Information Qecthreshold From Phi Ladder Qec Code Family Count

The five code families

Quantum error correction protects quantum information by spreading it across many physical qubits. The standard families each use a different pattern of redundancy: repetition codes duplicate the information, surface codes arrange qubits on a grid, colour codes use a more complex lattice, topological codes rely on global properties of the system, and concatenated codes layer one code inside another. These five families are the canonical approaches in the field.

The Recognition Science library contains a machine-checked collection of formal theorems. One declaration, qecCodeFamilyCount, proves that the count of these five families is exactly 5. The proof is direct: the library defines the family type, lists the five constructors, and the checker verifies the count by computation. The declaration is a theorem about a definitional choice, not a discovery about nature.

The same library defines a threshold function for code families. For a natural number k, the threshold is phi^(-k), where phi is the golden ratio. The library proves this threshold is always positive and that each step down the ladder divides the previous threshold by phi. These are formal facts about the defined function, not measurements of real quantum hardware.

In Recognition Science, the framework models the five families as the configurational dimension D = 5. It also notes that the surface code threshold of about 1.3% is close to phi^(-9), which is about 0.013. This agreement is a hypothesis with a named falsifier: if a measured surface code threshold moved far from the phi-ladder value, the prediction would fail. The count of five families is proved; the threshold agreement is not.

What the declaration does not claim is broader. It does not prove that real quantum error correction must have exactly five families, nor that measured thresholds must follow the phi-ladder. It does not establish that the golden ratio governs quantum hardware. The declaration only counts the families as defined in the library's own type.

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

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]

structure QECThresholdCert where
  five_families : Fintype.card QECCodeFamily = 5
  threshold_pos : ∀ k, 0 < codeThreshold k
  phi_decay : ∀ k, codeThreshold (k + 1) / codeThreshold k = phi⁻¹

What this page does not claim

The declaration does not prove that real quantum error correction must have exactly five families. The declaration does not establish that measured thresholds must follow the phi-ladder. The declaration does not claim that the golden ratio governs quantum hardware.

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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