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@article{Chomsky1956ThreeMF,
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@article{Darboux1882Pfaff,
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@book{doCarmo1976CurvesSurfaces,
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@book{LyndonSchupp1977Combinatorial,
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@book{Lee2018RiemannianManifolds,
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}
@book{KobayashiNomizu1963FDG1,
  author    = {Kobayashi, Shoshichi and Nomizu, Katsumi},
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  year      = {1963}
}
@book{Ahlfors1979ComplexAnalysis,
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}

@article{Rosen1954ContinuedFractions,
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  year    = {1954},
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@article{DoranEtAl2013TriangleForms,
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             Shokri, Khosro M.},
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  journal = {Communications in Number Theory and Physics},
  volume  = {7},
  number  = {4},
  pages   = {689--737},
  year    = {2013},
  doi     = {10.4310/CNTP.2013.v7.n4.a4},
  eprint  = {1307.4372},
  archivePrefix = {arXiv}
}

@article{LangLang2016HeckeGroups,
  author  = {Lang, Cheng Lien and Lang, Mong Lung},
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  pages   = {392--417},
  year    = {2016},
  doi     = {10.1016/j.jalgebra.2016.05.001}
}

@article{Dehornoy2015GeodesicFlow,
  author  = {Dehornoy, Pierre},
  title   = {Geodesic Flow, Left-Handedness, and Templates},
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  number  = {3},
  pages   = {1525--1597},
  year    = {2015},
  doi     = {10.2140/agt.2015.15.1525},
  eprint  = {1112.6296},
  archivePrefix = {arXiv}
}

@article{DehornoyPinsky2018Coding,
  author  = {Dehornoy, Pierre and Pinsky, Tali},
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  journal = {Ergodic Theory and Dynamical Systems},
  volume  = {38},
  number  = {3},
  pages   = {940--960},
  year    = {2018},
  doi     = {10.1017/etds.2016.59},
  eprint  = {1411.6857},
  archivePrefix = {arXiv}
}

@book{GrothendieckRaynaud1971SGA1,
  author    = {Grothendieck, Alexander and Raynaud, Mich\`ele},
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  series    = {Lecture Notes in Mathematics},
  volume    = {224},
  publisher = {Springer-Verlag},
  address   = {Berlin},
  year      = {1971},
  doi       = {10.1007/BFb0058656}
}

@book{Hartshorne1977AlgebraicGeometry,
  author    = {Hartshorne, Robin},
  title     = {Algebraic Geometry},
  series    = {Graduate Texts in Mathematics},
  volume    = {52},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1977},
  doi       = {10.1007/978-1-4757-3849-0}
}

@book{Neukirch1999AlgebraicNumberTheory,
  author     = {Neukirch, J{\"u}rgen},
  title      = {Algebraic Number Theory},
  translator = {Schappacher, Norbert},
  series     = {Grundlehren der mathematischen Wissenschaften},
  volume     = {322},
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  address    = {Berlin},
  year       = {1999},
  doi        = {10.1007/978-3-662-03983-0}
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@unpublished{Yuan2026AEGFoundations,
  author = {Yuan, Mingli},
  title  = {Arithmetic Expression Geometry I: Foundations---Sequential Histories,
            Projective Semantics, and Zero Geometry},
  note   = {Mathematical-review manuscript},
  year   = {2026}
}

@unpublished{Yuan2026AEGPaperZero,
  author = {Yuan, Mingli},
  title  = {Arithmetic Expression Geometry 0---Arithmetic Expression Spaces:
            Expansion and Motion, Eikonal Equation, Horocycle Pullbacks,
            Commutativization, Contact Structure, and Tearing},
  note   = {Mathematical-review manuscript.  Author, corresponding author, and
            sole holder of scientific responsibility: Mingli Yuan.
            Formalization and computational assistance by DeepSeek Harness
            (model \texttt{deepseek-v4-flash-vision-exp}) is credited on the
            title page and in the acknowledgements, not as authorship; see the
            repository authorship and provenance register},
  year   = {2026}
}

@article{Hormander1967Hypoelliptic,
  author  = {H\"ormander, Lars},
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  volume  = {119},
  pages   = {147--171},
  year    = {1967},
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@article{CaffarelliSilvestre2007Extension,
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  volume  = {32},
  number  = {7--9},
  pages   = {1245--1260},
  year    = {2007},
  doi     = {10.1080/03605300600987306}
}

