The information-rate bound for fiber-57 returns in Collatz orbits

Let $(n_0,n_1,\ldots)$ be a Collatz orbit with $n_0>1$. Let $\tau_1<\tau_2<\cdots$ be successive fiber-$57$ return times, set $q_j=\lfloor n_{\tau_j}/64\rfloor$, and let $I_r$ be the invariant core modulo $8^r$. Define the raw inter-chain return fraction

$$\widehat R_r(n_0)=\limsup_{N\to\infty}\frac{\#\{1\le j\le N:q_j\bmod8^r\in I_r,\ j\text{ inter-chain}\}}{\#\{1\le j\le N:j\text{ inter-chain}\}},$$

and the normalised ratio

$$R_r(n_0)=\frac{\widehat R_r(n_0)}{|I_r|/8^r}.$$

Thus $R_r=1$ is the baseline density and $R_r<1$ means under-visitation of $I_r$. Define the per-return memory supply by $c'(n_0)=-\log_2R_r(n_0)$. Information-rate bound. For every $n_0>1$ and every $r\ge2$ at which absorption holds,

$$c'(n_0)<c_0=\log_2\!\left(\frac{1024}{129}\right)\approx2.989.$$

The main reduction theorem states that this orbitwise bound would imply the Collatz conjecture. The statement is explicitly the sole unresolved dynamical input in the paper, and the numerical constant is based only on the known-gap partial kernel; including unresolved larger gaps would make the bound harder to satisfy.