% Bibliography for "An infinite family of irrational dilogarithm values at real quadratic units" (Theorem V, with
% Theorem U = pi^2 - 18 log^2 phi as corollary). Compiled 2026-11-07; HakimogluBrown2026 added for the V revision.
% Entries Hata1990 ... ViolaZudilin2018, SaffTotik, Lewin1981, Mathlib, Mathlib4, LeanFour, PNTplus, zeta5 are copied
% verbatim from lean_chi2/paper/references.bib, where each carries a verification comment (checked 2026-10-16/18).
% Own work: "Krisman, S." (rendered "S. Krisman"), as requested.
% 2026-11-07 whole-chain referee: every entry not copied from the chi_2 bibliography was checked against Crossref,
% arXiv or the publisher (comments at each entry); sample re-check of copied entries against Crossref: Hata1993,
% DHK2020, DHK2022, ViolaZudilin2018, LeanFour, SaffTotik -- all agree.

% ---------------------------------------------------------------------------
% Own work
% Hexagon record 2610.00372 checked through the Hexagon API on 2026-11-07 (title "chi_2(1/3) and chi_2(1/5) are
% irrational", sponsor Shawn Krisman, current version v2, 30 pp.). Hexagon 2611.00170 (Waldschmidt subgroup theorem in
% Lean) and 2611.00171 (four exponentials at CM and Beta periods) are not cited: this paper does not refer to that work.
% The chi_2(1/3) paper is in the public repository telaviv/chi2-irrationality as paper/chi2-irrationality.pdf
% (added in commit 55fa479, 2026-10-18). That paper cites its own repository as the entry Chi2Repo below (with
% commit 0d781d2). We cite commit 35b030d (2026-10-22), from which the ported files were taken; it was origin/main
% on 2026-11-07. Reference check 2026-11-09: the public main of telaviv/chi2-irrationality is now 184b987
% (2026-11-10 UTC, GitHub API); 35b030d is still in its history. The pinned commit is kept.
@misc{Chi2Paper,
  author       = {Krisman, S.},
  title        = {$\chi_2(1/3)$ and $\chi_2(1/5)$ are irrational},
  howpublished = {Hexagon 2610.00372v2, \url{https://hexagonmath.org/abs/2610.00372v2}; also file
                  \texttt{paper/chi2-irrationality.pdf} in \url{https://github.com/telaviv/chi2-irrationality}},
  year         = {2026},
  note         = {Not peer reviewed; produced largely by AI models, see its disclosure}
}

@misc{Chi2Repo,
  author       = {Krisman, S.},
  title        = {Lean formalisation: $\chi_2(1/3)$ and $\chi_2(1/5)$ are irrational},
  howpublished = {\url{https://github.com/telaviv/chi2-irrationality}},
  year         = {2026},
  note         = {Commit 35b030db9fdae70b97583f6ded0781bf1b379fdc}
}

% The repository of this paper: the Lean project (development name lean_chi2u), this paper as
% paper/chi2-quadratic-units-irrationality.pdf, and the computation scripts of Section 11 under scripts/paper/.
% Reference check 2026-11-09: the repository is public (created 2026-11-08 23:46 UTC; default branch master, at
% bfba108 = "V_i docs", 2026-11-09, GitHub API). Title = the README title. No commit hash in the entry: the paper is
% itself part of the repository.
@misc{PiTwoRepo,
  author       = {Krisman, S.},
  title        = {Irrationality of $\chi_2$ and $\mathrm{Ti}_2$ at units of real quadratic fields: a {L}ean~4 proof},
  howpublished = {\url{https://github.com/telaviv/chi2-quadratic-units-irrationality}},
  year         = {2026},
  note         = {Lean proof, this paper, and the computation scripts (folder \texttt{scripts/paper})}
}

% ---------------------------------------------------------------------------
% Copied from lean_chi2/paper/references.bib (verification comments there).
@article{Hata1990,
  author   = {Hata, Masayoshi},
  title    = {On the linear independence of the values of polylogarithmic functions},
  journal  = {J. Math. Pures Appl. (9)},
  volume   = {69},
  number   = {2},
  year     = {1990},
  pages    = {133--173},
  note     = {Zbl 0712.11040},
}

