e_to_phi
This definition introduces the real number e raised to the power of the golden ratio φ. Researchers exploring exponential relations between e and φ in Recognition Science reference it when examining summations or normalizations in the Euler module. The declaration is a direct noncomputable abbreviation that applies the standard real exponential to the imported phi constant.
claimDefine the real number $e^φ$ by the assignment $e^φ := (exp 1)^φ$, where $φ$ is the golden ratio fixed point.
background
The module MATH-003 targets derivation of Euler's number e from φ-related summations. It recalls that e is the base of the natural logarithm with series expansion Σ 1/n! and notes its emergence in Recognition Science via J-cost exponential decay together with 8-tick probability normalization. The imported phi constant originates from the PhiForcing module as the self-similar fixed point of the forcing chain.
proof idea
The declaration is a direct definition with an empty proof body; it simply equates the identifier to the expression (exp 1) raised to phi using the standard real exponential.
why it matters in Recognition Science
The definition supplies a basic exponential link that supports the MATH-003 exploration of e-φ connections through J-cost decay and 8-tick normalization. The module explicitly states there is no known simple algebraic relationship yet permits such constructions for further study. No downstream theorems depend on it in the current graph.
scope and limits
- Does not derive e from any summation or continued-fraction expression.
- Does not invoke the J-cost function or any forcing-chain step.
- Does not supply numerical values or computational reductions.
- Does not claim any physical or dimensional interpretation.
formal statement (Lean)
78noncomputable def e_to_phi : ℝ := (exp 1) ^ phi