[{"data":1,"prerenderedAt":4},["ShallowReactive",2],{"hzH1rDe305":3},"# A Proof of Conway’s Refinement Conjecture\n\n50 years ago, in his book *On Numbers and Games*, John Conway proposed a conjecture:\n\n\u003Cimg width=\"500\" height=\"267\" alt=\"Conway's refinement conjecture: Omnific integers have a refinement property, i.e. if ab = cd for omnific integers, then there are further integers e, f, g, h with a = ef, b = gh, c = eg, d = fh.\" src=\"https://github.com/user-attachments/assets/2d905459-91d8-4ebd-98e6-89f3dfc998f7\" />\n\nHe claims that if $a,b,c,d\\in\\mathbf{Oz}$ and $ab=cd$, then there are $e,f,g,h\\in\\mathbf{Oz}$ such that $a=ef$, $b=gh$, $c=eg$, and $d=fh$.\n\n(If you're not familiar with omnific integers aka $\\mathbf{Oz}$, I briefly describe them at the bottom.)\n\nI believe that this repository contains a [proof](https://github.com/gaearon/conway-refinement/blob/main/ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinementProof.lean) of this conjecture, which you can explore through its [proof map](https://gaearon.github.io/conway-refinement/#/map/conway-refinement).\n\n## About this proof\n\nThis proof was obtained in a somewhat unusual way.\n\nI am not a mathematician, but I have an amateur level of interest in mathematics. Initially, after seeing all the recent headlines, I wanted to try an \"AI proof\" as an experiment. I asked Claude to pick an open problem about [surreal numbers](https://www.infinitelymore.xyz/p/surreal-numbers), a strange and beautiful invention (discovery?) of John Conway. Claude suggested this little conjecture because Sonia L'Innocente and Vincenzo Mantova had [recently made some progress toward it](https://www.sciencedirect.com/science/article/pii/S0001870824000288), so new pathways might have become available.\n\nI couldn't \"one-shot\" the conjecture with AI, but this only got me more interested. I started wondering whether it was possible to make some progress if I steered the AI in some particular way. I would also have to figure out some way to know if I've actually made any progress, or if it's feeding me hallucinations. I thought it was funny and considerably absurd that I'm trying to do this without actually understanding the space, and I've limited my mathematical inquest to understanding what the *statement* says, but not anything about its proof methods.\n\nOver the course of the last month, I've spent all my free time on it, repeatedly maxed out my 20x Pro usage for [ChatGPT](https://chatgpt.com/) and [Claude](https://claude.ai/) to generate potential \"ideas\" and then steered the AI into writing [Lean](https://lean-lang.org/) to sort out which of these ideas \"agree\" with acceptable mathematics.\n\nThis is probably why this proof ended up so long and confusing.\n\n## Why I think it's correct\n\nAs anyone who's tried to prove things with Lean knows, there are two main dangers:\n\n1. You may transitively [rely on bad axioms](https://overreacted.io/the-math-is-haunted/) or `sorry`s.\n2. You may have proven a different statement than you intended.\n\nTo protect myself against these two dangers, I have taken two precautions.\n\n### The CombinatorialGames formulation\n\nThe harder question is whether the Lean statement actually says what Conway's conjecture says. To make this easier to check, I put the statement in a standalone file with as little of \"my\" code as possible. [`ConwayRefinement.lean`](ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinement.lean) uses only `Surreal` from [CombinatorialGames](https://github.com/vihdzp/combinatorial-games):\n\n```lean\nimport CombinatorialGames.Surreal.Multiplication\n\nuniverse u\n\n/-- `x` is an omnific integer when it is the cut with\nsole left option `x - 1` and sole right option `x + 1`. -/\nabbrev IsConwayOmnificInteger (x : Surreal.{u}) : Prop :=\n  x = !{{x - 1} | {x + 1}}' (by\n    simp only [Set.mem_singleton_iff]\n    rintro _ rfl _ rfl\n    simp [sub_eq_add_neg])\n\n/-- Conjecture: Every equality `a * b = c * d` of omnific integers\nhas an omnific refinement `a = e * f`, `b = g * h`, `c = e * g`, `d = f * h`. -/\nabbrev ConwayConjecture : Prop :=\n  ∀ a b c d : Surreal.{u},\n    IsConwayOmnificInteger a → IsConwayOmnificInteger b →\n    IsConwayOmnificInteger c → IsConwayOmnificInteger d → a * b = c * d →\n    ∃ e f g h : Surreal.{u},\n      IsConwayOmnificInteger e ∧ IsConwayOmnificInteger f ∧\n      IsConwayOmnificInteger g ∧ IsConwayOmnificInteger h ∧\n      a = e * f ∧ b = g * h ∧ c = e * g ∧ d = f * h\n```\n\nThen I offer a proof of exactly that statement in [`ConwayRefinementProof.lean`](ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinementProof.lean):\n\n```lean\ntheorem proof : ConwayConjecture.{u} := by\n  -- ...\n\n#print axioms proof\n```\n\nwhich prints:\n\n```text\n'ConwayConjecture.proof' depends on axioms: [propext, Classical.choice, Quot.sound]\n```\n\n*(CI also runs a linter [inspired by TauCeti](https://github.com/TauCetiProject/TauCeti/blob/main/scripts/Axioms.lean) that bans all other axioms from the entire project.)*\n\n### The Mathlib-only formulation\n\nOf course, you might not want to take the `Surreal` definition from CombinatorialGames on faith, so I also have a [Mathlib](https://github.com/leanprover-community/mathlib4)-only formulation in [`InlineConwayRefinement.lean`](ConwayRefinement/Standalone/Mathlib/InlineConwayRefinement.lean). It inlines every necessary definition from [CombinatorialGames](https://github.com/vihdzp/combinatorial-games/) into a ~250-line file.\n\nThe corresponding proof is in [`InlineConwayRefinementProof.lean`](ConwayRefinement/Standalone/Mathlib/InlineConwayRefinementProof.lean). It reports the same three axioms: `propext`, `Classical.choice`, and `Quot.sound`.\n\nIt is still quite possible that one or both of these statements do not correspond to Conway's conjecture in some small or big way, in which case this would not be a valid mathematical proof no matter what Lean says about it.\n\n## Navigating the proof\n\nWhile the above approach gave me some confidence in the validity of the proof, I am sure it is needlessly complicated and poorly explained. I've experimented with different ways to nudge the models into writing clearer mathematics, but it was difficult to get them to reliably write using standard terms, or to separate the core of the argument from bookkeeping or long detours that got fossilized in Lean.\n\nInitially, I tried to generate a \"paper\". Eventually, I switched to [a proof guide website](https://gaearon.github.io/conway-refinement/), which is closer to what this project actually is. The website has AI-written descriptions of some Lean theorems that contribute to the top-level result. AI also chose which theorems to include. The tooling verifies that each numbered result actually corresponds to a checked piece of Lean code:\n\n\u003Cimg width=\"2660\" height=\"1848\" alt=\"Proof guide: statements\" src=\"https://github.com/user-attachments/assets/c07b59ae-5a70-4216-9d93-60434e37f908\" />\n\nInside these snippets, you can Cmd/Ctrl+Click any Lean identifier to open its source code on GitHub. For each selected theorem, you can step through the captured proof state at every character to retrace that part of the proof:\n\nhttps://github.com/user-attachments/assets/04a4d2e8-913b-496c-95f2-b0decdec8811\n\nThe mathematical prose summarizing each Lean proof is written by AI and may contain mistakes. (There's an edit button if you'd like to contribute a fix.) I welcome pull requests that refine which theorems are selected for the website, or improve their factoring.\n\nYou can also select any result and see its \"proof map\", which traces its dependencies and consequences:\n\n\u003Cimg width=\"2660\" height=\"1848\" alt=\"Proof guide: map\" src=\"https://github.com/user-attachments/assets/b4405de8-14b1-4b91-9e76-4307976e5139\" />\n\nI thought this might help a mathematical reader make some sense of the AI's argument. If you can propose a more direct path, please open an issue or a pull request. I would be happy to delete any uninteresting intermediate results, to factor the results differently, or to try another path as long as your proposal compiles or AI can get it to compile. If there is a better route, I can merge it and regenerate the proof map.\n\nThe [interactive guide](https://gaearon.github.io/conway-refinement/) is also available in [PDF form](https://gaearon.github.io/conway-refinement/blueprint.pdf), which you can save offline. Source code links inside them are versioned and link to the commit that they were built from.\n\nIf the result is correct, it might eventually be useful to cite it. I don't think the proof in this repository is in an acceptable state for that yet, but hopefully putting it out will either build confidence in the Lean result or debunk it. I also hope it serves as an example of another AI-driven experiment in proof exploration. I don't consider this completed mathematical work.\n\n## Formalized results\n\nThe proof guide has a lot of intermediate steps. This is a shorter list of results and examples that I think are worth looking at. It includes every result from the guide's [Highlights page](https://gaearon.github.io/conway-refinement/#/highlights).\n\nEach standalone statement is paired with a proof file. Each map shows how the result depends on the rest of the proof. Only Lean declarations explicitly annotated for the proof guide appear in these maps.\n\n### The headline claim\n\n| Formulation | Links |\n|---|---|\n| CombinatorialGames formulation of Conway's refinement conjecture | [statement](ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinement.lean), [proof](ConwayRefinement/Standalone/CombinatorialGames/ConwayRefinementProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/conway-refinement) |\n| Mathlib-only formulation of Conway's refinement conjecture | [statement](ConwayRefinement/Standalone/Mathlib/InlineConwayRefinement.lean), [proof](ConwayRefinement/Standalone/Mathlib/InlineConwayRefinementProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/conway-refinement) |\n\n### Standalone statements over Mathlib\n\nEach statement file in this table imports only Mathlib. Its paired proof connects it to the rest of the proof.