phi_pos
The golden ratio φ is established to be strictly positive. Researchers deriving physical constants from the Recognition Science forcing chain would cite this result when bounding J-cost expressions or phi-ladder masses. The proof is a direct term-level reference to the corresponding field inside the Constants structure.
claim$φ > 0$, where $φ$ is the positive real solution to the fixed-point equation $φ = 1 + 1/φ$.
background
The Inequalities module assembles basic real-number lemmas that underwrite the Recognition Science derivation of constants and inequalities. The symbol φ denotes the golden ratio, introduced as the self-similar fixed point in the Constants structure of the Law of Existence module. That structure bundles the core CPM constants (Knet, Cproj, Ceng, Cdisp) together with their sign and ordering properties; the phi_pos field records the positivity of this fixed point.
proof idea
One-line wrapper that applies the phi_pos field of the Constants structure.
why it matters in Recognition Science
The result supplies the sign foundation for all subsequent phi-based inequalities in the module, including J_formula_pos and phi_gt_one. It directly supports the T6 step of the forcing chain where φ is forced as the unique positive self-similar fixed point, and thereby enables the eight-tick octave and D = 3 derivations downstream.
scope and limits
- Does not compute the numerical value of φ.
- Does not address sign properties of other roots of the fixed-point equation.
- Does not extend to complex or matrix generalizations.
formal statement (Lean)
85theorem phi_pos : φ > 0 := Constants.phi_pos
proof body
Term-mode proof.
86
87/-- φ > 1 -/