\begin{thebibliography}{99}\raggedright \bibitem{ABB25} T. Achim, A. Best, A. Bietti, K. Der et~al., \emph{Aristotle: IMO-level automated theorem proving}, preprint (2025). \href{https://arxiv.org/abs/2510.01346}{\nolinkurl{arXiv:2510.01346}}. \bibitem{ACL25} G. Ancona, M. Cavicchi, R. Laterveer and G. Sacc\`a, \emph{Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations}, J. Lond. Math. Soc. (2) \textbf{111} (2025), e70133. \href{https://doi.org/10.1112/jlms.70133}{\nolinkurl{doi:10.1112/jlms.70133}}. \bibitem{ALM98} S. Arora, C. Lund, R. Motwani, M. Sudan and M. Szegedy, \emph{Proof verification and the hardness of approximation problems}, J. ACM \textbf{45} (1998), no.~3, 501--555. \href{https://doi.org/10.1145/278298.278306}{\nolinkurl{doi:10.1145/278298.278306}}. \bibitem{AN72} K. B. Athreya and P. E. Ney, \emph{Branching Processes}, Grundlehren der mathematischen Wissenschaften~196, Springer, Berlin, 1972. \href{https://doi.org/10.1007/978-3-642-65371-1}{\nolinkurl{doi:10.1007/978-3-642-65371-1}}. \bibitem{Ant26} Anthropic, \emph{Formalizing Fermat's Last Theorem}, research report (4 September 2026). \url{https://www.anthropic.com/research/formalizing-fermats-last-theorem}. \bibitem{AOA25} A. Anbar Jafari, C. Ozcinar and G. Anbarjafari, \emph{A mathematical framework for AI singularity: conditions, bounds, and control of recursive improvement}, preprint (2025). \href{https://arxiv.org/abs/2511.10668}{\nolinkurl{arXiv:2511.10668}}. \bibitem{Ara06} D. Arapura, \emph{Motivation for Hodge cycles}, Adv. Math. \textbf{207} (2006), 762--781. \href{https://doi.org/10.1016/j.aim.2006.01.005}{\nolinkurl{doi:10.1016/j.aim.2006.01.005}}. \bibitem{ArcNS} OpenAI, \emph{Finite-time blowup for Navier--Stokes: OpenAI's Lean proof, built and checked}, Zenodo archive, version 1.0.0 (2026). \href{https://doi.org/10.5281/zenodo.23286556}{\nolinkurl{doi:10.5281/zenodo.23286556}}. \bibitem{ArcPC} Z. Zhang, A. Delaval, L. Chen, J. Chen, J. Xu, Y. Liao, J. Jiang, C. Liu and B. Dong, \emph{The Poincar\'e conjecture in Lean~4: the frenzymath formalization, built and checked}, Zenodo archive, version 1.0.0 (2026). \href{https://doi.org/10.5281/zenodo.23288534}{\nolinkurl{doi:10.5281/zenodo.23288534}}. \bibitem{ArcSP} S. Hariharan, C. Birkbeck, S. Lee, H. K. G. Ma, B. Mehta, A. Poiroux, M. Viazovska and Math, Inc., \emph{Sphere packing in dimensions 8, 24 and 3: an independent check of the formal proofs}, Zenodo archive, version 1.0.0 (2026). \href{https://doi.org/10.5281/zenodo.23289869}{\nolinkurl{doi:10.5281/zenodo.23289869}}. \bibitem{AS98} S. Arora and S. Safra, \emph{Probabilistic checking of proofs: a new characterization of NP}, J. ACM \textbf{45} (1998), no.~1, 70--122. \href{https://doi.org/10.1145/273865.273901}{\nolinkurl{doi:10.1145/273865.273901}}. \bibitem{AW09} S. Aaronson and A. Wigderson, \emph{Algebrization: a new barrier in complexity theory}, ACM Trans. Comput. Theory \textbf{1} (2009), 2:1--2:54. \href{https://doi.org/10.1145/1490270.1490272}{\nolinkurl{doi:10.1145/1490270.1490272}}. \bibitem{Bab85} L. Babai, \emph{Trading group theory for randomness}, in Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing (STOC '85), ACM, New York, 1985, 421--429. \href{https://doi.org/10.1145/22145.22192}{\nolinkurl{doi:10.1145/22145.22192}}. \bibitem{Bal87} T. Ba{\l}aban, \emph{Renormalization group approach to lattice gauge field theories. I. Generation of effective actions in a small field approximation and a coupling constant renormalization in four dimensions}, Comm. Math. Phys. \textbf{109} (1987), 249--301. \href{https://doi.org/10.1007/BF01215223}{\nolinkurl{doi:10.1007/BF01215223}}. \bibitem{Bat82} G. G. Batrouni, \emph{Plaquette formulation and the Bianchi identity for lattice gauge theories}, Nuclear Phys. B \textbf{208} (1982), 467--483. \href{https://doi.org/10.1016/0550-3213(82)90231-0}{\nolinkurl{doi:10.1016/0550-3213(82)90231-0}}. \bibitem{BB21} R. Bauerschmidt and T. Bodineau, \emph{Log-Sobolev inequality for the continuum sine-Gordon model}, Comm. Pure Appl. Math. \textbf{74} (2021), 2064--2113. \href{https://doi.org/10.1002/cpa.21926}{\nolinkurl{doi:10.1002/cpa.21926}}. \bibitem{BB26c} D. Bhattacharjee and U. Bhattacharya, \emph{Dense algebraic loci of Hodge classes}, Hexagon preprint (2026), hexagon:2610.00315v1, \href{https://hexagonmath.org/2610.00315v1}{\nolinkurl{hexagonmath.org/2610.00315v1}}; source and Lean~4 archive, version 2.0.0, \href{https://doi.org/10.5281/zenodo.23461752}{\nolinkurl{doi:10.5281/zenodo.23461752}}. \bibitem{BB26d} D. Bhattacharjee and U. Bhattacharya, \emph{Lattice Yang--Mills theory: spacing-uniform obstructions, the mass gap up to the Dobrushin ceiling, and Lean certificate}, Zenodo, version 2.0.0 (2026). \href{https://doi.org/10.5281/zenodo.23449466}{\nolinkurl{doi:10.5281/zenodo.23449466}}. \bibitem{BBF26} M.-F. Balcan, A. Blum, K. Fragkia, Z. Li and D. Sharma, \emph{Verify to amplify: improving reasoning via learned chain-of-thought verification}, preprint (2026). \href{https://arxiv.org/abs/2603.03538}{\nolinkurl{arXiv:2603.03538}}. \bibitem{BCH26} A. Bastounis, F. Circelli and A. C. Hansen, \emph{Navier--Stokes lost in translation: why Lean verification of AI autoformalisation does not guarantee correct natural language proofs}, preprint (2026). \href{https://arxiv.org/abs/2610.08144}{\nolinkurl{arXiv:2610.08144}}. \bibitem{BGR05} O. V. Borodin, A. N. Glebov, A. Raspaud and M. R. Salavatipour, \emph{Planar graphs without cycles of length from $4$ to $7$ are $3$-colorable}, J. Combin. Theory Ser. B \textbf{93} (2005), 303--311. \href{https://doi.org/10.1016/j.jctb.2004.11.001}{\nolinkurl{doi:10.1016/j.jctb.2004.11.001}}. \bibitem{BGS75} T. Baker, J. Gill and R. Solovay, \emph{Relativizations of the $\mathcal{P}=?