Logarithmic Escape Theorem

Statement

Theorem (Logarithmic Escape). For any Dropping Set $\text{Dset}_k$ with period $P = 2^{k-s}$, the maximum number of consecutive self-transitions (drops where both $n$ and $\text{dest}(n)$ belong to $\text{Dset}_k$) starting from $n$ is at most:

$$m \leq \log_P(n) - 1$$

Equivalently: no orbit can remain in the same dropping set for more than $O(\log n)$ consecutive drops.

Significance

The contraction ratio alone would allow long self-chains (e.g., Set$_{13}$ with ratio 0.949 could theoretically self-loop ~124 times from $n \sim 10^5$). The modular tightening bound is much stronger and independent of the contraction ratio — it depends only on the period $P$. This forces orbits to change dropping sets frequently.

Examples

Bounds by set:

$k$	$P$	Bound	$n = 10^6$ gives $m \leq$
3	4	$\log_4 n$	9
6	16	$\log_{16} n$	4
8	32	$\log_{32} n$	3
13	256	$\log_{256} n$	2

Verified chains for $\text{Dset}_3$ ($P = 4$):

$m$	Required modulus	Smallest $n$	Chain
1	$n \equiv 1 \pmod{16}$	17	$17 \to 13$
2	$n \equiv 1 \pmod{64}$	65	$65 \to 49 \to 37$
3	$n \equiv 1 \pmod{256}$	257	$257 \to 193 \to 145 \to 109$
4	$n \equiv 1 \pmod{1024}$	1025	$1025 \to 769 \to 577 \to 433 \to 325$
7	$n \equiv 1 \pmod{65536}$	65537	8-element chain

Proof

By induction on chain length $m$ (modular tightening).

Each self-transition multiplies the required modular constraint by $P$.

Claim: $m$ consecutive self-transitions require $n$ to satisfy a specific residue class mod $P^{m+1}$.

Base ($m = 0$): Membership in $\text{Dset}_k$ requires $n \equiv r \pmod{P}$.

Inductive step: Suppose $m$ self-transitions require $n \equiv r_m \pmod{P^{m+1}}$.

Write $n = P^{m+1} \cdot a + r_m$. By the Affine Orbit Structure:

$$\text{dest}(n) = \frac{3^s}{P} \cdot n + C = 3^s \cdot P^m \cdot a + (\text{const})$$

For dest to lie in $\text{Dset}_k$ mod $P^{m+1}$ (not just mod $P^m$), we need:

$$3^s \cdot P^m \cdot a \equiv (\text{target}) \pmod{P^{m+1}} \implies 3^s \cdot a \equiv (\text{target}) \pmod{P}$$

Since $\gcd(3^s, P) = \gcd(3^s, 2^{k-s}) = 1$, this uniquely determines $a \bmod P$, giving:

$$n \equiv r_{m+1} \pmod{P^{m+2}}$$

Since $n \geq P^{m+1}$ is required, the chain length satisfies $m \leq \log_P(n) - 1$.

Corollaries

Corollary 1. No orbit can "camp" in a slow-contracting set. Even Set$_{13}$ ($P = 256$, ratio 0.949) forces escape after ~2 steps for $n < 10^6$.

Corollary 2. Combined with the Bit Destruction Bound, this gives progress guarantees: orbits cannot accumulate many low-destruction drops from any single set.

Related Results

• Affine Orbit Structure — prerequisite: the affine formula that makes modular tightening work
• Bit Destruction Bound — uses this result to bound slow-drop windows
• 3-Adic Mixing — complements this: Log Escape bounds same-set chains, mixing bounds cross-set slow chains