IndisputableMonolith.Mathematics.NumberSystemsFromRS
This module defines number systems derived from Recognition Science and certifies that the rational system contains the J-cost domain of positive rationals. Researchers building the phi-ladder or mass formulas from the T5 J-uniqueness would cite it for the base setup. It consists of definitions for NumberSystem and NumberSystemCert, importing only Mathlib, with no proofs.
claimThe positive rationals contain the J-cost domain: let $J(x) = (x + x^{-1})/2 - 1$ for $x > 0$; then the rational system includes all such $x$ in its domain.
background
Recognition Science derives physics from the forcing chain with T5 J-uniqueness given by $J(x) = (x + x^{-1})/2 - 1$, also written as cosh(log x) - 1, and the Recognition Composition Law J(xy) + J(x/y) = 2J(x)J(y) + 2J(x) + 2J(y). This module introduces NumberSystem as the structure for RS-derived number systems and NumberSystemCert as the certificate that the rational system contains the J-cost domain (positive rationals). It relies on Mathlib for the underlying rational arithmetic.
proof idea
this is a definition module, no proofs
why it matters in Recognition Science
This module supplies the number-system foundation that feeds theorems on the phi-ladder, eight-tick octave, and D = 3 spatial dimensions. It directly supports the mass formula yardstick * phi^(rung - 8 + gap(Z)) by establishing the rational J-cost domain required for the Recognition Composition Law and the alpha inverse band.
scope and limits
- Does not derive any physical constants or the forcing chain steps T0-T8.
- Does not address irrational or real completions of the number system.
- Does not prove the Recognition Composition Law itself.