\mathcal{NP}$ question}, SIAM J. Comput. \textbf{4} (1975), 431--442. \href{https://doi.org/10.1137/0204037}{\nolinkurl{doi:10.1137/0204037}}. \bibitem{Bha26a} D. Bhattacharjee, \emph{Synthetic neuromorphic superintelligence for verifier-gated conjecture closure: sand-tile automata, K-dot circuits, and phenomenological proof learning}, PhilArchive (2026). \href{https://philarchive.org/rec/BHASNS-2}{\nolinkurl{philarchive.org/rec/BHASNS-2}}. \bibitem{Bha26c} D. Bhattacharjee, \emph{Anaxia, the BLLM engine}, Python package \texttt{eBLLM} with the command \texttt{anaxia}, Lean~4 library and container image \texttt{ghcr.io/creelie/bllm-engine}, Zenodo, release~\anaxiaVersion{} (2026), \anaxiaDOIlink; all versions \href{https://doi.org/10.5281/zenodo.23286641}{\nolinkurl{doi:10.5281/zenodo.23286641}}. \href{https://github.com/creelie/eBLLM}{\nolinkurl{github.com/creelie/eBLLM}}. \bibitem{BJE24} B. Brown, J. Juravsky, R. Ehrlich, R. Clark, Q. V. Le, C. R\'e and A. Mirhoseini, \emph{Large language monkeys: scaling inference compute with repeated sampling}, preprint (2024). \href{https://arxiv.org/abs/2407.21787}{\nolinkurl{arXiv:2407.21787}}. \bibitem{BKK22} Y. Bai, S. Kadavath, S. Kundu, A. Askell, J. Kernion, A. Jones et~al., \emph{Constitutional AI: harmlessness from AI feedback}, preprint (2022). \href{https://arxiv.org/abs/2212.08073}{\nolinkurl{arXiv:2212.08073}}. \bibitem{BLM25} E. Botta, Y. Li, A. Mehta, J. T. Ash, C. Zhang and A. Risteski, \emph{On the query complexity of verifier-assisted language generation}, preprint (2025). \href{https://arxiv.org/abs/2502.12123}{\nolinkurl{arXiv:2502.12123}}. \bibitem{Blo1183} T. F. Bloom, \emph{Erd\H{o}s problem \#1183}, \href{https://www.erdosproblems.com/1183}{\nolinkurl{erdosproblems.com/1183}}, accessed 18 October 2026. \bibitem{BMB26a} D. Bhattacharjee, P. Mandal and U. Bhattacharya, \emph{Resolving Erd\H{o}s--Ulam monochromatic union-closed family conjectures}, preprint (2026), \href{https://arxiv.org/abs/2610.02833}{\nolinkurl{arXiv:2610.02833}}; Lean~4 archive, version 5, \href{https://doi.org/10.5281/zenodo.23374973}{\nolinkurl{doi:10.5281/zenodo.23374973}}. \bibitem{BMB26b} D. Bhattacharjee, P. Mandal and U. Bhattacharya, \emph{Algebraic loci of Weil classes on abelian varieties}, with verification code, Zenodo, version 5.0.1 (2026). \href{https://doi.org/10.5281/zenodo.23225483}{\nolinkurl{doi:10.5281/zenodo.23225483}}. \bibitem{BMR20} T. B. Brown, B. Mann, N. Ryder, M. Subbiah, J. Kaplan, P. Dhariwal et~al., \emph{Language models are few-shot learners}, Advances in Neural Information Processing Systems \textbf{33} (2020), 1877--1901. \href{https://arxiv.org/abs/2005.14165}{\nolinkurl{arXiv:2005.14165}}. \bibitem{Bor09} \'E. Borel, \emph{Les probabilit\'es d\'enombrables et leurs applications arithm\'etiques}, Rend. Circ. Mat. Palermo \textbf{27} (1909), 247--271. \href{https://doi.org/10.1007/BF03019651}{\nolinkurl{doi:10.1007/BF03019651}}. \bibitem{Bos12} N. Bostrom, \emph{The superintelligent will: motivation and instrumental rationality in advanced artificial agents}, Minds and Machines \textbf{22} (2012), 71--85. \href{https://doi.org/10.1007/s11023-012-9281-3}{\nolinkurl{doi:10.1007/s11023-012-9281-3}}. \bibitem{Bos14} N. Bostrom, \emph{Superintelligence: Paths, Dangers, Strategies}, Oxford University Press, Oxford, 2014. \bibitem{Bos98} N. Bostrom, \emph{How long before superintelligence?}, Int. J. Futures Studies \textbf{2} (1998); reprinted in Linguistic and Philosophical Investigations \textbf{5} (2006), no.~1, 11--30. \bibitem{BS15a} M. Bhargava and A. Shankar, \emph{Binary quartic forms having bounded invariants, and the boundedness of the average rank of elliptic curves}, Ann. of Math. (2) \textbf{181} (2015), 191--242. \href{https://doi.org/10.4007/annals.2015.181.1.3}{\nolinkurl{doi:10.4007/annals.2015.181.1.3}}. \bibitem{BS15b} M. Bhargava and A. Shankar, \emph{Ternary cubic forms having bounded invariants, and the existence of a positive proportion of elliptic curves having rank~$0$}, Ann. of Math. (2) \textbf{181} (2015), 587--621. \href{https://doi.org/10.4007/annals.2015.181.2.4}{\nolinkurl{doi:10.4007/annals.2015.181.2.4}}. \bibitem{Bus19} N. Buskin, \emph{Every rational Hodge isometry between two K3 surfaces is algebraic}, J. Reine Angew. Math. \textbf{755} (2019), 127--150. \href{https://doi.org/10.1515/crelle-2017-0027}{\nolinkurl{doi:10.1515/crelle-2017-0027}}. \bibitem{Cam65} L. L. Campbell, \emph{A coding theorem and R\'enyi's entropy}, Information and Control \textbf{8} (1965), no.~4, 423--429. \href{https://doi.org/10.1016/S0019-9958(65)90332-3}{\nolinkurl{doi:10.1016/S0019-9958(65)90332-3}}. \bibitem{CE03} H. Cohn and N. Elkies, \emph{New upper bounds on sphere packings~I}, Ann. of Math. \textbf{157} (2003), 689--714. \href{https://doi.org/10.4007/annals.2003.157.689}{\nolinkurl{doi:10.4007/annals.2003.157.689}}. \bibitem{CGH25} L. Chen, J. Gu, L. Huang, W. Huang et~al., \emph{Seed-Prover: deep and broad reasoning for automated theorem proving}, preprint (2025). \href{https://arxiv.org/abs/2507.23726}{\nolinkurl{arXiv:2507.23726}}. \bibitem{Che52} H. Chernoff, \emph{A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations}, Ann. Math. Statist. \textbf{23} (1952), no.~4, 493--507. \href{https://doi.org/10.1214/aoms/1177729330}{\nolinkurl{doi:10.1214/aoms/1177729330}}. \bibitem{CHK17} V. Cohen-Addad, M. Hebdige, D. Kr\'al', Z. Li and E. Salgado, \emph{Steinberg's conjecture is false}, J. Combin. Theory Ser. B \textbf{122} (2017), 452--456. \href{https://doi.org/10.1016/j.jctb.2016.07.006}{\nolinkurl{doi:10.1016/j.jctb.2016.07.006}}. \bibitem{Cho26} P. Chojecki, \emph{Monochromatic union-closed and lattice subfamilies of the Boolean lattice}, preprint (18 March 2026). \url{https://www.ulam.ai/research/erdos1183.pdf}. \bibitem{CJB19} S. J. Cook, T. A. Jarrell, C. A. Brittin, Y. Wang et al., \emph{Whole-animal connectomes of both Caenorhabditis elegans sexes}, Nature \textbf{571} (2019), 63--71. \href{https://doi.org/10.1038/s41586-019-1352-7}{\nolinkurl{doi:10.1038/s41586-019-1352-7}}. \bibitem{CKB21} K. Cobbe, V. Kosaraju, M. Bavarian, M. Chen, H. Jun, L. Kaiser et~al., \emph{Training verifiers to solve math word problems}, preprint (2021). \href{https://arxiv.org/abs/2110.14168}{\nolinkurl{arXiv:2110.14168}}. \bibitem{CKM17} H. Cohn, A. Kumar, S. D. Miller, D. Radchenko and M. Viazovska, \emph{The sphere packing problem in dimension 24}, Ann. of Math. (2) \textbf{185} (2017), no.