{
  "id": "oai:papers.qnfo.org:idempotent-core",
  "slug": "idempotent-core",
  "title": "The Idempotent Core: Quantity as Broken Distinction and the Hidden Assumptions of Arithmetic and Algebra",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "Classical arithmetic treats the statement 1+1=2 as a primitive truth and quantity as a fundamental concept. This paper examines the hidden premise of that statement: the two instances of the numeral $1$ must denote distinct marks. In the calculus of indications, the law of calling — idempotence of the mark, $AA = A$ — states that repeating the same mark yields the same mark. If the two occurrences of $1$ were literally the same mark, addition would collapse to 1+1=1. Quantity therefore presupposes the creation of new distinctions, and the natural numbers are generated not by repetition but by successive acts of distinction. The paper presents this critique as its defensible claim, and then outlines a reconstruction program in which the exponential modality of linear logic provides the controlled duplication that breaks idempotence, yielding the integers as the group completion of the natural numbers object in the coKleisli category of that modality. The reconstruction is labeled conjecture: its central claims are unproven, and the evidential-weight boundary between the critique and the program is stated explicitly. Claims marked [my conjecture] carry falsifiability conditions and are not presented as established results.",
  "abstract_markdown": "Classical arithmetic treats the statement $1+1=2$ as a primitive truth and quantity as a fundamental concept. This paper examines the hidden premise of that statement: the two instances of the numeral $1$ must denote *distinct* marks. In the calculus of indications, the law of calling — idempotence of the mark, $AA = A$ — states that repeating the same mark yields the same mark. If the two occurrences of $1$ were literally the same mark, addition would collapse to $1+1=1$. Quantity therefore presupposes the creation of new distinctions, and the natural numbers are generated not by repetition but by successive acts of distinction. The paper presents this critique as its defensible claim, and then outlines a reconstruction program in which the exponential modality of linear logic provides the controlled duplication that breaks idempotence, yielding the integers as the group completion of the natural numbers object in the coKleisli category of that modality. The reconstruction is labeled conjecture: its central claims are unproven, and the evidential-weight boundary between the critique and the program is stated explicitly. Claims marked `[my conjecture]` carry falsifiability conditions and are not presented as established results.",
  "date": "2026-08-13",
  "first_published": "2026-08-13",
  "last_revised": "2026-08-13T00:00:00Z",
  "modified": "2026-08-13T00:00:00Z",
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  "url": "https://papers.qnfo.org/papers/idempotent-core",
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  "version_url": "https://papers.qnfo.org/papers/idempotent-core/v/1.3.0",
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "The Idempotent Core: Quantity as Broken Distinction and the Hidden Assumptions of Arithmetic and Algebra"
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    "publicationYear": "2026",
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        "subject": "idempotence"
      },
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        "subject": "laws of form"
      },
      {
        "subject": "calculus of indications"
      },
      {
        "subject": "foundations of arithmetic"
      },
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        "subject": "linear logic"
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