Information Polar Code Gap From Phi Gap At Succ Ratio

The one-step ratio

Polar codes, introduced by Erdal Arıkan in 2009, are error-correcting codes that provably achieve the Shannon capacity of a binary-input memoryless channel as the block length grows. In practical finite-length operation, they fall short of that ideal limit by a gap that shrinks as the block length increases. The classical result is that this gap decays roughly as O(2^{-N^{0.5}}), where N is the block length. That decay is fast but not exponential in the usual sense, and it has driven a large literature on how to close the remaining gap in practice.

Within the Recognition Science framework, the finite-length gap is modeled on a discrete ladder of block lengths. The framework defines a sequence of gap values gapAt(k), one for each nonnegative integer rung k, starting from a reference gap of 1 at the smallest block length. The key theorem, gapAt_succ_ratio, proves that moving from one rung to the next multiplies the gap by the reciprocal of the golden ratio phi, about 0.618. In symbols: gapAt(k+1) = gapAt(k) * phi⁻¹. A direct corollary states the same fact as a ratio: gapAt(k+1) / gapAt(k) = phi⁻¹. The proof is a short algebraic manipulation of the definition, and the library certifies it with zero axioms beyond the standard three.

What the theorem does not claim is that this golden-ratio ladder is a proven property of actual polar codes. The definition gapAt is a model, a choice of how to represent the gap in the framework's vocabulary. The theorem proves the arithmetic of that model, not the empirical or theoretical connection to Arıkan's codes. That connection, the bridge from the framework's discrete ledger to the classical channel-coding result, is not part of the declaration. The theorem also does not claim that the gap reaches zero at any finite rung; the positivity theorem gapAt_pos shows the gap is strictly positive for every rung, so the ladder never lands exactly on capacity.

What the theorem does give is a clean structural statement: within the framework, the gap-to-capacity for polar codes follows the same phi-ladder that appears in the framework's treatments of quantum channel capacity and LDPC code-rate gaps. The one-step ratio is the load-bearing fact that makes the ladder a ladder rather than an arbitrary list of numbers. It is the kind of result that lets the framework say, in its own terms, that the finite-length behavior of polar codes is not a collection of separate constants but a single recursive rule.

theorem gapAt_succ_ratio (k : ℕ) :
    gapAt (k + 1) = gapAt k * phi⁻¹ := by
  unfold gapAt
  have hphi_ne : phi ≠ 0 := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

theorem gapAt_adjacent_ratio (k : ℕ) :
    gapAt (k + 1) / gapAt k = phi⁻¹ := by
  rw [gapAt_succ_ratio]
  field_simp [(gapAt_pos k).ne']

theorem gapAt_pos (k : ℕ) : 0 < gapAt k := by
  unfold gapAt referenceGap
  have : 0 < phi ^ (-(k : ℤ)) := zpow_pos Constants.phi_pos _
  linarith [this]

What this page does not claim

The golden-ratio ladder is a proven property of actual polar codes. The gap reaches zero at any finite rung. The framework derives the classical polar code gap decay from first principles.

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/PolarCodeGapFromPhi.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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