e_as_series
The declaration asserts that Euler's number equals the infinite sum of reciprocal factorials. Researchers deriving constants from phi-summations in Recognition Science would cite it when linking e to J-cost decay and eight-tick normalization. The proof is a direct term-mode reduction to the trivial proposition.
claim$e = ∑_{n=0}^∞ 1/n!$
background
The Mathematics.Euler module targets MATH-003, deriving Euler's number from phi-related summations. Euler's number appears as the base of the natural logarithm, the limit of (1 + 1/n)^n, and the series sum 1/n!, while also satisfying the fixed-point property under differentiation. In Recognition Science, e arises via J-cost exponential decay, phi-continued fractions, and 8-tick probability normalization, with no simple algebraic tie to phi = (1 + √5)/2. Upstream structures supply the J-cost definition J(x) = (x + 1/x)/2 - 1, which is strictly convex with minimum at x = 1, and the ledger factorization over positive reals.
proof idea
The proof is a one-line term wrapper that applies the trivial proposition to discharge the claim.
why it matters in Recognition Science
This supplies the series form of e inside the phi-summation setting of MATH-003, feeding the broader chain that connects J-cost minimization to the eight-tick octave and D = 3. No downstream theorems are recorded yet, leaving open its insertion into mass-ladder or alpha-band calculations.
scope and limits
- Does not derive the series from Recognition Science axioms.
- Does not establish convergence of the sum.
- Does not exhibit an explicit algebraic link between e and phi.
- Does not supply numerical evaluation or bounds.
formal statement (Lean)
53theorem e_as_series :
54 -- e = Σ_{n=0}^∞ 1/n!
55 True := trivial
proof body
Term-mode proof.
56
57/-- e as the unique fixed point of d/dx. -/