Information Ldpccode Rate From Phi Gap Times N Invariant

The gap-times-length law

Low-Density Parity-Check (LDPC) codes are the error-correction workhorses of 5G, Wi-Fi 6/7, DVB-S2, and storage systems. They approach the Shannon capacity, the theoretical maximum information rate for a reliable channel, to within fractions of a decibel at moderate block lengths. The gap to capacity, measured in decibels, shrinks as the block length N grows. The statement gap_times_N_invariant in the Recognition Science framework's machine-checked library of formal theorems establishes a precise algebraic relationship for this gap: for any positive block length N, the gap multiplied by N equals 1 divided by the golden ratio φ, approximately 0.618.

In symbols, the statement says that gapToCapacity(N) * N = 1/φ. This is not an approximation or a statistical fit; it is a proved identity within the framework. The gap is positive for any positive N, it decreases monotonically as N increases, and doubling N exactly halves the gap. These properties follow from the definition of gapToCapacity(N) as 1/(φN). The framework's library proves these as formal statements, meaning they are derived from the axioms of the system without any unverified assumptions.

The framework's library also contains a certificate structure, LDPCRateCert, which packages these four properties (positivity, monotonicity, doubling-halves, and the N-invariant) into a single object. The certificate, named cert, is constructed from the individual statements, providing a single point of reference for the entire set of gap properties. This certificate is a formal object that can be passed to other parts of the framework that require these properties.

In Recognition Science, this invariant is presented as a consequence of the framework's foundational forcing chain, which derives constants and laws from a single cost function. The framework models the gap to capacity for LDPC codes as following the same φ-suppression law that appears elsewhere in its structure. The empirical observation that industry LDPC codes at block length N ≈ 10,000 show a gap of about 0.5 dB, which is close to 1/φ⁵ ≈ 0.09, is offered as a supporting check, not as a proof of the invariant.

The statement does not claim that real-world LDPC codes exactly follow this law. The framework's own docstring specifies a falsifier: any LDPC code at moderate-to-long block length with a stable gap-to-capacity outside the predicted 1/(φN) law, based on a corpus of at least 100 codes. The invariant is a formal statement about a defined function, not a universal empirical law. It does not address the design constraints (variable-node degree, check-node degree, girth) that the framework's docstring lists as conditions for the belief-propagation decoder to achieve this gap.

/-- 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

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

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 statement does not claim that any real LDPC code exactly follows the 1/(φN) law. The invariant does not address the code design conditions (degree distributions, girth) listed in the framework's docstring. The empirical gap values at N ≈ 10,000 and N ≈ 100,000 are supporting observations, not formal consequences of the statement.

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)

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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