Information Quantum Error Rate Rs

The threshold claim

In quantum computing, a fault-tolerance threshold is the error rate per operation below which error correction can, in principle, keep a computation reliable. The Recognition Science framework proposes a specific value for this threshold: 1.18 percent. That number comes from its cost function J, evaluated at the golden ratio phi and divided by ten. The framework's research note records this as a prediction, and it compares favorably with common surface code thresholds around 1.0 percent and color code thresholds near 1.1 percent.

The machine-checked library of formal theorems establishes something far narrower. It defines a cost function, domainCost m e = J(m/e), and establishes three general facts about it: the cost is zero when m equals e, the cost is never negative for positive inputs, and phi minus 1.5 is positive. These are true statements about the cost function itself, but they do not connect that function to quantum error rates. The module defines no mapping from physical error processes to the variables m and e. Without that mapping, the 1.18 percent figure is a research aspiration, not a derived result.

In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions, a theorem established in the framework's library. The framework models error rates as a ratio of two quantities, m and e, and applies J to that ratio. The threshold prediction then reads off a specific number from the golden ratio. But the proof stops at the general properties of J; it never shows that quantum error rates actually follow this ratio. The formal content is a template, shared across many modules, waiting for a definition of m and e in quantum error correction's own terms.

What the module does establish, in plain language, is modest but solid: the cost function has the three properties stated, and the golden ratio minus 1.5 is a positive number. These are kernel-checked facts. The 1.18 percent threshold is not among them. A reader should take the framework's prediction as a hypothesis with a clear falsifier: a quantum error correction code with a threshold measured well outside the 1.18 percent band, say below 0.9 percent or above 1.4 percent, would count against it. The framework's own note marks the result as a partial pass, not a proof.

The distinction matters for anyone using the framework. The library's theorems give you a cost function with clean properties. The threshold prediction gives you a number to test. The gap between them is the missing definition of m and e. Until that definition exists, the module is a promise about quantum error rates, not a demonstration of them.

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

noncomputable def cert : QECThresholdExactRS where
  cost_at_eq := domainCost_at_eq
  cost_nonneg := domainCost_nonneg
  threshold_pos := canonicalThreshold_pos

What this page does not claim

The 1.18 percent value is not a proved theorem about quantum error correction. The module does not establish that quantum error rates follow the ratio m/e. No claim is made that the threshold prediction has been empirically confirmed.

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

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Derived articles

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