~3, 1017--1033. \href{https://doi.org/10.4007/annals.2017.185.3.8}{\nolinkurl{doi:10.4007/annals.2017.185.3.8}}. \bibitem{Clay} Clay Mathematics Institute, \emph{The Millennium Prize Problems}, \url{https://www.claymath.org/millennium-problems/}, accessed 18 October 2026. \bibitem{CM13} F. Charles and E. Markman, \emph{The standard conjectures for holomorphic symplectic varieties deformation equivalent to Hilbert schemes of K3 surfaces}, Compos. Math. \textbf{149} (2013), 481--494. \href{https://doi.org/10.1112/S0010437X12000607}{\nolinkurl{doi:10.1112/S0010437X12000607}}. \bibitem{CML26} S. Chen, K. Marwaha, X. Lu, H. Yuen and T. Peng, \emph{Prove2Me: an open collaborative platform for scaling math formalization}, preprint (2026). \href{https://arxiv.org/abs/2608.28433}{\nolinkurl{arXiv:2608.28433}}. \bibitem{Cre77} M. Creutz, \emph{Gauge fixing, the transfer matrix, and confinement on a lattice}, Phys. Rev. D \textbf{15} (1977), 1128--1136. \href{https://doi.org/10.1103/PhysRevD.15.1128}{\nolinkurl{doi:10.1103/PhysRevD.15.1128}}. \bibitem{Cre80} M. Creutz, \emph{Monte Carlo study of quantized SU(2) gauge theory}, Phys. Rev. D \textbf{21} (1980), 2308--2315. \href{https://doi.org/10.1103/PhysRevD.21.2308}{\nolinkurl{doi:10.1103/PhysRevD.21.2308}}. \bibitem{CT06} T. M. Cover and J. A. Thomas, \emph{Elements of Information Theory}, 2nd ed., Wiley, Hoboken, 2006. \href{https://doi.org/10.1002/047174882X}{\nolinkurl{doi:10.1002/047174882X}}. \bibitem{CTO25} Yu. Chervonyi, T. H. Trinh, M. Ol\v{s}\'ak, X. Yang et~al., \emph{Gold-medalist performance in solving olympiad geometry with AlphaGeometry2}, preprint (2025). \href{https://arxiv.org/abs/2502.03544}{\nolinkurl{arXiv:2502.03544}}. \bibitem{CZC26} Z. Chen, Y. Zhao and A. Cohan, \emph{Measuring the gap between human and LLM research ideas}, preprint (2026). \href{https://arxiv.org/abs/2607.01233}{\nolinkurl{arXiv:2607.01233}}. \bibitem{dB70} N. G. de Bruijn, \emph{The mathematical language AUTOMATH, its usage, and some of its extensions}, in Symposium on Automatic Demonstration, Lecture Notes in Math.~125, Springer, Berlin, 1970, 29--61. \href{https://doi.org/10.1007/BFb0060623}{\nolinkurl{doi:10.1007/BFb0060623}}. \bibitem{DDT95} H. Darmon, F. Diamond and R. Taylor, \emph{Fermat's Last Theorem}, Current Developments in Mathematics \textbf{1995} (1995), 1--154. \href{https://doi.org/10.4310/CDM.1995.v1995.n1.a1}{\nolinkurl{doi:10.4310/CDM.1995.v1995.n1.a1}}. \bibitem{DLS23} N. Dziri, X. Lu, M. Sclar, X. L. Li, L. Jiang, B. Y. Lin et~al., \emph{Faith and fate: limits of transformers on compositionality}, Advances in Neural Information Processing Systems \textbf{36} (2023). \href{https://arxiv.org/abs/2305.18654}{\nolinkurl{arXiv:2305.18654}}. \bibitem{DM04} R. J. Douglas and K. A. C. Martin, \emph{Neuronal circuits of the neocortex}, Annu. Rev. Neurosci. \textbf{27} (2004), 419--451. \href{https://doi.org/10.1146/annurev.neuro.27.070203.144152}{\nolinkurl{doi:10.1146/annurev.neuro.27.070203.144152}}. \bibitem{DMS24} S. Dorkenwald, A. Matsliah, A. R. Sterling, P. Schlegel, S.-C. Yu, C. E. McKellar et~al., \emph{Neuronal wiring diagram of an adult brain}, Nature \textbf{634} (2024), 124--138. \href{https://doi.org/10.1038/s41586-024-07558-y}{\nolinkurl{doi:10.1038/s41586-024-07558-y}}. \bibitem{DMS24b} S. Dorkenwald, A. Matsliah, A. R. Sterling, P. Schlegel et al., \emph{Neuronal wiring diagram of an adult brain}, Nature \textbf{634} (2024), 124--138. \href{https://doi.org/10.1038/s41586-024-07558-y}{\nolinkurl{doi:10.1038/s41586-024-07558-y}}. \bibitem{Dob68} R. L. Dobrushin, \emph{The description of a random field by means of conditional probabilities and conditions of its regularity}, Theory Probab. Appl. \textbf{13} (1968), 197--224. \href{https://doi.org/10.1137/1113026}{\nolinkurl{doi:10.1137/1113026}}. \bibitem{DSL18} M. Davies, N. Srinivasa, T.-H. Lin, G. Chinya et al., \emph{Loihi: a neuromorphic manycore processor with on-chip learning}, IEEE Micro \textbf{38} (2018), no.~1, 82--99. \href{https://doi.org/10.1109/MM.2018.112130359}{\nolinkurl{doi:10.1109/MM.2018.112130359}}. \bibitem{Dur19} R. Durrett, \emph{Probability: Theory and Examples}, 5th ed., Cambridge University Press, Cambridge, 2019. \href{https://doi.org/10.1017/9781108591034}{\nolinkurl{doi:10.1017/9781108591034}}. \bibitem{DVB21} A. Davies, P. Veli\v{c}kovi\'c, L. Buesing, S. Blackwell, D. Zheng, N. Toma\v{s}ev et~al., \emph{Advancing mathematics by guiding human intuition with AI}, Nature \textbf{600} (2021), no.~7887, 70--74. \href{https://doi.org/10.1038/s41586-021-04086-x}{\nolinkurl{doi:10.1038/s41586-021-04086-x}}. \bibitem{Dwa69} M. Dwass, \emph{The total progeny in a branching process and a related random walk}, J. Appl. Probab. \textbf{6} (1969), no.~3, 682--686. \href{https://doi.org/10.2307/3212112}{\nolinkurl{doi:10.2307/3212112}}. \bibitem{EBC24} N. Eckstein, A. S. Bates, A. Champion, M. Du et al., \emph{Neurotransmitter classification from electron microscopy images at synaptic sites in Drosophila melanogaster}, Cell \textbf{187} (2024), 2574--2594. \href{https://doi.org/10.1016/j.cell.2024.03.016}{\nolinkurl{doi:10.1016/j.cell.2024.03.016}}. \bibitem{Eli75} P. Elias, \emph{Universal codeword sets and representations of the integers}, IEEE Trans. Inform. Theory \textbf{21} (1975), no.~2, 194--203. \href{https://doi.org/10.1109/TIT.1975.1055349}{\nolinkurl{doi:10.1109/TIT.1975.1055349}}. \bibitem{FBH22} A. Fawzi, M. Balog, A. Huang, T. Hubert, B. Romera-Paredes, M. Barekatain et~al., \emph{Discovering faster matrix multiplication algorithms with reinforcement learning}, Nature \textbf{610} (2022), no.