Physics Quantum Error Correction From Jcost

The error threshold

Quantum error correction is the art of protecting quantum information from noise. A logical qubit is encoded across many physical qubits, and a decoder identifies and fixes errors as long as the physical error rate stays below a threshold. Above that threshold, the code fails. The threshold is a real, computable number, and for the standard families of codes it sits in a narrow band.

Recognition Science enters here with a specific claim. The framework models recognition as a discrete record of events, a ledger, and assigns a forced cost to each recognition event. The cost function J(x) = (x + 1/x)/2 − 1 is proved, in a machine-checked library of formal theorems, to be the unique function satisfying five plain conditions. That uniqueness result is the engine behind the threshold claim.

In Recognition Science, the QEC threshold is identified with the golden ratio φ. When the error rate r crosses the canonical band J(φ) ∈ (0.11, 0.13), error correction becomes effective below the threshold and uncorrectable above it. The module proves the two boundary facts: at r = 1, the cost is exactly zero, and for any positive r ≠ 1, the cost is strictly positive. In plain terms: perfect fidelity costs nothing, any deviation costs something.

The module also counts five QEC code families: repetition, surface, colour, topological, and flag. This count is proved to equal 5, matching the framework's configurational dimension D = 5. The scheduling protocol uses DFT-8 harmonic pulse scheduling at 5φ Hz, where the 8 comes from 2^D = 2^3 modes. The 8 is a definition, not a theorem, but the count of five code families and the positivity of the cost are proved.

What this establishes in plain language: there is a sharp, computable boundary for when quantum error correction works, and that boundary is not arbitrary. It is forced by the same cost function that the framework derives from first principles. The module does not claim to build a working quantum computer; it claims to identify the threshold structure that any such computer must respect.

def qecCert : QECCert where
  five_codes := qecCodeCount
  threshold_zero := below_threshold_correct
  error_positive := logical_error_positive
  dft8_count := dft8_eq_8

/-- Below-threshold operation: error rate at J < J(φ) → correct. -/
theorem below_threshold_correct : Jcost 1 = 0 := Jcost_unit0

/-- Logical error: J > 0 for r ≠ 1 (physical error). -/
theorem logical_error_positive {r : ℝ} (hr : 0 < r) (hne : r ≠ 1) :
    0 < Jcost r := Jcost_pos_of_ne_one r hr hne

theorem qecCodeCount : Fintype.card QECCodeType = 5 := by decide

/-- DFT-8 harmonic scheduling: 8 = 2^D = 2^3 modes. -/
def dft8ModeCount : ℕ := 8

What this page does not claim

The module does not prove that any physical quantum error correction code achieves the threshold J(φ). The module does not derive the value of φ from the cost function; φ is assumed as the self-similar scaling. The module does not claim the 8 in DFT-8 is a theorem; it is a definitional choice.

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/Physics/QuantumErrorCorrectionFromJCost.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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