@unpublished{Yuan2026AEGFunctionTheory,
  author = {Yuan, Mingli},
  title  = {Arithmetic Expression Geometry II: Hyperbolic Real Function Theory---
            Horizontal Operators, Boundary Problems, and Arithmetic Holomorphicity},
  note   = {Mathematical-review manuscript},
  year   = {2026}
}

@incollection{Ehresmann1952Connexions,
  author    = {Ehresmann, Charles},
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  volume    = {1},
  pages     = {153--168},
  publisher = {Soci\'et\'e Math\'ematique de France},
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  year      = {1952},
  url       = {https://www.numdam.org/item/SB_1948-1951__1__153_0/}
}

@book{Hirsch1976DifferentialTopology,
  author    = {Hirsch, Morris W.},
  title     = {Differential Topology},
  series    = {Graduate Texts in Mathematics},
  volume    = {33},
  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1976},
  doi       = {10.1007/978-1-4684-9449-5}
}

@article{Artin1925TheorieDerZoepfe,
  author  = {Artin, Emil},
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  volume  = {4},
  pages   = {47--72},
  year    = {1925},
  doi     = {10.1007/BF02950718}
}

@article{FadellNeuwirth1962ConfigurationSpaces,
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  volume  = {10},
  pages   = {111--118},
  year    = {1962},
  doi     = {10.7146/math.scand.a-10517}
}

@book{Birman1974Braids,
  author    = {Birman, Joan S.},
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  year      = {1974},
  doi       = {10.1515/9781400881420}
}

@book{Brown1982CohomologyGroups,
  author    = {Brown, Kenneth S.},
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  publisher = {Springer-Verlag},
  address   = {New York},
  year      = {1982},
  doi       = {10.1007/978-1-4684-9327-6}
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@article{Alexander1923KnottedCurves,
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  year    = {1923},
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@article{Rudolph1983AlgebraicFunctions,
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  year    = {1983},
  doi     = {10.1016/0040-9383(83)90031-9}
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@article{CarterElhamdadiSaito2002TwistedQuandle,
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  volume        = {2},
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  year          = {2002},
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  eprint        = {math/0108051},
  archivePrefix = {arXiv}
}

@article{Looijenga1974Bifurcation,
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  year    = {1974},
  doi     = {10.1007/BF01405164}
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@article{Lyashko1984Bifurcation,
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@article{Denes1959TranspositionFactorizations,
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  year    = {1959},
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@article{BirmanHilden1973Isotopies,
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  year    = {1973},
  doi     = {10.2307/1970830}
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@article{ACampo1979Tresses,
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  year    = {1979},
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@article{Selmer1956IrreducibleTrinomials,
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@article{Osada1987GaloisTrinomials,
  author  = {Osada, Hiroyuki},
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@article{Zarhin2000HyperellipticJacobians,
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  doi     = {10.4310/MRL.2000.v7.n1.a11},
  eprint  = {math/9909052},
  archivePrefix = {arXiv}
}

@article{Penner1988PseudoAnosov,
  author  = {Penner, Robert C.},
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  volume  = {310},
  number  = {1},
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  year    = {1988},
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@article{DigneKassel2023SymplecticSteinberg,
  author        = {Digne, Fran{\c{c}}ois and Kassel, Christian},
  title         = {Braid Groups and Symplectic Steinberg Groups},
  journal       = {Annals of K-Theory},
  volume        = {8},
  number        = {4},
  pages         = {669--692},
  year          = {2023},
  doi           = {10.2140/akt.2023.8.669},
  eprint        = {2201.07153},
  archivePrefix = {arXiv}
}

@article{Igusa1960ArithmeticVariety,
  author  = {Igusa, Jun-ichi},
  title   = {Arithmetic Variety of Moduli for Genus Two},
  journal = {Annals of Mathematics},
  volume  = {72},
  number  = {3},
  pages   = {612--649},
  year    = {1960},
  doi     = {10.2307/1970233}
}

@article{FarbKisinWolfson2023Modular,
  author  = {Farb, Benson and Kisin, Mark and Wolfson, Jesse},
  title   = {Modular Functions and Resolvent Problems},
  journal = {Mathematische Annalen},
  volume  = {386},
  number  = {1--2},
  pages   = {113--150},
  year    = {2023},
  doi     = {10.1007/s00208-022-02395-8},
  eprint  = {1912.12536},
  archivePrefix = {arXiv}
}

@article{FarbWolfson2019Resolvent,
  author  = {Farb, Benson and Wolfson, Jesse},
  title   = {Resolvent Degree, {Hilbert}'s 13th Problem and Geometry},
  journal = {L'Enseignement Math\'ematique},
  volume  = {65},
  number  = {3--4},
  pages   = {303--376},
  year    = {2019},
  doi     = {10.4171/LEM/65-3/4-2},
  eprint  = {1803.04063},
  archivePrefix = {arXiv}
}

