Information Error Correction Codes From Jcost Ecccert

The ECC certificate

Error-correcting codes let a message survive noise. A sender adds redundancy, a receiver uses it to detect and fix flipped bits. Five families dominate practice: repetition codes, Hamming codes, BCH and Reed-Solomon codes, low-density parity-check (LDPC) codes, and polar codes. Each trades redundancy against correction power, and each has a threshold rate: the fraction of the channel's theoretical capacity, the Shannon limit, at which decoding becomes possible.

In the Recognition Science framework, a ledger (a discrete record of events) models this setting. The framework's central cost function J(x) = (x + 1/x)/2 - 1, proved unique in its library, assigns a cost to any rate below capacity. The library's ECCCert declaration collects three proved facts about this setup. First, exactly five canonical families exist. Second, every threshold gap, the distance from capacity, is positive. Third, these gaps strictly decrease as families descend the ladder: each deeper family sits closer to the Shannon limit, and adjacent families differ by the golden ratio 1/φ.

The certificate itself is a small structure in the framework's machine-checked library of formal theorems. It bundles the five-family count, the positivity proof, and the strict-decrease proof into one object. The proofs are checked by the library's kernel with no unproved assumptions. What the certificate does not do is claim that any physical code achieves these thresholds, or that the five families are the only possible codes. It establishes a formal ordering among named families, not an engineering guarantee.

A reader can take one concrete lesson: the framework predicts a precise spacing between code families, a spacing governed by the same golden ratio that appears in its cost function. Whether real codes obey this spacing is an empirical question the certificate does not settle.

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

noncomputable def eccCert : ECCCert where
  five_families := eccFamily_count
  threshold_always_pos := thresholdGap_pos
  threshold_strictly_decreasing := thresholdGap_strictDecr

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

What this page does not claim

No claim that any physical code achieves these thresholds. No claim that the five families are the only possible codes. No claim that the golden-ratio spacing has been measured in real systems.

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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