e_minus_phi
plain-language theorem explainer
The definition introduces the real number obtained by subtracting the golden ratio from the base of the natural exponential. Analysts examining constant interrelations in Recognition Science would reference this quantity during explorations of φ-summation derivations for e. It arises as a direct subtraction in the reals with no lemmas applied.
Claim. Define the real number $e - phi$ by $exp(1) - phi$, where $phi$ is the self-similar fixed point from the phi-forcing chain.
background
The module MATH-003 targets derivation of Euler's number e from φ-related summations. Euler's number appears as the limit of (1 + 1/n)^n, the series sum 1/n!, and the unique fixed point of the exponential derivative. In Recognition Science, e connects to J-cost exponential decay, φ-continued fractions, and 8-tick probability normalization. The declaration sits among sibling definitions that express e through various φ operations.
proof idea
The definition is a direct one-line subtraction of phi from exp(1) in the reals.
why it matters
This definition supplies a basic building block for the module's goal of linking e to φ-summations within Recognition Science. It supports investigation of the relationship noted in the module documentation, where no simple algebraic tie is known but connections via J-cost and continued fractions remain open for exploration. It aligns with the framework's use of phi as the self-similar fixed point in the forcing chain.
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