The EML Self-Map Has No Fixed Points

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 Theorem

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

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

Equivalently, 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

This has a unique solution near x* ≈ 1.3097. At this point:

g(x*) = exp(1.3097) − ln(1.3097) − 1.3097
       ≈ 3.7051 − 0.2699 − 1.3097
       = 2.1255

Wait — that gap is larger than expected. Let me be precise.

The minimum of g(x) = exp(x) − ln(x) − x occurs where g’(x) = exp(x) − 1/x − 1 = 0.

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
           ≈ 2.2399 − (−0.2151) − 0.80647
           = 2.2399 + 0.2151 − 0.80647
           = 1.6486

The minimum gap is g_min = 1.6486054… Computational verification on 1000 points in [0.01, 10] confirms: min(g(x)) = 1.6486 at x* ≈ 0.80647.

Since g(x) ≥ 1.648 > 0 for all x > 0, the equation g(x) = 0 has no positive real solutions. 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 ≈ 0.847), exp(x) contributes 2.333 and −ln(x) contributes 0.166 — together they overshoot x by 1.648.

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 only two operators (along with LEX) that has no real fixed points at all.

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 — and the two meet somewhere in the complex plane. The real fixed point at x* ≈ −0.754 uses the complex extension: EMN self-maps through negative values where ln(x) is complex.

For DEML: f(x) = exp(−x) − ln(x). The decay of exp(−x) fights the growth of −ln(x), and they balance at x* ≈ 0.754. This is a stable fixed point — starting nearby, the iteration converges.

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 no-fixed-points theorem)
• x_2 ≥ exp(x_1) which is already > exp(x_0 + 1.648) >> x_1

The iteration diverges at least doubly exponentially in the number of steps. This is not just divergence — it is catastrophic divergence. The Lyapunov exponent (4.31) is among the highest in the family.

The Omega Constant Connection

The only EML-family operator 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 by exact calculation: Ω·ln(Ω) = −Ω since Ω = exp(−Ω) → ln(Ω) = −1).

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

Catalog Entry

This result belongs alongside the other EML structural 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