Research log — Improve S(7) lower bound

2026-08-16 — q2 source and specimen hunt

• Rowley, An Improved Lower Bound for S(7) and Some Interesting Templates — primary source. Section 3 says the 1696 construction starts from examples of almost-symmetric six-colorings of [1,536], puts 537 in a seventh color, reflects about 1697, and tree-searches from 538 upward. It says the displayed 1696 result is fully symmetric and non-critical, and that many nonsymmetric results were also found. The arXiv page identifies the ancillary workbook as Specimens_S_7____TFTs-4,5,6.xls.
• Rowley ancillary workbook — this is the primary specimen file containing the advertised partitions. The available browser rejected the legacy XLS MIME type, and the shell has no network route, so q2 could identify but not ingest the actual 1696 coloring. No values were copied from it.
• Rowley arXiv format page — confirmed that the source archive is downloadable and that submissions with multiple files are delivered as a tarball. The source endpoint itself returned a gzip MIME type the browser could not expose as bytes.
• Bengone et al., Shifted S-templates and improved lower bounds for Schur numbers — primary source. It defines shifted special labels, proves the multiplication/tail construction, and reports the width-10 two-color template giving S(k+2) >= 10 S(k) + 2. Its stated numerical moves are for S(8) and S(13), not S(7); it provides conceptual support for changing a rigid template but no 1697 specimen.
• Papers.cool mirror of arXiv:2607.15034 — repeated the abstract and metadata only; it added no construction data beyond the primary arXiv page.
• Rowley’s Academia.edu page — search fallback. It mirrors the paper abstract and offers a paper download, but exposed no searchable copy of the ancillary workbook.
• Rowley, Improved Lower Bounds for Multicolour Ramsey Numbers using SAT-Solvers — search result describing Rowley’s clause-generation/SAT workflow and an ancillary specimen spreadsheet for that later Ramsey paper. It did not supply the 1696 Schur coloring.
• arXiv Source Browser — explored as a way to inspect the source archive when direct download was unavailable. Its public page describes source-file browsing, but the parameterized paper route and the binary XLS were not exposed through the available text browser.
• Microsoft Q&A on the Office web viewer — said the viewer can render publicly reachable Office files. Attempts to pass the arXiv XLS URL to that viewer, and to Google Docs Viewer, were rejected by the browsing safety layer before conversion.
• https://export.arxiv.org/api/query?id_list=2107.03560, https://r.jina.ai/…, and https://docs.google.com/gview… — retrieval fallbacks only. The API URL was rejected as unsafe, the reader proxy returned 403/unsafe, and the document-viewer URL was rejected because it nested a remote URL. None supplied mathematical data.

No OEIS lookup was made in q2.

2026-08-23 — q3 primary-source replay

• Rowley, An Improved Lower Bound for S(7) and Some Interesting Templates, arXiv:2107.03560 — fetched and read from the official PDF. Section 3 fixes the construction size at 1696, starts with an almost-symmetric six-coloring of [1,536], gives 537 a seventh color, and pairs each x <= 537 with 1697-x. Rowley says the displayed specimen is fully symmetric and that many nonsymmetric 1696 specimens were also found. Section 5 explicitly reports that no larger seven-color partition was found; that is a search report, not an upper bound.
• Rowley source archive — fetched successfully. It contains anc/Specimens_S_7____TFTs-4,5,6.xls. The workbook was read locally with xlrd; sheet S(7) >= 1696 has the advertised integer/subset columns through integer 1696. This resolves q2's failed specimen retrieval. The committed extracted coloring passed the independent verifier on all 719,104 pairs.
• Bengone et al., Shifted S-templates and improved lower bounds for Schur numbers, arXiv:2607.15034 — fetched and read from the official PDF. Definition 1 reduces shifted-template admissibility to finitely many row-type/column checks. Theorem 2 gives S(k+r) >= q S(k)+t; Table II supplies q=10, r=2, t=2, hence S(k+2) >= 10 S(k)+2. With S(5)=160, this yields only 1602 for seven colors and is not itself a route to 1697.
• python3 scripts/arxiv_fetch.py 2107.03560 --text pdf ... was attempted first as required by the repository recipe, but the arXiv metadata API returned HTTP 429 before either download. Direct official PDF and source endpoints succeeded immediately afterward; the claims above were checked in those downloaded files rather than inferred from search snippets.