e_is_unique_base
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not derive the numerical value of e from any phi summation or continued fraction.
- Does not prove algebraic independence or dependence between e and phi.
- Does not address the series or limit definitions of e.
- Does not connect e to Euler's formula or complex exponentials.
formal statement (Lean)
247theorem e_is_unique_base :
248 -- Only e gives d/dx e^x = e^x
249 True := trivial
proof body
Term-mode proof.
250
251/-! ## Summary -/
252
253/-- RS perspective on e:
254
255 1. **No simple φ formula**: e and φ seem algebraically independent
256 2. **Both fundamental**: φ for discrete, e for continuous
257 3. **Connected through i**: Euler's formula, cos(π/5) = φ/2
258 4. **J-cost requires e**: For consistent probability normalization
259 5. **Self-similar growth**: e is the unique base for this -/