e_gt_phi
plain-language theorem explainer
Euler's number strictly exceeds the golden ratio. Analysts comparing transcendental constants to algebraic irrationals in Recognition Science frameworks cite this bound when separating summation limits from self-similar fixed points. The argument chains the elementary inequalities phi less than 2 and exp(1) greater than 2 through linear arithmetic.
Claim. $e > phi$ where $phi = (1 + sqrt(5))/2$ is the golden ratio.
background
The module MATH-003 derives Euler's number e from phi-related summations, presenting e as the base of the natural logarithm, the limit of (1 + 1/n)^n, and the series sum 1/n!. In Recognition Science, e emerges from J-cost exponential decay and phi-related continued fractions within the eight-tick probability normalization. Upstream, phi_lt_two establishes phi < 2 by comparing sqrt(5) < 3, while e_gt_two shows exp(1) > 2 from the strict convexity of the exponential function via add_one_lt_exp.
proof idea
The proof obtains phi < 2 from the phi_lt_two lemma and exp(1) > 2 from e_gt_two, then concludes the desired inequality by linear arithmetic.
why it matters
This result feeds the distinctness theorem e_ne_phi, which asserts e ≠ phi. It supports the module's goal of exploring connections between e and phi without assuming algebraic relations, consistent with the absence of a simple closed-form link noted in the module documentation. Within the Recognition framework it separates the exponential base from the self-similar fixed point phi in summations and forcing chains.
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