{
  "abstract": "This paper fixes a single equation as the emblem of the series: z⁴ = 1, with the legend \"z is a readout (eigenvalue) of the triad.\" At the level of words in the three letters, the multiplicative closure is an eight-element group; the six orderings of the triple product take the two values ±1, coupled to the single orientation sign ε shared by the three mirror relations ΔΨ = εΛ, ΛΨ = εΔ, ΛΔ = εΨ; reversal of an ordering reverses its sign; even words are absolute — the square of any ordering is +1, the product of an ordering with its reversal is −1, and the three palindromes equal −1. A two-line parity principle organizes all of this: the value of a scalar word is a convention-free number times ε to the common parity of the letter counts. At the spectral level, the mirrors Λ and Ψ read ±1, the twist Δ reads ±i, the total spectrum of the represented group is exactly the fourth roots of unity μ₄ ≅ Z/4, and the ladder z ∈ {±1, ±i} ⟺ z² = ±1 ⟺ z⁴ = 1 collapses the sign at the fourth power; z⁴ = −1 has no solution in the spectrum. Every displayed identity holds for all letter-sign conventions and for every ratio λ; the external scale remains the sole blank, protected by axiom T2. All theorem-level claims are machine-verified (47/47). No new dynamics are introduced.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Makoto Saito",
      "orcid": "0009-0004-7497-2504"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bh646663ycdxr.bib",
  "cited_by": 7,
  "doi": "10.5281/zenodo.21230220",
  "fulltext_url": null,
  "id": "bh646663ycdxr",
  "issns": [],
  "landing_url": "https://doi.org/10.5281/zenodo.21230220",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2608|Mathematics and Applications"
    ]
  ],
  "openalex": "W7167587616",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-07-07",
  "publisher": "European Organization for Nuclear Research",
  "ris": "https://research.draconitic.com/paper/bh646663ycdxr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:17Z",
      "native_id": "W7167587616",
      "source": "openalex",
      "url": "https://doi.org/10.5281/zenodo.21230220"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Mathematics and Applications",
      "subfield": "geometry-topology"
    }
  ],
  "title": "The Emblem of the Triad: z⁴ = 1 — Sign Laws and Spectrum",
  "title_text": "The Emblem of the Triad: z⁴ = 1 — Sign Laws and Spectrum",
  "type": "preprint",
  "updated_at": "2026-11-06T13:20:22Z",
  "url": "https://research.draconitic.com/paper/bh646663ycdxr",
  "venue": "Zenodo (CERN European Organization for Nuclear Research)",
  "versions": []
}