Information Ldpccode Rate From Phi Gap Doubling Halves

The doubling law

Low-Density Parity-Check (LDPC) codes are the error-correction workhorses of modern digital communication, used in 5G, Wi-Fi 6/7, and digital video broadcasting. They approach the Shannon capacity, the theoretical maximum rate at which information can be sent reliably over a noisy channel, to within fractions of a decibel at moderate block lengths. The gap to capacity is the distance, measured in decibels, between a code's actual performance and this theoretical limit.

The framework's library of machine-checked formal theorems contains a definition of this gap for an N-bit code: gapToCapacity(N) = 1/(φN), where φ is the golden ratio, approximately 1.618. This is a definitional choice, not a derived result. The library then proves several properties of this defined gap. The gap_doubling_halves theorem states that doubling the block length N exactly halves the gap: gapToCapacity(2N) = gapToCapacity(N)/2. This follows directly from the definition by simple algebra.

The library also proves that the gap is always positive for any positive N, and that it decreases monotonically as N increases. These are formal theorems, checked by the machine, about the defined quantity. The definition and theorems together form a certificate, a package of properties that the framework's account of LDPC code rates asserts should hold.

In Recognition Science, this doubling law is presented as a consequence of a deeper structure involving the golden ratio. The framework models the gap to capacity as a φ-suppression law, where the golden ratio plays a fundamental role. The empirical observation that industry LDPC codes at block length around 10,000 have a gap of about 0.5 dB, which is close to 1/φ⁵, is cited as supporting evidence. This is a measured agreement, not a derived consequence of the framework's axioms.

The theorem gap_doubling_halves does not claim that real LDPC codes obey this law. It is a statement about a defined mathematical function. The framework's hypothesis, with a named falsifier, is that any LDPC code at moderate-to-long block length will have a stable gap-to-capacity outside the predicted 1/(φN) law. The theorem itself is a piece of pure algebra about a chosen definition, not an empirical claim about physical codes.

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

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'

What this page does not claim

The theorem does not prove that any real LDPC code achieves this gap. The definition of gapToCapacity is a model, not a derived physical law. The framework does not derive the empirical agreement with industry codes; it is a measured observation.

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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