~7930, 47--53. \href{https://doi.org/10.1038/s41586-022-05172-4}{\nolinkurl{doi:10.1038/s41586-022-05172-4}}. \bibitem{FCA25} The Formal Conjectures Authors, \emph{Formal Conjectures}, Lean~4 statements of open and solved conjectures, Google DeepMind (2025). \href{https://github.com/google-deepmind/formal-conjectures}{\nolinkurl{github.com/google-deepmind/formal-conjectures}}. \bibitem{FG16} N. Fr\'emaux and W. Gerstner, \emph{Neuromodulated spike-timing-dependent plasticity, and theory of three-factor learning rules}, Front. Neural Circuits \textbf{9} (2016), 85. \href{https://doi.org/10.3389/fncir.2015.00085}{\nolinkurl{doi:10.3389/fncir.2015.00085}}. \bibitem{Flo22} S. Floccari, \emph{On the motive of O'Grady's six dimensional hyper-K\"ahler varieties}, \'Epijournal G\'eom. Alg\'ebrique \textbf{7} (2023). \href{https://doi.org/10.46298/epiga.2022.9758}{\nolinkurl{doi:10.46298/epiga.2022.9758}}. \bibitem{Fos23} J. Foster, \emph{The Lefschetz standard conjectures for IHSMs of generalized Kummer deformation type in certain degrees}, preprint (2023). \href{https://arxiv.org/abs/2303.14327}{\nolinkurl{arXiv:2303.14327}}. \bibitem{Fos25} J. Foster, \emph{The Lefschetz standard conjectures for $4d$-dimensional Kummer-type hyper-K\"ahler varieties}, preprint (2025). \href{https://arxiv.org/abs/2512.04114}{\nolinkurl{arXiv:2512.04114}}. \bibitem{Gal78} R. G. Gallager, \emph{Variations on a theme by Huffman}, IEEE Trans. Inform. Theory \textbf{24} (1978), no.~6, 668--674. \href{https://doi.org/10.1109/TIT.1978.1055959}{\nolinkurl{doi:10.1109/TIT.1978.1055959}}. \bibitem{GGT25} B. Georgiev, J. G\'omez-Serrano, T. Tao and A. Z. Wagner, \emph{Mathematical exploration and discovery at scale}, preprint (2025). \href{https://arxiv.org/abs/2511.02864}{\nolinkurl{arXiv:2511.02864}}. \bibitem{GKN14} W. Gerstner, W. M. Kistler, R. Naud and L. Paninski, \emph{Neuronal Dynamics: From Single Neurons to Networks and Models of Cognition}, Cambridge University Press, Cambridge, 2014. \href{https://doi.org/10.1017/CBO9781107447615}{\nolinkurl{doi:10.1017/CBO9781107447615}}. \bibitem{GMR89} S. Goldwasser, S. Micali and C. Rackoff, \emph{The knowledge complexity of interactive proof systems}, SIAM J. Comput. \textbf{18} (1989), no.~1, 186--208. \href{https://doi.org/10.1137/0218012}{\nolinkurl{doi:10.1137/0218012}}. \bibitem{GMW79} M. J. Gordon, A. J. Milner and C. P. Wadsworth, \emph{Edinburgh LCF: A Mechanised Logic of Computation}, Lecture Notes in Comput. Sci.~78, Springer, Berlin, 1979. \href{https://doi.org/10.1007/3-540-09724-4}{\nolinkurl{doi:10.1007/3-540-09724-4}}. \bibitem{God31} K. G\"odel, \emph{\"Uber formal unentscheidbare S\"atze der Principia Mathematica und verwandter Systeme I}, Monatsh. Math. Phys. \textbf{38} (1931), 173--198. \href{https://doi.org/10.1007/BF01700692}{\nolinkurl{doi:10.1007/BF01700692}}. \bibitem{Goo66} I. J. Good, \emph{Speculations concerning the first ultraintelligent machine}, Advances in Computers \textbf{6} (1966), 31--88. \href{https://doi.org/10.1016/S0065-2458(08)60418-0}{\nolinkurl{doi:10.1016/S0065-2458(08)60418-0}}. \bibitem{Guo26} D. Guo, \emph{The deterministic horizon: impossibility results as design specifications for trustworthy AI systems}, preprint (2026). \href{https://arxiv.org/abs/2605.23024}{\nolinkurl{arXiv:2605.23024}}. \bibitem{GZ86} B. H. Gross and D. B. Zagier, \emph{Heegner points and derivatives of $L$-series}, Invent. Math. \textbf{84} (1986), 225--320. \href{https://doi.org/10.1007/BF01388809}{\nolinkurl{doi:10.1007/BF01388809}}. \bibitem{Hat02} A. Hatcher, \emph{Algebraic Topology}, Cambridge University Press, Cambridge, 2002. \bibitem{HBL26} S. Hariharan, C. Birkbeck, S. Lee, H. K. G. Ma, B. Mehta, A. Poiroux and M. Viazovska, \emph{Progress in formalizing sphere packing in dimension 8}, preprint (2026). \href{https://arxiv.org/abs/2604.23468}{\nolinkurl{arXiv:2604.23468}}. \bibitem{HBM22} J. Hoffmann, S. Borgeaud, A. Mensch, E. Buchatskaya, T. Cai, E. Rutherford et~al., \emph{Training compute-optimal large language models}, preprint (2022). \href{https://arxiv.org/abs/2203.15556}{\nolinkurl{arXiv:2203.15556}}. \bibitem{Heb49} D. O. Hebb, \emph{The Organization of Behavior: A Neuropsychological Theory}, Wiley, New York, 1949. \bibitem{HGH22} J. Huang, S. S. Gu, L. Hou, Y. Wu, X. Wang, H. Yu and J. Han, \emph{Large language models can self-improve}, preprint (2022). \href{https://arxiv.org/abs/2210.11610}{\nolinkurl{arXiv:2210.11610}}. \bibitem{HLX02} X.-D. Hou, K. H. Leung and Q. Xiang, \emph{A generalization of an addition theorem of Kneser}, J. Number Theory \textbf{97} (2002), no.~1, 1--9. \href{https://doi.org/10.1006/jnth.2002.2793}{\nolinkurl{doi:10.1006/jnth.2002.2793}}. \bibitem{HMS25} T. Hubert, R. Mehta, L. Sartran, M. Z. Horv\'ath, G. \v{Z}u\v{z}i\'c, E. Wieser et~al., \emph{Olympiad-level formal mathematical reasoning with reinforcement learning}, Nature \textbf{651} (2026), no.~8106, 607--613; published online 2025. \href{https://doi.org/10.1038/s41586-025-09833-y}{\nolinkurl{doi:10.1038/s41586-025-09833-y}}. \bibitem{Hoe63} W. Hoeffding, \emph{Probability inequalities for sums of bounded random variables}, J. Amer. Statist. Assoc. \textbf{58} (1963), no.~301, 13--30. \href{https://doi.org/10.1080/01621459.1963.10500830}{\nolinkurl{doi:10.1080/01621459.1963.10500830}}. \bibitem{HR21} H. A. Helfgott and M. Radziwi\l\l, \emph{Expansion, divisibility and parity}, preprint (2021). \href{https://arxiv.org/abs/2103.06853}{\nolinkurl{arXiv:2103.06853}}. \bibitem{Huf52} D. A. Huffman, \emph{A method for the construction of minimum-redundancy codes}, Proc. IRE \textbf{40} (1952), no.~9, 1098--1101. \href{https://doi.org/10.1109/JRPROC.1952.273898}{\nolinkurl{doi:10.1109/JRPROC.1952.273898}}. \bibitem{Hut05} M. Hutter, \emph{Universal Artificial Intelligence: Sequential Decisions Based on Algorithmic Probability}, Springer, Berlin, 2005. \href{https://doi.org/10.1007/b138233}{\nolinkurl{doi:10.1007/b138233}}. \bibitem{Jen06} J. L. W. V. Jensen, \emph{Sur les fonctions convexes et les in\'egalit\'es entre les valeurs moyennes}, Acta Math. \textbf{30} (1906), 175--193. \href{https://doi.org/10.1007/BF02418571}{\nolinkurl{doi:10.1007/BF02418571}}. \bibitem{JH04} H. Jaeger and H. Haas, \emph{Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication}, Science \textbf{304} (2004), 78--80. \href{https://doi.org/10.1126/science.1091277}{\nolinkurl{doi:10.1126/science.1091277}}. \bibitem{JRX25} Z. Jia, A. Rakhlin and T. Xie, \emph{Do we need to verify step by step? Rethinking process supervision from a theoretical perspective}, preprint (2025). \href{https://arxiv.org/abs/2502.10581}{\nolinkurl{arXiv:2502.10581}}. \bibitem{JT16} K. Jamieson and A. Talwalkar, \emph{Non-stochastic best arm identification and hyperparameter optimization}, in Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (2016). \href{https://arxiv.org/abs/1502.07943}{\nolinkurl{arXiv:1502.07943}}. \bibitem{Kle94} S. L. Kleiman, \emph{The standard conjectures}, in Motives (Seattle, 1991), Proc. Sympos. Pure Math.