@article{FarbLooijenga2021WimanEdge,
  author  = {Farb, Benson and Looijenga, Eduard},
  title   = {Arithmeticity of the Monodromy of the {Wiman--Edge} Pencil},
  journal = {Annales de l'Institut Fourier},
  volume  = {71},
  number  = {4},
  pages   = {1325--1361},
  year    = {2021},
  doi     = {10.5802/aif.3423},
  eprint  = {1911.01210},
  archivePrefix = {arXiv}
}

@article{Farb2024ModuliRigidity,
  author  = {Farb, Benson},
  title   = {Rigidity of Moduli Spaces and Algebro-Geometric Constructions},
  journal = {Surveys in Differential Geometry},
  volume  = {26},
  pages   = {31--49},
  year    = {2024},
  doi     = {10.4310/SDG.2021.v26.n1.a3},
  eprint  = {2302.06369},
  archivePrefix = {arXiv}
}

@misc{FarbWolfson2025RelativeEssentialDimension,
  author        = {Farb, Benson and Wolfson, Jesse},
  title         = {Essential Dimension Relative to Branched Covers of Degree at Most $n$},
  year          = {2025},
  eprint        = {2510.22786},
  archivePrefix = {arXiv},
  primaryClass  = {math.AG}
}

@unpublished{Yuan2026AEGZeroGeometry,
  author = {Yuan, Mingli},
  title  = {Arithmetic Expression Geometry III: Singular Zero Geometry and Tubes---
            Branched Pullbacks, Arithmetic Zero Networks, and Topological Transport},
  note   = {Mathematical-review manuscript},
  year   = {2026}
}

@article{Shannon1948,
  author  = {Shannon, Claude E.},
  title   = {A Mathematical Theory of Communication},
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  year    = {1948}
}

@article{Nerode1958,
  author  = {Nerode, Anil},
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  doi     = {10.1090/S0002-9939-1958-0135681-9}
}

@article{Newman1942,
  author  = {Newman, M. H. A.},
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  pages = {223--243}, year = {1942}, doi = {10.2307/1968867}
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@article{Gromov1981,
  author  = {Gromov, M.},
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@article{Bryant1986,
  author  = {Bryant, Randal E.},
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@book{Wegener2000,
  author = {Wegener, Ingo},
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}

@article{CooleyTukey1965,
  author = {Cooley, James W. and Tukey, John W.},
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@article{Bennett1989,
  author = {Bennett, Charles H.},
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}

@article{GriewankWalther2000,
  author = {Griewank, Andreas and Walther, Andrea},
  title = {Algorithm 799: {REVOLVE}: An Implementation of Checkpointing for the Reverse or Adjoint Mode of Computational Differentiation},
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  pages = {19--45}, year = {2000}, doi = {10.1145/347837.347846}
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@book{Wu2000MathematicsMechanization,
  author    = {Wu, Wen-tsun},
  title     = {Mathematics Mechanization: Mechanical Geometry Theorem-Proving,
               Mechanical Geometry Problem-Solving and Polynomial Equations-Solving},
  series    = {Mathematics and Its Applications},
  publisher = {Kluwer Academic Publishers},
  address   = {Dordrecht},
  year      = {2000},
  isbn      = {0-7923-5835-X}
}

@misc{Yuan2026ProcessGeometry,
  author       = {Yuan, Mingli and {Process Geometry contributors}},
  title        = {Process Geometry},
  howpublished = {\url{https://github.com/mountain/process-geometry}},
  year         = {2026},
  note         = {Research software and literate calibrations, version 0.0.4;
                  pre-alpha research preview.  Cited here only for the
                  research-local calibrations named in the text, which carry the
                  repository's own maturity labels.}
}

@unpublished{Yuan2026AEGHoleRegister,
  author = {Yuan, Mingli},
  title  = {The Inevitability of Holes: A Ledger of Six Computed Obstructions},
  note   = {Exploration register, retained as
            \texttt{notes/foundations-and-geometry/06-hole-obstructions-ledger.md}
            in the AEG paper repository.  Non-authoritative working record; not
            a paper claim.  English rendering of a Chinese working note of the
            same content.},
  year   = {2026}
}

@incollection{Etnyre2001IntroductoryLectures,
  author    = {Etnyre, John B.},
  title     = {Introductory Lectures on Contact Geometry},
  booktitle = {Topology and Geometry of Manifolds},
  series    = {Proceedings of Symposia in Pure Mathematics},
  volume    = {71},
  pages     = {81--90},
  publisher = {American Mathematical Society},
  address   = {Providence, RI},
  year      = {2003},
  note      = {arXiv:math/0111118}
}

