Information Qecthreshold From Phi Ladder Qeccode Family

The five code families

Quantum error correction protects quantum information by encoding it redundantly, so that local errors can be detected and corrected without destroying the underlying state. Each error-correcting code has an error threshold: the maximum physical error rate below which the code's logical error rate can be suppressed arbitrarily well by scaling up the code. Different code families have very different thresholds. The repetition code, the simplest family, tolerates roughly 50% errors. The surface code, the basis of most current hardware roadmaps, has a threshold near 1%. The colour code sits near 0.5%.

In the Recognition Science framework, a ledger (a discrete record of recognition events) forces a specific scaling constant: the golden ratio φ ≈ 1.618. The framework's machine-checked library of formal theorems defines five canonical code families: repetition, surface, colour, topological, and concatenated. Its inductive declaration QECCodeFamily lists exactly these five, and a theorem proves the count is five. The library then defines a threshold function codeThreshold(k) = φ^(-k), so the threshold for family k is the k-th power of the golden ratio, inverted.

The decisive property is the decay ratio. A theorem proves that for any step k, codeThreshold(k+1) divided by codeThreshold(k) equals φ^(-1) ≈ 0.618. In plain words: each successive code family's threshold is about 61.8% of the previous one. The surface code, as the second family (k=9 in the framework's indexing), gets a predicted threshold of φ^(-9) ≈ 0.013 ≈ 1.3%, which is consistent with the measured ~1% figure. The repetition code at k=0 has threshold 1, which the framework reads as its ~50% practical tolerance.

In Recognition Science, this is not a fitted curve. The framework derives φ from its cost function, then defines the threshold ladder as a structural consequence. The five-family count matches the framework's configDim D = 5, a number that appears elsewhere in its derivations. The library packages all of this into a single certificate structure QECThresholdCert, which bundles the count, the positivity of every threshold, and the decay law into one machine-checked object.

What the declaration does not claim is important. The framework does not prove that real quantum hardware must have thresholds in this ratio; it defines a model and checks it against one known value. The ~1.3% surface code prediction is a consistency check, not a derivation from first principles of the measured threshold. The framework does not claim that only five code families exist, only that its model chooses these five as canonical. And the framework does not claim that φ^(-k) thresholds are optimal or necessary for error correction; it claims that its model predicts this pattern.

inductive QECCodeFamily where
  | repetition | surface | colour | topological | concatenated
  deriving DecidableEq, Repr, BEq, Fintype

theorem qecCodeFamilyCount : Fintype.card QECCodeFamily = 5 := by decide

theorem codeThreshold_decay (k : ℕ) :
    codeThreshold (k + 1) / codeThreshold k = phi⁻¹ := by
  unfold codeThreshold
  have hk := (pow_pos phi_pos k).ne'
  rw [pow_succ, mul_inv]
  field_simp [hk, phi_ne_zero]

noncomputable def codeThreshold (k : ℕ) : ℝ := (phi ^ k)⁻¹

What this page does not claim

The framework does not prove that real quantum hardware thresholds must follow the φ^(-k) ratio. The framework does not claim that only five quantum error correction code families exist. The framework does not derive the surface code's ~1% threshold from first principles; it checks consistency against the measured value.

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

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