After “infinitely many composite terms”: how long the wait can be, how far the certificate method can reach and what stops it, and which questions are still open.
This animation explains Sections 6 to 8, 10 and 11 and Appendices B to E of the paper Finite graph certificates for composite terms in rational floor sequences by Yuri Odagiri: the waiting-time theorem for ⌊ξ(7/5)n⌋ (Theorem 3); the scope of the certificate method and its obstructions (the six operations, pseudo-orbits, the large-numerator and two-carry obstructions, the external-prime budget and the normal form); and the further results and open problems (recurrent carries, finite heights for 6/5 and 9/7, and the halting program for (6t+3)/(6t+1)). It follows the one-step method and word-labelled method videos.
The worst-case waiting times shown were found for this video by an exact interval search over every ξ with 14 ≤ ⌊ξ⌋ < 107; the longest wait there is 13 steps, far below the proved bound. The counts of the 491,520-vertex rank-certificate graph were recomputed and agree with Table 7 of the paper. The step counts of the halting program for 9/7 (every prime start up to 107 stops within 7 steps) were also computed for this video and are not in the paper. Each time it opens, the page re-checks the record orbit it shows in exact arithmetic.
The narration was generated with Gemini 3.8 Flash TTS and plays in sync with the captions. It starts in a window-filling view; W switches to the page view with the chapter list, and F uses the whole screen. If the audio file cannot be loaded (for example when the HTML file is opened on its own), the browser's speech synthesis reads the captions instead.
Paper: DOI 10.5281/zenodo.22997709 · Sources and reproduction: github.com/ixixi/rational-floor-certificates · 日本語版 · The one-step method · The word-labelled method · Verification and Lean