Information Quantum Error Correction Eight Tick Encodes Redundancy

The declaration's scope

Quantum error correction protects quantum information from noise by spreading one logical qubit across several physical qubits. The best-known codes include the Shor code, the Steane code, and surface codes. They work because extra physical qubits carry redundancy: if one qubit flips, the others still identify the original state. The Recognition Science library contains a declaration named eight_tick_encodes_redundancy, which is a discrete record of eight phases, that states the eight phases provide three bits of redundancy and that this is enough for single-error correction. The declaration is a theorem whose conclusion is the tautology True, so it establishes nothing beyond the existence of that statement.

The library defines supporting structures: a code with eight physical qubits and one logical qubit, a repetition code with distance three, a syndrome structure whose bits uniquely identify an error, and a depolarizing error model with equal probabilities for bit, phase, and combined flips. These are definitions, not results. The declaration's own docstring calls the module a sketch and warns that every declaration concludes in True or another tautology. The intent is not the establishment.

In Recognition Science, the intended account is that the eight phases provide natural redundancy, errors correspond to phase shifts, and correction restores proper phase alignment. The library also lists predictions: an error threshold related to a tau-zero over gate-time ratio, holographic error correction from ledger projection, and novel code families from phi-geometry. These are predictions, not theorems. A falsifier structure exists that would be falsified if no eight-tick connection holds and codes perform poorly.

What the declaration does not claim is the central fact. It does not prove that any quantum error correction code works, that the eight-tick structure yields a working code, or that any threshold value is correct. The surface code threshold is defined as 0.01, but that is a number assigned by definition, not a derived result. The declaration's own documentation says the upgrade path is to replace the sketch with a real claim and then prove it. That proof has not been supplied.

/-- The 8-tick phases naturally encode redundancy:

    Phase k ↦ e^{ikπ/4} for k = 0, 1, ..., 7

    A Z error adds π to the phase (shifts by 4 ticks).
    An X error cycles through phases differently.

    The 8-fold structure provides natural syndrome detection. -/
theorem eight_tick_encodes_redundancy :
    -- The 8 phases provide 3 bits of redundancy
    -- This is enough for single-error correction
    True := trivial

/-- The "8-tick code": A natural QEC code from RS structure.

    Encode logical qubit in 8-tick phase pattern:
    |0_L⟩ = (|0⟩ + |4⟩)/√2  (even phases)
    |1_L⟩ = (|2⟩ + |6⟩)/√2  (other even phases)

    Or more sophisticated encodings using all 8 phases. -/
def eightTickLogicalCode : EightTickCode := {
  n_physical := 8,
  n_logical := 1,
  uses_8tick := true,
  rate := 1/8
}

/-- The 3-qubit repetition code.

    |0⟩ → |000⟩
    |1⟩ → |111⟩

    Corrects single bit-flip errors. -/
def repetitionCode3 : ClassicalCode := {
  n := 3,
  k := 1,
  d := 3,
  k_le_n := by norm_num,
  d_pos := by norm_num
}

/-- The syndrome measurement.

    Different errors produce different syndromes.
    The syndrome tells us WHICH error occurred without
    revealing the encoded information! -/
structure Syndrome where
  /-- Syndrome bits -/
  bits : List Bool
  /-- Syndrome uniquely identifies error -/
  unique : Bool := true

/-- The depolarizing channel with error probability p.
    All errors equally likely. -/
noncomputable def depolarizing (p : ℝ) (hp : 0 ≤ p ∧ p ≤ 1) : ErrorModel := {
  p_I := 1 - p,
  p_X := p / 3,
  p_Y := p / 3,
  p_Z := p / 3,
  nonneg_I := by linarith [hp.right],
  nonneg_X := by linarith [hp.left],
  nonneg_Y := by linarith [hp.left],
  nonneg_Z := by linarith [hp.left],
  normalized := by ring
}

/-- Surface code threshold: p_threshold ≈ 1%.

    Below this error rate, arbitrarily long computation is possible.
    Above it, errors accumulate faster than correction. -/
noncomputable def surfaceCodeThreshold : ℝ := 0.01

What this page does not claim

The declaration does not prove that any quantum error correction code works. The declaration does not establish that the eight-tick structure yields a working code. The declaration does not derive the value 0.01 for any error threshold.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

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