{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:26:53.781Z",
  "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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  "notes": [
    {
      "id": "n_njMFOHxv3Ybo",
      "number": 3,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/3",
      "title": "Free rieg elements as games",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:kvbsp5",
      "body": "## Question\n\nLet \\(F_1\\) be the free rieg on one generator \\(x\\). Can its elements be interpreted as combinatorial games? Can the unique rieg homomorphism\n\\[\nF_1\\longrightarrow\\mathbb N,\\qquad x\\longmapsto0\n\\]\nbe interpreted in terms of those games?\n\n## What an answer must provide\n\nSpecify the games and their equality, interpret \\(0,1,+,\\times\\), and exponentiation, and show that the rieg equations hold. A full model of \\(F_1\\) must also identify when two game expressions represent the same element; a map out of \\(F_1\\) alone need not be injective.\n\n## Next step\n\nGive a concrete candidate for exponentiation of games, then test the exponent laws and the evaluation \\(x\\mapsto0\\). The existing recursive-coalgebra model of impartial games does not by itself supply these rieg operations.\n\nSources: [original problem 7.0.1](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L953-L962); [Classical vs quantum computation](https://golem.ph.utexas.edu/category/2006/10/classical_vs_quantum_computati_3.html).",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
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        "updatedInSumire": "2026-10-30T04:48:13.344Z",
        "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
        "legacyUpdatedAt": "2026-10-30T04:48:13.344Z"
      },
      "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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