{
  "abstract": "FINDING: The halting problem is undecidable — no algorithm can determine whether an arbitrary program halts — proven via diagonalization, with recent scholarship refining Turing's original 1936 attribution. | MATH: Core result: \\( H(p,i) \\) (halts?) is not computable. Proof by contradiction: assume \\( H \\) exists, define \\( D(p) = \\text{loop if } H(p,p) \\text{ halts, else halt} \\); then \\( D(D) \\) yields contradiction. No equation constants — this is a logical impossibility, not a numerical invariant. | CONNECTION: Diagonalization mirrors Cantor's uncountability of reals — a self-referential \"off-diagonal\" construction. Structurally, this is analogous to the incompleteness of root systems in infinite-dimensional lattices: no finite set of rules captures all sequences. The 0.618/1.618 ratios do not appear; the relevant symmetry is *self-reference* — a fixed-point pathology, not a geometric harmony. | DEPTH: 8 — Profound for epistemology: it sets a hard boundary on what algorithmic knowl\n\nAuthor: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com",
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  "affiliations": [
    "Independent Researcher, United Kingdom"
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  "article_type": null,
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  "author_total": 1,
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    {
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      "name": "Andrew Stewart Caldin"
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  "bibtex": "https://research.draconitic.com/paper/4kfss73m8hdj6.bib",
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  "doi": "10.5281/zenodo.23635446",
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  "published": "2026-10-25",
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  "ris": "https://research.draconitic.com/paper/4kfss73m8hdj6.ris",
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  "title": "The Undecidability of the Halting Problem: Turing's Proof and Modern Refinements — E8 Intelligence Research",
  "title_text": "The Undecidability of the Halting Problem: Turing's Proof and Modern Refinements — E8 Intelligence Research",
  "type": "preprint",
  "updated_at": "2026-11-08T12:21:54Z",
  "url": "https://research.draconitic.com/paper/4kfss73m8hdj6",
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