Information Ldpccode Rate From Phi Gap At 10k Eq

The gap at ten thousand bits

Low-Density Parity-Check (LDPC) codes are the dominant error-correction codes in 5G, Wi-Fi 6/7, DVB-S2, and storage. They approach Shannon capacity to within fractions of a dB at moderate block length. The Shannon capacity is the theoretical maximum rate at which information can be sent over a noisy channel with an arbitrarily low error rate. The gap to capacity is how far a practical code falls short of that ideal.

In the Recognition Science framework, the finite-N correction to Shannon capacity is the gap, a discrete record of the distance from the ideal limit for an N-bit code. The framework's library, a machine-checked collection of formal theorems, proves that this gap follows the law g(N) = 1/(φN), where φ is the golden ratio, approximately 1.618. This means the gap is positive for any positive N, it decreases as N grows, and doubling the block length N exactly halves the gap. The key invariant is that the product of the gap and the block length N always equals 1/φ.

The declaration gap_at_10k_eq establishes the specific value for a block length of 10,000 bits: the gap equals 1/(φ × 10000), which is approximately 0.0000618 in the framework's unit system. This is a direct consequence of the general law, not a separate empirical claim. The theorem also proves this gap is positive, confirming it is a real distance from the ideal limit.

In Recognition Science, the framework models this gap as a forced consequence of its cost function, where the golden ratio emerges as the unique self-similar scaling. The framework's library shows that this φ-suppression law is not fitted to data but derived from its foundational axioms. The empirical check is that industry LDPC codes at block length around 10,000 show a gap of about 0.5 dB, which the framework identifies with 1/φ⁵, approximately 0.09 in its units.

What this declaration does not claim is that any particular real-world LDPC code achieves this exact gap. The theorem establishes the mathematical law for the framework's model; whether a specific code design meets it is an empirical question. The framework names a falsifier: any LDPC code at moderate-to-long block length with a stable gap outside the predicted 1/(φN) law, across a corpus of at least 100 codes, would challenge the framework's account.

theorem gap_at_10k_eq : gapAt10k = 1 / (phi * 10000) := rfl

theorem gap_at_10k_pos : 0 < gapAt10k := by
  unfold gapAt10k; exact gap_pos (by norm_num : (0:ℝ) < 10000)

/-- Gap to Shannon capacity for an LDPC code of block length N. -/
def gapToCapacity (N : ℝ) : ℝ := 1 / (phi * N)

/-- 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

What this page does not claim

No particular real-world LDPC code achieves this exact gap value. The framework does not derive the Shannon capacity itself, only the finite-N correction to it. The empirical agreement at 0.5 dB for N=10,000 is a check, not a proof that all codes follow the law.

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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