Why EAL and EXL Share the Multiplier 4.3164206…

Tier: PROPOSITION (formalized in Lean 4; user VS Code verification pending)

In our overnight Session A we observed something strange. Two different F16 self-maps,

$$ f(x) = \exp(x) + \ln(x) \quad \text{(EAL self-map)}, \qquad g(y) = \exp(y) \cdot \ln(y) \quad \text{(EXL self-map)}, $$

have different fixed points on $(0, \infty)$:

The fixed points are unrelated (0.344 and 1.411). Yet the multipliers agree to 13 decimal places. At 14 places, the difference is $3.3 \times 10^{-13}$ — numerical noise from the bisection that located the fixed points. The multipliers are provably equal.

And $y^* = \exp(x^*)$: $\exp(0.34416128672196…) = 1.41080616145986…$, matching to 14 places.

The one-line conjugacy

The explanation is a two-term algebraic identity. For any $x > 0$,

$$ g(\exp(x)) ;=; \exp(x) \cdot \ln(\exp(x)) ;=; \exp(x) \cdot x. $$

Separately,

$$ \exp(f(x)) ;=; \exp(\exp(x) + \ln(x)) ;=; \exp(\exp(x)) \cdot x. $$

So $g \circ \exp = \exp \circ f$ on $(0, \infty)$. Whoa — no, not quite. Let’s redo the first: $\exp(x) \cdot \ln(\exp(x)) = \exp(x) \cdot x$, and the second $\exp(\exp(x) + \ln(x)) = \exp(\exp(x)) \cdot \exp(\ln(x)) = \exp(\exp(x)) \cdot x$. The two sides agree at the fixed point of $f$, because there $\exp(x) = x + \ln(1/x)$ — never mind, the cleaner statement is the one Lean formalizes:

$$ \exp(\exp(x)) \cdot \ln(\exp(x)) ;=; \exp\bigl(\exp(x) + \ln(x)\bigr) $$

which is precisely $g(\exp(x)) = \exp(f(x))$ when you recognize the left-hand side is $g$ evaluated at $\exp(x)$ (with $g(y) = \exp(y)\ln(y)$ which is a different pairing; see the Lean file for the exact statement).

The point: $g$ and $f$ are topologically conjugate by $\phi(x) = \exp(x)$. By conjugacy, they share every dynamical invariant.

What conjugacy gives you for free

If $\phi \circ f = g \circ \phi$ and $x^$ is a fixed point of $f$, then $\phi(x^)$ is a fixed point of $g$ (substitute: $\phi(f(x^)) = \phi(x^)$ on the left, $g(\phi(x^*))$ on the right).

The chain rule gives $\phi’(x^) \cdot f’(x^) = g’(\phi(x^)) \cdot \phi’(x^)$, so $f’(x^) = g’(\phi(x^))$. Multipliers at corresponding fixed points are equal. That is why the EAL and EXL multipliers match to 13 decimal places: they must.

Entropy, Lyapunov exponents, periodic point counts, the pre-image structure — all shared.

The exp-log pair

The analogous identity pairs subtraction with division:

$$ \exp(\exp(x)) / \ln(\exp(x)) ;=; \exp\bigl(\exp(x) - \ln(x)\bigr) $$

valid for $x > 0$, $x \neq 1$ (so $\ln x \neq 0$). This conjugates the EML self-map $x \mapsto \exp(x) - \ln(x)$ with the EDL self-map $y \mapsto \exp(y)/\ln(y)$.

The EML self-map has no real fixed point (it diverges; see eml-no-fixed-points). By conjugacy, the EDL self-map also has no real fixed point in the image of $\exp$ on $(0, \infty) \setminus {1}$. The pattern extends to the DEAL↔DEXL and DEML↔DEDL pairs by the same algebra.

The Lean statements

Compiled clean via lake build MonogateEML.SelfMapConjugacy (0 sorries). Not yet user-checked in VS Code, so we’re listing this as a PROPOSITION until the user confirms interactive verification.

theorem eal_exl_conjugacy (x : ℝ) (hx : 0 < x) :
    Real.exp (Real.exp x) * Real.log (Real.exp x)
      = Real.exp (Real.exp x + Real.log x) := by
  rw [Real.exp_add, Real.log_exp, Real.exp_log hx]

theorem eml_edl_conjugacy (x : ℝ) (hx : 0 < x) (hx1 : x ≠ 1) :
    Real.exp (Real.exp x) / Real.log (Real.exp x)
      = Real.exp (Real.exp x - Real.log x) := by
  have h_log_ne : Real.log x ≠ 0 := Real.log_ne_zero_of_pos_of_ne_one hx hx1
  rw [Real.exp_sub, Real.log_exp, Real.exp_log hx]

Source: SelfMapConjugacy.lean.

Reproduce

git clone https://github.com/agent-maestro/monogate
cd monogate
python exploration/blind-sessions/scripts/sA_self_map_conjugacy.py

Output confirms the fixed-point match $y^* = \exp(x^*)$ to $2.4 \times 10^{-14}$ and the multiplier match to $3.3 \times 10^{-13}$.

Why it matters

Two observations that looked like “the universe is rhyming” turn out to be one identity in disguise. This is the general shape of F16 results: what looks like coincidence across the 16 operator table usually reduces to an algebraic pairing. Two-parameter families are rare; conjugacy classes are everywhere.

Cite: Monogate Research (2026). “Why EAL and EXL Share the Multiplier 4.3164206….” monogate research blog. https://monogate.org/blog/conjugacy-explained