Information Error Correction Bounds Eight Tick Corrects 3

The eight-tick correction claim

Error correction is the art of adding redundancy so that noise can be undone. A sender transmits a codeword, noise flips some of its symbols, and the receiver uses the redundancy to reconstruct the original message. The classical theory, built by Claude Shannon in 1948, sets a hard limit: the channel capacity is the maximum rate at which information can be transmitted reliably over a noisy channel. For a binary symmetric channel that flips each bit with probability p, the capacity is 1 + p log2(p) + (1-p) log2(1-p).

Within this classical framework, a code is described by three numbers: n, the codeword length; k, the message length; and d, the minimum distance between any two codewords. A code with minimum distance d can detect up to d-1 errors and correct up to (d-1)/2 errors. The Hamming bound gives the maximum number of codewords possible for given length and error-correction ability. For a code of length 8 that corrects 3 errors, the bound checks that 2^1 times the sum of binomial coefficients for 0, 1, 2, and 3 errors is at most 2^8, which holds.

In Recognition Science, the framework models an eight-phase recognition cycle as a natural source of redundancy. Its library of machine-checked formal theorems defines an eight-tick code as a code with length 8, message length 1, and minimum distance 8. The theorem eight_tick_corrects_3 proves that this code corrects up to 3 errors, since (8-1)/2 = 3. A companion theorem shows the same code can detect up to 7 errors. The framework also records that majority voting works below an error rate of 3/8, and that the maximum rate for single-error correction is 7/8.

What the theorem does not claim is broader than what it proves. It does not show that eight-tick codes achieve Shannon capacity, nor that they outperform classical codes. It does not establish that the eight-phase structure is physically realized in any communication system. The theorem is a formal statement about a specific code definition within the framework's library, not an empirical result about real-world error correction. The framework's own falsifier structure acknowledges this: it lists as open questions whether error correction could exceed Shannon's bound, whether eight-tick redundancy is necessary, and whether quantum error correction from eight-tick phase coherence works.

The practical consequence is modest but real. For a reader familiar with coding theory, the theorem confirms that the framework's eight-tick code sits inside the classical bounds: it corrects 3 errors in a length-8 code, exactly as the distance formula requires. The framework's contribution is not a new bound but a vocabulary: it names the eight phases as a recognition structure and asks whether that structure yields useful codes. That question remains open, and the theorem does not pretend to answer it.

theorem eight_tick_corrects_3 :
    -- 8-tick code corrects up to 3 errors
    (eightTickCode.d - 1) / 2 = 3 := by rfl

/-- 8-tick: detect up to d-1 = 7 errors, correct up to ⌊(d-1)/2⌋ = 3. -/
theorem detect_vs_correct :
    (eightTickCode.d - 1 = 7) ∧ ((eightTickCode.d - 1) / 2 = 3) := by
  constructor <;> rfl

/-- The Hamming bound (sphere-packing bound):

    For t-error-correcting code:
    2^k × Σᵢ₌₀ᵗ C(n,i) ≤ 2ⁿ

    Codes meeting this bound are "perfect" (e.g., Hamming codes).

    Formalized as: for our 8-tick code with d=8, t=3,
    the volume of radius-3 balls around 2^1 codewords fits inside {0,1}^8. -/
theorem hamming_bound_8tick :
    2 ^ 1 * (Nat.choose 8 0 + Nat.choose 8 1 +
      Nat.choose 8 2 + Nat.choose 8 3)
    ≤ 2 ^ 8 := by
  native_decide

What this page does not claim

The theorem does not show that eight-tick codes achieve Shannon capacity. The theorem does not establish that the eight-phase structure is physically realized. The theorem is a formal statement about a code definition, not an empirical result about real-world 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/ErrorCorrectionBounds.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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