@book{Khinchin1964ContinuedFractions,
  author    = {Khinchin, A. Ya.},
  title     = {Continued Fractions},
  publisher = {University of Chicago Press},
  address   = {Chicago},
  year      = {1964},
  note      = {Translated from the third Russian edition; reprinted by Dover,
               1997.  Cited for the standard convergence theory of continued
               fractions, not for the specific arithmetic reading used here}
}

@article{Wolpert2003Completion,
  author = {Wolpert, Scott A.},
  title = {Geometry of the {Weil--Petersson} Completion of {Teichm\"uller} Space},
  journal = {Surveys in Differential Geometry},
  volume = {8},
  pages = {357--393},
  year = {2003},
  doi = {10.4310/SDG.2003.v8.n1.a13},
  url = {https://www.math.umd.edu/~swolpert/preprints/WPcompletion.pdf}
}

@article{GrahamEtAl2005Apollonian,
  author = {Graham, Ronald L. and Lagarias, Jeffrey C. and Mallows, Colin L.
            and Wilks, Allan R. and Yan, Catherine H.},
  title = {Apollonian Circle Packings: Geometry and Group Theory {I}.
           The {Apollonian} Group},
  journal = {Discrete \& Computational Geometry},
  volume = {34},
  pages = {547--585},
  year = {2005},
  doi = {10.1007/s00454-005-1196-9},
  url = {https://arxiv.org/abs/math/0010298}
}

@book{Hatcher2002,
  author = {Hatcher, Allen},
  title = {Algebraic Topology},
  publisher = {Cambridge University Press},
  year = {2002},
  url = {https://pi.math.cornell.edu/~hatcher/AT/ATch1.pdf}
}

@unpublished{Yuan2026AEGCondensation,
  author = {Yuan, Mingli},
  title = {Arithmetic Expression Geometry {IV}: Projective Condensation
           and Computational Complexity},
  year = {2026},
  note = {Companion manuscript, repository path \texttt{paper-4/}.}
}

@misc{Yuan2026AdvaProgramProcessCore,
  author = {Yuan, Mingli},
  title = {Technical Report: The Program-Process Core of Adva},
  year = {2026},
  url = {https://github.com/mountain/adva/blob/cce73004c2b4fbfb87d9ba1ccc66820423273cf6/docs/TECHNICAL_REPORT_PROGRAM_PROCESS_CORE.md},
  note = {Pinned research report at commit cce7300; finite program substitution
          and program/value distinction, Sections 1.2--1.3.}
}

@misc{Yuan2026AdvaTriadicFrontiers,
  author = {Yuan, Mingli},
  title = {Neutral Carrier and Triadic Mechanism Frontiers},
  year = {2026},
  url = {https://github.com/mountain/adva/blob/cce73004c2b4fbfb87d9ba1ccc66820423273cf6/docs/research/0109-neutral-carrier-triadic-mechanism-frontiers.md},
  note = {Research note 0109, pinned at commit cce7300; outer arity and inner
          holes are distinct.}
}

@misc{Yuan2026AdvaArithmeticCalibration,
  author = {Yuan, Mingli},
  title = {Three-Sided Trace-Arithmetic Calibration},
  year = {2026},
  url = {https://github.com/mountain/adva/blob/cce73004c2b4fbfb87d9ba1ccc66820423273cf6/docs/research/0114-three-sided-trace-arithmetic-calibration.md},
  note = {Research note 0114, pinned at commit cce7300; all three opposite-side
          characteristic witnesses remain open in the recorded version.}
}

@article{Richardson1968UndecidableProblems,
  author = {Richardson, Daniel},
  title = {Some Undecidable Problems Involving Elementary Functions of a Real Variable},
  journal = {The Journal of Symbolic Logic},
  volume = {33},
  number = {4},
  pages = {514--520},
  year = {1968},
  doi = {10.2307/2271358}
}

@misc{Yuan2022AEGEvaluableExpressions,
  author = {Yuan, Mingli},
  title = {Geometry of Arithmetic Expressions: I. Basics and Problems},
  year = {2022},
  url = {https://github.com/mountain/aeg-paper/blob/339e0ce/aeg-paper.tex},
  note = {Historical draft, 23 December 2022, commit 339e0ce;
          arithmetic-expression subsection defines partial evaluation and evaluability.}
}

