{
  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:12:14.116Z",
  "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"
    },
    "goals": [],
    "goalDependencies": []
  },
  "notes": [
    {
      "id": "n_zpg854Oo6Xx_",
      "number": 2,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/2",
      "title": "Connected compact Hausdorff riegs",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:dyu9gy",
      "body": "## Question\n\nDoes a nontrivial connected compact Hausdorff topological rieg exist? All three binary operations—addition, multiplication, and exponentiation—are required to be continuous.\n\n## Current result\n\n[Compact rieg no-go](https://sumire.aiformath.jp/p/2609.00007/n/9) proves that every compact Hausdorff topological rieg with cancellative addition is trivial. Connectedness is not needed for that result. Thus a positive answer must have noncancellative addition; the general existence question remains open here.\n\nThe original motivation compares connected topological algebra with profinite algebra. A path-connected topological rig is contractible: a path \\(p\\) from \\(0\\) to \\(1\\) gives \\(H(t,x)=p(t)x\\). This observation concerns path-connectedness and does not by itself settle the connected compact case.\n\n## Next step\n\nChoose a compact connected noncancellative topological semiring and test whether it admits a jointly continuous exponentiation satisfying all the rieg equations, especially \\(x^0=1\\), \\(x^1=x\\), and \\(x^{y+z}=x^yx^z\\). Failure of a particular candidate is not a general nonexistence theorem.\n\n## Source\n\n[Original problem 7.0.5](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L996-L1017).",
      "dates": {
        "inspiration": null,
        "sourceRecorded": null,
        "evidence": null,
        "registeredInSumire": null,
        "updatedInSumire": "2026-10-30T14:29:31.307Z",
        "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
        "legacyUpdatedAt": "2026-10-30T14:29:31.307Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "open-questions",
      "problemState": "solved",
      "resolution": "Yes. Every De Morgan algebra \\((A,\\vee,\\wedge,0,1,\\neg)\\) is a rieg with \\(x+y=x\\vee y\\), \\(xy=x\\wedge y\\) and \\(x^y=x\\vee\\neg y\\): the laws \\(x^0=1\\), \\(x^1=x\\), \\(1^x=1\\), \\(x^{y+z}=x^yx^z\\), \\((xy)^z=x^zy^z\\) and \\((x^y)^z=x^{yz}\\) follow from distributivity and \\(\\neg0=1\\), \\(\\neg1=0\\), \\(\\neg(y\\vee z)=\\neg y\\wedge\\neg z\\), \\(\\neg(y\\wedge z)=\\neg y\\vee\\neg z\\). For a Boolean algebra this is the rieg \\(x^y=(y\\Rightarrow x)\\) of [Notes on Rieg Theory](https://sumire.aiformath.jp/p/2609.00007?view=main), Example 5.0.2 and Proposition 5.3.1. In particular \\([0,1]\\) with \\(\\max\\), \\(\\min\\) and \\(x^y=\\max(x,1-y)\\) is a nontrivial connected compact Hausdorff topological rieg; all three operations are continuous. Its addition is idempotent, hence not cancellative, as [[2609.00007/9]] requires of every example. Not yet verified by a human.",
      "verificationEvidence": {
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        "lean": "",
        "human": ""
      },
      "aiClaimModel": "",
      "formalization": "unknown",
      "humanReview": "unreviewed",
      "verification": "unverified",
      "ai": {
        "use": "assisted",
        "tools": "Codex",
        "additions": ""
      },
      "originalQuestion": "",
      "references": []
    }
  ],
  "noteDependencies": [],
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