Information Quantum Error Correction Threshold Qec Threshold At Pos

The threshold ladder

Quantum error correction is the art of protecting fragile quantum information by spreading it across many physical qubits. Every such code has a fault-tolerance threshold: the error rate per physical operation below which the code suppresses logical errors exponentially, and above which it fails. The surface code, a leading candidate for real quantum computers, has a threshold near 1%; the colour code, a related family, sits near 1.7%. These numbers are not arbitrary; they stand in a ratio of about 1.55, close to the golden ratio φ ≈ 1.618.

The Recognition Science framework models this pattern explicitly. Its ledger, a discrete record of recognition events, generates a φ-ladder of characteristic scales. The framework defines a function qecThresholdAt(k) = φ^(-k)/2, placing the threshold for the k-th code family on rung k of this ladder. Rung 9 gives φ^(-9)/2 ≈ 0.011, matching the surface code's ~1% threshold; rung 8 gives φ^(-8)/2 ≈ 0.017, matching the colour code's ~1.7%. The framework calls this the φ-ladder structure for QEC thresholds.

The theorem qecThresholdAt_pos establishes one precise fact: for every natural number k, the threshold qecThresholdAt(k) is strictly positive. That is, every rung of the ladder names a positive error rate. The proof is a short chain: φ is positive, so any power of φ is positive, and dividing by 2 keeps it positive. The theorem is machine-checked in the framework's library of formal theorems, with zero axioms beyond the standard three logical postulates.

Two companion theorems complete the ladder's structure. qecThresholdAt_succ_ratio shows that moving one rung down multiplies the threshold by φ^(-1), and qecThresholdAt_adjacent_ratio shows that the ratio of adjacent thresholds is exactly φ^(-1). Together they certify that the ladder is geometric with ratio φ. The framework packages these three facts into a certificate structure, QECThresholdCert, ready for use in larger derivations.

What the theorem does not claim is equally important. It does not prove that any real quantum code achieves these thresholds; the match to surface and colour code values is an empirical observation, not a derivation. It does not predict a new code family's threshold from first principles. It only establishes the internal consistency of the framework's model: if thresholds sit on the φ-ladder, then each rung names a positive rate, and adjacent rungs shrink by exactly φ. The physical bridge from recognition events to actual quantum error correction remains an open question, not a proved result.

theorem qecThresholdAt_pos (k : ℕ) : 0 < qecThresholdAt k := by
  unfold qecThresholdAt
  exact div_pos (zpow_pos Constants.phi_pos _) (by norm_num)

theorem qecThresholdAt_succ_ratio (k : ℕ) :
    qecThresholdAt (k + 1) = qecThresholdAt k * phi⁻¹ := by
  unfold qecThresholdAt
  have hphi_ne := Constants.phi_ne_zero
  have : phi ^ (-((k : ℤ) + 1)) = phi ^ (-(k : ℤ)) * phi⁻¹ := by
    rw [show (-((k : ℤ) + 1)) = -(k : ℤ) + (-1 : ℤ) by ring]
    rw [zpow_add₀ hphi_ne]; simp
  have hcast : ((k + 1 : ℕ) : ℤ) = (k : ℤ) + 1 := by push_cast; ring
  rw [hcast, this]; ring

theorem qecThresholdAt_adjacent_ratio (k : ℕ) :
    qecThresholdAt (k + 1) / qecThresholdAt k = phi⁻¹ := by
  rw [qecThresholdAt_succ_ratio]
  field_simp [(qecThresholdAt_pos k).ne']

theorem qecThresholdAt_pos (k : ℕ) : 0 < qecThresholdAt k := by
  unfold qecThresholdAt
  exact div_pos (zpow_pos Constants.phi_pos _) (by norm_num)

What this page does not claim

The theorem does not prove that any real quantum code achieves these thresholds. The framework does not derive the surface or colour code thresholds from recognition principles alone. The physical bridge from recognition events to quantum error correction is not established.

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

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