The EML Self-Map Has No Fixed Points

Correction (2026-09-13): this post called its result a theorem; /theorems lists it as a proposition (T11), and its complete proof is below. It said a 1000-point grid confirmed the minimum at x* ≈ 0.80647, but that grid's spacing is 0.01 and its smallest value, 1.648629, is at x = 0.81. The operator table gave EML a Lyapunov exponent of 4.31 with no source; the iteration has no finite one. The catalog table at the end listed the Weierstrass and EMN results without their status: one argument has a gap, and the other is a conjecture.

Define the EML self-map:

$$f(x) = \text{eml}(x, x) = \exp(x) - \ln(x)$$

A fixed point would be a value x* where f(x*) = x*, i.e., exp(x*) − ln(x*) = x*.

There are none.

The Proposition

Proposition (EML No Fixed Points, T11): For all x > 0,

$$\exp(x) - \ln(x) > x$$

Consequently, the iteration x_{n+1} = exp(x_n) − ln(x_n) diverges for every starting point.

Proof:

Define g(x) = exp(x) − ln(x) − x. We show g(x) > 0 for all x > 0.

Find the minimum of g:

g'(x) = exp(x) − 1/x − 1 = 0

Since g”(x) = exp(x) + 1/x² > 0, g is convex on (0, ∞), so g’ has at most one root, and a root of g’ is the minimum of g.

At x = 0.5: g’(0.5) = exp(0.5) − 2 − 1 = 1.649 − 3 = −1.351 < 0. At x = 1.0: g’(1.0) = e − 1 − 1 = e − 2 ≈ 0.718 > 0.

So the minimum is in (0.5, 1.0). Numerically: x ≈ 0.80647* (root of exp(x) = 1 + 1/x).

At x* ≈ 0.80647:

g(0.80647) = exp(0.80647) − ln(0.80647) − 0.80647
           ≈ 2.2400 − (−0.2151) − 0.80647
           = 2.2400 + 0.2151 − 0.80647
           = 1.6486

The minimum gap is g_min = 1.6486054… A 1000-point grid on [0.01, 10] has spacing 0.01; its smallest value is 1.648629, at x = 0.81, and a grid that coarse cannot place the minimum more finely. Solving eˣ = 1 + 1/x directly gives x* = 0.8064659942 and g(x*) = 1.6486054403.

Since g(x) ≥ 1.648 > 0 for all x > 0, the equation g(x) = 0 has no positive real solutions. (Positivity needs no numerics: for x > 0, exp(x) ≥ 1 + x + x²/2 and ln(x) ≤ x − 1 give g(x) ≥ (x − 1)²/2 + 3/2 ≥ 3/2.) QED.

For x ≤ 0: ln(x) is undefined over ℝ (the operator has no real fixed points at all, not just no positive ones). □

The Gap Table

The gap never closes. At the minimum (x* ≈ 0.8065), exp(x) contributes 2.2400 and −ln(x) contributes 0.2151 — together they overshoot x by 1.6486.

The Operator Zoo Comparison

Each operator in the family defines a self-map op(x, x). Which ones have real fixed points?

EML is one of three operators here (with EMN and EDL) that have no real fixed points at all, and the only one of the three whose iteration stays defined and diverges from every starting point.

Why EML Is Different

For EMN: f(x) = ln(x) − exp(x). This is the negation of the EML self-map. Where EML always overshoots, EMN always undershoots: f(x) < 0 < x for every x > 0, so it has no real fixed point, and one step leaves the domain of ln. (The x* ≈ −0.754 this table first listed solves ln|x| − exp(x) = x, not f(x) = x.)

For DEML: f(x) = exp(−x) − ln(x). The decay of exp(−x) fights the growth of −ln(x), and they balance at x* ≈ 0.7536. This fixed point is unstable: |f′(x*)| = exp(−x*) + 1/x* ≈ 1.80 > 1, so starts near it are pushed away.

For EML: exp(x) grows too fast and ln(x) doesn’t slow it down enough. Both terms push f(x) above x. There’s no crossover.

Dynamical Consequences

EML(x,x) iteration: x_{n+1} = exp(x_n) − ln(x_n).

Starting from any x > 0:

• x_1 ≥ x_0 + 1.648 (by the proposition above)
• x_2 = exp(x_1) − ln(x_1) > exp(x_1) − x_1, and exp(x_1) > exp(x_0 + 1.648)

The iteration diverges at least doubly exponentially in the number of steps. This is not just divergence — it is catastrophic divergence. It has no finite Lyapunov exponent: along an orbit, ln|f′(xₙ)| grows without bound (from x₀ = 1 it is 0.54, 2.69, 14.2, then about 1.4 × 10⁶), and so does its running average. The 4.31 this post’s table gave has no source.

The Omega Constant Connection

One EML-family map with a globally stable attractor is:

$$x_{n+1} = \exp(-x_n)$$

This is the DEML self-map with y=1: deml(x, 1) = exp(−x).

Its unique fixed point is the Omega constant:

$$\Omega = W(1) \approx 0.5671432904…$$

where W is the Lambert W function. Every starting point in (0, ∞) converges to Ω.

Lyapunov exponent at Ω: −0.5671 (equal to −Ω exactly: f′(Ω) = −exp(−Ω) = −Ω, and ln(Ω) = −Ω since Ω = exp(−Ω)).

The EML operator diverges; its flipped cousin converges globally to a transcendental constant.

Catalog Entry

This result belongs alongside these other EML structural claims (their status is on /theorems):

The minimum gap 1.6486054… — is this a known constant? PSLQ against {e, π, ln(2), γ, √2} finds no relation at 15 digits. It is the unique positive minimum of exp(x) − ln(x) − x, defined by the transcendental equation exp(x*) = 1 + 1/x*. Numerically x* = 0.80646599… and g(x*) = 1.64860544…

Session M2 · Direction 13 of the Research Roadmap