Information Polar Code Gap From Phi Gap At Adjacent Ratio

The gap 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. The gap between the code's achievable rate and the channel capacity decays roughly as O(2^-N0.5) in block length N. This is a classical result in information theory, independent of any particular framework.

The Recognition Science framework models this finite-length gap as a sequence of values indexed by an integer rung k. The framework defines a reference gap of 1 at rung 0, and then defines the gap at rung k as the reference gap multiplied by phi raised to the power -k, where phi is the golden ratio, approximately 1.618. This is a definitional choice, not a derivation from first principles.

What the framework's machine-checked library of formal theorems proves is a property of this defined sequence. The theorem gapAt_adjacent_ratio establishes that for every natural number k, the ratio of the gap at rung k+1 to the gap at rung k equals phi^-1, which is approximately 0.618. In plain language, each step up the ladder multiplies the gap by about 0.618, so the gap shrinks by a fixed proportion at every step. The proof is a direct algebraic consequence of the definition, verified with zero axioms beyond the standard logical ones.

What the theorem does not claim is that this sequence describes any actual polar code. The framework's own documentation describes the polar-code gap as sitting on the phi-ladder, and notes the same 1/phi structure appears in models of quantum-channel capacity and LDPC code-rate gaps. But the theorem itself is about the defined sequence, not about physical codes. Whether real polar codes exhibit this exact ratio at finite block lengths is an empirical question the theorem does not address.

The value of the theorem is structural: it shows the framework's chosen definition is internally consistent and has the geometric property one would expect from a ladder. The ratio being constant means the sequence is a geometric progression with ratio phi^-1, a clean and memorable fact. But the step from this formal consistency to a claim about actual coding performance is not made by this declaration.

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

/-- Gap-to-capacity at φ-ladder rung `k`. -/
def gapAt (k : ℕ) : ℝ := referenceGap * phi ^ (-(k : ℤ))

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

What this page does not claim

The theorem does not claim that any physical polar code achieves this exact gap ratio. The theorem does not derive the phi-ladder from channel properties; it only proves a property of a defined sequence. The theorem does not claim the gap decays as a function of block length N, only as a function of the rung index k.

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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