\n\n| Result | Links |\n|---|---|\n| Four-factor refinement in saturated Hahn integer parts with integer constants | [statement](ConwayRefinement/Standalone/Mathlib/HahnIntegerPartRefinement.lean), [proof](ConwayRefinement/Standalone/Mathlib/HahnIntegerPartRefinementProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/integer-hahn-refinement-of-saturation) |\n| Polynomial presentation and four-factor refinement of Hahn germ rings over Cauchy-complete exponent groups | [statement](ConwayRefinement/Standalone/Mathlib/CompleteHahnGerm.lean), [proof](ConwayRefinement/Standalone/Mathlib/CompleteHahnGermProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/complete-hahn-germ-refinement) |\n| Polynomial presentation of $K((\\mathbb R^{\\le 0}))$ over its finite-support subring | [statement](ConwayRefinement/Standalone/Mathlib/HahnSeriesPolynomialRing.lean), [proof](ConwayRefinement/Standalone/Mathlib/HahnSeriesPolynomialRingProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/hahn-series-polynomial-algebra) |\n| GCDs, every series primal, irreducibles prime, and uniqueness of irreducible factorisations in $K((\\mathbb R^{\\le 0}))$ | [statement](ConwayRefinement/Standalone/Mathlib/HahnSeriesGCD.lean), [proof](ConwayRefinement/Standalone/Mathlib/HahnSeriesGCDProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/hahn-series-gcd-domain) |\n| Berarducci's germ ring is a polynomial ring and a UFD | [statement](ConwayRefinement/Standalone/Mathlib/GermPolynomialRing.lean), [proof](ConwayRefinement/Standalone/Mathlib/GermPolynomialRingProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/ordinal-value-quotient) |\n| An explicit degree-two prime in $K((\\mathbb R^{\\le 0}))$ | [statement](ConwayRefinement/Standalone/Mathlib/Examples/DegreeTwoPrime.lean), [proof](ConwayRefinement/Standalone/Mathlib/Examples/DegreeTwoPrimeProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/degree-two-series-prime) |\n\n### A standalone statement over Mathlib and CombinatorialGames\n\n| Result | Links |\n|---|---|\n| Minimal homogeneous families in $\\widehat{\\mathrm P}$ are algebraically independent | [statement](ConwayRefinement/Standalone/CombinatorialGames/PrincipalRVAlgebraicIndependence.lean), [proof](ConwayRefinement/Standalone/CombinatorialGames/PrincipalRVAlgebraicIndependenceProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/polynomial) |\n\n### Other results and examples\n\nThese also have short statement and proof files, but their statements use code from this repository.\n\n| Result | Links |\n|---|---|\n| General conditions guaranteeing Conway's four-factor property for certain rings of infinite series | [statement](ConwayRefinement/Standalone/Mathlib/Examples/HahnIntegerPartRefinementCriterion.lean), [proof](ConwayRefinement/Standalone/Mathlib/Examples/HahnIntegerPartRefinementCriterionProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/hahn-integer-part-refinement) |\n| Normal-form identification, every omnific integer primal, irreducibles prime, and uniqueness of irreducible factorisations in $\\mathbf{Oz}$ | [statement](ConwayRefinement/Standalone/CombinatorialGames/Examples/OmnificFactorization.lean), [proof](ConwayRefinement/Standalone/CombinatorialGames/Examples/OmnificFactorizationProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/omnific-factorisation) |\n| An explicit degree-two prime in $\\mathbf{Oz}$, with a coefficient-doubled foil of the same support order type that factors nontrivially | [statement](ConwayRefinement/Standalone/CombinatorialGames/Examples/DegreeTwoPrime.lean), [proof](ConwayRefinement/Standalone/CombinatorialGames/Examples/DegreeTwoPrimeProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/explicit-omnific-prime) |\n| Conway's one-row prime in $\\mathbf{Oz}$ | [statement](ConwayRefinement/Standalone/CombinatorialGames/Examples/OneRowPrime.lean), [proof](ConwayRefinement/Standalone/CombinatorialGames/Examples/OneRowPrimeProof.lean), [map](https://gaearon.github.io/conway-refinement/#/map/omnific-factorisation) |\n\nConway's one-row prime is a concrete example of the general theorem about omnific integers above, so both link to the same proof map.\n\n### Results without separate statement files\n\nThese results don't have separate standalone statement files, but AI chose them as useful on their own or especially important to the proof.\n\n| Result | Links |\n|---|---|\n| Multiplicativity of the Cantor–Bendixson value | [source](ConwayRefinement/HahnSeries/Germ/AlgebraicIndependence/CantorBendixsonValueMultiplicativity.lean), [map](https://gaearon.github.io/conway-refinement/#/map/cantor-bendixson-value-multiplicative) |\n| Algebraic independence in the Cantor–Bendixson associated graded ring | [source](ConwayRefinement/HahnSeries/Germ/AlgebraicIndependence/AlgebraicIndependence.lean), [map](https://gaearon.github.io/conway-refinement/#/map/cantor-bendixson-minimal-generators-independent) |\n| Random series of finite Cantor degree, and their lower-order perturbations, are prime | [source](ConwayRefinement/HahnSeries/Primality/Random.lean), [map](https://gaearon.github.io/conway-refinement/#/map/random-series-prime) |\n| Every reduced nonordinary omnific integer is primal | [source](ConwayRefinement/Surreal/OmnificInteger/Primality/OmnificIntegers.lean), [map](https://gaearon.github.io/conway-refinement/#/map/reduced-omnific-primal) |\n| Refinement of $Z+K((G^{\u003C0}))$ forces $K=\\mathrm{Frac}(Z)$ | [source](ConwayRefinement/HahnSeries/IntegerPart/Refinement/TruncationIntegerPartFractionField.lean), [map](https://gaearon.github.io/conway-refinement/#/map/refinement-forces-coefficient-fraction-field) |\n\nPlease open an issue if any of these are wrong or could be clearer.\n\n## Proof maps\n\nThe proof map on the website is based on a dependency graph from specially marked theorems. I originally used [LeanArchitect](https://github.com/hanwenzhu/LeanArchitect) and [subverso](https://github.com/leanprover/subverso), but ended up slopforking both, so the tooling for the site is currently custom.\n\nI also have a script that turns the proof map into the Mermaid diagrams below:\n\n\u003C!-- blueprint-mermaid:start -->\n### Proof overview\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Edge labels count visible result dependencies.\nflowchart TB\n  p0[\"Algebraic and ordinal preliminaries\u003Cbr/>28 results\"]:::phase\n  p1[\"Ordinal value and degree\u003Cbr/>7 results\"]:::phase\n  p2[\"Cantor–Bendixson ranks of supports\u003Cbr/>12 results\"]:::phase\n  p3[\"Algebraic independence in graded rings\u003Cbr/>21 results\"]:::phase\n  p4[\"Translated truncations\u003Cbr/>26 results\"]:::phase\n  p5[\"Limit ordinals in the degree induction\u003Cbr/>14 results\"]:::phase\n  p6[\"Principal RV-elements\u003Cbr/>4 results\"]:::phase\n  p7[\"Polynomial presentations\u003Cbr/>15 results\"]:::phase\n  p8[\"Primality and factorisation for real exponents\u003Cbr/>10 results\"]:::phase\n  p9[\"Reduction along Archimedean classes\u003Cbr/>5 results\"]:::phase\n  p10[\"Refinement and transfinite induction\u003Cbr/>17 results\"]:::phase\n  p11[\"A cut criterion for Cauchy completeness\u003Cbr/>2 results\"]:::phase\n  p12[\"Bounded generalised-power-series integer parts\u003Cbr/>10 results\"]:::phase\n  p13[\"Surreal numbers and omnific integers\u003Cbr/>17 results\"]:::phase\n  p1 -->|\"1\"| p2\n  p2 -->|\"7\"| p3\n  p0 -->|\"20\"| p3\n  p1 -->|\"13\"| p4\n  p0 -->|\"3\"| p4\n  p0 -->|\"9\"| p5\n  p4 -->|\"8\"| p5\n  p1 -->|\"1\"| p5\n  p2 -->|\"1\"| p6\n  p4 -->|\"1\"| p6\n  p5 -->|\"1\"| p6\n  p3 -->|\"2\"| p6\n  p1 -->|\"2\"| p7\n  p0 -->|\"4\"| p7\n  p3 -->|\"3\"| p7\n  p6 -->|\"7\"| p7\n  p4 -->|\"2\"| p7\n  p1 -->|\"1\"| p8\n  p0 -->|\"2\"| p8\n  p7 -->|\"1\"| p8\n  p8 -->|\"1\"| p9\n  p7 -->|\"2\"| p10\n  p9 -->|\"3\"| p10\n  p0 -->|\"1\"| p10\n  p11 -->|\"2\"| p12\n  p9 -->|\"1\"| p12\n  p10 -->|\"1\"| p12\n  p8 -->|\"3\"| p13\n  p9 -->|\"4\"| p13\n  p11 -->|\"2\"| p13\n  p10 -->|\"1\"| p13\n  click p0 \"https://gaearon.github.io/conway-refinement/#/chapter/preliminaries\" \"Open Algebraic and ordinal preliminaries\"\n  click p1 \"https://gaearon.github.io/conway-refinement/#/chapter/ordinal-value\" \"Open Ordinal value and degree\"\n  click p2 \"https://gaearon.github.io/conway-refinement/#/chapter/cantor-bendixson-ranks\" \"Open Cantor–Bendixson ranks of supports\"\n  click p3 \"https://gaearon.github.io/conway-refinement/#/chapter/associated-graded-algebraic-independence\" \"Open Algebraic independence in graded rings\"\n  click p4 \"https://gaearon.github.io/conway-refinement/#/chapter/translated-truncations\" \"Open Translated truncations\"\n  click p5 \"https://gaearon.github.io/conway-refinement/#/chapter/limit-degree\" \"Open Limit ordinals in the degree induction\"\n  click p6 \"https://gaearon.github.io/conway-refinement/#/chapter/principal-rv-algebraic-independence\" \"Open Principal RV-elements\"\n  click p7 \"https://gaearon.github.io/conway-refinement/#/chapter/polynomial-presentations\" \"Open Polynomial presentations\"\n  click p8 \"https://gaearon.github.io/conway-refinement/#/chapter/real-exponent-primality\" \"Open Primality and factorisation for real exponents\"\n  click p9 \"https://gaearon.github.io/conway-refinement/#/chapter/finite-archimedean-support\" \"Open Reduction along Archimedean classes\"\n  click p10 \"https://gaearon.github.io/conway-refinement/#/chapter/archimedean-class-refinement\" \"Open Refinement and transfinite induction\"\n  click p11 \"https://gaearon.github.io/conway-refinement/#/chapter/cauchy-completeness\" \"Open A cut criterion for Cauchy completeness\"\n  click p12 \"https://gaearon.github.io/conway-refinement/#/chapter/bounded-integer-parts\" \"Open Bounded generalised-power-series integer parts\"\n  click p13 \"https://gaearon.github.io/conway-refinement/#/chapter/conway-refinement\" \"Open Surreal numbers and omnific integers\"\n\n  classDef phase fill:#f8fafc,stroke:#475569,stroke-width:2px,color:#111827\n```\n\n\u003Ca id=\"phase-preliminaries\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Algebraic and ordinal preliminaries\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph current[\"Algebraic and ordinal preliminaries\"]\n    direction TB\n    n001[\"(1) A primal zero-residue element forces the fraction-field condition\"]:::lemma\n    n002[\"(2) Extension of a degreewise independent family\"]:::lemma\n    n003[\"(3) Homogeneous decomposition in a finitely generated graded ideal\"]:::lemma\n    n004[\"(4) Weighted-homogeneous polynomial representatives\"]:::lemma\n    n005[\"(5) Linear part of a weighted-homogeneous polynomial\"]:::lemma\n    n006[\"(6) Vanishing of the linear part of a homogeneous relation\"]:::lemma\n    n007[\"(7) Weights of variables in a homogeneous relation\"]:::lemma\n    n008[\"(8) Weighted Euler decomposition at the ω^0 coefficient\"]:::lemma\n    n009[\"(9) Partial-derivative decomposition of a homogeneous ideal relation\"]:::lemma\n    n010[\"(10) Greatest common divisors in multivariate polynomial rings\"]:::lemma\n    n011[\"(11) Partial-derivative obstruction to a linear variable\"]:::lemma\n    n012[\"(12) Vanishing criterion