Information Qecthreshold From Phi Ladder Code Threshold Decay

The decay law

Quantum error correction codes have a threshold: a maximum error rate below which the code can suppress errors arbitrarily well by scaling up. Different code families have different thresholds, from about 50% for the simplest repetition code down to about 1% for the surface code and lower for more complex families. The Recognition Science framework models these thresholds as a ladder: each rung is a code family, and the threshold at rung k is defined as phi raised to the power k, then inverted, where phi is the golden ratio (1 + sqrt(5))/2 ≈ 1.618.

The declaration codeThreshold_decay proves, in the framework's machine-checked library of formal theorems, that the ratio of consecutive thresholds is exactly the reciprocal of the golden ratio. In symbols: codeThreshold(k+1) / codeThreshold(k) = phi^(-1). This means each step down the ladder multiplies the threshold by about 0.618, a constant decay factor. The proof is a short algebraic identity: since codeThreshold(k) = phi^(-k), the ratio of consecutive terms is phi^(-1) by the laws of exponents. The theorem holds for every natural number k, and the framework also proves every threshold is positive.

The framework's library also certifies that there are exactly five canonical QEC code families (repetition, surface, colour, topological, concatenated), matching the dimension count of five in the framework's configuration space. The surface code threshold at roughly 1.3% lands near phi^(-9) ≈ 0.013, which the framework cites as consistent with the measured value. But that consistency is an empirical check, not a derived theorem: the framework does not prove that real quantum codes must have these thresholds, nor that the five families are the only ones that exist.

What the declaration establishes is a formal relationship inside a model. If you accept the definition that the k-th threshold equals phi^(-k), then the decay ratio follows necessarily. The theorem does not derive the threshold values from first principles, does not identify which rung corresponds to which physical code, and does not claim the golden ratio is the only possible decay factor for real error correction. It proves the arithmetic of the ladder, nothing more and nothing less.

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]

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

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

What this page does not claim

The theorem does not derive the absolute threshold values from first principles. The theorem does not prove that real quantum codes must follow the phi-ladder. The theorem does not identify which rung corresponds to which physical code family.

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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