IndisputableMonolith.Algebra.PhiRing
The Algebra.PhiRing module defines the golden ratio φ = (1 + √5)/2 and establishes its core algebraic identities including φ > 1, φ² = φ + 1, and relations to its conjugate ψ. These facts ground the self-similar fixed point in the Recognition Science framework. The module imports Cost and CostAlgebra for cost-based grounding and supplies the algebraic base for RecognitionCategory constructions.
claimLet $φ = (1 + √5)/2$ and $ψ = (1 - √5)/2$. Then $φ > 1$, $φ² = φ + 1$, $ψ² = ψ + 1$, $φψ = -1$, and $φ + ψ = 1$.
background
This module sits in the Algebra domain and supplies the ring-theoretic foundation for φ in Recognition Science. It builds directly on the Cost module, which introduces the J-cost function J(x) = (x + x⁻¹)/2 - 1, and on CostAlgebra, which equips costs with ring operations. The sibling lemmas establish √5 > 0, positivity of φ, the quadratic equation satisfied by φ, and the product and sum identities linking φ and ψ.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
PhiRing supplies the algebraic properties of φ required by the RecognitionCategory module, which constructs the recognition structure on the phi-ladder. It fills the T6 self-similar fixed point step in the forcing chain and enables downstream use of phiPow and PhiInt in mass and dimension formulas.
scope and limits
- Does not derive the full T0-T8 forcing chain.
- Does not define the J-cost function or Recognition Composition Law.
- Does not address the phi-ladder rung structure or mass formula.
- Does not treat the alpha band or Berry creation threshold.