for partial derivatives of weighted-homogeneous polynomials\"]:::lemma\n    n013[\"(13) Weighted degree of polynomial derivations\"]:::lemma\n    n014[\"(14) Weighted Euler identity\"]:::lemma\n    n015[\"(15) Finite generation of polynomial syzygies\"]:::lemma\n    n016[\"(16) Lifting generation from the associated graded ring\"]:::lemma\n    n017[\"(17) Separation below the last Cantor term\"]:::lemma\n    n018[\"(18) Well-ordering of a countable ordered union\"]:::lemma\n    n019[\"(19) Order type of a countable ordered union\"]:::lemma\n    n020[\"(20) Countable ordered unions below ω^e\"]:::lemma\n    n021[\"(21) Cofinality below a Hessenberg sum\"]:::lemma\n    n022[\"(22) Uniform bound for simultaneous decreases in a Hessenberg sum\"]:::lemma\n    n023[\"(23) Weighted-degree bound for terms with two translated truncations\"]:::lemma\n    n024[\"(24) Rigidity of upper Cantor terms under simultaneous decreases\"]:::lemma\n    n025[\"(25) Hessenberg-sum decomposition with unequal last Cantor terms\"]:::lemma\n    n026[\"(26) Hessenberg-sum decomposition with equal last Cantor terms\"]:::lemma\n    n027[\"(27) Upper Cantor-term bound for a Hessenberg sum\"]:::lemma\n    n028[\"(28) Intermediate ordinals below a Hessenberg sum\"]:::lemma\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Algebraic independence in graded rings\u003Cbr/>15 results used downstream\"]:::boundary\n    out1[\"Translated truncations\u003Cbr/>3 results used downstream\"]:::boundary\n    out2[\"Limit ordinals in the degree induction\u003Cbr/>8 results used downstream\"]:::boundary\n    out3[\"Polynomial presentations\u003Cbr/>2 results used downstream\"]:::boundary\n    out4[\"Primality and factorisation for real exponents\u003Cbr/>2 results used downstream\"]:::boundary\n    out5[\"Refinement and transfinite induction\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n005 --> n006\n  n006 --> n007\n  n003 --> n009\n  n008 --> n009\n  n018 --> n019\n  n018 --> n020\n  n022 --> n023\n  n024 --> n025\n  n027 --> n028\n  n003 --> out0\n  n017 --> out0\n  n028 --> out0\n  n004 --> out0\n  n021 --> out0\n  n007 --> out0\n  n011 --> out0\n  n023 --> out0\n  n009 --> out0\n  n013 --> out0\n  n014 --> out0\n  n015 --> out0\n  n012 --> out0\n  n025 --> out0\n  n026 --> out0\n  n004 --> out1\n  n019 --> out1\n  n012 --> out1\n  n026 --> out2\n  n025 --> out2\n  n018 --> out2\n  n020 --> out2\n  n003 --> out2\n  n021 --> out2\n  n028 --> out2\n  n017 --> out2\n  n002 --> out3\n  n016 --> out3\n  n002 --> out4\n  n010 --> out4\n  n001 --> out5\n  click n001 \"https://gaearon.github.io/conway-refinement/#/result/primal-zero-residue-fraction-field\" \"Open the statement for lemma 1\"\n  click n002 \"https://gaearon.github.io/conway-refinement/#/result/extend-to-minimal-system\" \"Open the statement for lemma 2\"\n  click n003 \"https://gaearon.github.io/conway-refinement/#/result/homogeneous-element-of-generated-ideal\" \"Open the statement for lemma 3\"\n  click n004 \"https://gaearon.github.io/conway-refinement/#/result/generate\" \"Open the statement for lemma 4\"\n  click n005 \"https://gaearon.github.io/conway-refinement/#/result/homogeneous-polynomial-linear-term\" \"Open the statement for lemma 5\"\n  click n006 \"https://gaearon.github.io/conway-refinement/#/result/minimal-generators-relation-has-no-linear-term\" \"Open the statement for lemma 6\"\n  click n007 \"https://gaearon.github.io/conway-refinement/#/result/variables-of-homogeneous-relation-have-smaller-degree\" \"Open the statement for lemma 7\"\n  click n008 \"https://gaearon.github.io/conway-refinement/#/result/relation-shape\" \"Open the statement for lemma 8\"\n  click n009 \"https://gaearon.github.io/conway-refinement/#/result/successor-relation-decomposition\" \"Open the statement for lemma 9\"\n  click n010 \"https://gaearon.github.io/conway-refinement/#/result/multivariate-polynomial-gcd\" \"Open the statement for lemma 10\"\n  click n011 \"https://gaearon.github.io/conway-refinement/#/result/relation-at-limit-ordinal-partial-contradiction\" \"Open the statement for lemma 11\"\n  click n012 \"https://gaearon.github.io/conway-refinement/#/result/partial-derivative-vanishes\" \"Open the statement for lemma 12\"\n  click n013 \"https://gaearon.github.io/conway-refinement/#/result/polynomial-vector-field-lowers-degree\" \"Open the statement for lemma 13\"\n  click n014 \"https://gaearon.github.io/conway-refinement/#/result/weighted-euler-identity\" \"Open the statement for lemma 14\"\n  click n015 \"https://gaearon.github.io/conway-refinement/#/result/syzygies-finite-variables\" \"Open the statement for lemma 15\"\n  click n016 \"https://gaearon.github.io/conway-refinement/#/result/initial-forms-generate-subalgebra\" \"Open the statement for lemma 16\"\n  click n017 \"https://gaearon.github.io/conway-refinement/#/result/separation\" \"Open the statement for lemma 17\"\n  click n018 \"https://gaearon.github.io/conway-refinement/#/result/increasing-union\" \"Open the statement for lemma 18\"\n  click n019 \"https://gaearon.github.io/conway-refinement/#/result/increasing-union-order-type\" \"Open the statement for lemma 19\"\n  click n020 \"https://gaearon.github.io/conway-refinement/#/result/increasing-union-below-principal-ordinal\" \"Open the statement for lemma 20\"\n  click n021 \"https://gaearon.github.io/conway-refinement/#/result/natural-sum-approach\" \"Open the statement for lemma 21\"\n  click n022 \"https://gaearon.github.io/conway-refinement/#/result/two-lowerings-have-strict-bound\" \"Open the statement for lemma 22\"\n  click n023 \"https://gaearon.github.io/conway-refinement/#/result/two-truncations-below\" \"Open the statement for lemma 23\"\n  click n024 \"https://gaearon.github.io/conway-refinement/#/result/two-lowering-bound-high-part\" \"Open the statement for lemma 24\"\n  click n025 \"https://gaearon.github.io/conway-refinement/#/result/unequal-last-terms-hessenberg-decomposition\" \"Open the statement for lemma 25\"\n  click n026 \"https://gaearon.github.io/conway-refinement/#/result/equal-last-terms-hessenberg-decomposition\" \"Open the statement for lemma 26\"\n  click n027 \"https://gaearon.github.io/conway-refinement/#/result/separation-bounds-high-part\" \"Open the statement for lemma 27\"\n  click n028 \"https://gaearon.github.io/conway-refinement/#/result/intermediate-ordinal-hessenberg-decomposition\" \"Open the statement for lemma 28\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-ordinal-value\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Ordinal value and degree\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph current[\"Ordinal value and degree\"]\n    direction TB\n    n029[\"(29) Convolution formula for translated truncations\"]:::lemma\n    n030[\"(30) Principal representatives of P_α (LM24, Remark 7.2.4)\"]:::fact\n    n031[\"(31) Support-tail characterization of the ordinal value (Ber00, Definition 5.2)\"]:::fact\n    n032[\"(32) Multiplicative property of the ordinal value (Ber00, Theorem 9.7)\"]:::fact\n    n033[\"(33) Multiplicativity of order type for weakly principal series (Ber00, Corollary 9.9; LM24, Fact 3.4.1)\"]:::fact\n    n034[\"(34) Multiplicativity of the degree (LM24, Theorem D)\"]:::fact\n    n035[\"(35) Decrease of the ordinal value under translated truncation\"]:::lemma\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Cantor–Bendixson ranks of supports\u003Cbr/>1 results used downstream\"]:::boundary\n    out1[\"Translated truncations\u003Cbr/>4 results used downstream\"]:::boundary\n    out2[\"Limit ordinals in the degree induction\u003Cbr/>1 results used downstream\"]:::boundary\n    out3[\"Polynomial presentations\u003Cbr/>2 results used downstream\"]:::boundary\n    out4[\"Primality and factorisation for real exponents\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n029 --> n032\n  n031 --> n032\n  n032 --> n033\n  n033 --> n034\n  n031 --> n035\n  n031 --> out0\n  n031 --> out1\n  n035 --> out1\n  n030 --> out1\n  n029 --> out1\n  n031 --> out2\n  n030 --> out3\n  n034 --> out3\n  n032 --> out4\n  click n029 \"https://gaearon.github.io/conway-refinement/#/result/convolution-formula\" \"Open the statement for lemma 29\"\n  click n030 \"https://gaearon.github.io/conway-refinement/#/result/principal-series-representatives\" \"Open the statement for fact 30\"\n  click n031 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-support-tail\" \"Open the statement for fact 31\"\n  click n032 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-multiplicativity\" \"Open the statement for fact 32\"\n  click n033 \"https://gaearon.github.io/conway-refinement/#/result/weakly-principal-order-type-multiplicativity\" \"Open the statement for fact 33\"\n  click n034 \"https://gaearon.github.io/conway-refinement/#/result/degree-multiplicativity\" \"Open the statement for fact 34\"\n  click n035 \"https://gaearon.github.io/conway-refinement/#/result/truncation-drop\" \"Open the statement for lemma 35\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-cantor-bendixson-ranks\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Cantor–Bendixson ranks of supports\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Ordinal value and degree\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Cantor–Bendixson ranks of supports\"]\n    direction TB\n    n036[\"(36) The Cantor–Bendixson value at exponent zero\"]:::definition\n    n037[\"(37) Finite convolution of translated truncations\"]:::lemma\n    n038[\"(38) Reconstruction from translated truncations of fixed rank\"]:::lemma\n    n039[\"(39) Cantor–Bendixson derivatives of sums of well-ordered sets\"]:::lemma\n    n040[\"(40) Product upper bound for the Cantor–Bendixson value\"]:::lemma\n    n041[\"(41) Pure-power cancellation for the Cantor–Bendixson value\"]:::lemma\n    n042[\"(42) Power-times-factor cancellation for the Cantor–Bendixson value\"]:::lemma\n    n043[\"(43) Multiplicativity of the Cantor–Bendixson value\"]:::theorem\n    n044[\"(44) Multiplicativity of the Cantor–Bendixson degree\"]:::theorem\n    n045[\"(45) Cantor–Bendixson rank at a strict supremum\"]:::lemma\n    n046[\"(46) Cantor–Bendixson formula for the ordinal value\"]:::lemma\n    n047[\"(47) Cantor–Bendixson formula for deg_J\"]:::lemma\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Algebraic independence in graded rings\u003Cbr/>1 results used downstream\"]:::boundary\n    out1[\"Principal RV-elements\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n036 --> n037\n  n036 --> n038\n  n036 --> n040\n  n039 --> n040\n  n038 --> n041\n  n040 --> n041\n  n038 --> n042\n  n040 --> n042\n  n041 --> n043\n  n042 --> n043\n  n043 --> n044\n  n036 --> n046\n  n045 --> n046\n  