{
  "abstract": "Under the stated regularity assumptions, the full orthogonality functionals of the five finite para-families considered here are classical for the respective affine, quadratic, $q$-quadratic, $q$-linear and alternating calculi. For each of the para-Krawtchouk, para-Racah, $q$-para-Racah and $q$-para-Krawtchouk families, two finite solutions of one global structural equation have $N+1$ nodes altogether and common moments through degree $N-1$. Their relative mass changes only the central recurrence coefficients. We prove regularity for every normalised combination with both component masses non-zero and recover the full published functionals and recurrences. The alternating construction identifies para-Bannai-Ito within the same structural framework. For the four non-alternating families, equal positive combinations give persymmetric Jacobi matrices; the known spectral gap criterion then decides perfect state transfer. Explicit parameter correspondences locate the classical components and resonant limits of the full systems. A positive seven-point example shows how further named families can be manufactured without a new governing principle. Counterexamples to mathematical assertions in three of four recent preprints expose the consequences of insufficient hypotheses for finite representations and physical models. In an era of AI-assisted symbolic production, we call for a standard that rewards structural explanation and verification rather than the proliferation of examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16704",
  "author_total": 2,
  "authors": [
    {
      "name": "K. Castillo"
    },
    {
      "name": "G. Gordillo-Núñez"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/09hkq61d750tc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16704",
  "fulltext_url": null,
  "id": "09hkq61d750tc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16704",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Mathematical functions and polynomials"
    ]
  ],
  "openalex": "W7221781134",
  "pdf_url": "https://arxiv.org/pdf/2610.16704",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/09hkq61d750tc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16704",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16704"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The manufacture of examples II: the \"para-classical\" orthogonal polynomials",
  "title_text": "The manufacture of examples II: the \"para-classical\" orthogonal polynomials",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/09hkq61d750tc",
  "venue": "arXiv",
  "versions": []
}