@article{Hata1993,
  author   = {Hata, Masayoshi},
  title    = {Rational approximations to the dilogarithm},
  journal  = {Trans. Amer. Math. Soc.},
  volume   = {336},
  number   = {1},
  year     = {1993},
  pages    = {363--387},
  doi      = {10.1090/S0002-9947-1993-1147401-5},
  note     = {MR 1147401},
}

@article{DHK2022,
  author        = {David, Sinnou and Hirata-Kohno, Noriko and Kawashima, Makoto},
  title         = {Linear independence criteria for generalized polylogarithms with distinct shifts},
  journal       = {Acta Arith.},
  volume        = {206},
  number        = {2},
  year          = {2022},
  pages         = {127--169},
  doi           = {10.4064/aa220307-18-10},
  eprint        = {2202.13931},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
}

@article{DHK2020,
  author        = {David, Sinnou and Hirata-Kohno, Noriko and Kawashima, Makoto},
  title         = {Can polylogarithms at algebraic points be linearly independent?},
  journal       = {Mosc. J. Comb. Number Theory},
  volume        = {9},
  number        = {4},
  year          = {2020},
  pages         = {389--406},
  doi           = {10.2140/moscow.2020.9.389},
  eprint        = {1912.03811},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
}

@misc{Prellberg2026,
  author        = {Prellberg, Thomas},
  title         = {Three irrationality results for the dilogarithm: {R}hin--{V}iola and {V}iola--{Z}udilin constructions at $-1/4$, $1/5$, and $-1/3$},
  year          = {2026},
  howpublished  = {arXiv:2609.06895},
  eprint        = {2609.06895},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  doi           = {10.48550/arXiv.2609.06895},
  note          = {Preprint, September 7, 2026; not peer reviewed (AI-assisted, see the paper's statement on AI assistance)},
}

@misc{LiuLi2,
  author       = {Liu, Si-Qi},
  title        = {{Li2Rational}: {L}ean 4 proofs of the irrationality of $\mathrm{Li}_2(r)$ at certain rational points $r$},
  year         = {2026},
  howpublished = {GitHub repository, \url{https://github.com/siqiliu-tsinghua/Li2Rational}},
  note         = {Accessed 2026-10-18 (commit b2a61c4). Lean formalisation, not peer reviewed; AI-assisted per README},
}

@misc{TangLi2,
  author       = {Tang, Qian},
  title        = {Irrationality of $\mathrm{Li}_2(1/2)$: a {L}ean 4 proof},
  year         = {2026},
  howpublished = {GitHub repository, \url{https://github.com/dtq1997/li2-half-irrationality}},
  note         = {Accessed 2026-10-18 (commit 950ee9d). Lean formalisation, no accompanying paper found, not peer reviewed},
}

@book{SaffTotik,
  author    = {Saff, Edward B. and Totik, Vilmos},
  title     = {Logarithmic Potentials with External Fields},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {316},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1997},
  doi       = {10.1007/978-3-662-03329-6},
}

@book{Lewin1981,
  author    = {Lewin, Leonard},
  title     = {Polylogarithms and Associated Functions},
  publisher = {North-Holland},
  address   = {New York and Oxford},
  year      = {1981},
  note      = {Zbl 0465.33001},
}

@article{RhinViola2005,
  author   = {Rhin, Georges and Viola, Carlo},
  title    = {The permutation group method for the dilogarithm},
  journal  = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
  volume   = {4},
  number   = {3},
  year     = {2005},
  pages    = {389--437},
  note     = {MR 2185863, Zbl 1170.11316},
}

@article{ViolaZudilin2018,
  author   = {Viola, Carlo and Zudilin, Wadim},
  title    = {Linear independence of dilogarithmic values},
  journal  = {J. Reine Angew. Math.},
  volume   = {736},
  year     = {2018},
  pages    = {193--223},
  doi      = {10.1515/crelle-2015-0030},
}

@inproceedings{Mathlib,
  author    = {{The mathlib Community}},
  title     = {The {L}ean mathematical library},
  booktitle = {Proceedings of the 9th ACM SIGPLAN International Conference on Certified Programs and Proofs (CPP 2020)},
  publisher = {ACM},
  year      = {2020},
  pages     = {367--381},
  doi       = {10.1145/3372885.3373824},
  note      = {arXiv:1910.09336}
}

@misc{Mathlib4,
  author       = {{The mathlib Community}},
  title        = {mathlib4},
  howpublished = {\url{https://github.com/leanprover-community/mathlib4}},
  year         = {2026},
  note         = {Tag v4.35.0, commit 56a005855ea17e0b0958f73ec801ae140b0a1679}
}

