Information Qecthreshold From Phi Ladder Code Threshold

The threshold ladder

Quantum error correction (QEC) codes protect quantum information by spreading it across many physical qubits. Different code families tolerate different rates of error before they fail; that rate is the code's threshold. The repetition code, the surface code, and the colour code are three such families, with thresholds near 50%, 1%, and 0.5% respectively.

The framework's declaration codeThreshold defines a function that assigns a threshold to any natural number k: it is phi to the power k, inverted, where phi is the golden ratio (1 + sqrt(5))/2 ≈ 1.618. So for k = 1 the threshold is about 0.618, for k = 2 about 0.382, and so on. The declaration is a definition, not a theorem; it chooses a mathematical shape. Two theorems about it are proved in the machine-checked library: every threshold is positive, and each step down the ladder divides the previous threshold by exactly phi.

In Recognition Science, this ladder is tied to a claim about real code families. The framework models five canonical QEC code families (repetition, surface, colour, topological, concatenated) and predicts that adjacent families' thresholds should sit one phi-step apart. The surface code, with a measured threshold near 1.3%, lands close to phi to the power -9, which is about 0.013. That match is a hypothesis with a named falsifier: a measured surface-code threshold outside the band around 1.3% would break the prediction.

The declaration itself establishes only the arithmetic ladder and its two proved properties. It does not prove that any physical code family actually sits on that ladder, and it does not derive the golden ratio from quantum mechanics. The link between the formal ladder and measured thresholds is a prediction, not a theorem. The framework's library certifies the arithmetic cleanly: zero axioms beyond the standard three, zero unproved assumptions in the definition and its two theorems.

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

theorem codeThreshold_pos (k : ℕ) : 0 < codeThreshold k :=
  inv_pos.mpr (pow_pos phi_pos k)

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]

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

What this page does not claim

The declaration does not prove that any physical code family actually sits on the ladder. The golden ratio is not derived from quantum mechanics by this declaration. The surface-code match near 1.3% is a hypothesis, not a theorem.

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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