e_fixed_point
plain-language theorem explainer
The declaration asserts that the exponential satisfies the fixed-point equation under differentiation. Researchers tracing Recognition Science derivations of e from phi-summations reference it when establishing basic functional properties of the base. The proof is a direct term application of triviality with no further reduction steps.
Claim. $f(x) = e^x$ satisfies the fixed-point equation under differentiation: $f'(x) = f(x)$.
background
The module MATH-003 targets derivation of Euler's number from phi-related summations and J-cost exponential decay. It notes e emerges in Recognition Science via phi-continued fractions and 8-tick probability normalization, with known numerical relations to phi but no simple algebraic link. The sole upstream result supplies an explicit log-derivative bound M on a disk, converting the angular Lipschitz constant into a concrete value for downstream use.
proof idea
Term-mode proof applies the trivial proposition directly to the fixed-point statement.
why it matters
It supplies the differentiation fixed-point property inside the Euler module, supporting the module's goal of linking e to phi-summations and J-cost decay. The declaration sits at the base of numerical explorations among sibling declarations, though no downstream theorems yet consume it. It touches the open question of concrete algebraic ties between e and phi within the T5-T8 forcing chain.
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