Information Qecthreshold From Phi Ladder Qecthreshold Cert

The threshold certificate

Quantum error correction is the art of protecting quantum information against noise. A quantum computer's memory is fragile; error correction encodes the information redundantly so that small disturbances can be detected and fixed. Each family of error-correcting codes has a characteristic error threshold: the noise rate below which the code can protect information, and above which it cannot. The repetition code, the simplest family, has a threshold near 50 percent; the surface code, a workhorse of modern proposals, sits near 1 percent; the colour code near 0.5 percent. These numbers are measured or simulated in the laboratory and in classical computers, not derived from first principles.

The Recognition Science framework offers a different kind of statement. Its machine-checked library of formal theorems contains a small package called QECThresholdCert. The certificate does not measure any physical threshold. Instead, it packages three formal facts into a single reusable object. First, it records that there are exactly five canonical code families in the framework's classification: repetition, surface, colour, topological, and concatenated. Second, it states that every threshold value in its model is positive. Third, it states that the ratio of thresholds for adjacent families is exactly the inverse of the golden ratio, phi to the minus one power. In symbols: codeThreshold (k + 1) / codeThreshold k = phi⁻¹.

The golden ratio phi here is not a free parameter. It emerges from the framework's forcing chain, the sequence of theorems that derives constants from the cost of recognition. The threshold model chooses to define codeThreshold k as (phi^k)⁻¹, which means the thresholds decay geometrically as k grows. The surface code, with k = 9, lands at phi⁻⁹, approximately 0.013, or 1.3 percent, which is consistent with the measured surface code threshold near 1 percent. The certificate assembles the formal facts: the five families, the positivity of thresholds, and the phi decay ratio, all checked by the machine with no axioms beyond the standard three.

What the certificate does not claim is just as important as what it proves. It does not assert that real quantum hardware has thresholds that follow this exact ratio. The measured values for repetition, surface, and colour codes are consistent with the pattern, but the certificate itself contains no experimental data and no claim about the physical world. It is a statement about the framework's model of code families: a definitional choice of five families, a definition of threshold as a phi power, and the formal consequences of those choices. The prediction that adjacent code families ratio by 1/phi in reality is a hypothesis with a named falsifier: a measurement of two canonical families whose threshold ratio departs from phi⁻¹ by more than experimental uncertainty.

The value of the certificate is architectural. It gives the framework a single, compact object that any later theorem can import and use, without re-deriving the five-family count or the decay ratio each time. It also marks a boundary: the formal machinery can package a prediction cleanly, but it cannot make the prediction true. The physics remains an empirical question, and the certificate is the container, not the evidence.

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]

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

What this page does not claim

The certificate does not assert that real quantum hardware thresholds follow the phi ratio. The certificate contains no experimental data and no claim about the physical world. The framework does not derive the fine-structure constant alpha from its forcing chain.

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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