@inproceedings{LeanFour,
  author    = {de Moura, Leonardo and Ullrich, Sebastian},
  title     = {The {L}ean 4 Theorem Prover and Programming Language},
  booktitle = {Automated Deduction -- CADE 28},
  series    = {Lecture Notes in Comput. Sci.},
  volume    = {12699},
  publisher = {Springer},
  year      = {2021},
  pages     = {625--635},
  doi       = {10.1007/978-3-030-79876-5_37}
}

@misc{PNTplus,
  author       = {Kontorovich, Alex and Tao, Terence and others},
  title        = {{PrimeNumberTheoremAnd}},
  howpublished = {\url{https://github.com/AlexKontorovich/PrimeNumberTheoremAnd}},
  year         = {2026},
  note         = {Lean project, Apache License 2.0; the formalisation uses files from commit
                  a5154676af9aa3095150ee410cdda80555aa0642 (30 August 2026)}
}

@misc{zeta5,
  author       = {Del Solar, C\'esar},
  title        = {$\zeta(5)$ is irrational: a {L}ean~4 formalization},
  howpublished = {\url{https://github.com/domino14/zeta5}},
  year         = {2026},
  note         = {Source of the port of the prime number theorem used here
                  (commit 3da8eb1b91f9a90a1c0b7959e845c060dbd16256, 23 September 2026)}
}

% ---------------------------------------------------------------------------
% Not in the chi_2 bibliography.
% Standard reference for the Gel'fond--Schneider theorem (Chapter 2). Checked against Crossref 2026-11-07
% (doi 10.1017/CBO9780511565977, Cambridge University Press, 1975).
@book{Baker1975,
  author    = {Baker, Alan},
  title     = {Transcendental Number Theory},
  publisher = {Cambridge University Press},
  address   = {Cambridge},
  year      = {1975},
  doi       = {10.1017/CBO9780511565977},
}

% Checked against Crossref 2026-11-07 (doi below): IEEE Trans. Comput. 66 (2017), no. 8, 1281--1292.
@article{Arb,
  author  = {Johansson, Fredrik},
  title   = {Arb: efficient arbitrary-precision midpoint-radius interval arithmetic},
  journal = {IEEE Trans. Comput.},
  volume  = {66},
  number  = {8},
  year    = {2017},
  pages   = {1281--1292},
  doi     = {10.1109/TC.2017.2690633},
}

% ---------------------------------------------------------------------------
% Copied from lean_chi2/paper/references.bib (checked there 2026-10-18: arXiv v2, no journal publication found).
@misc{CDT,
  author        = {Calegari, Frank and Dimitrov, Vesselin and Tang, Yunqing},
  title         = {The linear independence of $1$, $\zeta(2)$, and ${L}(2,\chi_{-3})$},
  year          = {2024},
  howpublished  = {arXiv:2408.15403},
  eprint        = {2408.15403},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  note          = {Preprint},
}

% ---------------------------------------------------------------------------
% The following entries are taken from pilot_U/novelty/NOVELTY.md (2026-11-07), whose author read the statements in
% downloaded copies (Kawashima, DHK 2022b, CDT, Prellberg, Tan--Zhang), the abstract (Orr, Rhin--Viola 2019), or only
% zbMATH reviews / snippets (Hata 1998, Lewin 1982).
% Bibliographic data checked 2026-11-07 by the whole-chain referee: DHK 2022b against Crossref and the journal page
% (journals.sns.it, article 5693: vol. 23, no. 3); Rhin--Viola 2019, Hata 1998, Lewin 1982 against Crossref (issue
% numbers and DOIs added); Kawashima, Orr, Tan--Zhang against the arXiv API (Tan--Zhang: Hu Tan, Ying Zhang).
@article{DHK2022b,
  author        = {David, Sinnou and Hirata-Kohno, Noriko and Kawashima, Makoto},
  title         = {Linear forms in polylogarithms},
  journal       = {Ann. Sc. Norm. Super. Pisa Cl. Sci. (5)},
  volume        = {23},
  number        = {3},
  year          = {2022},
  pages         = {1447--1490},
  doi           = {10.2422/2036-2145.202010_047},
  eprint        = {2010.09167},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
}

