e_gt_one
exp(1) exceeds 1. Researchers deriving Euler's number from phi-summations in Recognition Science reference this positivity bound when establishing growth model foundations. The argument reduces directly to the companion inequality exp(1) > 2 via linear arithmetic.
claimThe base of the natural exponential satisfies $e > 1$ where $e = exp(1)$.
background
The MATH-003 module targets derivation of Euler's number from phi-related summations. Euler's number is the base of the natural logarithm, the limit of (1 + 1/n)^n as n approaches infinity, and the sum of the series 1/n!. Recognition Science links e to J-cost exponential decay, phi-related continued fractions, and 8-tick probability normalization.
proof idea
The proof is a one-line wrapper that applies linear arithmetic to the stronger bound exp(1) > 2.
why it matters in Recognition Science
This inequality anchors the MATH-003 exploration of phi and e connections. It supports positivity in growth models from J-cost minimization, which is strictly convex with unique minimum at argument 1. The module notes both constants appear in growth processes yet lack a simple algebraic relation, with e tied to continuous rates. It aligns with the phi fixed point and eight-tick octave in the forcing chain.
scope and limits
- Does not compute the numerical value of e.
- Does not derive any algebraic relation between e and phi.
- Does not address the series or limit definitions of e.
- Does not extend the bound to complex arguments.
formal statement (Lean)
181theorem e_gt_one : Real.exp 1 > 1 := by
proof body
Term-mode proof.
182 linarith [e_gt_two]
183
184/-! ## φ and e: A Deeper Connection? -/
185
186/-- Is there a deep connection between φ and e?
187
188 Both are transcendental.
189 Both appear in growth processes.
190
191 φ: Discrete (Fibonacci recursion)
192 e: Continuous (differential equations)
193
194 They represent two sides of growth:
195 - φ: Optimal discrete packing/ratios
196 - e: Optimal continuous rates -/