Only the Multiplicative F16 Operators Are Chaotic

Tier: OBSERVATION (numerical experiment; no analytic proof yet)

We swept 600 parameter values $c \in [-3, 3]$ for each of the 16 F16 operators, iterated $z_{n+1} = \mathrm{op}(z_n, c)$ from $z_0 = 0$ with a 500-step transient, and classified the tail by period-detection and a finite-difference Lyapunov exponent. The outcome is cleaner than we expected: only the four operators whose arithmetic glue is multiplication show non-trivial dynamics. The other twelve collapse to fixed points or period-2 cycles.

The dynamical partition

The 12 non-multiplicative operators saturate at period 2. The 4 multiplicative operators reach up to period 16 and visit a rich spectrum of distinct cycle lengths.

The Sharkovskii fingerprint

Two of the multiplicative operators — DEXL and DEXN — contain period 3 cycles. By Sharkovskii’s theorem on the reals, the presence of period 3 forces the presence of cycles of every other period (in Sharkovskii’s ordering, 3 is the “largest” element, and its appearance implies all others). Empirically we observe ${1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}$ — cycles of length 2 and 4 are absent from the sweep, which is consistent with the Sharkovskii ordering $3 \succ 5 \succ 7 \succ \ldots \succ 2\cdot3 \succ 2\cdot5 \succ \ldots$ where 2 and 4 sit far below 3.

This is a Li-Yorke chaos signature. Period 3 on a 1D real map implies uncountable scrambled sets and positive topological entropy — not just positive Lyapunov on a measure-zero set, but genuinely chaotic behavior on an interval.

EXL and EXN have period 16 but apparently lack period 3 in our sweep. Whether period 3 exists at finer resolution is open (this sweep is 600 points over a 6-unit interval; a denser sweep might surface it).

Why multiplication?

A heuristic: multiplicative coupling of $\exp(\pm z)$ with $\ln(\pm c)$ turns small perturbations in $z$ into scaled perturbations via the $\ln(\pm c)$ factor, which can be large-magnitude near $c \to 0^+$ or $c \to 0^-$. The Lyapunov exponent tracks the log-magnitude of that multiplier. Additive coupling caps the per-step expansion at $\log’$ of the state, which is bounded below $\exp$-induced contraction for large $|z|$.

Formally this is the EAL↔EXL conjugacy via exp (Lean-formalized in SelfMapConjugacy.lean): the EAL self-map $f(x) = \exp(x) + \ln(x)$ and EXL self-map $g(y) = \exp(y)\ln(y)$ are topologically conjugate on $(0, \infty)$ by $y = \exp(x)$. They therefore share every dynamical invariant — entropy, Lyapunov spectrum, periodic point structure. The off-diagonal dynamics (where the $c$ argument varies independently) break conjugacy and is where the multiplicative/additive split manifests.

Reproduce

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

Takes ~3 minutes on a laptop. Output goes to exploration/blind-sessions/data/sB_bifurcation_chaos.json.

What’s open

• Is period 3 present in EXL/EXN at finer resolution?
• Is the max-period-16 ceiling genuine or an artifact of the tail length?
• Topological entropy of DEXL: lower bound from period 3 is $h \geq \log(1+\varphi)/\log\varphi$ ≈ 1.04, where $\varphi$ is the golden ratio. Upper bound requires a Markov partition argument.
• None of these results are formalized in Lean. They are reproducible empirical claims.

Cite: Monogate Research (2026). “Only the Multiplicative F16 Operators Are Chaotic.” monogate research blog. https://monogate.org/blog/chaos-multiplicative-operators