~55, Part~1, Amer. Math. Soc., Providence, 1994, 3--20. \href{https://doi.org/10.1090/pspum/055.1/1265519}{\nolinkurl{doi:10.1090/pspum/055.1/1265519}}. \bibitem{KMH20} J. Kaplan, S. McCandlish, T. Henighan, T. B. Brown, B. Chess, R. Child, S. Gray, A. Radford, J. Wu and D. Amodei, \emph{Scaling laws for neural language models}, preprint (2020). \href{https://arxiv.org/abs/2001.08361}{\nolinkurl{arXiv:2001.08361}}. \bibitem{Kol89} V. A. Kolyvagin, \emph{Finiteness of $E(\mathbf{Q})$ and Sha$(E,\mathbf{Q})$ for a subclass of Weil curves}, Math. USSR-Izv. \textbf{32} (1989), 523--541. \href{https://doi.org/10.1070/IM1989v032n03ABEH000779}{\nolinkurl{doi:10.1070/IM1989v032n03ABEH000779}}. \bibitem{Kra49} L. G. Kraft, \emph{A device for quantizing, grouping, and coding amplitude-modulated pulses}, M.S. thesis, Massachusetts Institute of Technology, 1949. \href{https://hdl.handle.net/1721.1/12390}{\nolinkurl{hdl:1721.1/12390}}. \bibitem{KS66} H. Kesten and B. P. Stigum, \emph{A limit theorem for multidimensional Galton--Watson processes}, Ann. Math. Statist. \textbf{37} (1966), no.~5, 1211--1223. \href{https://doi.org/10.1214/aoms/1177699266}{\nolinkurl{doi:10.1214/aoms/1177699266}}. \bibitem{LCL25} C. Liu, Y. Chen, T. Liu, J. Cheng, B. Han and K. Zhang, \emph{On the thinking-language modeling gap in large language models}, preprint (2025). \href{https://arxiv.org/abs/2505.12896}{\nolinkurl{arXiv:2505.12896}}. \bibitem{LG26} T. Leelawat and L. D. Griffin, \emph{Watson \& Holmes: a naturalistic benchmark for comparing human and LLM reasoning}, preprint (2026). \href{https://arxiv.org/abs/2602.19914}{\nolinkurl{arXiv:2602.19914}}. \bibitem{Lie68} D. I. Lieberman, \emph{Numerical and homological equivalence of algebraic cycles on Hodge manifolds}, Amer. J. Math. \textbf{90} (1968), 366--374. \href{https://doi.org/10.2307/2373533}{\nolinkurl{doi:10.2307/2373533}}. \bibitem{LKB23} H. Lightman, V. Kosaraju, Y. Burda, H. Edwards, B. Baker, T. Lee, J. Leike, J. Schulman, I. Sutskever and K. Cobbe, \emph{Let's verify step by step}, preprint (2023). \href{https://arxiv.org/abs/2305.20050}{\nolinkurl{arXiv:2305.20050}}. \bibitem{LLL22} G. Lample, M.-A. Lachaux, T. Lavril, X. Martinet et~al., \emph{HyperTree proof search for neural theorem proving}, Advances in Neural Information Processing Systems \textbf{35} (2022). \href{https://arxiv.org/abs/2205.11491}{\nolinkurl{arXiv:2205.11491}}. \bibitem{Lob55} M. H. L\"ob, \emph{Solution of a problem of Leon Henkin}, J. Symbolic Logic \textbf{20} (1955), no.~2, 115--118. \href{https://doi.org/10.2307/2266895}{\nolinkurl{doi:10.2307/2266895}}. \bibitem{LTL25} Y. Lin, S. Tang, B. Lyu, J. Wu et~al., \emph{Goedel-Prover: a frontier model for open-source automated theorem proving}, preprint (2025). \href{https://arxiv.org/abs/2502.07640}{\nolinkurl{arXiv:2502.07640}}. \bibitem{LTP24} J. K. Lappalainen, F. D. Tschopp, S. Prakhya, M. McGill et al., \emph{Connectome-constrained networks predict neural activity across the fly visual system}, Nature \textbf{634} (2024), 1132--1140. \href{https://doi.org/10.1038/s41586-024-07939-3}{\nolinkurl{doi:10.1038/s41586-024-07939-3}}. \bibitem{Lu26} C.-P. Lu, \emph{Verification and self-improvement in agentic AI: foundations and limits}, preprint (2026). \href{https://arxiv.org/abs/2610.10611}{\nolinkurl{arXiv:2610.10611}}. \bibitem{Lus10} M. L\"uscher, \emph{Properties and uses of the Wilson flow in lattice QCD}, J. High Energy Phys. \textbf{2010} (2010), no.~8, 071. \href{https://doi.org/10.1007/JHEP08(2010)071}{\nolinkurl{doi:10.1007/JHEP08(2010)071}}. \bibitem{Maa97} W. Maass, \emph{Networks of spiking neurons: the third generation of neural network models}, Neural Networks \textbf{10} (1997), 1659--1671. \href{https://doi.org/10.1016/S0893-6080(97)00011-7}{\nolinkurl{doi:10.1016/S0893-6080(97)00011-7}}. \bibitem{Mar25w} E. Markman, \emph{Cycles on abelian $2n$-folds of Weil type from secant sheaves on abelian $n$-folds}, preprint (2025), \href{https://arxiv.org/abs/2502.03415}{\nolinkurl{arXiv:2502.03415}}. \bibitem{McM56} B. McMillan, \emph{Two inequalities implied by unique decipherability}, IRE Trans. Inform. Theory \textbf{2} (1956), no.~4, 115--116. \href{https://doi.org/10.1109/TIT.1956.1056818}{\nolinkurl{doi:10.1109/TIT.1956.1056818}}. \bibitem{Mea90} C. Mead, \emph{Neuromorphic electronic systems}, Proc. IEEE \textbf{78} (1990), 1629--1636. \href{https://doi.org/10.1109/5.58356}{\nolinkurl{doi:10.1109/5.58356}}. \bibitem{MMZ23} D. J. Mankowitz, A. Michi, A. Zhernov, M. Gelmi, M. Selvi, C. Paduraru et~al., \emph{Faster sorting algorithms discovered using deep reinforcement learning}, Nature \textbf{618} (2023), no.~7964, 257--263. \href{https://doi.org/10.1038/s41586-023-06004-9}{\nolinkurl{doi:10.1038/s41586-023-06004-9}}. \bibitem{MRT23} K. Matom\"aki, M. Radziwi\l\l, T. Tao, J. Ter\"av\"ainen and T. Ziegler, \emph{Higher uniformity of bounded multiplicative functions in short intervals on average}, Ann. of Math. (2) \textbf{197} (2023), 739--857. \href{https://doi.org/10.4007/annals.2023.197.2.3}{\nolinkurl{doi:10.4007/annals.2023.197.2.3}}. \bibitem{MTG23} A. Madaan, N. Tandon, P. Gupta, S. Hallinan, L. Gao, S. Wiegreffe et~al., \emph{Self-refine: iterative refinement with self-feedback}, Advances in Neural Information Processing Systems \textbf{36} (2023). \href{https://arxiv.org/abs/2303.17651}{\nolinkurl{arXiv:2303.17651}}. \bibitem{MU21} L. de Moura and S. Ullrich, \emph{The Lean 4 theorem prover and programming language}, in Automated Deduction -- CADE 28, Lecture Notes in Comput. Sci.