NumericalMethod
NumericalMethod enumerates five canonical numerical algorithms that Recognition Science equates with configuration dimension D equal to 5. A researcher certifying a complete solver set for RS dynamics would cite this enumeration when checking that the method collection matches the required cardinality. The declaration is a bare inductive type whose Fintype instance is obtained automatically by derivation.
claimLet $M$ be the finite set of numerical methods consisting of Newton's method, Euler integration, Runge-Kutta integration, Gaussian elimination, and the fast Fourier transform.
background
The module states that five canonical numerical methods correspond to configDim D = 5. In RS these are Newton's method, Euler integration, Runge-Kutta, Gaussian elimination, and FFT. The same setting identifies DFT-8 as the canonical algorithm with 8 = 2^D modes and FFT as its fast implementation requiring 24 = 3 × 8 operations per tick.
proof idea
The declaration is a direct inductive definition that introduces five constructors and derives the instances DecidableEq, Repr, BEq, and Fintype automatically. No separate proof body is supplied.
why it matters in Recognition Science
The enumeration supplies the five_methods field of NumericalAnalysisCert and is counted by the numericalMethodCount theorem. It realizes the RS identification of numerical method cardinality with configDim D = 5 and links the eight-mode DFT to the eight-tick octave of the forcing chain. The definition therefore closes the interface between abstract RS dynamics and concrete numerical implementation.
scope and limits
- Does not supply convergence rates or stability conditions for any method.
- Does not specify the underlying vector space or field of application.
- Does not connect the methods to the J-cost functional or defectDist.
- Does not prove that the listed methods are exhaustive for all RS equations.
formal statement (Lean)
22inductive NumericalMethod where
23 | newton | eulerIntegration | rungeKutta | gaussElimination | fft
24 deriving DecidableEq, Repr, BEq, Fintype
25