Information Ldpccode Rate From Phi Gap At 10k Pos

The gap at ten thousand bits

Low-Density Parity-Check (LDPC) codes are the error-correction workhorses of 5G, Wi-Fi 6/7, and digital video broadcasting. They let a receiver correct corrupted data by exchanging messages across a sparse graph of parity checks, approaching the theoretical Shannon capacity to within fractions of a decibel at moderate block lengths. The gap to capacity is the remaining distance, measured in decibels, between what a code actually achieves and the absolute limit set by information theory.

In the Recognition Science framework, a machine-checked library of formal theorems defines this gap for an N-bit code as g(N) = 1/(φN), where φ is the golden ratio, about 1.618. The library proves the gap is positive for any positive N, that it decreases as N grows, and that doubling the block length exactly halves the gap. For N = 10,000, the declaration gap_at_10k_pos establishes the concrete instance: the gap at ten thousand bits is a positive real number, specifically 1/(φ × 10,000), roughly 0.0000618 in the framework's normalized units. This is not an empirical measurement; it is a theorem derived from the definition of the gap function.

The same library packages these properties into a certificate structure, a formal object that bundles the positivity, monotonicity, doubling, and invariant laws into one machine-checked unit. The certificate exists to be reused by other proofs, not to make a claim about any physical LDPC code in the world. The framework's docstring notes that industry codes at N ≈ 10,000 show gaps around 0.5 dB, which is numerically close to 1/φ⁵, but that comparison is an empirical observation, not a theorem in the library.

What the declaration does not claim is equally precise. It does not assert that any real LDPC code achieves this gap; real codes depend on degree distributions, graph girth, and decoder behavior, all of which the theorem ignores. It does not claim that 1/(φ × 10,000) equals 0.5 dB, because the framework's units are not decibels and the conversion is not part of the proof. It does not claim that Shannon's capacity itself is derived from the framework; the gap function takes Shannon's limit as a given input. The theorem is a statement about a defined mathematical function, not about any physical communication system.

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_at_10k_pos : 0 < gapAt10k := by
  unfold gapAt10k; exact gap_pos (by norm_num : (0:ℝ) < 10000)

/-- 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

def cert : LDPCRateCert where
  gap_pos := gap_pos
  gap_monotone := gap_decreasing
  doubling_halves := gap_doubling_halves
  gap_N_invariant := gap_times_N_invariant

What this page does not claim

The theorem does not assert that any real LDPC code achieves this gap. The theorem does not claim that 1/(φ × 10,000) equals 0.5 dB in physical units. The theorem does not derive Shannon's capacity from the framework; it takes the limit as given.

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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