IndisputableMonolith.Mathematics.NumberSystemsFromRS
The module defines number systems in Recognition Science and certifies that the rational system contains the J-cost domain of positive rationals. Researchers formalizing RS number foundations cite it to ground J-cost on rationals before extending to reals. The module supplies definitions and certificates using only Mathlib, with no internal proofs.
claimThe rational number system contains the J-cost domain: positive rationals form a valid domain for $J(x) = (x + x^{-1})/2 - 1$.
background
Recognition Science uses the J-cost function on positive reals as the core of its functional equation. This module introduces NumberSystem as a structure whose domain supports J-cost operations and NumberSystemCert as its certificate. The module documentation states that the rational system supplies this domain via positive rationals, relying on Mathlib for the underlying rational arithmetic.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
The module supplies NumberSystem and NumberSystemCert definitions that feed the mathematics layer of the Recognition framework. It anchors the J-cost domain for later steps in the forcing chain, including T5 J-uniqueness.
scope and limits
- Does not prove any properties of the J-cost function.
- Does not extend the domain beyond positive rationals.
- Does not import Recognition Science constants or physics results.
- Does not contain theorem statements or proof bodies.