n043 --> n047\n  n046 --> n047\n  in0 --> n046\n  n043 --> out0\n  n047 --> out1\n  click n036 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-value\" \"Open the statement for definition 36\"\n  click n037 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-convolution\" \"Open the statement for lemma 37\"\n  click n038 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-rank-reconstruction\" \"Open the statement for lemma 38\"\n  click n039 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-derivative-of-sum\" \"Open the statement for lemma 39\"\n  click n040 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-value-product-upper-bound\" \"Open the statement for lemma 40\"\n  click n041 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-pure-power-cancellation\" \"Open the statement for lemma 41\"\n  click n042 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-power-factor-cancellation\" \"Open the statement for lemma 42\"\n  click n043 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-value-multiplicative\" \"Open the statement for theorem 43\"\n  click n044 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-degree-multiplicative\" \"Open the statement for theorem 44\"\n  click n045 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-rank-of-strict-supremum\" \"Open the statement for lemma 45\"\n  click n046 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-cantor-bendixson\" \"Open the statement for lemma 46\"\n  click n047 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-degree-is-cantor-bendixson-rank\" \"Open the statement for lemma 47\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-associated-graded-algebraic-independence\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Algebraic independence in graded rings\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 10 results\"]:::boundary\n    in1[\"Cantor–Bendixson ranks of supports\u003Cbr/>used by 7 results\"]:::boundary\n  end\n  subgraph current[\"Algebraic independence in graded rings\"]\n    direction TB\n    n048[\"(48) Discrete finite unions of Cantor–Bendixson rank sets\"]:::lemma\n    n049[\"(49) Cofactors from local data by well-founded induction\"]:::lemma\n    n050[\"(50) Local ideal membership in the associated graded ring\"]:::lemma\n    n051[\"(51) Polynomial syzygies from homogeneous ideal membership\"]:::lemma\n    n052[\"(52) Local-to-global principle for homogeneous ideal membership\"]:::lemma\n    n053[\"(53) The successor formula for the Cantor–Bendixson derivation\"]:::lemma\n    n054[\"(54) The Leibniz rule for the Cantor–Bendixson derivation\"]:::lemma\n    n055[\"(55) Prescribing the Cantor–Bendixson derivative on one exact-rank set\"]:::lemma\n    n056[\"(56) The derivative criterion for successor ideal membership\"]:::lemma\n    n057[\"(57) Prescribing the Cantor–Bendixson derivative on a discrete set\"]:::lemma\n    n058[\"(58) Leading-coefficient obstruction for homogeneous relations\"]:::lemma\n    n059[\"(59) Injectivity of evaluation when the degree is a limit ordinal\"]:::lemma\n    n060[\"(60) Simultaneous Cantor–Bendixson derivatives of homogeneous tuples\"]:::lemma\n    n061[\"(61) Successor step for Cantor–Bendixson homogeneous evaluation\"]:::lemma\n    n062[\"(62) Degreewise injectivity of homogeneous evaluation\"]:::lemma\n    n063[\"(63) Algebraic independence of a minimal homogeneous generating system\"]:::theorem\n    n064[\"(64) Local Jacobian ideal membership for translated partial derivatives\"]:::lemma\n    n065[\"(65) Maximal-variable linearity for the Cantor–Bendixson degree\"]:::lemma\n    n066[\"(66) Partial-derivative decomposition when the degree is a limit ordinal\"]:::lemma\n    n067[\"(67) Well-founded Archimedean-ball bases\"]:::lemma\n    n068[\"(68) Algebraic independence in the Cantor–Bendixson associated graded ring\"]:::theorem\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Principal RV-elements\u003Cbr/>2 results used downstream\"]:::boundary\n    out1[\"Polynomial presentations\u003Cbr/>3 results used downstream\"]:::boundary\n  end\n\n  n049 --> n050\n  n050 --> n052\n  n051 --> n052\n  n053 --> n054\n  n053 --> n055\n  n054 --> n056\n  n055 --> n056\n  n053 --> n057\n  n048 --> n060\n  n054 --> n060\n  n057 --> n060\n  n056 --> n061\n  n060 --> n061\n  n058 --> n062\n  n059 --> n062\n  n061 --> n062\n  n062 --> n063\n  n064 --> n065\n  n052 --> n066\n  n064 --> n066\n  n062 --> n068\n  n065 --> n068\n  n066 --> n068\n  n067 --> n068\n  in1 --> n048\n  in0 --> n049\n  in1 --> n049\n  in0 --> n051\n  in1 --> n051\n  in0 --> n052\n  in1 --> n053\n  in0 --> n056\n  in0 --> n058\n  in1 --> n058\n  in0 --> n059\n  in1 --> n059\n  in0 --> n061\n  in0 --> n064\n  in1 --> n064\n  in0 --> n065\n  in0 --> n066\n  n063 --> out0\n  n065 --> out0\n  n068 --> out1\n  n063 --> out1\n  n065 --> out1\n  click n048 \"https://gaearon.github.io/conway-refinement/#/result/discrete-finite-union-cantor-bendixson-rank-sets\" \"Open the statement for lemma 48\"\n  click n049 \"https://gaearon.github.io/conway-refinement/#/result/well-founded-cofactor-construction\" \"Open the statement for lemma 49\"\n  click n050 \"https://gaearon.github.io/conway-refinement/#/result/local-ideal-membership-associated-graded\" \"Open the statement for lemma 50\"\n  click n051 \"https://gaearon.github.io/conway-refinement/#/result/homogeneous-ideal-membership-polynomial-syzygy\" \"Open the statement for lemma 51\"\n  click n052 \"https://gaearon.github.io/conway-refinement/#/result/eventual-local-ideal-membership-gives-syzygy\" \"Open the statement for lemma 52\"\n  click n053 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-derivation-successor-formula\" \"Open the statement for lemma 53\"\n  click n054 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-derivation-leibniz\" \"Open the statement for lemma 54\"\n  click n055 \"https://gaearon.github.io/conway-refinement/#/result/prescribed-cantor-bendixson-derivative-exact-rank\" \"Open the statement for lemma 55\"\n  click n056 \"https://gaearon.github.io/conway-refinement/#/result/successor-ideal-membership-from-cantor-bendixson-derivative\" \"Open the statement for lemma 56\"\n  click n057 \"https://gaearon.github.io/conway-refinement/#/result/prescribed-cantor-bendixson-derivative-discrete-set\" \"Open the statement for lemma 57\"\n  click n058 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-leading-coefficient\" \"Open the statement for lemma 58\"\n  click n059 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-limit-ordinal-evaluation\" \"Open the statement for lemma 59\"\n  click n060 \"https://gaearon.github.io/conway-refinement/#/result/simultaneous-cantor-bendixson-derivative-representatives\" \"Open the statement for lemma 60\"\n  click n061 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-successor-step\" \"Open the statement for lemma 61\"\n  click n062 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-degree-induction\" \"Open the statement for lemma 62\"\n  click n063 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-polynomiality\" \"Open the statement for theorem 63\"\n  click n064 \"https://gaearon.github.io/conway-refinement/#/result/local-jacobian-ideal-membership\" \"Open the statement for lemma 64\"\n  click n065 \"https://gaearon.github.io/conway-refinement/#/result/linear-occurrence\" \"Open the statement for lemma 65\"\n  click n066 \"https://gaearon.github.io/conway-refinement/#/result/limit-ordinal-partial-derivative-decomposition\" \"Open the statement for lemma 66\"\n  click n067 \"https://gaearon.github.io/conway-refinement/#/result/well-founded-archimedean-ball-basis\" \"Open the statement for lemma 67\"\n  click n068 \"https://gaearon.github.io/conway-refinement/#/result/cantor-bendixson-minimal-generators-independent\" \"Open the statement for theorem 68\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-translated-truncations\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Translated truncations\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 3 results\"]:::boundary\n    in1[\"Ordinal value and degree\u003Cbr/>used by 9 results\"]:::boundary\n  end\n  subgraph current[\"Translated truncations\"]\n    direction TB\n    n069[\"(69) Translated truncations of successor ordinal value\"]:::lemma\n    n070[\"(70) Injectivity of the translated-truncation map on P_α\"]:::proposition\n    n071[\"(71) Initial form of a weighted-homogeneous evaluation\"]:::lemma\n    n072[\"(72) Existence of polynomial representatives modulo J\"]:::proposition\n    n073[\"(73) Ordinal value of a polynomial evaluation\"]:::proposition\n    n074[\"(74) Omission of a maximal-weight variable from proper truncation representatives\"]:::lemma\n    n075[\"(75) Ordinal value of a polynomial representative\"]:::proposition\n    n076[\"(76) Leibniz rule for the lowering derivation\"]:::theorem\n    n077[\"(77) Local finiteness of translated truncations at successor ordinal value\"]:::lemma\n    n078[\"(78) Countable-support representatives of P_α\"]:::lemma\n    n079[\"(79) Detection of support order type by a translated truncation\"]:::lemma\n    n080[\"(80) Detection of interval support order type by a translated truncation\"]:::lemma\n    n081[\"(81) Support-order bound from uniformly small translated truncations\"]:::corollary\n    n082[\"(82) Ordinal-value bound for proper translated truncations of a principal series\"]:::lemma\n    n083[\"(83) Homogeneous component representing a class in P_β\"]:::lemma\n    n084[\"(84) Translated truncations of monomials in principal series\"]:::lemma\n    n085[\"(85) Translated truncations of weighted-homogeneous polynomials\"]:::lemma\n    n086[\"(86) High-degree components of translated truncations of a polynomial multiple\"]:::lemma\n    n087[\"(87) Prescribed P_δ classes at translated truncations\"]:::proposition\n    n088[\"(88) Derivative criterion for homogeneous ideal membership\"]:::proposition\n    n089[\"(89) Convolution formula for polynomial representatives\"]:::lemma\n    n090[\"(90) Leading coefficient after translated truncation\"]:::lemma\n    n091[\"(91) Polynomial Leibniz rule for translated truncations\"]:::lemma\n    n092[\"(92) Differentiated polynomial Leibniz rule\"]:::lemma\n    n093[\"(93) High-degree Jacobian ideal membership\"]:::lemma\n    n094[\"(94) Ideal membership in P_{τ + 1} from translated truncations\"]:::lemma\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Limit ordinals in the degree induction\u003Cbr/>6 results used downstream\"]:::boundary\n    out1[\"Principal RV-elements\u003Cbr/>1 results used downstream\"]:::boundary\n    out2[\"Polynomial presentations\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n069 --> n070\n  n071 --> n072\n  n071 --> n073\n  n072 --> n074\n  n073 --> n074\n  n072 --> n075\n  n073 --> n075\n  n070 --> n078\n  n077 --> n078\n  n079 --> n080\n  n079 --> n081\n  n072 --> n083\n  n073 --> n083\n  n071 --> n084\n  n082 --> n084\n  n084 --> n085\n  n072 --> n086\n  n073 --> n086\n  n085 --> n086\n  n076 --> n088\n  n078 --> n088\n  n087 --> n088\n  n072 --> n089\n  n073 --> n089\n  n074 --> n090\n  n089 --> n090\n  n089 --> n091\n  n091 --> n092\n  n082 --> n093\n  n092 --> n093\n  n083 --> n094\n  n088 --> n094\n  in1 --> n069\n  in1 --> n070\n  in0 --> n072\n  in1 --> n072\n  in1 --> n074\n  in1 --> n076\n  in1 --> n078\n  in1 --> n084\n  in0 --> n087\n  in1 --> n087\n  in1 --> n089\n  in0 --> n093\n  n081 --> out0\n  n085 --> out0\n  n094 --> out0\n  n075 --> out0\n  n093 --> out0\n  n086 --> out0\n  n082 --> out1\n  n072 --> out2\n  click n069 \"https://gaearon.github.io/conway-refinement/#/result/successor-truncation-value\" \"Open the statement for lemma 69\"\n  click n070 \"https://gaearon.github.io/conway-refinement/#/result/successor-principal-rv-injective\" \"Open the statement for proposition 70\"\n  click n071 \"https://gaearon.github.io/conway-refinement/#/result/homogeneous-evaluation-represents\" \"Open the statement for lemma 71\"\n  click n072 \"https://gaearon.github.io/conway-refinement/#/result/polynomial-representative-exists\" \"Open the statement for proposition 72\"\n  click n073 \"https://gaearon.github.io/conway-refinement/#/result/polynomial-evaluation-ordinal-value\" \"Open the statement for proposition 73\"\n  click n074 \"https://gaearon.github.io/conway-refinement/#/result/proper-truncation-omits-variable\" \"Open the statement for lemma 74\"\n  click n075 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-of-polynomial-representative\" \"Open the statement for proposition 75\"\n  click n076 \"https://gaearon.github.io/conway-refinement/#/result/leibniz-rule-lowering-derivation\" \"Open the statement for theorem 76\"\n  click n077 \"https://gaearon.github.io/conway-refinement/#/result/finite-successor-value-cutoffs\" \"Open the statement for lemma 77\"\n  click n078 \"https://gaearon.github.io/conway-refinement/#/result/successor-principal-rv-countable-support\" \"Open the statement for lemma 78\"\n  click n079 \"https://gaearon.github.io/conway-refinement/#/result/cutoff-detects-support-order\" \"Open the statement for lemma 79\"\n  click n080 \"https://gaearon.github.io/conway-refinement/#/result/cutoff-detects-support-order-in-interval\" \"Open the statement for lemma 80\"\n  click n081 \"https://gaearon.github.io/conway-refinement/#/result/small-truncations-small-support\" \"Open the statement for corollary 81\"\n  click n082 \"https://gaearon.github.io/conway-refinement/#/result/principal-truncations-lower-value\" \"Open the statement for lemma 82\"\n  click n083 \"https://gaearon.github.io/conway-refinement/#/result/polynomial-homogeneous-component-represents-class\" \"Open the statement for lemma 83\"\n  click n084 \"https://gaearon.github.io/conway-refinement/#/result/principal-representatives-truncation\" \"Open the statement for lemma 84\"\n  click n085 \"https://gaearon.github.io/conway-refinement/#/result/principal-representatives-homogeneous-polynomial-truncation\" \"Open the statement for lemma 85\"\n  click n086 \"https://gaearon.github.io/conway-refinement/#/result/term-truncation-condition\" \"Open the statement for lemma 86\"\n  click n087 \"https://gaearon.github.io/conway-refinement/#/result/realise-derivative\" \"Open the statement for proposition 87\"\n  click n088 \"https://gaearon.github.io/conway-refinement/#/result/ideal-from-derivative\" \"Open the statement for proposition 88\"\n  click n089 \"https://gaearon.github.io/conway-refinement/#/result/polynomial-convolution-formula\" \"Open the statement for lemma 89\"\n  click n090 \"https://gaearon.github.io/conway-refinement/#/result/leading-coefficient-of-truncated-product\" \"Open the statement for lemma 90\"\n  click n091 \"https://gaearon.github.io/conway-refinement/#/result/leibniz-remainder\" \"Open the statement for lemma 91\"\n  click n092 \"https://gaearon.github.io/conway-refinement/#/result/differentiated-leibniz-remainder\" \"Open the statement for lemma 92\"\n  click n093 \"https://gaearon.github.io/conway-refinement/#/result/differentiated-relation\" \"Open the statement for lemma 93\"\n  click n094 \"https://gaearon.github.io/conway-refinement/#/result/lower-below-successor\" \"Open the statement for lemma 94\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-limit-degree\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Limit ordinals in the degree induction\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 8 results\"]:::boundary\n    in1[\"Ordinal value and degree\u003Cbr/>used by 1 results\"]:::boundary\n    in2[\"Translated truncations\u003Cbr/>used by 4 results\"]:::boundary\n  end\n  subgraph current[\"Limit ordinals in the degree induction\"]\n    direction TB\n    n095[\"(95) Linearity when (w_B)_{< β} ⊕ ν ≠ λ_0 for every ν\"]:::lemma\n    n096[\"(96) Separation of last Cantor terms under a two-factor bound\"]:::lemma\n    n097[\"(97) Upper Cantor terms at the complementary low weight\"]:::lemma\n    n098[\"(98) Translated truncations of interval pieces\"]:::lemma\n    n099[\"(99) Translated truncations of shifted truncation sums\"]:::lemma\n    n100[\"(100) Order type of a sum along cutoffs\"]:::lemma\n    n101[\"(101) Finiteness of translated truncations above an ordinal-value bound\"]:::lemma\n    n102[\"(102) Support-order reduction at a successor degree\"]:::lemma\n    n103[\"(103) Realisation of lower ordinal values by translated truncations\"]:::lemma\n    n104[\"(104) Maximal-variable linearity for the ordinal-value degree\"]:::lemma\n    n105[\"(105) Support-order reduction by interval decomposition\"]:::lemma\n    n106[\"(106) Translated-truncation approximation at intermediate degrees\"]:::lemma\n    n107[\"(107) Ideal membership from translated truncations\"]:::proposition\n    n108[\"(108) Partial-derivative syzygy when (w_{B'})_{< β} ⊕ η = λ_0 for some η\"]:::proposition\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Principal RV-elements\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n095 --> n097\n  n096 --> n097\n  n101 --> n102\n  n103 --> n104\n  n098 --> n105\n  n099 --> n105\n  n100 --> n105\n  n101 --> n105\n  n103 --> n105\n  n102 --> n106\n  n105 --> n106\n  n106 --> n107\n  n097 --> n108\n  n107 --> n108\n  in0 --> n095\n  in0 --> n096\n  in0 --> n099\n  in0 --> n100\n  in0 --> n102\n  in2 --> n102\n  in1 --> n103\n  in0 --> n104\n  in2 --> n104\n  in2 --> n105\n  in0 --> n106\n  in0 --> n108\n  in2 --> n108\n  n108 --> out0\n  click n095 \"https://gaearon.github.io/conway-refinement/#/result/low-degree-part-outside-algebraic-bound-occurs-linearly\" \"Open the statement for lemma 95\"\n  click n096 \"https://gaearon.github.io/conway-refinement/#/result/later-cantor-terms-outside-algebraic-bound\" \"Open the statement for lemma 96\"\n  click n097 \"https://gaearon.github.io/conway-refinement/#/result/complementary-cantor-tail\" \"Open the statement for lemma 97\"\n  click n098 \"https://gaearon.github.io/conway-refinement/#/result/window-truncation\" \"Open the statement for lemma 98\"\n  click n099 \"https://gaearon.github.io/conway-refinement/#/result/cutoff-sum-truncation\" \"Open the statement for lemma 99\"\n  click n100 \"https://gaearon.github.io/conway-refinement/#/result/cutoff-sum-support\" \"Open the statement for lemma 100\"\n  click n101 \"https://gaearon.github.io/conway-refinement/#/result/successor-large-truncations-finite\" \"Open the statement for lemma 101\"\n  click n102 \"https://gaearon.github.io/conway-refinement/#/result/successor-support-lowering\" \"Open the statement for lemma 102\"\n  click n103 \"https://gaearon.github.io/conway-refinement/#/result/truncation-values\" \"Open the statement for lemma 103\"\n  click n104 \"https://gaearon.github.io/conway-refinement/#/result/relation-at-limit-ordinal-maximal-variable-linear\" \"Open the statement for lemma 104\"\n  click n105 \"https://gaearon.github.io/conway-refinement/#/result/lower-by-pieces\" \"Open the statement for lemma 105\"\n  click n106 \"https://gaearon.github.io/conway-refinement/#/result/induction-over-degrees\" \"Open the statement for lemma 106\"\n  click n107 \"https://gaearon.github.io/conway-refinement/#/result/ideal-from-truncations\" \"Open the statement for proposition 107\"\n  click n108 \"https://gaearon.github.io/conway-refinement/#/result/partials-when-low-degree-part-is-algebraically-bounded\" \"Open the statement for proposition 108\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-principal-rv-algebraic-independence\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Principal RV-elements\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Cantor–Bendixson ranks of supports\u003Cbr/>used by 1 results\"]:::boundary\n    in1[\"Algebraic independence in graded rings\u003Cbr/>used by 1 results\"]:::boundary\n    in2[\"Translated truncations\u003Cbr/>used by 1 results\"]:::boundary\n    in3[\"Limit ordinals in the degree induction\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Principal RV-elements\"]\n    direction TB\n    n109[\"(109) Cantor–Bendixson grading of P̂\"]:::lemma\n    n110[\"(110) Principal series representatives for the Cantor–Bendixson grading\"]:::lemma\n    n111[\"(111) Transport of the Jacobian syzygy to the Cantor–Bendixson grading\"]:::lemma\n    n112[\"(112) Minimal homogeneous generators of P̂ are algebraically independent\"]:::theorem\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Polynomial presentations\u003Cbr/>3 results used downstream\"]:::boundary\n  end\n\n  n109 --> n110\n  n109 --> n111\n  n110 --> n112\n  n111 --> n112\n  in0 --> n109\n  in2 --> n110\n  in3 --> n111\n  in1 --> n112\n  n112 --> out0\n  n110 --> out0\n  n111 --> out0\n  click n109 \"https://gaearon.github.io/conway-refinement/#/result/principal-subring-cantor-bendixson\" \"Open the statement for lemma 109\"\n  click n110 \"https://gaearon.github.io/conway-refinement/#/result/principal-representatives-cantor-bendixson\" \"Open the statement for lemma 110\"\n  click n111 \"https://gaearon.github.io/conway-refinement/#/result/real-translated-truncation-partials\" \"Open the statement for lemma 111\"\n  click n112 \"https://gaearon.github.io/conway-refinement/#/result/polynomial\" \"Open the statement for theorem 112\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-polynomial-presentations\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Polynomial presentations\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 