Information Error Correction Codes From Jcost

Error correction codes from J-cost

Error-correcting codes let a message survive noise. A sender adds redundancy, a receiver uses it to detect and fix flipped bits. The classical theory, founded by Claude Shannon in 1948, sets a hard ceiling: for a given noise level there is a maximum rate, the Shannon capacity, beyond which reliable communication is impossible. Codes are judged by how close their practical rate comes to that ceiling.

Five canonical families dominate practice: repetition codes, Hamming codes, BCH and Reed-Solomon codes, low-density parity-check (LDPC) codes, and polar codes. Each family has a characteristic threshold rate, the best fraction of the Shannon limit it can reach. The families form a natural progression: repetition is the simplest and farthest from capacity, polar codes are the deepest and closest. In the Recognition Science framework, the golden ratio φ ≈ 1.618 organizes this ladder. The threshold gap, the distance 1 − r between a family's rate and the perfect rate 1, is defined as 1/φ^k for successive integer steps k. The framework proves this gap is always positive and strictly shrinks as k grows, so each deeper family provably gets closer to the Shannon limit. The ratio between adjacent families' thresholds is exactly 1/φ.

In Recognition Science, the framework derives this structure from its central cost function J(x) = (x + 1/x)/2 − 1. The cost function, forced by five plain conditions, produces the golden ratio as the unique self-similar scaling. The error-correction module then identifies the five families as the five rungs of a φ-ladder, a discrete scale of thresholds where each step multiplies the gap by 1/φ. The framework's machine-checked library of formal theorems proves there are exactly five families, that every threshold gap is positive, and that the gaps strictly decrease. The library also states the decoding condition: any rate r below 1 is decodable exactly when J(1/(1 − r)) falls within the canonical band J(φ) between about 0.11 and 0.13. That band is the framework's expression of the Shannon limit itself.

What this means in practice is a clean prediction: the five standard code families are not an arbitrary toolbox but a complete, ordered set, with polar codes as the natural endpoint nearest the Shannon limit. The framework does not design new codes; it explains why the known families exist and why they sit where they do. The mathematical claims are proved with no unverified axioms in the library, though the mapping from the abstract ladder to real-world code performance remains a modeling choice, not a theorem.

theorem eccFamily_count : Fintype.card ECCFamily = 5 := by decide

theorem thresholdGap_pos (k : ℕ) : 0 < thresholdGap k := by
  unfold thresholdGap
  exact div_pos one_pos (pow_pos phi_pos k)

theorem thresholdGap_strictDecr (k : ℕ) :
    thresholdGap (k + 1) < thresholdGap k := by
  unfold thresholdGap
  have hpos_k : (0 : ℝ) < phi ^ k := pow_pos phi_pos k
  have h_growth : phi ^ k < phi ^ (k + 1) := by
    rw [pow_succ]
    have h1 : 1 < phi := one_lt_phi
    nlinarith
  exact one_div_lt_one_div_of_lt hpos_k h_growth

/-- Decoding threshold gap 1 - r on the φ-ladder: deeper family = smaller gap. -/
noncomputable def thresholdGap (k : ℕ) : ℝ := 1 / phi ^ k

structure ECCCert where
  five_families : Fintype.card ECCFamily = 5
  threshold_always_pos : ∀ k, 0 < thresholdGap k
  threshold_strictly_decreasing : ∀ k, thresholdGap (k + 1) < thresholdGap k

What this page does not claim

The framework does not prove that real-world codes achieve these thresholds; the mapping from the ladder to physical code performance is a modeling choice. The module does not derive the Shannon limit from first principles; it identifies the canonical band J(φ) with the limit as a definitional choice. The five families are not proved to be the only possible error-correcting codes, only the canonical set recognized by the framework.

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

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