Affine Orbit Structure

Statement

Theorem (Affine Orbit Structure). For any integer $n$ with dropping time $k$ and orbital oddity $s$, within each residue-class subgroup mod $2^{k-s}$:

$$\text{dest}(n) = \frac{3^s}{2^{k-s}} \cdot n + C$$

where $C$ is a constant depending only on the subgroup (residue class), not on $n$. The slope $3^s / 2^{k-s}$ — the contraction ratio — is the same across all subgroups of $\text{Dset}_k$. Only the intercept $C$ varies.

Similarly, the orbit sum $\sum_{j=0}^{k-1} f^j(n)$ and orbit maximum $\max_j f^j(n)$ are exact affine functions of $n$ within each subgroup.

Significance

This theorem reveals that Collatz dynamics are piecewise-affine: within each residue class of a Dropping Set, the destination, orbit sum, and orbit maximum are all linear functions of the starting value. Every drop is a linear map, making orbits algebraically tractable rather than chaotic. The universal contraction ratio $3^s/2^{k-s}$ connects directly to the Odd Stopping Time Spectrum: $3^s/2^{k-s} < 1$ iff $k > s \cdot \log_2 6$.

Examples

Table of verified slopes:

$k$	$s$	Slope $3^s/2^{k-s}$	Decimal	Example
1	0	$1/2$	0.500	$\text{dest} = n/2$
3	1	$3/4$	0.750	$\text{dest} = \frac{3}{4}n + \frac{1}{4}$
6	2	$9/16$	0.5625	$\text{dest} = \frac{9}{16}n + \frac{5}{16}$
8	3	$27/32$	0.844	Two subgroups, same slope
11	4	$81/128$	0.633	Three subgroups, same slope
13	5	$243/256$	0.949	Seven subgroups, same slope

Worked example for $\text{Dset}_3$: All members satisfy $n \equiv 1 \pmod{4}$. For $n = 5$: $\text{dest} = \frac{3}{4}(5) + \frac{1}{4} = \frac{16}{4} = 4$. Verify: $5 \to 16 \to 8 \to 4$. ✓

Verified exactly (using Fraction arithmetic) for all $k$ up to 37.

Proof

By induction on $k$ (number of Collatz steps).

Base case ($k = 0$): $f^0(n) = n = \frac{3^0}{2^0} n + 0$. ✓

Inductive step ($k \to k+1$): Assume after $k$ steps with parity word $w = (w_0, \ldots, w_{k-1})$ containing $s$ ones:

$$f^k(n) = \frac{3^s}{2^{k-s}} n + C_w$$

We must show this holds at step $k+1$.

Case 1: $f^k(n)$ is even (halving step, $s$ unchanged):

$$f^{k+1}(n) = \frac{f^k(n)}{2} = \frac{3^s}{2^{k-s+1}} n + \frac{C_w}{2}$$

Slope = $3^s / 2^{(k+1)-s}$. ✓

Case 2: $f^k(n)$ is odd ($3x+1$ step, $s \to s+1$):

$$f^{k+1}(n) = 3 \cdot f^k(n) + 1 = \frac{3^{s+1}}{2^{k-s}} n + 3C_w + 1$$

Slope = $3^{s+1} / 2^{(k+1)-(s+1)}$. ✓

Key insight (bit consumption): The parity of $f^k(n)$ is determined by $n \bmod 2^{k-s}$:

• Each even step consumes one bit of $n$ (requires knowing one more bit of $n$ to determine the next parity)
• Each odd step consumes no additional bits (the result of $3x+1$ is always even, so the next parity is determined for free)

After $k$ steps with $s$ odd steps and $k-s$ even steps, exactly $k-s$ bits of $n$ have been consumed. Therefore, the parity word — and hence the entire affine map — is determined by $n \bmod 2^{k-s}$.

Corollaries

Corollary 1 (Orbital Oddity Invariance). All members of $\text{Dset}_k$ in the same residue class share the same parity word, hence the same number of odd steps $s$. Since all residue classes defining $\text{Dset}_k$ have the same $k$ and the same $s$ (otherwise they'd belong to a different set), orbital oddity is constant within $\text{Dset}_k$.

Corollary 2 (Affine Orbit Sums). Each intermediate value $f^j(n)$ for $j = 0, \ldots, k-1$ is affine in $n$ (by the theorem at step $j$). Therefore the orbit sum $\sum_{j=0}^{k-1} f^j(n)$ and orbit max $\max_j f^j(n)$ are also affine in $n$ within each subgroup.

Corollary 3 (Period of $\text{Dset}_k$). Members of $\text{Dset}_k$ are exactly the integers in certain residue classes mod $2^{k-s}$. The period is $2^{k-s}$.

Corollary 4 (Dropping Condition). The contraction ratio $3^s / 2^{k-s} < 1$ is equivalent to $k > s \cdot \log_2 6$, linking to the Odd Stopping Time Spectrum: $k = \lceil s \cdot \log_2 6 \rceil$.

Related Results

• Logarithmic Escape Theorem — uses the affine structure to bound self-chains
• Bit Destruction Bound — derives bit destruction rate from the contraction ratio
• 3-Adic Mixing — the $3^s$ multiplication scrambles residue classes