4 results\"]:::boundary\n    in1[\"Ordinal value and degree\u003Cbr/>used by 1 results\"]:::boundary\n    in2[\"Algebraic independence in graded rings\u003Cbr/>used by 2 results\"]:::boundary\n    in3[\"Translated truncations\u003Cbr/>used by 2 results\"]:::boundary\n    in4[\"Principal RV-elements\u003Cbr/>used by 6 results\"]:::boundary\n  end\n  subgraph current[\"Polynomial presentations\"]\n    direction TB\n    n113[\"(113) Scalar extension from P̂ to RV̂ (LM24, Proposition 6.1.2)\"]:::fact\n    n114[\"(114) Polynomial presentation of the germ ring over a Cauchy-complete exponent group\"]:::theorem\n    n115[\"(115) Weighted degree under evaluation at series lifts\"]:::theorem\n    n116[\"(116) Algebraic independence of series lifts\"]:::theorem\n    n117[\"(117) Generation of the series ring by series lifts\"]:::theorem\n    n118[\"(118) Polynomial presentation of the series ring\"]:::theorem\n    n119[\"(119) Polynomial presentation of P̂\"]:::corollary\n    n120[\"(120) Algebraic independence of homogeneous families in P̂\"]:::corollary\n    n121[\"(121) Algebraic independence in additively principal degree\"]:::corollary\n    n122[\"(122) Prime elements of additively principal degree\"]:::corollary\n    n123[\"(123) Surjectivity of evaluation modulo J\"]:::lemma\n    n124[\"(124) Injectivity of evaluation modulo J\"]:::lemma\n    n125[\"(125) Polynomial presentation of the quotient by J\"]:::theorem\n    n126[\"(126) Polynomial representative of a germ\"]:::theorem\n    n127[\"(127) Polynomial representatives below an ordinal-value bound\"]:::theorem\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Primality and factorisation for real exponents\u003Cbr/>1 results used downstream\"]:::boundary\n    out1[\"Refinement and transfinite induction\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n113 --> n115\n  n113 --> n116\n  n113 --> n117\n  n116 --> n118\n  n117 --> n118\n  n120 --> n121\n  n119 --> n122\n  n123 --> n125\n  n124 --> n125\n  n125 --> n126\n  in1 --> n113\n  in0 --> n114\n  in2 --> n114\n  in4 --> n115\n  in4 --> n116\n  in0 --> n117\n  in4 --> n117\n  in4 --> n119\n  in0 --> n120\n  in4 --> n120\n  in0 --> n122\n  in3 --> n123\n  in2 --> n124\n  in4 --> n124\n  in3 --> n127\n  n118 --> out0\n  n114 --> out1\n  click n113 \"https://gaearon.github.io/conway-refinement/#/result/principal-subring-tensor-decomposition\" \"Open the statement for fact 113\"\n  click n114 \"https://gaearon.github.io/conway-refinement/#/result/complete-hahn-germ-polynomial-algebra\" \"Open the statement for theorem 114\"\n  click n115 \"https://gaearon.github.io/conway-refinement/#/result/series-polynomial-degree\" \"Open the statement for theorem 115\"\n  click n116 \"https://gaearon.github.io/conway-refinement/#/result/series-lifts-algebraically-independent\" \"Open the statement for theorem 116\"\n  click n117 \"https://gaearon.github.io/conway-refinement/#/result/series-lifts-generate-series-ring\" \"Open the statement for theorem 117\"\n  click n118 \"https://gaearon.github.io/conway-refinement/#/result/hahn-series-polynomial-algebra\" \"Open the statement for theorem 118\"\n  click n119 \"https://gaearon.github.io/conway-refinement/#/result/principal-subring-polynomial-algebra\" \"Open the statement for corollary 119\"\n  click n120 \"https://gaearon.github.io/conway-refinement/#/result/independent-homogeneous-family\" \"Open the statement for corollary 120\"\n  click n121 \"https://gaearon.github.io/conway-refinement/#/result/principal-degree-linear-independence\" \"Open the statement for corollary 121\"\n  click n122 \"https://gaearon.github.io/conway-refinement/#/result/prime-at-principal-degree\" \"Open the statement for corollary 122\"\n  click n123 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-quotient-evaluation-surjective\" \"Open the statement for lemma 123\"\n  click n124 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-quotient-evaluation-injective\" \"Open the statement for lemma 124\"\n  click n125 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-quotient\" \"Open the statement for theorem 125\"\n  click n126 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-quotient-polynomial-representative\" \"Open the statement for theorem 126\"\n  click n127 \"https://gaearon.github.io/conway-refinement/#/result/ordinal-value-quotient-filtration\" \"Open the statement for theorem 127\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-real-exponent-primality\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Primality and factorisation for real exponents\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 1 results\"]:::boundary\n    in1[\"Ordinal value and degree\u003Cbr/>used by 1 results\"]:::boundary\n    in2[\"Polynomial presentations\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Primality and factorisation for real exponents\"]\n    direction TB\n    n128[\"(128) Irreducibility from translated truncation dimension\"]:::fact\n    n129[\"(129) Greatest common divisors of finite-support series\"]:::fact\n    n130[\"(130) Greatest common divisors in K((ℝ^{≤ 0}))\"]:::theorem\n    n131[\"(131) Primality of generalised power series\"]:::theorem\n    n132[\"(132) Primality criterion for series of degree one\"]:::corollary\n    n133[\"(133) Irreducible series are prime\"]:::corollary\n    n134[\"(134) Random series of finite Cantor degree are prime\"]:::corollary\n    n135[\"(135) Lower-order perturbations of random series are prime\"]:::corollary\n    n136[\"(136) Random principal series of positive finite degree are prime\"]:::corollary\n    n137[\"(137) Unique factorisation under support-order conditions\"]:::corollary\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Reduction along Archimedean classes\u003Cbr/>1 results used downstream\"]:::boundary\n    out1[\"Surreal numbers and omnific integers\u003Cbr/>2 results used downstream\"]:::boundary\n  end\n\n  n129 --> n130\n  n130 --> n131\n  n131 --> n132\n  n131 --> n133\n  n133 --> n134\n  n133 --> n135\n  n133 --> n136\n  n133 --> n137\n  in1 --> n128\n  in0 --> n130\n  in2 --> n130\n  n131 --> out0\n  n128 --> out1\n  n133 --> out1\n  click n128 \"https://gaearon.github.io/conway-refinement/#/result/ps06-irreducibility\" \"Open the statement for fact 128\"\n  click n129 \"https://gaearon.github.io/conway-refinement/#/result/finite-support-hahn-gcd\" \"Open the statement for fact 129\"\n  click n130 \"https://gaearon.github.io/conway-refinement/#/result/hahn-series-gcd-domain\" \"Open the statement for theorem 130\"\n  click n131 \"https://gaearon.github.io/conway-refinement/#/result/hahn-series-primality\" \"Open the statement for theorem 131\"\n  click n132 \"https://gaearon.github.io/conway-refinement/#/result/degree-one\" \"Open the statement for corollary 132\"\n  click n133 \"https://gaearon.github.io/conway-refinement/#/result/hahn-series-irreducible-is-prime\" \"Open the statement for corollary 133\"\n  click n134 \"https://gaearon.github.io/conway-refinement/#/result/random-series-prime\" \"Open the statement for corollary 134\"\n  click n135 \"https://gaearon.github.io/conway-refinement/#/result/random-series-small-perturbation\" \"Open the statement for corollary 135\"\n  click n136 \"https://gaearon.github.io/conway-refinement/#/result/random-principal-series-prime\" \"Open the statement for corollary 136\"\n  click n137 \"https://gaearon.github.io/conway-refinement/#/result/lm17-support-order\" \"Open the statement for corollary 137\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-finite-archimedean-support\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Reduction along Archimedean classes\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Primality and factorisation for real exponents\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Reduction along Archimedean classes\"]\n    direction TB\n    n138[\"(138) Fraction fields of bounded Hahn integer parts\"]:::theorem\n    n139[\"(139) Convexity of the supports of factors\"]:::lemma\n    n140[\"(140) Primality transfer at the leading Archimedean class\"]:::fact\n    n141[\"(141) Primality of reduced bounded Hahn integer-part series\"]:::corollary\n    n142[\"(142) Primality for finitely many Archimedean support classes\"]:::theorem\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Refinement and transfinite induction\u003Cbr/>2 results used downstream\"]:::boundary\n    out1[\"Bounded generalised-power-series integer parts\u003Cbr/>1 results used downstream\"]:::boundary\n    out2[\"Surreal numbers and omnific integers\u003Cbr/>3 results used downstream\"]:::boundary\n  end\n\n  n138 --> n140\n  n140 --> n141\n  n141 --> n142\n  in0 --> n141\n  n139 --> out0\n  n142 --> out0\n  n138 --> out1\n  n138 --> out2\n  n141 --> out2\n  n142 --> out2\n  click n138 \"https://gaearon.github.io/conway-refinement/#/result/bounded-hahn-integer-part-fraction-field\" \"Open the statement for theorem 138\"\n  click n139 \"https://gaearon.github.io/conway-refinement/#/result/convex-support-of-factors\" \"Open the statement for lemma 139\"\n  click n140 \"https://gaearon.github.io/conway-refinement/#/result/leading-class-primality-transfer\" \"Open the statement for fact 140\"\n  click n141 \"https://gaearon.github.io/conway-refinement/#/result/reduced-hahn-integer-part-primal\" \"Open the statement for corollary 141\"\n  click n142 \"https://gaearon.github.io/conway-refinement/#/result/finite-support-classes-primality\" \"Open the statement for theorem 142\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-archimedean-class-refinement\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Refinement and transfinite induction\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Algebraic and ordinal preliminaries\u003Cbr/>used by 1 results\"]:::boundary\n    in1[\"Polynomial presentations\u003Cbr/>used by 2 results\"]:::boundary\n    in2[\"Reduction along Archimedean classes\u003Cbr/>used by 3 results\"]:::boundary\n  end\n  subgraph current[\"Refinement and transfinite induction\"]\n    direction TB\n    n143[\"(143) Normalization of a closed-class refinement in a Hahn integer part\"]:::lemma\n    n144[\"(144) Closed-ball restriction under quotient regrouping\"]:::lemma\n    n145[\"(145) Four-factor refinement in the germ ring over a Cauchy-complete exponent group\"]:::theorem\n    n146[\"(146) Divisibility from a convex restriction\"]:::lemma\n    n147[\"(147) Transport of refinement from a quotient Archimedean class\"]:::lemma\n    n148[\"(148) Factorisation by a convex restriction\"]:::lemma\n    n149[\"(149) Support classes beyond a quotient Archimedean ball\"]:::lemma\n    n150[\"(150) An irreducible germ with zero constant