@misc{Kawashima2026,
  author        = {Kawashima, Makoto},
  title         = {Linear independence of periods related to polylogarithms},
  year          = {2026},
  howpublished  = {arXiv:2605.17756},
  eprint        = {2605.17756},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  note          = {Preprint, May 2026; not peer reviewed},
}

@misc{Orr2026,
  author        = {Orr, Martin},
  title         = {Explicit height bounds for {G}-functions and unlikely intersections with lines in tori},
  year          = {2026},
  howpublished  = {arXiv:2607.09565},
  eprint        = {2607.09565},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  note          = {Preprint, July 2026; not peer reviewed},
}

@misc{TanZhang2026,
  author        = {Tan, Hu and Zhang, Ying},
  title         = {Uniform cubic bounds for irrational values of the dilogarithm},
  year          = {2026},
  howpublished  = {arXiv:2611.01066},
  eprint        = {2611.01066},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  note          = {Preprint; not peer reviewed (AI-assisted, see the paper's acknowledgments)},
}

@article{RhinViola2019,
  author   = {Rhin, Georges and Viola, Carlo},
  title    = {Linear independence of $1$, ${\rm Li}_1$ and ${\rm Li}_2$},
  journal  = {Mosc. J. Comb. Number Theory},
  volume   = {8},
  number   = {1},
  year     = {2019},
  pages    = {81--96},
  doi      = {10.2140/moscow.2019.8.81},
}

@article{Hata1998,
  author   = {Hata, Masayoshi},
  title    = {The irrationality of $\log(1+1/q)\log(1-1/q)$},
  journal  = {Trans. Amer. Math. Soc.},
  volume   = {350},
  number   = {6},
  year     = {1998},
  pages    = {2311--2327},
  doi      = {10.1090/S0002-9947-98-01764-4},
}

@article{Lewin1982,
  author   = {Lewin, Leonard},
  title    = {The dilogarithm in algebraic fields},
  journal  = {J. Austral. Math. Soc. Ser. A},
  volume   = {33},
  number   = {3},
  year     = {1982},
  pages    = {302--330},
  doi      = {10.1017/S1446788700018747},
}

% Added 2026-11-07 for the Theorem V revision. Checked against the first page of the downloaded arXiv PDF
% (pilot_V/downloads/arxiv_2608.23792/2608.23792.pdf: "arXiv:2608.23792v1 [math.NT] 24 Aug 2026", author and title as
% below); Corollary 14 (bases a = (sqrt17 - 3)/4, b = sqrt17 - 4) located in its text. Unrefereed.
@misc{HakimogluBrown2026,
  author        = {Hakimoglu-Brown, Cetin},
  title         = {Collapsing fifty dilogarithm arguments to five terms over $\mathbb{Q}(u,\sqrt{4-3u^2})$},
  year          = {2026},
  howpublished  = {arXiv:2608.23792},
  eprint        = {2608.23792},
  archivePrefix = {arXiv},
  primaryClass  = {math.NT},
  note          = {Preprint, v1, August 24, 2026; not peer reviewed},
}

% Version 0.9.0 is the one installed in the environment that ran the scripts (checked with python, 2026-11-07).
@misc{pythonflint,
  author       = {{The FLINT team}},
  title        = {python-flint: {P}ython bindings for {FLINT}},
  howpublished = {\url{https://github.com/flintlib/python-flint}},
  year         = {2026},
  note         = {Version 0.9.0}
}

% Added 2026-11-08 (extensions revision). Cited for the rational values of sine at rational multiples of pi (PROOF_TI2
% Section 0 states the fact; pilot_w3/RESULT.md Section 3.2 names Niven). Title, author, series, publisher and year checked
% against library catalogue records and the MAA/AMS bookstore listing (Carus Mathematical Monographs 11); the corollary
% number 3.12 (p. 41) is taken from secondary sources (Wikipedia, "Niven's theorem"); the book text was not seen.
% Reference check 2026-11-09: zbMATH 0070.27101 gives "The Carus Mathematical Monographs. No. 11. New York: John Wiley
% and Sons, Inc., Publ. by The Mathematical Association of America, xii+164 p. (1956)"; address set accordingly.
@book{Niven1956,
  author    = {Niven, Ivan},
  title     = {Irrational Numbers},
  series    = {The Carus Mathematical Monographs},
  number    = {11},
  publisher = {Mathematical Association of America},
  address   = {New York (distributed by John Wiley and Sons)},
  year      = {1956},
}