{
  "id": "oai:papers.qnfo.org:the-zero-point-phase-as-an-archimedean-factor-of-weil-index-reciprocity-a-formal",
  "slug": "the-zero-point-phase-as-an-archimedean-factor-of-weil-index-reciprocity-a-formal",
  "title": "The Zero-Point Phase as an Archimedean Factor of Weil-Index Reciprocity: A Formal Verification and Its Limits",
  "authors": [
    "Rowan Brad Quni-Gudzinas"
  ],
  "abstract": "The vacuum phase of the harmonic oscillator — the Archimedean Maslov (metaplectic) phase e^iπ/4 accompanying the zero-point energy (1)/(2)ℏω — is conventionally treated as a purely real-place, analytic structure. We investigate the conjecture that this phase is instead one local factor γ_∞ in the global Weil-index product ∏_v γ_v = 1 over all places v of Q, so that the zero-point phase is adelically constrained by quadratic reciprocity. We carry out the conjecture's proposed formal test for the quadratic form q(x) = x²: we compute the Archimedean factor γ_∞(x²) = e^iπ/4 from the metaplectic action on chirp states, verify the metaplectic double-cover consistency (e^iπ/4)⁴ = -1, compute the odd-prime factors γ_p(x²) = 1 via explicit normalized Gauss sums (G₃ = i, G₅ = 1, with full arithmetic), and derive γ₂(x²) = e^-iπ/4 from the reciprocity product. The product formula is thereby verified for x². [...]",
  "abstract_markdown": "The vacuum phase of the harmonic oscillator — the Archimedean Maslov (metaplectic) phase $e^{i\\pi/4}$ accompanying the zero-point energy $\\tfrac{1}{2}\\hbar\\omega$ — is conventionally treated as a purely real-place, analytic structure. We investigate the conjecture that this phase is instead one local factor $\\gamma_\\infty$ in the global Weil-index product $\\prod_v \\gamma_v = 1$ over all places $v$ of $\\mathbb{Q}$, so that the zero-point phase is adelically constrained by quadratic reciprocity. We carry out the conjecture's proposed formal test for the quadratic form $q(x) = x^2$: we compute the Archimedean factor $\\gamma_\\infty(x^2) = e^{i\\pi/4}$ from the metaplectic action on chirp states, verify the metaplectic double-cover consistency $(e^{i\\pi/4})^4 = -1$, compute the odd-prime factors $\\gamma_p(x^2) = 1$ via explicit normalized Gauss sums ($G_3 = i$, $G_5 = 1$, with full arithmetic), and derive $\\gamma_2(x^2) = e^{-i\\pi/4}$ from the reciprocity product. The product formula is thereby verified for $x^2$. [...]",
  "date": "2026-10-10",
  "first_published": "2026-10-10",
  "last_revised": "2026-10-10T00:00:00Z",
  "modified": "2026-10-10T00:00:00Z",
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  "peer_review_status": "Not independently peer-reviewed",
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  "doi": null,
  "doi_url": null,
  "url": "https://papers.qnfo.org/papers/the-zero-point-phase-as-an-archimedean-factor-of-weil-index-reciprocity-a-formal",
  "license_url": "https://legal.qnfo.org/",
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  "sha256": "e1ad8ae810cb36498c593c00ced463e819575118331a216c5c7fb682535819ba",
  "version_url": "https://papers.qnfo.org/papers/the-zero-point-phase-as-an-archimedean-factor-of-weil-index-reciprocity-a-formal/v/1.0.0",
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    "creators": [
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        "name": "Rowan Brad Quni-Gudzinas",
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    ],
    "titles": [
      {
        "title": "The Zero-Point Phase as an Archimedean Factor of Weil-Index Reciprocity: A Formal Verification and Its Limits"
      }
    ],
    "publisher": "QNFO",
    "publicationYear": "2026",
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        "date": "2026-10-10",
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        "description": "The vacuum phase of the harmonic oscillator — the Archimedean Maslov (metaplectic) phase e^iπ/4 accompanying the zero-point energy (1)/(2)ℏω — is conventionally treated as a purely real-place, analytic structure. We investigate the conjecture that this phase is instead one local factor γ_∞ in the global Weil-index product ∏_v γ_v = 1 over all places v of Q, so that the zero-point phase is adelically constrained by quadratic reciprocity. We carry out the conjecture's proposed formal test for the quadratic form q(x) = x²: we compute the Archimedean factor γ_∞(x²) = e^iπ/4 from the metaplectic action on chirp states, verify the metaplectic double-cover consistency (e^iπ/4)⁴ = -1, compute the odd-prime factors γ_p(x²) = 1 via explicit normalized Gauss sums (G₃ = i, G₅ = 1, with full arithmetic), and derive γ₂(x²) = e^-iπ/4 from the reciprocity product. The product formula is thereby verified for x². [...]",
        "descriptionType": "Abstract"
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