{ "format": "sumire-research-record", "version": 1, "exportedAt": "2026-11-04T08:12:14.118Z", "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_1Kk7ESJ_I_Ho", "number": 7, "url": "https://sumire.aiformath.jp/p/2609.00007/n/7", "title": "Rieg theory: the research programme", "author": { "name": "hora", "handle": "ryuya" }, "source": "kobin:u_EMlmNonOMJFd:idea:suw4hw", "body": "The aim is to develop the theory of semirings with exponentiation across four subjects: logic, finite and profinite arithmetic, enumerative combinatorics, and category theory.\n\nThe central object is a rieg: a commutative semiring with an exponentiation operation satisfying the exponent laws, with the convention \\(0^0=1\\).\n\nThe concrete questions in this project are:\n\n- [Rational subrieg of the nonnegative reals](https://sumire.aiformath.jp/p/2609.00007/n/1): determine the relations introduced by exponentiation.\n- [Free rieg elements as games](https://sumire.aiformath.jp/p/2609.00007/n/3): find a game interpretation compatible with the rieg operations and evaluation at zero.\n- [Unary functions represented by terms](https://sumire.aiformath.jp/p/2609.00007/n/4): characterize the functions on natural numbers obtained from a free rieg.\n- [Connected compact Hausdorff riegs](https://sumire.aiformath.jp/p/2609.00007/n/2): answered positively by the unit interval with \\(\\max\\), \\(\\min\\) and \\(x^y=\\max(x,1-y)\\); by the [compact no-go theorem](https://sumire.aiformath.jp/p/2609.00007/n/9) every example has noncancellative addition.\n- [Finite-valued presheaf categories](https://sumire.aiformath.jp/p/2609.00007/n/5): identify when categorical counting has the topos structure motivating Burnside riegs.\n\nThe target is a single mathematical account that states the definitions and examples precisely and separates established results from these remaining questions. The proposed title is *Notes on Rieg theory: semiring with exponentials in logic, profinite arithmetic, enumerative combinatorics, and category theory*.", "dates": { "inspiration": null, "sourceRecorded": null, "evidence": null, "registeredInSumire": null, "updatedInSumire": "2026-10-30T14:29:34.425Z", "legacyRecordedAt": "2026-07-14T00:00:00.000Z", "legacyUpdatedAt": "2026-10-30T14:29:34.425Z" }, "language": null, "projectLanguage": "ja", "languageSource": "not_recorded", "legacyStage": "idea", "classification": "unclassified", "classificationStatus": "unclassified", "importance": "unspecified", "section": "goals", "problemState": "unknown", "resolution": "", "verificationEvidence": { "ai": "", "lean": "", "human": "" }, "aiClaimModel": "", "formalization": "unknown", "humanReview": "unreviewed", "verification": "unverified", "ai": { "use": "assisted", "tools": "Codex, Claude", "additions": "" }, "originalQuestion": "", "references": [] } ], "noteDependencies": [], "guides": { "arxiv": "https://info.arxiv.org/help/submit/index.html", "formal_conjectures": "https://github.com/google-deepmind/formal-conjectures/blob/main/CONTRIBUTING.md", "mathlib": "https://leanprover-community.github.io/contribute/index.html" }, "search": { "arxiv": "https://arxiv.org/search/", "formal_conjectures": "https://google-deepmind.github.io/formal-conjectures/browse/", "mathlib": "https://leanprover-community.github.io/mathlib4_docs/" } }