Information Ldpccode Rate From Phi Gap Pos

The gap stays positive

Low-Density Parity-Check (LDPC) codes are the error-correction workhorses of 5G, Wi-Fi 6, satellite television, and data storage. They approach the Shannon capacity, the theoretical maximum information rate a noisy channel can carry, to within fractions of a decibel. The gap to capacity is the small remaining distance between what an LDPC code actually achieves and that fundamental limit. For a code with block length N, the framework's machine-checked library of formal theorems defines this gap as g(N) = 1 / (φ · N), where φ is the golden ratio, about 1.618.

The declaration gap_pos proves a simple but essential fact: for any positive block length N, the gap is strictly greater than zero. In other words, no matter how long the code block becomes, the gap never collapses to zero. The proof is direct: since φ is positive and N is positive, their product is positive, and the reciprocal of a positive number is positive. This is not an empirical observation or a heuristic; it is a theorem checked by the framework's kernel with no unproven assumptions.

This positivity is the foundation for the other properties the library proves about the gap. The gap decreases monotonically as N grows, so longer codes get closer to capacity. Doubling the block length halves the gap exactly. And the product of the gap and the block length is always the constant 1/φ, meaning the gap shrinks in precise proportion to the block length. These results together form a certificate, a packaged set of guarantees about how the gap behaves.

In Recognition Science, this is one thread in a larger story where the golden ratio appears throughout information theory. The framework's library proves the finite-N correction to Shannon capacity is 1/(φN), and the LDPC results follow from that. The empirical agreement is striking: for N around 10,000, industry codes show a gap near 0.5 dB, which matches 1/φ⁵; for N around 100,000, the gap drops to near 0.1 dB, matching 1/φ⁹. The framework models this behavior with its φ-suppression law.

The declaration does not claim that real LDPC codes always achieve this exact gap. The framework's own docstring specifies conditions: variable-node mean degree at least 3, check-node mean degree at least 4, and Tanner graph girth at least 6. Without those design constraints, the theorem does not apply. Nor does gap_pos claim that the gap can be made arbitrarily small; it proves positivity, not convergence to zero. The empirical match at specific block lengths is a measured observation, not a theorem. And the framework does not claim that the golden ratio itself is derived from information theory; φ comes from the broader Recognition Science framework, and its appearance here is a consequence of that framework's structure.

theorem gap_pos {N : ℝ} (hN : 0 < N) : 0 < gapToCapacity N := by
  unfold gapToCapacity
  apply one_div_pos.mpr
  exact mul_pos phi_pos hN

theorem gap_decreasing {N₁ N₂ : ℝ} (h₁ : 0 < N₁) (h_lt : N₁ < N₂) :
    gapToCapacity N₂ < gapToCapacity N₁ := by
  unfold gapToCapacity
  have h₂ : 0 < N₂ := lt_trans h₁ h_lt
  have hp : 0 < phi := phi_pos
  have hphi_N1 : 0 < phi * N₁ := mul_pos hp h₁
  have hphi_N2 : 0 < phi * N₂ := mul_pos hp h₂
  -- 1 / (phi * N₂) < 1 / (phi * N₁) since phi * N₁ < phi * N₂
  have h_lt' : phi * N₁ < phi * N₂ := mul_lt_mul_of_pos_left h_lt hp
  exact one_div_lt_one_div_of_lt hphi_N1 h_lt'

/-- Doubling N halves the gap. -/
theorem gap_doubling_halves {N : ℝ} (hN : 0 < N) :
    gapToCapacity (2 * N) = gapToCapacity N / 2 := by
  unfold gapToCapacity
  have hp : phi ≠ 0 := phi_ne_zero
  have hN' : N ≠ 0 := ne_of_gt hN
  field_simp

/-- Gap-vs-block-length monotone law: gap(N) · N = 1/φ. -/
theorem gap_times_N_invariant {N : ℝ} (hN : 0 < N) :
    gapToCapacity N * N = 1 / phi := by
  unfold gapToCapacity
  have h : N ≠ 0 := ne_of_gt hN
  field_simp

What this page does not claim

The theorem does not claim that every LDPC code achieves this exact gap; it requires specific degree and girth conditions. The theorem does not claim the gap converges to zero; it proves only that the gap is positive for finite N. The empirical agreement at N = 10,000 and N = 100,000 is a measured observation, not a 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/LDPCCodeRateFromPhi.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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