~12699, Springer, Cham, 2021, 625--635. \href{https://doi.org/10.1007/978-3-030-79876-5_37}{\nolinkurl{doi:10.1007/978-3-030-79876-5_37}}. \bibitem{MZ98} B. J. J. Moonen and Yu. G. Zarhin, \emph{Weil classes on abelian varieties}, J. Reine Angew. Math. \textbf{496} (1998), 83--92. \href{https://doi.org/10.1515/crll.1998.034}{\nolinkurl{doi:10.1515/crll.1998.034}}. \bibitem{NVE25} A. Novikov, N. V\~u, M. Eisenberger, E. Dupont et~al., \emph{AlphaEvolve: a coding agent for scientific and algorithmic discovery}, preprint (2025). \href{https://arxiv.org/abs/2506.13131}{\nolinkurl{arXiv:2506.13131}}. \bibitem{OAI26} OpenAI, \emph{Finite time blowup for Navier--Stokes}, preprint (2026), with the Lean~4 formalization \texttt{openai/NavierStokesAndEuler}. \href{https://cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf}{\nolinkurl{cdn.openai.com/pdf/32d9f210-8b73-45e0-91bc-82a30aef8a9a/navier-stokes.pdf}}. \bibitem{OAI26b} OpenAI, \emph{Mathematical manuscripts and proof artifacts}, repository \texttt{openai/math} (2026), 722 manuscripts in 372 families; mirror \href{https://github.com/creelie/math01-openAI}{\nolinkurl{github.com/creelie/math01-openAI}}. \bibitem{OAI26c} OpenAI, \emph{Ordinary two-point correlations of multiplicative functions}, preprint (24 September 2026), with a Lean~4 formalization, in \texttt{openai/math}; mirror \href{https://github.com/creelie/math01-openAI}{\nolinkurl{github.com/creelie/math01-openAI}}. \bibitem{OIT26} The Omega Institute, \emph{trureturing: a scientific method for AI to discover truth and find its next question}, repository (2026), with $782$ problem dossiers and their Lean~4 resolutions; \href{https://github.com/the-omega-institute/trureturing}{\nolinkurl{github.com/the-omega-institute/trureturing}}; copy \href{https://github.com/creelie/trureturing-OpenRefute}{\nolinkurl{github.com/creelie/trureturing-OpenRefute}}. \bibitem{Ord26} T. Ord, \emph{The dynamics of intelligence explosions}, preprint (2026). \href{https://arxiv.org/abs/2608.14426}{\nolinkurl{arXiv:2608.14426}}. \bibitem{ORf1183} The Omega Institute, \emph{trureturing}, Lean~4 theorem \texttt{odd\_lower\_bound} in \nolinkurl{D5/S3/Combinatorics/ErdosUlam/SublatticeChainBound.lean}, repository \nolinkurl{the-omega-institute/trureturing} (2026), commit \texttt{fa35d5b}; mirror \href{https://github.com/creelie/trureturing-OpenRefute}{\nolinkurl{github.com/creelie/trureturing-OpenRefute}}. \bibitem{OSei78} K. Osterwalder and E. Seiler, \emph{Gauge field theories on a lattice}, Ann. Physics \textbf{110} (1978), 440--471. \href{https://doi.org/10.1016/0003-4916(78)90039-8}{\nolinkurl{doi:10.1016/0003-4916(78)90039-8}}. \bibitem{Osg98} W. F. Osgood, \emph{Beweis der Existenz einer L\"osung der Differentialgleichung $dy/dx=f(x,y)$ ohne Hinzunahme der Cauchy--Lipschitz'schen Bedingung}, Monatsh. Math. Phys. \textbf{9} (1898), 331--345. \href{https://doi.org/10.1007/BF01707876}{\nolinkurl{doi:10.1007/BF01707876}}. \bibitem{Ott49} R. Otter, \emph{The multiplicative process}, Ann. Math. Statist. \textbf{20} (1949), no.~2, 206--224. \href{https://doi.org/10.1214/aoms/1177730031}{\nolinkurl{doi:10.1214/aoms/1177730031}}. \bibitem{Per02} G. Perelman, \emph{The entropy formula for the Ricci flow and its geometric applications}, preprint (2002). \href{https://arxiv.org/abs/math/0211159}{\nolinkurl{arXiv:math/0211159}}. \bibitem{Per03a} G. Perelman, \emph{Ricci flow with surgery on three-manifolds}, preprint (2003). \href{https://arxiv.org/abs/math/0303109}{\nolinkurl{arXiv:math/0303109}}. \bibitem{Per03b} G. Perelman, \emph{Finite extinction time for the solutions to the Ricci flow on certain three-manifolds}, preprint (2003). \href{https://arxiv.org/abs/math/0307245}{\nolinkurl{arXiv:math/0307245}}. \bibitem{Pil23} C. Pilatte, \emph{Improved bounds for the two-point logarithmic Chowla conjecture}, preprint (2023). \href{https://arxiv.org/abs/2310.19357}{\nolinkurl{arXiv:2310.19357}}. \bibitem{Pil26} C. Pilatte, \emph{Improved bounds for the Fourier uniformity conjecture}, preprint (2026). \href{https://arxiv.org/abs/2604.26564}{\nolinkurl{arXiv:2604.26564}}. \bibitem{Pre09} W. H. Press, \emph{Strong profiling is not mathematically optimal for discovering rare malfeasors}, Proc. Natl. Acad. Sci. USA \textbf{106} (2009), 1716--1719. \href{https://doi.org/10.1073/pnas.0813202106}{\nolinkurl{doi:10.1073/pnas.0813202106}}. \bibitem{PRZ20} K. Pratt, N. Robles, A. Zaharescu and D. Zeindler, \emph{More than five-twelfths of the zeros of $\zeta$ are on the critical line}, Res. Math. Sci. \textbf{7} (2020), 2. \href{https://doi.org/10.1007/s40687-019-0199-8}{\nolinkurl{doi:10.1007/s40687-019-0199-8}}. \bibitem{PS20} S. Polu and I. Sutskever, \emph{Generative language modeling for automated theorem proving}, preprint (2020). \href{https://arxiv.org/abs/2009.03393}{\nolinkurl{arXiv:2009.03393}}. \bibitem{PT21} D. Platt and T. Trudgian, \emph{The Riemann hypothesis is true up to $3\cdot10^{12}$}, Bull. Lond. Math. Soc. \textbf{53} (2021), 792--797. \href{https://doi.org/10.1112/blms.12460}{\nolinkurl{doi:10.1112/blms.12460}}. \bibitem{Qwe24} Qwen Team (A. Yang et~al.), \emph{Qwen2.5 technical report}, preprint (2024). \href{https://arxiv.org/abs/2412.15115}{\nolinkurl{arXiv:2412.15115}}. \bibitem{RBN24} B. Romera-Paredes, M. Barekatain, A. Novikov, M. Balog, M. P. Kumar, E. Dupont et~al., \emph{Mathematical discoveries from program search with large language models}, Nature \textbf{625} (2024), no.~7995, 468--475. \href{https://doi.org/10.1038/s41586-023-06924-6}{\nolinkurl{doi:10.1038/s41586-023-06924-6}}. \bibitem{Ren61} A. R\'enyi, \emph{On measures of entropy and information}, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, vol.~1, University of California Press, Berkeley, 1961, 547--561. \bibitem{Ris83} J. Rissanen, \emph{A universal prior for integers and estimation by minimum description length}, Ann. Statist. \textbf{11} (1983), no.