Some problems amplify tiny errors into huge ones no matter how good your method is. The CONDITION NUMBER measures that fragility: feed the problem a whisker of input error, how much output error comes out? A well-conditioned problem (κ near 1) is sturdy; an ill-conditioned one (κ huge) is a house of cards — nearly-parallel equations, a near-singular matrix. It’s the honest ceiling on accuracy. Slide from a sturdy problem to a fragile one.
The condition number κ(A)=‖A‖·‖A⁻¹‖ (for a linear system, the ratio of largest to smallest singular value) measures how much a problem AMPLIFIES relative input error into output error: ‖δx‖/‖x‖ ≤ κ · ‖δb‖/‖b‖. A well-conditioned problem has κ≈1 (sturdy); an ill-conditioned one has κ⋙1 (fragile) — geometrically, nearly-parallel equations / a near-singular matrix, where the solution point is barely pinned down. Crucially it is a property of the PROBLEM, not the algorithm: no method beats it, and you lose ~log₁₀κ digits of accuracy. It bounds the reliability of every solve here (pairs with [[the-machine-epsilon]]). A fail-loud self-check throws unless a near-singular matrix reports κ⋙ that of the identity. ◆ real numerical methods, node-verified.
κ bounds WORST-case amplification (a property of the problem, method-independent); a specific input may fare better. This is the honest ceiling — with [[the-machine-epsilon]], the floor under every method here. The κ = σ₁/σₙ definition is exact.