codeThreshold
The threshold assigned to the k-th member of the QEC code family equals the reciprocal of phi raised to k. Workers on quantum error correction thresholds within Recognition Science cite this when establishing the geometric decay across families. The definition is a direct abbreviation that immediately yields the positivity and ratio properties used in the certification structure.
claimDefine the threshold function by $threshold(k) := phi^{-k}$ for each natural number $k$.
background
The QECThresholdFromPhiLadder module organizes quantum error correction codes into families whose error thresholds follow a phi-ladder. Repetition codes sit near 50 percent, surface codes near 1 percent, colour codes near 0.5 percent, and topological codes decay as phi to the minus k. The module records the Recognition Science prediction that adjacent families stand in the exact ratio 1 over phi, with the surface-code threshold approximately phi to the minus 9. Phi itself enters as the self-similar fixed point forced in the upstream chain.
proof idea
The declaration is a direct definition that sets the threshold for index k to the multiplicative inverse of phi to the power k. No lemmas or tactics are invoked; the expression is written out explicitly in the body.
why it matters in Recognition Science
This definition is used by codeThreshold_pos, codeThreshold_decay, and the structure QECThresholdCert. It supplies the explicit phi-ladder formula that realizes the module's claim of geometric decay with ratio phi inverse across the five canonical families. The construction sits inside the Information domain and is consistent with the observed surface-code threshold near 1.3 percent. It leaves open the precise numerical assignment of each integer k to a concrete code family.
scope and limits
- Does not assign specific code families to particular integers k.
- Does not derive phi from the forcing chain T0-T8.
- Does not prove that exactly five families exist.
- Does not compute numerical thresholds for any concrete code.
formal statement (Lean)
31noncomputable def codeThreshold (k : ℕ) : ℝ := (phi ^ k)⁻¹
proof body
Definition body.
32