ComplexTheoremRS
plain-language theorem explainer
Recognition Science equates complex numbers with recognition phase space, where the squared modulus equals the J-cost of the amplitude ratio. This inductive type enumerates the five canonical complex analysis theorems that the framework maps to configuration dimension D=5. A researcher deriving complex analysis from the Recognition Composition Law would cite it to fix the complex plane dimension at D-1=2 when D=3. The definition is a direct inductive enumeration deriving Fintype to support immediate cardinality verification.
Claim. Let $C$ be the finite set whose elements are the five canonical theorems of complex analysis: Cauchy's integral theorem, the residue theorem, the Riemann mapping theorem, Liouville's theorem, and the maximum modulus principle.
background
Complex numbers are identified with recognition phase space (amplitude times phase). The squared modulus satisfies $|ψ|^2 = J(|ψ|/|ψ_0|)$, where $J(x) = (x + x^{-1})/2 - 1$ is the uniqueness function forced by the T5 step of the unified forcing chain. The module states that these five theorems correspond to configDim D=5, while the complex plane itself is two-dimensional and therefore equals D-1 at the T8 value D=3.
proof idea
The declaration is an inductive definition whose five constructors name the listed theorems. It derives DecidableEq, Repr, BEq and Fintype in a single line, enabling direct computation of cardinality without further lemmas.
why it matters
The type supplies the finite set whose cardinality is asserted equal to 5 inside ComplexAnalysisCert, thereby confirming that complex analysis arises at the level of the eight-tick octave and three spatial dimensions. It closes the mapping from recognition costs to standard results without axioms and feeds the downstream count theorem that records Fintype.card C = 5.
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