{
  "id": "oai:papers.qnfo.org:exchange-phase-logical-scalar",
  "slug": "exchange-phase-logical-scalar",
  "title": "The Exchange Phase as a Logical Scalar: R = e^(2 pi i s) from the Re-Entrant Calculus",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The boson/fermion dichotomy is conventionally presented as a primitive classification of nature. Recent work established that the underlying invariant is the relation between the exchange phase of identical particles and their topological spin, R = e^{2πis}, with the dichotomy as its shadow in three spatial dimensions where the involutive braiding quantizes the spin parameter. This paper examines whether a calculus whose only primitive is the act of drawing a distinction can generate this invariant rather than importing it as an axiom. The re-entrant mark of the calculus of indications, disciplined by linearity, already generates the constants e and π as logical scalars: e as the fixed point of the differential equation D f = f, and π as the trace of the identity on the circle type. The central observation of this paper is that the exchange phase is the (2s)-fold half-turn of the re-entrant mark: R = (e^{iπ})^{2s} = e^{2πis} = (−1)^{2s}, so that the boson/fermion dichotomy is the parity of 2s. The arithmetic identity is established; the identification of the exchange monodromy with a power of the mark's half-turn is a model of the re-entrant phase; the claim that the calculus derives the invariant as a logical scalar within a single formal system is stated as a conjecture with explicit falsifiability conditions. The framework is distinguished from the nearest prior work deriving a Z₂ exchange phase from self-referential scattering, which lacks both the power structure for arbitrary spin and the unification of e, π, and R as a single scalar family.",
  "abstract_markdown": "The boson/fermion dichotomy is conventionally presented as a primitive classification of nature. Recent work established that the underlying invariant is the relation between the exchange phase of identical particles and their topological spin, R = e^{2πis}, with the dichotomy as its shadow in three spatial dimensions where the involutive braiding quantizes the spin parameter. This paper examines whether a calculus whose only primitive is the act of drawing a distinction can generate this invariant rather than importing it as an axiom. The re-entrant mark of the calculus of indications, disciplined by linearity, already generates the constants e and π as logical scalars: e as the fixed point of the differential equation D f = f, and π as the trace of the identity on the circle type. The central observation of this paper is that the exchange phase is the (2s)-fold half-turn of the re-entrant mark: R = (e^{iπ})^{2s} = e^{2πis} = (−1)^{2s}, so that the boson/fermion dichotomy is the parity of 2s. The arithmetic identity is established; the identification of the exchange monodromy with a power of the mark's half-turn is a model of the re-entrant phase; the claim that the calculus derives the invariant as a logical scalar within a single formal system is stated as a conjecture with explicit falsifiability conditions. The framework is distinguished from the nearest prior work deriving a Z₂ exchange phase from self-referential scattering, which lacks both the power structure for arbitrary spin and the unification of e, π, and R as a single scalar family.",
  "date": "2026-08-14",
  "first_published": "2026-08-14",
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  "language": "en",
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    "exchange phase",
    "spin-statistics",
    "topological spin",
    "laws of form",
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    "logical scalar",
    "anyons"
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        "name": "Rowan Brad Quni-Gudzinas",
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    "titles": [
      {
        "title": "The Exchange Phase as a Logical Scalar: R = e^(2 pi i s) from the Re-Entrant Calculus"
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        "subject": "spin-statistics"
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      {
        "subject": "topological spin"
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      {
        "subject": "laws of form"
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      {
        "subject": "re-entrant mark"
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      {
        "subject": "logical scalar"
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