NumberSystemCert
NumberSystemCert certifies that the Recognition Science framework contains exactly five number systems, enumerated as natural, integer, rational, real, and complex, together with the positivity of the rational unit. A researcher deriving number systems from the J-cost domain or the recognition composition law would cite this structure to confirm the count matches configDim D = 5. The declaration is a plain structure definition that states the two fields directly with no reduction steps or lemma calls.
claimLet $N$ be the inductive type whose constructors are natural numbers, integers, rationals, reals, and complexes. A certificate asserts that the cardinality of $N$ equals 5 and that $1 > 0$ holds in the rationals.
background
The module defines NumberSystem as an inductive type with five constructors: natural, integer, rational, real, and complex. The module documentation states that these five systems equal configDim D = 5, each corresponding to a recognition depth, with the J-cost function defined on positive reals (a subset of the reals) and therefore on the rationals where ratios are formed. An upstream result in ElementaryRegularNumberSystems supplies an analogous structure that asserts the same cardinality claim for its own NumberSystem inductive type.
proof idea
NumberSystemCert is a structure definition. It introduces the two fields directly and contains no proof body, no tactics, and no applied lemmas.
why it matters in Recognition Science
The structure supplies the type for the concrete instance numberSystemCert in the same module, which populates five_systems via numberSystemCount and rational_pos via rational_contains_jcost_domain. It thereby anchors the five number systems to the Recognition Science requirement that they represent distinct recognition depths and support the domain of the J-cost function, linking to the overall derivation of constants from the forcing chain without invoking the RCL or phi-ladder itself.
scope and limits
- Does not derive the five systems from the J-uniqueness formula or the phi fixed point.
- Does not verify the Fintype instance beyond the deriving clause on the inductive type.
- Does not relate the number systems to the mass formula or Berry creation threshold.
- Does not address the eight-tick octave or the emergence of three spatial dimensions.
formal statement (Lean)
33structure NumberSystemCert where
34 five_systems : Fintype.card NumberSystem = 5
35 rational_pos : (1 : ℚ) > 0
36