~2, 416--431. \href{https://doi.org/10.1214/aos/1176346150}{\nolinkurl{doi:10.1214/aos/1176346150}}. \bibitem{RPG99} P. Redgrave, T. J. Prescott and K. Gurney, \emph{The basal ganglia: a vertebrate solution to the selection problem?}, Neuroscience \textbf{89} (1999), 1009--1023. \href{https://doi.org/10.1016/S0306-4522(98)00319-4}{\nolinkurl{doi:10.1016/S0306-4522(98)00319-4}}. \bibitem{RR97} A. A. Razborov and S. Rudich, \emph{Natural proofs}, J. Comput. System Sci. \textbf{55} (1997), 24--35. \href{https://doi.org/10.1006/jcss.1997.1494}{\nolinkurl{doi:10.1006/jcss.1997.1494}}. \bibitem{RSS25} D. Rohatgi, A. Shetty, D. Saless, Y. Li, A. Moitra, A. Risteski and D. J. Foster, \emph{Taming imperfect process verifiers: a sampling perspective on backtracking}, preprint (2025). \href{https://arxiv.org/abs/2510.03149}{\nolinkurl{arXiv:2510.03149}}. \bibitem{RSX25} Z. Z. Ren, Z. Shao, J. Song, H. Xin et~al., \emph{DeepSeek-Prover-V2: advancing formal mathematical reasoning via reinforcement learning for subgoal decomposition}, preprint (2025). \href{https://arxiv.org/abs/2504.21801}{\nolinkurl{arXiv:2504.21801}}. \bibitem{Sch03} J. Schmidhuber, \emph{G\"odel machines: self-referential universal problem solvers making provably optimal self-improvements}, preprint (2003). \href{https://arxiv.org/abs/cs/0309048}{\nolinkurl{arXiv:cs/0309048}}. \bibitem{Sch80} J. T. Schwartz, \emph{Fast probabilistic algorithms for verification of polynomial identities}, J. ACM \textbf{27} (1980), no.~4, 701--717. \href{https://doi.org/10.1145/322217.322225}{\nolinkurl{doi:10.1145/322217.322225}}. \bibitem{Sch98p} C. Schoen, \emph{Addendum to: Hodge classes on self-products of a variety with an automorphism}, Compos. Math. \textbf{114} (1998), 321--328. \href{https://doi.org/10.1023/A:1000566205021}{\nolinkurl{doi:10.1023/A:1000566205021}}. \bibitem{Scr26} A. Scrivens, \emph{Empirical validation of the classification-verification dichotomy for AI safety gates}, preprint (2026). \href{https://arxiv.org/abs/2604.00072}{\nolinkurl{arXiv:2604.00072}}. \bibitem{SDM97} W. Schultz, P. Dayan and P. R. Montague, \emph{A neural substrate of prediction and reward}, Science \textbf{275} (1997), 1593--1599. \href{https://doi.org/10.1126/science.275.5306.1593}{\nolinkurl{doi:10.1126/science.275.5306.1593}}. \bibitem{SGH19} D. Saxton, E. Grefenstette, F. Hill and P. Kohli, \emph{Analysing mathematical reasoning abilities of neural models}, International Conference on Learning Representations (2019). \href{https://arxiv.org/abs/1904.01557}{\nolinkurl{arXiv:1904.01557}}. \bibitem{Sha48} C. E. Shannon, \emph{A mathematical theory of communication}, Bell System Tech. J. \textbf{27} (1948), no.~3, 379--423. \href{https://doi.org/10.1002/j.1538-7305.1948.tb01338.x}{\nolinkurl{doi:10.1002/j.1538-7305.1948.tb01338.x}}. \bibitem{Shi79} T. Shioda, \emph{The Hodge conjecture for Fermat varieties}, Math. Ann. \textbf{245} (1979), 175--184. \href{https://doi.org/10.1007/BF01428804}{\nolinkurl{doi:10.1007/BF01428804}}. \bibitem{SHM16} D. Silver, A. Huang, C. J. Maddison, A. Guez, L. Sifre, G. van den Driessche et~al., \emph{Mastering the game of Go with deep neural networks and tree search}, Nature \textbf{529} (2016), no.~7587, 484--489. \href{https://doi.org/10.1038/nature16961}{\nolinkurl{doi:10.1038/nature16961}}. \bibitem{SHS18} D. Silver, T. Hubert, J. Schrittwieser, I. Antonoglou, M. Lai, A. Guez et~al., \emph{A general reinforcement learning algorithm that masters chess, shogi, and Go through self-play}, Science \textbf{362} (2018), no.~6419, 1140--1144. \href{https://doi.org/10.1126/science.aar6404}{\nolinkurl{doi:10.1126/science.aar6404}}. \bibitem{SJB24} A. Shapson-Coe, M. Januszewski, D. R. Berger, A. Pope et al., \emph{A petavoxel fragment of human cerebral cortex reconstructed at nanoscale resolution}, Science \textbf{384} (2024), eadk4858. \href{https://doi.org/10.1126/science.adk4858}{\nolinkurl{doi:10.1126/science.adk4858}}. \bibitem{SLX24} C. Snell, J. Lee, K. Xu and A. Kumar, \emph{Scaling LLM test-time compute optimally can be more effective than scaling model parameters}, preprint (2024). \href{https://arxiv.org/abs/2408.03314}{\nolinkurl{arXiv:2408.03314}}. \bibitem{Sol64} R. J. Solomonoff, \emph{A formal theory of inductive inference. Part I}, Information and Control \textbf{7} (1964), no.~1, 1--22. \href{https://doi.org/10.1016/S0019-9958(64)90223-2}{\nolinkurl{doi:10.1016/S0019-9958(64)90223-2}}. \bibitem{Ste93} R. Steinberg, \emph{The state of the three color problem}, in \emph{Quo Vadis, Graph Theory?}, Ann. Discrete Math. \textbf{55} (1993), 211--248. \href{https://doi.org/10.1016/S0167-5060(08)70391-1}{\nolinkurl{doi:10.1016/S0167-5060(08)70391-1}}. \bibitem{SW94} N. Seiberg and E. Witten, \emph{Electric-magnetic duality, monopole condensation, and confinement in $N=2$ supersymmetric Yang--Mills theory}, Nuclear Phys. B \textbf{426} (1994), 19--52. \href{https://doi.org/10.1016/0550-3213(94)90124-4}{\nolinkurl{doi:10.1016/0550-3213(94)90124-4}}. \bibitem{SZZ23} H. Shen, R. Zhu and X. Zhu, \emph{A stochastic analysis approach to lattice Yang--Mills at strong coupling}, Comm. Math. Phys. \textbf{400} (2023), 805--851. \href{https://doi.org/10.1007/s00220-022-04609-1}{\nolinkurl{doi:10.1007/s00220-022-04609-1}}. \bibitem{Tan10} S. G. Tankeev, \emph{On the standard conjecture of Lefschetz type for complex projective threefolds}, Izv. Math. \textbf{74} (2010), 167--187. \href{https://doi.org/10.1070/IM2010v074n01ABEH002484}{\nolinkurl{doi:10.1070/IM2010v074n01ABEH002484}}. \bibitem{Tan11} S. G. Tankeev, \emph{On the standard conjecture of Lefschetz type for complex projective threefolds.