ComplexTheoremRS
Recognition Science encodes the five core theorems of complex analysis as constructors of an inductive type. A researcher deriving complex analysis from the recognition functional equation would cite this enumeration to fix the complex plane dimension at D-1. The definition lists the theorems explicitly and derives decidable equality plus finite cardinality from the five cases.
claimThe inductive type enumerating the five canonical theorems of complex analysis consists of the constructors Cauchy’s integral theorem, the residue theorem, the Riemann mapping theorem, Liouville’s theorem, and the maximum modulus principle.
background
Complex numbers are treated as recognition phase space with amplitude and phase, where |ψ|² equals the J-cost of the normalized amplitude. The module equates the five theorems to configuration dimension D=5 and sets the complex dimension to D-1=2 at spatial dimension D=3. The upstream Liouville function λ(n)=(-1)^Ω(n) supplies the arithmetic ingredient for one of the listed theorems.
proof idea
This is an inductive definition that enumerates the five theorems by name. Lean derives the DecidableEq, Repr, BEq, and Fintype instances automatically from the finite list of constructors.
why it matters in Recognition Science
The type supplies the enumeration required by ComplexAnalysisCert to certify that exactly five theorems exist and that complex dimension equals 3-1. It anchors the complex-analysis block of the RS derivation to the spatial dimension D=3 fixed by forcing-chain step T8.
scope and limits
- Does not contain proofs of any listed theorem.
- Does not derive the theorems from the J-uniqueness or recognition composition law.
- Does not address analytic continuation or contour deformation.
- Does not compute explicit residues or conformal maps.
formal statement (Lean)
22inductive ComplexTheoremRS where
23 | cauchy | residue | riemannMapping | liouville | maximumModulus
24 deriving DecidableEq, Repr, BEq, Fintype
25