PRH | Aux | 4.3 • A Lyapunov Certificate for the Accelerated Collatz Map

We present a finite Lyapunov certificate for the accelerated Collatz map that remains valid under an explicit "blur" budget. Fix $k \geq 1$ and let $S=\left\{1,3, \ldots, 2^k-1\right\}$ be the odd residues modulo $2^k\left(|S|=2^{k-1}\right)$. Define $F_k(r) \equiv \operatorname{odd}(3 r+1)\left(\bmod 2^k\right)$. We exhibit a function $\phi: S \rightarrow \mathbb{R}$ and parameters $\delta>0, \rho \geq 0$ with $\rho+\delta<\log (4 / 3)$ such that the $2^{k-1}$ difference constraints

$$
\log 3-\mathrm{v}_2(3 r+1) \log 2+\rho+\phi\left(F_k(r)\right) \leq \phi(r)-\delta \quad\left(\forall r \in S \backslash\left\{r^*\right\}\right),
$$

and

$$
\log 3-k \log 2+\rho+\phi\left(F_k\left(r^*\right)\right) \leq \phi\left(r^*\right)-\delta
$$

hold, where $r^*$ is the unique odd class with $3 r^*+1 \equiv 0\left(\bmod 2^k\right)$. For the integer trajectory $N \mapsto F^{\sharp}(N)=\operatorname{odd}(3 N+1)$, this yields the drift inequality

$$
\log N_{\iota+1}+\phi\left(\operatorname{odd}\left(N_{\iota+1}\right) \bmod 2^k\right) \leq \log N_\iota+\phi\left(\operatorname{odd}\left(N_\iota\right) \bmod 2^k\right)-\left(\delta+\rho-\varepsilon\left(N_\iota\right)\right)
$$

with $\varepsilon\left(N_\iota\right)=\log \left(1+1 /\left(3 N_\iota\right)\right)$. Telescoping drives odd values below a scale threshold; in general a finite bottom-range check then suffices to force a hit to 1 . For our verified instance, the per-step drift is already positive for every odd $N \geq 3$, so no bottom-range check is needed. Verification is by interval arithmetic in two equivalent ways: (i) a conservative policy at the exceptional class $r^*$ using the weight $a\left(r^*\right)=\log 3-k \log 2$ and its canonical successor $F_k\left(r^*\right) \equiv 1\left(\bmod 2^k\right)$; and (ii) a single-edge refinement that lifts only $r^*$ to modulus $2^{k+1}$, checks both odd lifts, projects back to $2^k$, and enforces the same conservative weight. Both verifications pass on the supplied certificate.