In 1974 Michael Shub conjectured that every smooth map's topological entropy dominates the logarithm of its homological spectral radius, $h_{top}(f)≥logρ(f_{∗})$. For half a century the inequality held in every sufficiently smooth category — until a fifteen-page preprint (OpenAI, September 25, 2026) built a $C^1$ self-map with zero entropy whose homology still doubles every step.
Disproving Shub's Entropy Conjecture: A Step-by-Step Study Companion & Proof Audit takes that construction apart bolt by bolt. Watch the clock speed crawl toward its stall at $t=50$ without ever arriving; watch two register channels compute a readout 67 iterates in advance; watch a single well-timed push hurl the main coordinate to infinity — erasing all entropy while homology remembers the doubling. Every proof is expanded with zero skipped steps, every definition is instantiated in explicit numbers, and every quantitative claim is re-verified by an independently executed Python suite whose output appears verbatim, including an honest audit of where floating-point arithmetic must fail and why the theorem is untouched.
The volume closes with a fully audited treatment of Andrey Gogolev's $C^r$ simplification on $S^1\times S^2\times S^2\times S^2$. Written for graduate students and researchers in dynamical systems — and for anyone who wants to see, line by line, exactly how a fifty-year-old conjecture falls.