attempt3
plain-language theorem explainer
The declaration assigns the real value 2 + 1/phi^2 as a candidate approximation to e. Researchers testing algebraic links between the golden ratio and the base of the natural exponential in Recognition Science cite this expression among several trial forms. Construction is a direct substitution of phi into the algebraic term without intermediate lemmas or reductions.
Claim. Define $a_3 := 2 + 1/phi^2$, where $phi = (1 + sqrt(5))/2$ satisfies $phi^2 = phi + 1$.
background
Module MATH-003 targets derivation of Euler's number e from phi-summations. Euler's number is introduced as the limit of (1 + 1/n)^n and the series sum 1/n! from n = 0 to infinity. In Recognition Science, e emerges from J-cost exponential decay, phi-related continued fractions, and 8-tick probability normalization. The module records the absence of any known simple algebraic relation between e and phi.
proof idea
The definition is a direct assignment of the expression 2 + 1/phi^2.
why it matters
This definition supplies one concrete candidate in the suite of phi-based trials inside MATH-003. It probes possible ties to the phi-ladder and 8-tick normalization without invoking the Recognition Composition Law or the T0-T8 forcing chain. The module leaves open whether deeper connections exist through J-cost decay or continued-fraction expansions.
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