coefficient\"]:::lemma\n    n151[\"(151) Refinement forces K = Frac(Z)\"]:::theorem\n    n152[\"(152) Strict decrease of support-class order type\"]:::lemma\n    n153[\"(153) Factorisation at a quotient Archimedean class\"]:::lemma\n    n154[\"(154) Well-ordered subsets of closed rational spans\"]:::lemma\n    n155[\"(155) Cardinal-bounded refinement modulo series bounded away from zero\"]:::theorem\n    n156[\"(156) Exact refinement over a Cauchy-complete common-tail quotient\"]:::theorem\n    n157[\"(157) Refinement at a quotient Archimedean class\"]:::theorem\n    n158[\"(158) Transfinite extension of finite-class primality\"]:::theorem\n    n159[\"(159) Primality under finite-class and common-tail hypotheses\"]:::theorem\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Bounded generalised-power-series integer parts\u003Cbr/>1 results used downstream\"]:::boundary\n    out1[\"Surreal numbers and omnific integers\u003Cbr/>1 results used downstream\"]:::boundary\n  end\n\n  n144 --> n147\n  n146 --> n147\n  n150 --> n151\n  n148 --> n153\n  n149 --> n153\n  n152 --> n153\n  n145 --> n155\n  n154 --> n155\n  n155 --> n156\n  n143 --> n157\n  n156 --> n157\n  n147 --> n158\n  n153 --> n158\n  n157 --> n158\n  n158 --> n159\n  in1 --> n145\n  in2 --> n147\n  in1 --> n150\n  in0 --> n151\n  in2 --> n157\n  in2 --> n159\n  n159 --> out0\n  n158 --> out1\n  click n143 \"https://gaearon.github.io/conway-refinement/#/result/closed-class-refinement-normalization\" \"Open the statement for lemma 143\"\n  click n144 \"https://gaearon.github.io/conway-refinement/#/result/quotient-regrouping-closed-ball-restriction\" \"Open the statement for lemma 144\"\n  click n145 \"https://gaearon.github.io/conway-refinement/#/result/complete-hahn-germ-refinement\" \"Open the statement for theorem 145\"\n  click n146 \"https://gaearon.github.io/conway-refinement/#/result/divisibility-from-convex-restriction\" \"Open the statement for lemma 146\"\n  click n147 \"https://gaearon.github.io/conway-refinement/#/result/closed-class-refinement-transport\" \"Open the statement for lemma 147\"\n  click n148 \"https://gaearon.github.io/conway-refinement/#/result/factorisation-by-convex-restriction\" \"Open the statement for lemma 148\"\n  click n149 \"https://gaearon.github.io/conway-refinement/#/result/tail-quotient-class-bounds-support-classes\" \"Open the statement for lemma 149\"\n  click n150 \"https://gaearon.github.io/conway-refinement/#/result/irreducible-zero-constant-germ\" \"Open the statement for lemma 150\"\n  click n151 \"https://gaearon.github.io/conway-refinement/#/result/refinement-forces-coefficient-fraction-field\" \"Open the statement for theorem 151\"\n  click n152 \"https://gaearon.github.io/conway-refinement/#/result/support-class-order-type-strict-decrease\" \"Open the statement for lemma 152\"\n  click n153 \"https://gaearon.github.io/conway-refinement/#/result/support-class-factorisation\" \"Open the statement for lemma 153\"\n  click n154 \"https://gaearon.github.io/conway-refinement/#/result/well-ordered-subset-closed-rational-span-cardinality\" \"Open the statement for lemma 154\"\n  click n155 \"https://gaearon.github.io/conway-refinement/#/result/cardinal-bounded-germ-refinement\" \"Open the statement for theorem 155\"\n  click n156 \"https://gaearon.github.io/conway-refinement/#/result/complete-tail-quotient-refinement\" \"Open the statement for theorem 156\"\n  click n157 \"https://gaearon.github.io/conway-refinement/#/result/support-class-refinement\" \"Open the statement for theorem 157\"\n  click n158 \"https://gaearon.github.io/conway-refinement/#/result/limit-tail-primality\" \"Open the statement for theorem 158\"\n  click n159 \"https://gaearon.github.io/conway-refinement/#/result/hahn-integer-part-primality\" \"Open the statement for theorem 159\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-cauchy-completeness\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>A cut criterion for Cauchy completeness\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph current[\"A cut criterion for Cauchy completeness\"]\n    direction TB\n    n160[\"(160) Cut filling under monotone order-reflecting surjections\"]:::lemma\n    n161[\"(161) A cut criterion for Cauchy completeness\"]:::lemma\n  end\n  subgraph outputs[\"Used by other sections\"]\n    direction TB\n    out0[\"Bounded generalised-power-series integer parts\u003Cbr/>2 results used downstream\"]:::boundary\n    out1[\"Surreal numbers and omnific integers\u003Cbr/>2 results used downstream\"]:::boundary\n  end\n\n  n160 --> out0\n  n161 --> out0\n  n160 --> out1\n  n161 --> out1\n  click n160 \"https://gaearon.github.io/conway-refinement/#/result/cut-filling-order-reflecting-surjection\" \"Open the statement for lemma 160\"\n  click n161 \"https://gaearon.github.io/conway-refinement/#/result/complete-of-coinitial-scales-and-cut-filling\" \"Open the statement for lemma 161\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-bounded-integer-parts\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Bounded generalised-power-series integer parts\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Reduction along Archimedean classes\u003Cbr/>used by 1 results\"]:::boundary\n    in1[\"Refinement and transfinite induction\u003Cbr/>used by 1 results\"]:::boundary\n    in2[\"A cut criterion for Cauchy completeness\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Bounded generalised-power-series integer parts\"]\n    direction TB\n    n162[\"(162) Archimedean strata of saturated ordered groups\"]:::lemma\n    n163[\"(163) Cauchy completeness of common-tail quotients\"]:::lemma\n    n164[\"(164) Cofinality of common tails in saturated ordered groups\"]:::lemma\n    n165[\"(165) Cofinality of Archimedean inner balls in saturated ordered groups\"]:::lemma\n    n166[\"(166) Common-tail conditions in saturated ordered groups\"]:::lemma\n    n167[\"(167) Pre-Schreier property of bounded Hahn integer parts\"]:::theorem\n    n168[\"(168) Pre-Schreier Hahn integer parts over saturated exponent groups\"]:::corollary\n    n169[\"(169) Refinement of bounded generalised-power-series integer parts\"]:::theorem\n    n170[\"(170) Refinement over saturated exponent groups\"]:::corollary\n    n171[\"(171) Refinement of saturated Hahn integer parts with integer constants\"]:::corollary\n  end\n\n  n163 --> n166\n  n164 --> n166\n  n162 --> n168\n  n165 --> n168\n  n166 --> n168\n  n167 --> n168\n  n167 --> n169\n  n162 --> n170\n  n165 --> n170\n  n166 --> n170\n  n169 --> n170\n  n170 --> n171\n  in2 --> n163\n  in0 --> n166\n  in1 --> n167\n  click n162 \"https://gaearon.github.io/conway-refinement/#/result/saturated-archimedean-strata-real\" \"Open the statement for lemma 162\"\n  click n163 \"https://gaearon.github.io/conway-refinement/#/result/saturated-common-tail-quotient-complete\" \"Open the statement for lemma 163\"\n  click n164 \"https://gaearon.github.io/conway-refinement/#/result/saturated-common-tail-cofinality\" \"Open the statement for lemma 164\"\n  click n165 \"https://gaearon.github.io/conway-refinement/#/result/saturated-inner-ball-cofinality\" \"Open the statement for lemma 165\"\n  click n166 \"https://gaearon.github.io/conway-refinement/#/result/saturated-common-tail-conditions\" \"Open the statement for lemma 166\"\n  click n167 \"https://gaearon.github.io/conway-refinement/#/result/hahn-integer-part-pre-schreier\" \"Open the statement for theorem 167\"\n  click n168 \"https://gaearon.github.io/conway-refinement/#/result/hahn-integer-part-pre-schreier-of-saturation\" \"Open the statement for corollary 168\"\n  click n169 \"https://gaearon.github.io/conway-refinement/#/result/hahn-integer-part-refinement\" \"Open the statement for theorem 169\"\n  click n170 \"https://gaearon.github.io/conway-refinement/#/result/hahn-integer-part-refinement-of-saturation\" \"Open the statement for corollary 170\"\n  click n171 \"https://gaearon.github.io/conway-refinement/#/result/integer-hahn-refinement-of-saturation\" \"Open the statement for corollary 171\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\n\u003Ca id=\"phase-conway-refinement\">\u003C/a>\n\u003Cdetails>\n\u003Csummary>Surreal numbers and omnific integers\u003C/summary>\n\n```mermaid\n%% Generated by scripts/blueprint.sh; do not edit.\n%% Other sections are compressed to boundary nodes.\n%% Solid arrows are proof dependencies; dashed arrows are statement dependencies.\nflowchart TB\n  subgraph inputs[\"Inputs from other sections\"]\n    direction TB\n    in0[\"Primality and factorisation for real exponents\u003Cbr/>used by 2 results\"]:::boundary\n    in1[\"Reduction along Archimedean classes\u003Cbr/>used by 4 results\"]:::boundary\n    in2[\"Refinement and transfinite induction\u003Cbr/>used by 1 results\"]:::boundary\n    in3[\"A cut criterion for Cauchy completeness\u003Cbr/>used by 1 results\"]:::boundary\n  end\n  subgraph current[\"Surreal numbers and omnific integers\"]\n    direction TB\n    n172[\"(172) Primality of an explicit degree-two series\"]:::theorem\n    n173[\"(173) Equivalence of the cut and subring formulations of Conway's refinement conjecture\"]:::theorem\n    n174[\"(174) Real structure of surreal Archimedean strata\"]:::fact\n    n175[\"(175) Cofinality of surreal Archimedean balls\"]:::fact\n    n176[\"(176) Cofinality of common surreal Archimedean tails\"]:::lemma\n    n177[\"(177) Fraction fields of surreal common-tail integer parts\"]:::theorem\n    n178[\"(178) The simplicity theorem for small cuts\"]:::theorem\n    n179[\"(179) The signed normal-form isomorphism for omnific integers\"]:::theorem\n    n180[\"(180) Primality for finitely many Archimedean classes\"]:::corollary\n    n181[\"(181) Primality of reduced omnific integers outside ℤ\"]:::theorem\n    n182[\"(182) Irreducible omnific integers are prime\"]:::theorem\n    n183[\"(183) Primality of an explicit degree-two omnific integer\"]:::theorem\n    n184[\"(184) A coinitial family of positive elements in surreal common-tail quotients\"]:::lemma\n    n185[\"(185) Cauchy completeness of surreal common-tail quotients\"]:::lemma\n    n186[\"(186) Primality of the surreal Hahn integer part\"]:::theorem\n    n187[\"(187) Refinement property of Oz_u\"]:::theorem\n    n188[\"(188) Conway's refinement theorem for omnific integers\"]:::theorem\n  end\n\n  n176 --> n177\n  n174 --> n180\n  n175 --> n180\n  n179 --> n180\n  n174 --> n181\n  n175 --> n181\n  n179 --> n181\n  n181 --> n182\n  n182 --> n183\n  n178 --> n185\n  n184 --> n185\n  n174 --> n186\n  n175 --> n186\n  n177 --> n186\n  n185 --> n186\n  n179 --> n187\n  n186 --> n187\n  n173 --> n188\n  n187 --> n188\n  in0 --> n172\n  in1 --> n177\n  in1 --> n180\n  