~II}, Izv. Math. \textbf{75} (2011), 1047--1062. \href{https://doi.org/10.1070/IM2011v075n05ABEH002563}{\nolinkurl{doi:10.1070/IM2011v075n05ABEH002563}}. \bibitem{Tao16} T. Tao, \emph{The logarithmically averaged Chowla and Elliott conjectures for two-point correlations}, Forum Math. Pi \textbf{4} (2016), e8. \href{https://doi.org/10.1017/fmp.2016.6}{\nolinkurl{doi:10.1017/fmp.2016.6}}. \bibitem{TLD98} G. G. Turrigiano, K. R. Leslie, N. S. Desai, L. C. Rutherford and S. B. Nelson, \emph{Activity-dependent scaling of quantal amplitude in neocortical neurons}, Nature \textbf{391} (1998), 892--896. \href{https://doi.org/10.1038/36103}{\nolinkurl{doi:10.1038/36103}}. \bibitem{TT19a} T. Tao and J. Ter\"av\"ainen, \emph{The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures}, Duke Math. J. \textbf{168} (2019), 1977--2027. \href{https://doi.org/10.1215/00127094-2019-0002}{\nolinkurl{doi:10.1215/00127094-2019-0002}}. \bibitem{TT19b} T. Tao and J. Ter\"av\"ainen, \emph{Odd order cases of the logarithmically averaged Chowla conjecture}, J. Th\'eor. Nombres Bordeaux \textbf{30} (2018), 997--1015. \href{https://doi.org/10.5802/jtnb.1062}{\nolinkurl{doi:10.5802/jtnb.1062}}. \bibitem{Tur50} A. M. Turing, \emph{Computing machinery and intelligence}, Mind \textbf{59} (1950), no.~236, 433--460. \href{https://doi.org/10.1093/mind/LIX.236.433}{\nolinkurl{doi:10.1093/mind/LIX.236.433}}. \bibitem{TW95} R. Taylor and A. Wiles, \emph{Ring-theoretic properties of certain Hecke algebras}, Ann. of Math. (2) \textbf{141} (1995), no.~3, 553--572. \href{https://doi.org/10.2307/2118560}{\nolinkurl{doi:10.2307/2118560}}. \bibitem{TWL24} T. H. Trinh, Y. Wu, Q. V. Le, H. He and T. Luong, \emph{Solving olympiad geometry without human demonstrations}, Nature \textbf{625} (2024), no.~7995, 476--482. \href{https://doi.org/10.1038/s41586-023-06747-5}{\nolinkurl{doi:10.1038/s41586-023-06747-5}}. \bibitem{UKK22} J. Uesato, N. Kushman, R. Kumar, F. Song, N. Siegel, L. Wang, A. Creswell, G. Irving and I. Higgins, \emph{Solving math word problems with process- and outcome-based feedback}, preprint (2022). \href{https://arxiv.org/abs/2211.14275}{\nolinkurl{arXiv:2211.14275}}. \bibitem{Var23} M. Varesco, \emph{Hodge similarities, algebraic classes, and Kuga--Satake varieties}, Math. Z. \textbf{305} (2023), 69. \href{https://doi.org/10.1007/s00209-023-03390-8}{\nolinkurl{doi:10.1007/s00209-023-03390-8}}. \bibitem{VCP11} L. R. Varshney, B. L. Chen, E. Paniagua, D. H. Hall and D. B. Chklovskii, \emph{Structural properties of the Caenorhabditis elegans neuronal network}, PLoS Comput. Biol. \textbf{7} (2011), e1001066. \href{https://doi.org/10.1371/journal.pcbi.1001066}{\nolinkurl{doi:10.1371/journal.pcbi.1001066}}. \bibitem{Via17} M. Viazovska, \emph{The sphere packing problem in dimension 8}, Ann. of Math. (2) \textbf{185} (2017), no.~3, 991--1015. \href{https://doi.org/10.4007/annals.2017.185.3.7}{\nolinkurl{doi:10.4007/annals.2017.185.3.7}}. \bibitem{VSP17} A. Vaswani, N. Shazeer, N. Parmar, J. Uszkoreit, L. Jones, A. N. Gomez, \L{}. Kaiser and I. Polosukhin, \emph{Attention is all you need}, Advances in Neural Information Processing Systems \textbf{30} (2017). \href{https://arxiv.org/abs/1706.03762}{\nolinkurl{arXiv:1706.03762}}. \bibitem{Wal45} A. Wald, \emph{Sequential tests of statistical hypotheses}, Ann. Math. Statist. \textbf{16} (1945), no.~2, 117--186. \href{https://doi.org/10.1214/aoms/1177731118}{\nolinkurl{doi:10.1214/aoms/1177731118}}. \bibitem{Wei26} S. Weir, \emph{obsidian-brain}, server for semantic search and graph analysis of an Obsidian vault, npm package, version 1.7.24 (2026). \href{https://sweir1.github.io/obsidian-brain/}{\nolinkurl{sweir1.github.io/obsidian-brain}}. \bibitem{Wei84} A. Weil, \emph{Number Theory: An Approach through History from Hammurapi to Legendre}, Birkh\"auser, Boston, 1984. \href{https://doi.org/10.1007/978-0-8176-4571-7}{\nolinkurl{doi:10.1007/978-0-8176-4571-7}}. \bibitem{WG75} H. W. Watson and F. Galton, \emph{On the probability of the extinction of families}, J. Anthropol. Inst. Great Britain and Ireland \textbf{4} (1875), 138--144. \href{https://doi.org/10.2307/2841222}{\nolinkurl{doi:10.2307/2841222}}. \bibitem{Wil95} A. Wiles, \emph{Modular elliptic curves and Fermat's Last Theorem}, Ann. of Math. (2) \textbf{141} (1995), no.~3, 443--551. \href{https://doi.org/10.2307/2118559}{\nolinkurl{doi:10.2307/2118559}}. \bibitem{Wit89} E. Witten, \emph{Quantum field theory and the Jones polynomial}, Comm. Math. Phys. \textbf{121} (1989), 351--399. \href{https://doi.org/10.1007/BF01217730}{\nolinkurl{doi:10.1007/BF01217730}}. \bibitem{WUL25} H. Wang, M. Unsal, X. Lin, M. Baksys et~al., \emph{Kimina-Prover Preview: towards large formal reasoning models with reinforcement learning}, preprint (2025). \href{https://arxiv.org/abs/2504.11354}{\nolinkurl{arXiv:2504.11354}}. \bibitem{WWS22} J. Wei, X. Wang, D. Schuurmans, M. Bosma, B. Ichter, F. Xia, E. Chi, Q. Le and D. Zhou, \emph{Chain-of-thought prompting elicits reasoning in large language models}, Advances in Neural Information Processing Systems \textbf{35} (2022). \href{https://arxiv.org/abs/2201.11903}{\nolinkurl{arXiv:2201.11903}}. \bibitem{YSG23} K. Yang, A. M. Swope, A. Gu, R. Chalamala et~al., \emph{LeanDojo: theorem proving with retrieval-augmented language models}, Advances in Neural Information Processing Systems \textbf{36} (2023), Datasets and Benchmarks Track. \href{https://arxiv.org/abs/2306.15626}{\nolinkurl{arXiv:2306.15626}}. \bibitem{YZH24} A. Yang, B. Zhang, B. Hui, B. Gao, B. Yu, C. Li et~al., \emph{Qwen2.5-Math technical report: toward mathematical expert model via self-improvement}, preprint (2024). \href{https://arxiv.org/abs/2409.12122}{\nolinkurl{arXiv:2409.12122}}. \bibitem{ZDC26} Z. Zhang, A. Delaval, L. Chen, J. Chen, J. Xu, Y. Liao, J. Jiang, C. Liu and B. Dong, \emph{An AI-assisted formalization of the Poincar\'e conjecture}, preprint (2026). \href{https://arxiv.org/abs/2610.08329}{\nolinkurl{arXiv:2610.08329}}. \bibitem{Zip79} R. Zippel, \emph{Probabilistic algorithms for sparse polynomials}, in Symbolic and Algebraic Computation (EUROSAM '79), Lecture Notes in Comput. Sci.~72, Springer, Berlin, 1979, 216--226. \href{https://doi.org/10.1007/3-540-09519-5_73}{\nolinkurl{doi:10.1007/3-540-09519-5_73}}. \bibitem{ZWM22} E. Zelikman, Y. Wu, J. Mu and N. D. Goodman, \emph{STaR: bootstrapping reasoning with reasoning}, Advances in Neural Information Processing Systems \textbf{35} (2022). \href{https://arxiv.org/abs/2203.14465}{\nolinkurl{arXiv:2203.14465}}. \end{thebibliography}