@misc{Yuan2023AEGFoundationalProblem,
  author = {Yuan, Mingli},
  title = {Geometry of Arithmetic Expressions: I. Basic Concepts and Unsolved Problems},
  year = {2023},
  url = {https://github.com/mountain/aeg-paper/blob/b7a4d3f4bc007cde3dc3a2f15e8c444e0d16d059/aeg-paper.tex},
  note = {Historical draft, 2 January 2023; the Foundational problem
          connects rational arithmetic, division guards and semantic decision questions.}
}

@misc{Yuan2026ObserverMetricCompletion,
  author = {Yuan, Mingli},
  title = {Process Geometry Mathematical Core: Why the {Bolza} Surface Appears},
  year = {2026},
  url = {https://github.com/mountain/process-geometry/blob/c47c96fa79123c677172278be59d67ca1cc891b1/docs/MATHEMATICAL_CORE.md},
  note = {Section 2.6, pinned at commit c47c96f. Product-sign quotient after
          an independently declared metric square-root layer; not a canonical
          compactification of the hyperbolic plane.}
}

@misc{Yuan2023AEGAlternatingExpressions,
  author = {Yuan, Mingli},
  title = {Geometry of Arithmetic Expressions: Alternating Threadlike Expressions},
  year = {2023},
  url = {https://github.com/mountain/aeg-paper/blob/6aeef1729cd8b8951a49cdb5e080c7ebb774216a/sections/sec02.tex},
  note = {Historical draft, 21 February 2023; alternating words, distributive
          expansion and exact arithmetic propagation of an initial perturbation.}
}

@misc{Yuan2025AEGScaleRenormalization,
  author = {Yuan, Mingli},
  title = {Arithmetic Expression Geometry: Flow, Accumulation, Dual Timescales,
           and a Minimalist Realization of Renormalization},
  year = {2025},
  url = {https://github.com/mountain/aeg-paper/blob/010719c05b2895a3fd11c331489ad4e2d61e8b1e/notes/rg_en.tex},
  note = {Exploratory note, 20 May 2025, Section 3; cited for the historical
          interpretation of perturbation gain as accumulated scale,
          not as a proof of a general renormalization construction.}
}

@article{BauerTaylor2009Dedekind,
  author = {Bauer, Andrej and Taylor, Paul},
  title = {The {Dedekind} Reals in Abstract {Stone} Duality},
  journal = {Mathematical Structures in Computer Science},
  volume = {19},
  pages = {757--838},
  year = {2009},
  doi = {10.1017/S0960129509007695},
  url = {https://www.paultaylor.eu/ASD/dedras/}
}

@misc{Taylor2006IntervalWithoutIntervals,
  author = {Taylor, Paul},
  title = {Interval Analysis Without Intervals},
  year = {2006},
  url = {https://www.paultaylor.eu/ASD/intawi.pdf},
  note = {Manuscript dated 20 February 2006; exact real calculus and its
          translation to interval computation.}
}

@book{Conway1976NumbersGames,
  author = {Conway, John Horton},
  title = {On Numbers and Games},
  publisher = {Academic Press},
  address = {London},
  series = {London Mathematical Society Monographs},
  number = {6},
  year = {1976},
  isbn = {0-12-186350-6}
}

@article{Ehrlich2011ConwayNames,
  author = {Ehrlich, Philip},
  title = {{Conway} Names, the Simplicity Hierarchy and the Surreal Number Tree},
  journal = {Journal of Logic and Analysis},
  volume = {3},
  number = {1},
  pages = {1--26},
  year = {2011},
  doi = {10.4115/jla.2011.3.1}
}

@misc{Adva2026GroundedThrough,
  author = {{Adva contributors}},
  title = {Grounded Multi-Hole Through Adapter {V0}},
  year = {2026},
  url = {https://github.com/mountain/adva/blob/7a57d8d1cf13f683ff15ed856795b020b2ea7083/docs/research/0075-grounded-multi-hole-through-adapter-v0.md},
  note = {Research note 0075, pinned at commit 7a57d8d; bounded research
          adapter over checked program incidences, not a stable native operation.}
}

@misc{Adva2026CutObjectification,
  author = {{Adva contributors}},
  title = {Surreal Cut Presentations, Objectification, and a {PSP}-Star No-Go},
  year = {2026},
  url = {https://github.com/mountain/adva/blob/7a57d8d1cf13f683ff15ed856795b020b2ea7083/docs/research/0021-surreal-cut-objectification-no-go.md},
  note = {Research note 0021, pinned at commit 7a57d8d; finite presentation
          quotient and explicit failure to commute with option-wise scaling.}
}