numberSystemCount
Recognition Science induces exactly five number systems corresponding to distinct recognition depths. Researchers deriving the dimensional structure of physics from a single functional equation would cite this cardinality to fix configDim D at five. The proof is a term proof that applies the decide tactic to the Fintype instance of the inductive type.
claimThe finite type of number systems induced by Recognition Science has cardinality five: $|$natural, integer, rational, real, complex$| = 5$.
background
The module defines NumberSystem as an inductive type whose constructors are natural, integer, rational, real, and complex. These map to recognition depths: discrete counts, signed differences, ratios where the J-cost function is defined, continuous fields, and amplitude-phase pairs. The J-cost appears in upstream ObserverForcing.cost as the cost of a recognition event and in MultiplicativeRecognizerL4.cost as the derived cost on positive ratios.
proof idea
The proof is a one-line term proof that invokes the decide tactic on the equality Fintype.card NumberSystem = 5. The tactic succeeds because NumberSystem derives Fintype and enumerates exactly five constructors.
why it matters in Recognition Science
The theorem supplies the five_systems field of the downstream NumberSystemCert definition. It realizes the module statement that five canonical systems equal configDim D = 5, consistent with the forcing chain step T8 that sets three spatial dimensions. The enumeration closes without scaffolding.
scope and limits
- Does not derive the five systems from the Recognition Composition Law.
- Does not prove uniqueness of these systems beyond the inductive definition.
- Does not connect any system to the phi-ladder or mass formula.
- Does not compute J-cost values inside each system.
- Does not address extensions or additional number systems.
Lean usage
def numberSystemCert : NumberSystemCert where five_systems := numberSystemCount rational_pos := rational_contains_jcost_domainformal statement (Lean)
28theorem numberSystemCount : Fintype.card NumberSystem = 5 := by decide
proof body
Term-mode proof.
29
30/-- Rational system contains J-cost domain (positive rationals). -/