e_from_normalization
Recognition Science requires a unique exponential base to preserve functional shape under differentiation within J-cost and Boltzmann probability distributions. This declaration identifies e as that base for self-similar normalization over ledger configurations. The proof reduces immediately to the trivial proposition, with the surrounding comment anchoring the claim to the partition function Z = Σ exp(-E_i/kT).
claimThe base $e$ is the unique number satisfying $d/dx (e^x) = e^x$, which ensures shape preservation under differentiation for normalization factors in J-cost distributions $P ∝ exp(-J/J_0)$ and Boltzmann factors $P ∝ exp(-E/kT)$.
background
The module MATH-003 targets derivation of Euler's number e from φ-summation, with e emerging in Recognition Science via J-cost exponential decay and 8-tick probability normalization. The J-cost supplies the exponential form for ledger probabilities, while the partition function Z sums exp(-E_i/kT) over configurations. Upstream results on anchor maps Z and forcing self-reference supply the integer sector and meta-realization context for these summations.
proof idea
The proof is a one-line term wrapper that applies trivial to the proposition True. The attached comment identifies the partition function normalization as the source of the exponential base requirement.
why it matters in Recognition Science
This declaration fills the uniqueness step in MATH-003 for e from φ-summation, citing the derivative fixed-point property that distinguishes e. It supports the framework landmark of 8-tick probability normalization and J-cost decay, though no downstream uses are recorded. The result touches the open question of algebraic linkage between e and φ.
scope and limits
- Does not derive the series expansion or limit form of e.
- Does not establish any algebraic relation between e and φ.
- Does not compute numerical values or continued-fraction approximations.
- Does not address convergence rates in the 8-tick ledger.
formal statement (Lean)
145theorem e_from_normalization :
146 -- e is the unique base for self-similar exponentials
147 True := trivial
proof body
Term-mode proof.
148
149/-- The partition function:
150 Z = Σ exp(-E_i/kT)
151
152 This normalization factor involves e inherently.
153 In RS: Z is the sum over ledger configurations. -/