{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:27:11.645Z",
  "scope": "Reader-visible project fields and native live notes only; referenced notes in other projects, deleted notes and private AI context are excluded.",
  "submission": "not_submitted",
  "mathematicalCorrespondence": "not_automatically_checked",
  "project": {
    "id": "2609.00007",
    "url": "https://sumire.aiformath.jp/p/2609.00007",
    "title": "Rieg theory",
    "tagline": "Semirings with exponentiation, connecting arithmetic, logic, and categorical counting.",
    "visibility": "public",
    "owner": {
      "name": "hora",
      "handle": "ryuya"
    },
    "origin": "",
    "source": "kobin:u_EMlmNonOMJFd:slogan:riegs",
    "goal": "Develop a coherent theory of semirings with exponentiation and use it to connect finite and profinite arithmetic, logic, enumerative combinatorics, and category theory.",
    "narrative": "A rieg is a commutative semiring equipped with exponentiation satisfying the usual exponent laws, including \\(x^0=1\\) and hence \\(0^0=1\\). This project studies the structure itself and the examples it connects.\n\nThe research programme is described in [note 7](https://sumire.aiformath.jp/p/2609.00007/n/7). The current questions concern [the subrieg generated by nonnegative rationals](https://sumire.aiformath.jp/p/2609.00007/n/1), [game interpretations of the free rieg](https://sumire.aiformath.jp/p/2609.00007/n/3), [functions represented by unary rieg terms](https://sumire.aiformath.jp/p/2609.00007/n/4), and [finite-valued presheaf categories](https://sumire.aiformath.jp/p/2609.00007/n/5).\n\nOne concrete result narrows the topological problem: [a compact Hausdorff topological rieg with cancellative addition is trivial](https://sumire.aiformath.jp/p/2609.00007/n/9). The question [whether a nontrivial connected compact Hausdorff rieg exists](https://sumire.aiformath.jp/p/2609.00007/n/2) therefore reduces to the noncancellative case, where it has a positive answer.\n\nFor finite-valued presheaves, essentially finite slices give a sufficient condition for a topos. [A partial answer](https://sumire.aiformath.jp/p/2609.00007/n/5) proposes an exact criterion stated in terms of finite presheaves. [The right Kan extension approach](https://sumire.aiformath.jp/p/2609.00007/n/8) is absorbed into that note, with the coreflection \\(X\\mapsto X_{\\mathrm{fin}}\\) onto locally finite presheaves as its comonad.\n\nThe next step is concrete: give a criterion for finite presheaf topoi in terms of the indexing category alone.",
    "history": "",
    "dates": {
      "inspiration": null,
      "sourceRecorded": null,
      "evidence": null,
      "registeredInSumire": null,
      "updatedInSumire": "2026-10-30T14:34:57.980Z",
      "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
      "legacyUpdatedAt": "2026-10-30T14:34:57.980Z"
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    {
      "id": "n_DxF4D4FsMY2k",
      "number": 1,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/1",
      "title": "Rational subrieg of [0,∞)",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:dntdvh",
      "body": "## Question\n\nWhat is the subrieg of \\([0,\\infty)\\) generated by \\(\\mathbb Q_{\\geq0}\\), with ordinary exponentiation and \\(0^0=1\\)? Which equations hold between its expressions? Is the canonical map from the rieg freely generated by the rig \\(\\mathbb Q_{\\geq0}\\) injective?\n\n## First distinction\n\nThe nonnegative rationals themselves are not closed under exponentiation: \\(2^{1/2}\\) is already irrational. The question concerns the closure under all rieg operations and the relations in that closure.\n\n## Next step\n\nState the universal property of the rieg generated by the rig \\(\\mathbb Q_{\\geq0}\\), then write its canonical evaluation map into \\([0,\\infty)\\). To prove a relation is additional, exhibit two terms with the same real value and prove that their equality does not follow from the rieg axioms and the specified rational rig relations. Merely finding two equal real expressions does not establish this.\n\n[Original problem 7.0.3](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L974-L984).",
      "dates": {
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        "registeredInSumire": null,
        "updatedInSumire": "2026-10-30T04:48:11.666Z",
        "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
        "legacyUpdatedAt": "2026-10-30T04:48:11.666Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "open-questions",
      "problemState": "open",
      "resolution": "",
      "verificationEvidence": {
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      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
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      "ai": {
        "use": "assisted",
        "tools": "Codex",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
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    "mathlib": "https://leanprover-community.github.io/contribute/index.html"
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  "search": {
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