Information Qecthreshold From Phi Ladder

Error thresholds on a golden ladder

Quantum error correction codes protect fragile quantum information by spreading it across many physical qubits. Each code family has a threshold: the error rate below which adding more qubits helps, and above which it hurts. The repetition code, the simplest scheme, tolerates errors up to roughly 50 percent. The surface code, a modern favorite, works below about 1 percent. The colour code sits near 0.5 percent. These numbers are not arbitrary; they are measured properties of each code family.

The framework's ledger, a discrete record of recognition events, suggests a pattern. In Recognition Science, the golden ratio φ, about 1.618, appears as the unique self-similar scaling. The prediction is that adjacent code-family thresholds differ by a factor of 1/φ. The surface code threshold ≈ φ^(-9) ≈ 0.013 ≈ 1.3 percent, consistent with the measured range. The framework identifies five canonical families: repetition, surface, colour, topological, and concatenated.

The module QECThresholdFromPhiLadder formalizes this. It defines a type with exactly those five families, proved by computation. It defines a threshold function codeThreshold(k) = (φ^k)^(-1), which is always positive. The central theorem is the decay law: codeThreshold(k+1) / codeThreshold(k) = φ^(-1). Each step down the ladder divides the threshold by φ. The certificate bundles these three facts: five families, positive thresholds, and the φ decay ratio.

In plain language, the framework establishes a clean structural claim: if error thresholds follow the ladder, then they descend by a fixed golden ratio at each step. The machine-checked library proves the arithmetic of that ladder, not the physical claim that real codes obey it. The comparison to measured thresholds is an empirical check, not a theorem.

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]

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

What this page does not claim

The physical claim that real quantum error correction codes obey the φ ladder is not proved; only the arithmetic of the ladder is formalized. The surface code threshold is not derived from first principles; the match is an empirical consistency check. No claim that this ladder predicts the performance of any specific quantum computer.

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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