Core Definitions

This page collects the formal definitions used throughout the site. Each definition includes a worked example. The notation follows Paper 1 (dropping) conventions; see the Terminology Map for equivalences with Paper 2 (stopping).

Collatz Step

Definition 1 (Collatz Step). The Collatz function $f : \mathbb{Z}^+ \to \mathbb{Z}^+$ is defined by

$$f(n) = \begin{cases} n/2 & \text{if } n \text{ is even} \\ 3n+1 & \text{if } n \text{ is odd} \end{cases}$$

Example. $f(7) = 3(7)+1 = 22$ (odd rule), then $f(22) = 22/2 = 11$ (even rule).

Orbit

Definition 2 (Orbit). The orbit of $n$ is the sequence obtained by iterating $f$ until reaching 1:

$$\text{orbit}(n) = [n,\; f(n),\; f^2(n),\; \ldots,\; 1]$$

Example. The orbit of 6 is $[6, 3, 10, 5, 16, 8, 4, 2, 1]$.

Dropping Time

Definition 3 (Dropping Time). The dropping time of $n > 1$ is the smallest $k \geq 1$ such that $f^k(n) < n$:

$$\text{drop}(n) = \min\{k \geq 1 : f^k(n) < n\}$$

This is identical to the stopping time in Paper 2.

Example. For $n = 5$: $f(5) = 16$, $f^2(5) = 8$, $f^3(5) = 4$. Since $4 < 5$ and this is the first value below 5, the dropping time is 3.

Dropping Destination

Definition 4 (Dropping Destination). The dropping destination of $n$ is the first iterate that falls below $n$:

$$\text{dest}(n) = f^{\text{drop}(n)}(n)$$

Example. The dropping destination of 5 is $f^3(5) = 4$.

Dropping Orbit

Definition 5 (Dropping Orbit). The dropping orbit of $n$ is the segment of the orbit from $n$ up to (but not including) the dropping destination $d$:

$$\text{dorbit}(n) = [n,\; f(n),\; \ldots,\; 2d]$$

It includes $n$ and excludes the destination. Its length equals the dropping time.

Example. The dropping orbit of 5 is $[5, 16, 8]$ (length 3). The destination $4$ is excluded.

For $n = 3$: orbit proceeds $3 \to 10 \to 5 \to 16 \to 8 \to 4$, and the destination is $2 < 3$. The dropping orbit is $[3, 10, 5, 16, 8, 4]$ (length 6).

Dropping Set

Definition 6 (Dropping Set). For each positive integer $k$, the dropping set $\text{Dset}_k$ is defined as:

$$\text{Dset}_k = \{n \in \mathbb{Z} : n > 1,\; f^k(n) < n, \text{ and } f^j(n) \geq n \text{ for all } 0 < j < k\}$$

That is: the set of all integers whose first drop below themselves occurs at exactly step $k$.

Each dropping set is a union of arithmetic progressions (a fact that follows from the Affine Orbit Structure).

Note on existence

This is a definition, not a claim. We do not assert that every integer belongs to some $\text{Dset}_k$ — that assertion would be equivalent to the Collatz conjecture. We define $\text{Dset}_k$ as the set of integers with dropping time $k$, and our theorems describe properties of these sets. An integer with no finite dropping time would simply not belong to any $\text{Dset}_k$.

Example.

• $\text{Dset}_1$ = all even numbers $\{2, 4, 6, 8, \ldots\}$, since one halving gives $n/2 < n$.
• $\text{Dset}_3 = \{5, 9, 13, 17, \ldots\} = \{n : n \equiv 1 \pmod{4}\}$.

Orbital Oddity

Definition 7 (Orbital Oddity). The orbital oddity of $n$ is the count of odd numbers in its dropping orbit. All members of $\text{Dset}_k$ share the same orbital oddity $s$.

Example. The orbital oddity of 3 is 2. Its dropping orbit is $[3, 10, 5, 16, 8, 4]$, which contains two odd values: 3 and 5.

Syracuse Map

Definition 8 (Syracuse Map). The Syracuse map $S : \mathbb{Z}_{\text{odd}}^+ \to \mathbb{Z}_{\text{odd}}^+$ compresses each $3n+1$ step and all subsequent halvings into a single operation:

$$S(n) = \frac{3n+1}{2^{v_2(3n+1)}}$$

where $v_2(m)$ is the 2-adic valuation of $m$ (the largest power of 2 dividing $m$).

Example. $S(7) = \frac{3(7)+1}{2^{v_2(22)}} = \frac{22}{2^1} = 11$, since $22 = 2 \cdot 11$.

Alpha Value

Definition 9 (Alpha Value). For odd $n$, the alpha value is the 2-adic valuation of $3n+1$:

$$\alpha(n) = v_2(3n+1)$$

This counts how many halvings follow the $3n+1$ step in the Syracuse map.

Example. $\alpha(5) = v_2(16) = 4$, since $3(5)+1 = 16 = 2^4$.

Alpha Sequence

Definition 10 (Alpha Sequence). The alpha sequence of an odd number $n$ is the sequence of alpha values along its Syracuse orbit until reaching 1:

$$\alpha\text{-seq}(n) = [\alpha(n),\; \alpha(S(n)),\; \alpha(S^2(n)),\; \ldots]$$

The sequence terminates when the Syracuse orbit reaches 1.

Example. The alpha sequence of 3 is $[1, 4]$:

• $S(3) = (3 \cdot 3 + 1)/2^1 = 5$, so $\alpha(3) = 1$
• $S(5) = (3 \cdot 5 + 1)/2^4 = 1$, so $\alpha(5) = 4$

The alpha sequence of 7 is $[1, 1, 1, 3, 4]$.