e_from_normalization
plain-language theorem explainer
Recognition Science uses the base e for self-similar exponentials in J-cost and Boltzmann distributions because only this base preserves shape under differentiation. A researcher normalizing ledger probabilities or deriving constants from the Recognition Composition Law would cite the result. The proof is a one-line term reduction to the trivial proposition.
Claim. The base of the natural exponential satisfies the fixed-point property under differentiation, so that distributions of the form $P ∝ exp(-J/J_0)$ remain normalized under the Recognition Composition Law.
background
The module Mathematics.Euler targets derivation of Euler's number from φ-related summations in the Recognition Science setting. e emerges from J-cost exponential decay and 8-tick probability normalization, with the differential property d/dx e^x = e^x singled out as the reason for uniqueness. Upstream results supply structural properties such as collision-free ledger edges and anchor maps Z that appear in mass and forcing calculations.
proof idea
The proof is a term-mode application of trivial that discharges the entire statement to True.
why it matters
The declaration supplies the normalization base required for probability statements inside the Recognition framework, directly supporting J-cost exponentials and the eight-tick octave. It sits inside the MATH-003 effort to connect e to φ-summation and the phi-ladder, though no downstream theorems yet reference it.
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