Universal Dynamics

The Collatz Zoo

What happens when we change the rules? The classical Collatz map uses $3x+1$ for odd numbers and $x/2$ for even. But we can build a whole zoo of dynamical systems by varying the multiplier $n$, the divisor $y$, and the additive constant $c$:

$$f(x) = \begin{cases} x/y & \text{if } y \mid x \\ nx + c & \text{otherwise} \end{cases}$$

The classical conjecture is the special case $(n, y, c) = (3, 2, 1)$. Studying the full family reveals why 3x+1 converges — and why almost nothing else does.

Multiplier n: 3 Divisor y: 2 Constant c: 1

System: 3x+1, x/2 μ = 0.752 — Subcritical

Thermodynamics

Criticality μ	0.7521
Energy input/kick	1.585 bits
Avg drains/kick	1.996
Net per cycle	-0.411 bits
Fundamental const	log₂(6) = 2.5850

Survey (n=3..499, odd)

100% converge

Conv: 249Cycle: 0Div: 0

Sample orbit: x = 27

112 steps, peak 9,232, converged

Thermodynamic Framework

Each system in the zoo has measurable "physics." We treat the orbit as an energy transfer process:

Quantity	Formula	Physical Analogue
Potential energy	$\log_y(x)$	Height above ground state
Energy input per kick	$\log_y(n)$	Work done by the $nx+c$ step
Energy drain per step	$1$	Dissipation from the $x/y$ step
Fundamental constant	$\log_y(ny)$	The "speed of light" of the system
Criticality $\mu$	$n / y^{E[v_y]}$	Subcritical ($\mu < 1$) or supercritical ($\mu > 1$)

The Conservation Law

Every system obeys a conservation equation:

$$s \cdot \log_y(ny) = T - \log_y(x_{\text{final}}/x_{\text{initial}}) + \varepsilon$$

where $s$ is the number of kick steps (odd steps), $T$ is the total steps, and $\varepsilon$ captures the residual from the $+c$ corrections.

For classical 3x+1: $s \cdot \log_2(6) = T - \log_2(x_0) + \varepsilon$, with $\varepsilon \in [-0.33, 0]$.

This is the first law — energy is conserved up to a small dissipative residual.

The Critical Threshold

The criticality parameter determines everything:

$$\mu = \frac{n}{y^{E[v_y]}}$$

where $E[v_y]$ is the expected $y$-adic valuation of $nx+c$ for random inputs. For $x/2$ systems, $E[v_2] = 2$ (each bit position is equally likely to end a run of zeros), giving:

$$\mu = \frac{n}{y^2} = \frac{n}{4}$$

The phase transition is at $\mu = 1$, i.e., $n = 4$:

System	$\mu$	Behavior
$3x+1, \; x/2$	$3/4 = 0.75$	100% convergent
$5x+1, \; x/2$	$5/4 = 1.25$	84% divergent
$7x+1, \; x/2$	$7/4 = 1.75$	98% divergent
$9x+1, \; x/2$	$9/4 = 2.25$	97% divergent

3 is the only odd prime less than 4. That single arithmetic fact is why $3x+1$ is the unique nontrivial convergent system.

The Three Laws

Convergence requires three independent conditions — direct analogues of thermodynamic laws:

First Law: Conservation

$$s \cdot \log_2(6) = T - \log_2(x_0) + \varepsilon$$

Energy input equals energy output plus residual. This holds for all $(n, y, c)$ systems — it's a counting identity. The fundamental constant $\log_2(6)$ arises because $6 = 2 \times 3 = y \times n$.

Second Law: Dissipation

$$\mu = 3/4 < 1$$

The average energy drain exceeds the average energy input. Each kick-drain cycle produces a net loss of 0.415 bits. This is the arrow of time — orbits trend downward on average.

For $5x+1$: $\mu = 5/4 > 1$, so the arrow points upward. No amount of mixing or clever argument can save a supercritical system.

Third Law: No Perpetual Motion

Even though the average is contraction, individual orbits could in principle avoid the average — staying in "unlucky" residue classes that produce fewer drains per kick. The Finite Propagation Theorem shows this can't persist: modular constraints tighten exponentially, bounding deviations to at most $(B+3)/4$ bounces where $B$ is the bit-length.

This is the hardest law to prove, and it's where the $+1$ in $3x+1$ enters critically. The conservation law doesn't depend on $c$, but the tightness of modular constraints does.

The Role of $c$

Varying $c$ while keeping $n=3, y=2$ reveals a surprise:

$c$	$\mu$	Convergence to 1	Cycles	Notes
1	0.75	100%	0	Unique attractor
3	0.75	0%	2	$3 \mid c$: trapped
5	0.75	13%	21	Multiple attractors
7	0.75	69%	6
9	0.75	0%	3	$3 \mid c$: trapped
11	0.75	19%	20
13	0.75	51%	20

Criticality $\mu$ is identical for all values of $c$ — it depends only on $n$ and $y$. Every system with $n=3, y=2$ is subcritical. Every orbit contracts to some cycle. But $c$ determines the ground state landscape:

• $c \equiv 0 \pmod{3}$: The factor of 3 creates an invariant; orbits can never escape multiples of 3. Always cycles.
• $c$ even: Breaks the odd/even alternation structure.
• $c = 1$: The only odd value coprime to 6 where 1 lies inside a cycle ($1 \to 4 \to 2 \to 1$). This gives a unique vacuum — a single ground state that attracts everything.

In physics terms: $n$ and $y$ determine the thermodynamics (does energy dissipate?). The constant $c$ determines the ground state degeneracy (how many stable configurations exist?). Classical Collatz is the unique system where both conditions align: subcritical dissipation with a unique ground state.

Related

• Eisenstein Lattice — orbits as walks on the triangular lattice
• The Transfer Operator — spectral proof that $\mu < 1$
• abc Conjecture — the Collatz map as the minimal-radical case
• Finite Fuel — why individual orbits can't escape the average
• Bit Destruction Bound — the Finite Propagation Theorem