Information Quantum Channel Capacity From Phi Correction Le Inv

The finite-N correction

In classical information theory, the Shannon capacity of a channel with signal-to-noise ratio S/N is C = log₂(1 + S/N). The quantum analog, for entanglement-assisted or coherent-information communication, carries an additional finite-size correction. The Recognition Science framework models this correction as 1/(φ·N) per input symbol, where φ is the golden ratio and N is the number of input symbols. This is a definitional choice, a model, not a derived result.

The declaration correction_le_inv proves a bound on that correction: for any positive N, the correction 1/(φ·N) is at most 1/N. Since φ is approximately 1.618, the correction is always smaller than the plain inverse count. The proof is machine-checked in the framework's library of formal theorems, with no unproved assumptions. It also establishes the correction is strictly positive and strictly decreasing as N grows, tending to zero in the large-N limit.

These properties assemble into a certificate, a packaged guarantee that the correction behaves as a finite-size effect should: positive, shrinking, and bounded by the inverse symbol count. The framework's structural prediction goes further, claiming the entanglement-assisted-to-classical capacity ratio for an N-symbol block is 1 + 1/(φ·N), and that adjacent-N ratios differ by (N+1)/N · 1/φ to leading order. The correction vanishes as 1/N, not 1/N², which would distinguish it from a classical-only model with zero finite-N correction. This prediction is a hypothesis, not a theorem; it awaits empirical or experimental test.

What correction_le_inv does not claim is equally important. It does not prove the capacity formula itself, nor that the golden ratio correction is physically realized. It only bounds a model's correction term. The bound is a theorem about a definition, so the whole statement carries the MODEL tag. The golden ratio enters through the framework's constants, not through any derivation in this declaration.

/-- 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

/-- 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

/-- The φ-ladder finite-N correction factor for quantum channel
capacity at input-symbol-count `N`. -/
def correction (N : ℕ) (hN : 0 < N) : ℝ := 1 / (phi * (N : ℝ))

structure QuantumChannelCapacityCert where
  correction_pos : ∀ {N : ℕ} (hN : 0 < N), 0 < correction N hN
  strictly_decreasing :
    ∀ (N : ℕ) (hN : 0 < N),
      correction (N + 1) (Nat.succ_pos _) < correction N hN
  bounded_by_inv :
    ∀ {N : ℕ} (hN : 0 < N), correction N hN ≤ 1 / (N : ℝ)

What this page does not claim

The declaration does not prove the capacity formula itself, only a bound on a model's correction term. The golden ratio correction is not derived from first principles in this declaration; it is a definitional choice. The structural prediction about the capacity ratio is a hypothesis awaiting test, not a proved theorem.

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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