IndisputableMonolith.Mathematics.NumberTheoryFromRS
The module derives number theoretic identities from Recognition Science, starting from the golden ratio equation φ² = φ + 1 and extending to Fibonacci relations at phi powers. Researchers connecting RS foundations to algebraic sequences would cite it for the phi-ladder backbone. It structures content as targeted algebraic lemmas and count certifications built on the imported constants.
claimThe module centers on the identity $φ^2 = φ + 1$, with derived results including $φ^5$-Fibonacci and $φ^8$-Fibonacci relations plus RS key count certifications.
background
The module imports Mathlib and the Constants module, whose doc-comment states: 'The fundamental RS time quantum (RS-native). τ₀ = 1 tick.' It introduces phi-based sequences and counts in the Recognition Science setting, where phi arises as the self-similar fixed point. The local theoretical setting extracts number theory directly from the unified forcing chain.
proof idea
The module organizes a sequence of lemmas: the phi square identity, higher-power Fibonacci relations, and certification counts. It applies algebraic reductions from the defining phi property, with no single overarching proof but a collection of targeted identities.
why it matters in Recognition Science
This module supplies the algebraic number theory layer for the Recognition framework, enabling later derivations on the phi-ladder and mass formulas. It fills the T6 phi step and supports certification objects such as NumberTheoryCert, though no direct used_by parents are listed.
scope and limits
- Does not derive general number theory outside phi identities.
- Does not compute physical constants beyond RS-native units.
- Does not address applications to quantum mechanics or gravity.
- Does not include external number theory results beyond Mathlib basics.