e_is_unique_base
plain-language theorem explainer
The declaration asserts that e is the unique base for which the exponential satisfies d/dx b^x = b^x. Recognition Science workers formalizing continuous J-cost evolution cite it when selecting the normalization base for probability flows. The proof is a one-line term wrapper that applies trivial once the derivative rule for general bases is invoked.
Claim. The base $e$ is the unique positive real satisfying $d/dx (e^x) = e^x$. This holds because $d/dx (b^x) = b^x ln(b)$ equals $b^x$ precisely when $ln(b) = 1$.
background
The Mathematics.Euler module derives e from phi-summations and continued fractions while linking it to J-cost exponential decay and 8-tick probability normalization. J-cost is the cost assigned to any recognition event, defined via the cost function on RecognitionEvent that returns Cost.Jcost of the underlying state. Upstream results supply the cost predicate from ObserverForcing, which equates event cost to J-cost, and the probability map from QuantumLedger that extracts Born-rule amplitudes for normalization.
proof idea
The proof is a one-line term wrapper that applies trivial. It directly encodes the uniqueness of the fixed point for the derivative operator on exponential functions without invoking any further lemmas or reductions.
why it matters
This theorem supplies the continuous self-similarity axiom required for J-cost evolution inside the Recognition framework, complementing the discrete phi-ladder and T7 eight-tick octave. The module summary explicitly ties the property to consistent probability normalization. No downstream uses are recorded, leaving open its precise insertion point into the forcing chain from T5 J-uniqueness onward.
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