IndisputableMonolith.Mathematics.NumberSystemsFromRS
The module defines NumberSystem and NumberSystemCert structures and certifies that positive rationals contain the J-cost domain. Researchers building RS models or the phi-ladder would cite these definitions when selecting a concrete domain. The module consists of type definitions plus one certification statement linking rationals to the J functional equation.
claimA NumberSystem is a domain $D$ equipped with a J-cost function satisfying the Recognition Composition Law. The module certifies that $D = {Q}^+$ (positive rationals) contains the J-cost domain.
background
Recognition Science starts from the J functional equation whose solutions yield the self-similar fixed point phi and the eight-tick octave. This module supplies the first concrete number system by showing that the positive rationals form a valid carrier for the J-cost. It imports Mathlib and introduces the sibling declarations NumberSystem, numberSystemCount, rational_contains_jcost_domain, NumberSystemCert and numberSystemCert.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the rational carrier used by later constructions of the phi-ladder and mass formulas. It directly supports the J-uniqueness step (T5) in the forcing chain by exhibiting an explicit domain closed under the required operations.
scope and limits
- Does not derive values of physical constants.
- Does not prove the Recognition Composition Law inside the rationals.
- Does not address the transition to three spatial dimensions.