Information Quantum Channel Capacity From Phi

The capacity correction

Claude Shannon's 1948 formula C = log₂(1 + S/N) gives the maximum rate at which information can be sent through a noisy classical channel. The quantum analog, which handles entanglement-assisted communication or coherent information, inherits the same logarithmic shape. In Recognition Science, a correction term appears in this quantum capacity when the channel carries a finite number N of input symbols. The correction is 1/(φ·N), where φ is the golden ratio, about 1.618. It shrinks as the symbol count grows, vanishing entirely in the infinite limit where Shannon's classical result is recovered.

This finite-N correction is not a free parameter. The framework derives it from the same cost function that forces φ elsewhere, a chain that begins with a ledger, a discrete record of recognition events, and ends with the golden ratio as the unique self-similar scaling. The correction's structure is what makes it testable: for a block of N symbols, the ratio of entanglement-assisted capacity to classical capacity is predicted to be 1 + 1/(φ·N). Adjacent block sizes differ by (N+1)/N · 1/φ to leading order. The term falls off as 1/N, not 1/N², which distinguishes it from any classical-only model that has no finite-size correction at all.

The machine-checked library of formal theorems proves three properties of this correction. It is strictly positive for every positive N, so the quantum capacity always exceeds the classical bound by a small amount. It strictly decreases as N grows, meaning larger blocks approach the classical limit from above. And it is bounded above by 1/N, which pins down its rate of decay. These three facts together form a certificate, a packaged proof that the correction behaves as the framework claims. The library records zero axioms beyond the standard three and zero unfinished proofs.

What this means in practice is a concrete, falsifiable prediction. A quantum channel experiment with a finite symbol block should see a capacity slightly above the classical Shannon limit, with the excess shrinking like one over the block size. The specific φ in the numerator gives the correction a numerical fingerprint that no classical model reproduces. Measuring that 1/N scaling, and the φ factor in front of it, would be direct evidence for the framework's claim that the golden ratio is not a coincidence but a forced consequence of how recognition costs are structured.

/-- 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))

/-- 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 correction is a derived theorem of quantum mechanics itself; it is a framework prediction awaiting experimental confirmation. The 1/N scaling has been observed in any laboratory experiment. The golden ratio appears in the classical Shannon formula; it enters only through the finite-N quantum correction.

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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