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  "format": "sumire-research-record",
  "version": 1,
  "exportedAt": "2026-11-04T08:26:54.316Z",
  "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_ZNzJjsDrnFEf",
      "number": 4,
      "url": "https://sumire.aiformath.jp/p/2609.00007/n/4",
      "title": "Functions ℕ→ℕ from F1",
      "author": {
        "name": "hora",
        "handle": "ryuya"
      },
      "source": "kobin:u_EMlmNonOMJFd:idea:pcaa53",
      "body": "## Question\n\nWhich functions \\(\\mathbb N\\to\\mathbb N\\) can be represented by a term in the free rieg \\(F_1\\) on one generator?\n\nHere a term is built from \\(0,1,x\\) using addition, multiplication, and binary exponentiation. Evaluating \\(x=n\\) in the natural-number rieg gives its function, with \\(0^0=1\\).\n\n## Two distinct tasks\n\n1. Characterize the functions obtained by this evaluation.\n2. Determine when two terms give the same function on \\(\\mathbb N\\).\n\nThe second question must distinguish equality of functions on this particular rieg from equality forced by the axioms in the free rieg.\n\n## Next step\n\nIdentify a property shared by all representable functions and prove it by induction on terms, or exhibit a function together with a rigorous nonrepresentability argument. A table of small inputs alone cannot establish representability or equality for all natural numbers.\n\n[Original problem 7.0.2](https://github.com/hora-algebra/ryuya-hora-homepage/blob/1b8b74c2b3447bdc744a0cac256c5d5806588adc/scripts/site.js#L964-L972).",
      "dates": {
        "inspiration": null,
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        "registeredInSumire": null,
        "updatedInSumire": "2026-10-30T14:29:32.517Z",
        "legacyRecordedAt": "2026-03-30T00:00:00.000Z",
        "legacyUpdatedAt": "2026-10-30T14:29:32.517Z"
      },
      "language": null,
      "projectLanguage": "ja",
      "languageSource": "not_recorded",
      "legacyStage": "conjecture",
      "classification": "unclassified",
      "classificationStatus": "unclassified",
      "importance": "unspecified",
      "section": "open-questions",
      "problemState": "answer-proposed",
      "resolution": "Partial answer to the second task. (a) Equality of the functions on \\(\\mathbb N\\) is decidable. For \\(x\\geq1\\) every subterm is either identically \\(0\\) or \\(\\geq1\\), and which case occurs is read off from the term; rewriting from the inside out by \\(u+0\\mapsto u\\), \\(u\\cdot0\\mapsto0\\), \\(u^0\\mapsto1\\) and \\(0^v\\mapsto0\\) for \\(v\\neq0\\) turns a term, without changing its values at \\(x\\geq1\\), into \\(0\\) or into a term without \\(0\\). Two \\(0\\)-free terms agree on \\(\\mathbb N_{>0}\\) exactly when the one-variable identity holds in \\((\\mathbb N_{>0},+,\\cdot,\\uparrow,1)\\), which is decidable (Richardson, Z. Math. Logik Grundlagen Math. 15 (1969); see Burris–Yeats, *The saga of the high school identities*, Algebra Universalis 52, §3), and the values at \\(x=0\\) are computed directly. (b) The evaluation \\(F_1\\to\\mathbb N^{\\mathbb N}\\) is not injective: \\(x\\cdot0^x\\) and \\(0\\) define the same function, but in the three-element rieg \\(\\{0,\\tfrac12,1\\}\\) with \\(\\max\\), \\(\\min\\) and \\(y^z=\\max(y,1-z)\\) (one of the four exponentiations on the three-element chain in [Notes on Rieg Theory](https://sumire.aiformath.jp/p/2609.00007?view=main), Appendix B.2, Figure 8; see also [[2609.00007/2]]) the term \\(x\\cdot0^x\\) takes the value \\(\\tfrac12\\) at \\(x=\\tfrac12\\). Without \\(0\\) the analogous question is Tarski's high school algebra problem (Wilkie; Gurevič). The first task, a characterization of the functions, remains open; on \\(\\mathbb N_{>0}\\) the one-variable term functions without \\(0\\) form Skolem's class, which is well ordered by eventual domination (Ehrenfeucht, Algebra Universalis 3 (1973)). 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": []
    }
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