in1 --> n181\n  in0 --> n183\n  in3 --> n185\n  in1 --> n186\n  in2 --> n186\n  click n172 \"https://gaearon.github.io/conway-refinement/#/result/degree-two-series-prime\" \"Open the statement for theorem 172\"\n  click n173 \"https://gaearon.github.io/conway-refinement/#/result/conway-cut-subring-equivalence\" \"Open the statement for theorem 173\"\n  click n174 \"https://gaearon.github.io/conway-refinement/#/result/surreal-archimedean-strata\" \"Open the statement for fact 174\"\n  click n175 \"https://gaearon.github.io/conway-refinement/#/result/surreal-archimedean-ball-cofinality\" \"Open the statement for fact 175\"\n  click n176 \"https://gaearon.github.io/conway-refinement/#/result/surreal-common-tail-cofinality\" \"Open the statement for lemma 176\"\n  click n177 \"https://gaearon.github.io/conway-refinement/#/result/surreal-common-tail-integer-part-fraction-field\" \"Open the statement for theorem 177\"\n  click n178 \"https://gaearon.github.io/conway-refinement/#/result/surreal-simplicity-small-cuts\" \"Open the statement for theorem 178\"\n  click n179 \"https://gaearon.github.io/conway-refinement/#/result/signed-normal-form-omnific-integer-equivalence\" \"Open the statement for theorem 179\"\n  click n180 \"https://gaearon.github.io/conway-refinement/#/result/omnific-finite-classes\" \"Open the statement for corollary 180\"\n  click n181 \"https://gaearon.github.io/conway-refinement/#/result/reduced-omnific-primal\" \"Open the statement for theorem 181\"\n  click n182 \"https://gaearon.github.io/conway-refinement/#/result/omnific-factorisation\" \"Open the statement for theorem 182\"\n  click n183 \"https://gaearon.github.io/conway-refinement/#/result/explicit-omnific-prime\" \"Open the statement for theorem 183\"\n  click n184 \"https://gaearon.github.io/conway-refinement/#/result/surreal-common-tail-quotient-coinitial-scales\" \"Open the statement for lemma 184\"\n  click n185 \"https://gaearon.github.io/conway-refinement/#/result/surreal-common-tail-quotient-complete\" \"Open the statement for lemma 185\"\n  click n186 \"https://gaearon.github.io/conway-refinement/#/result/surreal-hahn-integer-part-primality\" \"Open the statement for theorem 186\"\n  click n187 \"https://gaearon.github.io/conway-refinement/#/result/omnific-integer-refinement-property\" \"Open the statement for theorem 187\"\n  click n188 \"https://gaearon.github.io/conway-refinement/#/result/conway-refinement\" \"Open the statement for theorem 188\"\n\n  classDef theorem fill:#fff3bf,stroke:#e67700,stroke-width:2px,color:#111827\n  classDef definition fill:#f3f4f6,stroke:#4b5563,color:#111827\n  classDef lemma fill:#dbeafe,stroke:#2563eb,color:#111827\n  classDef proposition fill:#ede9fe,stroke:#7c3aed,color:#111827\n  classDef corollary fill:#dcfce7,stroke:#16a34a,color:#111827\n  classDef fact fill:#fce7f3,stroke:#db2777,color:#111827\n  classDef boundary fill:#f8fafc,stroke:#64748b,stroke-dasharray:4 3,color:#334155\n```\n\u003C/details>\n\u003C!-- blueprint-mermaid:end -->\n\nThese diagrams can also help AI refactor or simplify the overall proof path.\n\n## But what was the conjecture about?\n\nThis conjecture is about *omnific integers*.\n\nOmnific integers were invented (or discovered?) by John Conway. Intuitively, I'd describe them as an extension of integers as we know them ($-3$, $10$, $1840$, $-437349$) to infinity and beyond. Recall that there is no biggest integer: after $1$ comes $2$, after $2$ comes $3$, and so on. So if you say some $X$ is the biggest integer, I would point to $X + 1$, which is bigger, and you would be wrong. However, omnific integers *do* contain a number bigger than every regular integer. It's called $\\omega$, \"omega\", and in a sense it represents an infinity: $\\omega$ is \"what stands after\" every regular integer.\n\nYou might think that it ends there, but it doesn't.\n\nYou can add $1$ to get $\\omega + 1$, then add $1$ again to get $\\omega + 2$, and so on. Curiously, you can also go *backwards* — to $\\omega - 1$, $\\omega - 2$, $\\omega - 800$, and so on. No matter how many finite steps you go down, you will never reach the \"regular\" integers. If you started off from $\\omega$, you're just *too high already* to ever reach the regular integers. And it doesn't end here either. You can divide $\\omega$, for example $\\omega / 2$, and it would *still* be bigger than all regular integers. (Curiously, $\\omega$ is even — it is divisible by 2 — but it is also divisible by 3, 4, or really by any integer except zero.)\n\nYou could also keep going up — perhaps, after another infinity of steps, reaching $\\omega + \\omega = \\omega \\cdot 2$, or further up to $\\omega \\cdot 3$, or, after making infinite jumps, to $\\omega \\cdot \\omega = \\omega^2$. You could keep going (or rather, \"jumping\" — walking can't even get you to $\\omega$), stacking powers of omega into ever taller towers:\n\n$$\n\\omega,\\qquad \\omega^\\omega,\\qquad \\omega^{\\omega^\\omega},\\qquad \\ldots\n$$\n\nNow \"jump\" through doing *that* forever, and you'll reach a tower so tall that raising $\\omega$ to it doesn't change it:\n\n$$\\omega^{\\left(\\omega^{\\omega^{\\omega^{\\cdots}}}\\right)} = \\omega^{\\omega^{\\omega^{\\cdots}}}.$$\n\nAnd yet you could still keep on going.\n\nAll of this is very strange, but what is stranger is that it all \"falls out\" of a very simple theory.\n\nForget all the numbers you know. Define a single number called \"zero\" and then go step by step. At every step, invent a new number at each distinct gap between the numbers you already have (\"to the left of all\" and \"to the right of all\" also count as gaps). Continue doing this forever, and then forevermore forever, and so on.\n\nTurns out, if you do this forever, you'll get a fractal tree of [all numbers great and small](https://upload.wikimedia.org/wikipedia/commons/4/49/Surreal_number_tree.svg):\n\n\u003Cimg width=\"500\" height=\"618\" alt=\"Surreal tree illustration by Joel David Hamkins\" src=\"https://github.com/user-attachments/assets/5bcec33f-e41b-41af-b736-515df166388b\" />\n\n*(Illustration by Joel David Hamkins, go [read his posts](https://www.infinitelymore.xyz/p/surreal-numbers) and [buy his book!](https://jdh.hamkins.org/the-book-of-infinity/))*\n\nThis simple recursive binary tree turns out to produce exactly the numbers we commonly use in life — naturals, integers, rationals, reals — as well as weirder numbers such as infinite \"ordinals\" like $\\omega$, \"infinitesimals\" like $\\frac{1}{\\omega}$, and all sorts of combinations between them, such as $\\omega^2+\\pi\\omega-7+\\frac{3}{\\omega}+\\frac{1}{\\omega^2}$.\n\nThese beautiful and strange numbers are called *surreal numbers*.\n\nOmnific integers are the \"whole\" part of the surreal number tree. Turns out, every surreal number can be seen as a result of division $A/B$ of two omnific integers $A$ and $B$. This is true even for familiar irrationals like $\\sqrt{2}$. No ratio of regular integers can equal $\\sqrt{2}$, but you get it if you divide $\\sqrt{2}\\omega$ by $\\omega$, and both of these are omnific integers. (Yep, $\\sqrt{2}\\omega$ is a \"whole\" number! The bigness of $\\omega$ \"swallows up\" the infinite precision of a real number like $\\sqrt{2}$, leaving no fractional part.) That's why Conway called them omnific, i.e. \"all-creating\". Their ratios \"create\" every number.\n\nFinally, the conjecture.\n\nConway claims that if $ab = cd$, we can \"break down\" $a$ and $b$ into pieces, and then $c$ and $d$ will turn out to be the same pieces recombined. With regular integers, we take this for granted. Say 210 = 10 × 21. We can break 10 down as 2 × 5 and 21 as 3 × 7, then reshuffle them into 2 × 3 = 6 and 5 × 7 = 35. The product is still 6 × 35 = 210.\n\nHowever, when you deal with infinities, things don't always turn out as we expect. So the conjecture means Conway thought omnific integers had, in a sense, enough \"structure\" to keep this property. It happens that every surreal number can also be written as a certain kind of infinite series, with omnific integers being a special case. This lets us study how omnific integers break apart by studying how these series break apart. Over the years, Berarducci, Pitteloud, Pommersheim and Shahriari, and L'Innocente and Mantova developed increasingly strong results about exactly that. This proof builds on all of that work and claims to finish the refinement conjecture.\n\nOr at least so I would hope!\n\n## Discussion\n\nIf you want to say or ask something, feel free to use Issues or Discussions.\n\nYou can also find me on Bluesky as [`@danabra.mov`](https://bsky.app/profile/danabra.mov).\n\nThank you for reading!\n\n## Build\n\n```text\nlake build\nlake exe palomar-compatibility\nlake exe module-system\nlake exe axioms\nlake exe standalone-mathlib\nlake exe standalone-combinatorial-games\nlake exe proof-links\nlake exe style\nlake exe documentation\nlake exe layering\nscripts/lint-env.sh\nscripts/blueprint.sh build\n```\n\nFor release, run `scripts/publish-blueprint.sh` from a clean commit.\n\nThis runs the release gates and generates the PDF and the website.\n\n## License\n\nApache License Version 2.0\n\n## References\n\n- J. H. Conway, *On Numbers and Games*, London Mathematical Society Monographs, no. 6, Academic Press, 1976.\n- M. Zafrullah, “[On a property of pre-Schreier domains](https://doi.org/10.1080/00927878708823512),” *Comm. Algebra* **15** (1987), no. 9, 1895–1920.\n- A. Berarducci, “[Factorization in generalized power series](https://doi.org/10.1090/S0002-9947-99-02172-8),” *Trans. Amer. Math. Soc.* **352** (2000), no. 2, 553–577.\n- D. Pitteloud, “[Existence of prime elements in rings of generalized power series](https://doi.org/10.2307/2695102),” *J. Symbolic Logic* **66** (2001), no. 3, 1206–1216.\n- D. Pitteloud, “[Algebraic properties of rings of generalized power series](https://doi.org/10.1016/S0168-0072(01)00099-9),” *Ann. Pure Appl. Logic* **116** (2002), nos. 1–3, 39–66.\n- J. Pommersheim and S. Shahriari, “[Unique factorization in generalized power series rings](https://doi.org/10.1090/S0002-9939-05-08162-1),” *Proc. Amer. Math. Soc.* **134** (2006), no. 5, 1277–1287.\n- S. L'Innocente and V. Mantova, “[Factorisation of germ-like series](https://doi.org/10.4115/jla.2017.9.3),” *J. Log. Anal.* **9** (2017), no. 3, 1–16.\n- S. L'Innocente and V. Mantova, “[A factorisation theory for generalised power series and omnific integers](https://doi.org/10.1016/j.aim.2024.109513),” *Adv. Math.* **442** (2024), 109513.\n- A. Fornasiero, N. Lavi, S. L'Innocente and V. Mantova, “[Irreducibility in generalized power series](https://arxiv.org/abs/2405.13815),” arXiv:2405.13815v1 (2024).\n",1792350319618]