symbol adjustment. Second, the correction is bounded above by 1/N, which means it tends to zero as N approaches infinity. These properties are collected in a certificate structure, a formal package that bundles the positivity, monotonicity, and vanishing behavior into a single object the framework can pass to other proofs.

In Recognition Science, this correction is not a free parameter. It emerges from the same φ-ladder that structures other quantities in the framework, where the golden ratio appears as a forced scaling constant. The framework's claim is that this finite-N correction distinguishes quantum channel capacity from any classical-only model, which would have no such term. The theorem itself, however, only establishes the arithmetic properties of the correction; it does not prove that this correction matches any experimental measurement of a real quantum channel.

What the declaration does not claim is just as important. It does not prove that the correction is the correct physical description of any actual channel. It does not derive the φ-ladder from first principles in this file; that derivation lives elsewhere in the framework. And it does not compare the correction to empirical data. The theorem is a piece of pure mathematics: given the definition, the positivity, monotonicity, and limit behavior follow. Whether that definition corresponds to nature is a separate question, one this declaration leaves open.

/-- 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

The theorem does not prove the correction matches any experimental measurement of a real quantum channel. The declaration does not derive the φ-ladder from first principles; that derivation lives elsewhere in the framework. The positivity proof does not establish that the correction term is physically meaningful for any actual communication system.

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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