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  "slug": "flat-schmidt-chains",
  "title": "Same local spectra, different quantum gaps",
  "shortTitle": "Flat Schmidt-rank-two chains",
  "url": "https://evidencepress.org/releases/flat-schmidt-chains/",
  "oneLine": "A geometric classification separates every gapped and gapless chain in a precise entanglement class—even when familiar short-chain measurements agree.",
  "abstract": "For homogeneous open nearest-neighbour chains with a normalised rank-one local projector whose forbidden vector has exactly two equal nonzero Schmidt probabilities, a written proof candidate classifies gappedness in every finite local dimension. Gaplessness is equivalent to a balanced-marker form and has exact finite-length gap 1−cos(π/N). All other interactions in the class have explicit uniform lower bounds. A complete qutrit short-spectrum fibre has fixed Schmidt probabilities, two- and three-site spectra and ground-space dimension at every length, yet only its endpoint is gapless. The proof distinguishes compatible propagation from an obstruction to four-site saturation.",
  "datePublished": "2026-09-28",
  "dateModified": "2026-09-28",
  "version": "1.0.1-candidate",
  "doi": "10.5281/zenodo.23024569",
  "doiUrl": "https://doi.org/10.5281/zenodo.23024569",
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  "repoUrl": "https://github.com/ipitchford/flat-schmidt-chains",
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    "Anonymous"
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  "problem": {
    "name": "Which flat Schmidt-rank-two forbidden states give gapped open chains?",
    "url": "https://arxiv.org/abs/1503.04035"
  },
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  "metadataProvenance": "Supplied v1.0 manuscript and review; v1.0.1 repairs the symbolic classifier contract and clarifies antecedents and the full range-Gram decomposition. Main theorem statements unchanged.",
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  "keywords": [
    "spectral gap",
    "quantum spin chains",
    "Schmidt rank",
    "frustration free",
    "qutrit",
    "isospectral fibre"
  ],
  "keyResults": [
    "Complete gapped/gapless classification for the stated flat Schmidt-rank-two forbidden-vector class, in every finite local dimension.",
    "Every gapless member has exact finite-chain gap 1−cos(π/N); all other members have explicit uniform positive lower bounds.",
    "The specified qutrit short-spectrum fibre has a complete one-parameter normal form and gap lower bound cos²(θ)/6 away from its endpoint.",
    "Ground-space dimension F_(2N+2) stays fixed throughout that fibre, including the gapless endpoint."
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  "evidencePackage": "Written analytic proofs, seven exact finite positivity certificates, finite full-Hamiltonian diagnostics, a parameter-free exact classifier and semantic mutation controls.",
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  "relatedWorks": [
    {
      "citation": "Bravyi and Gosset (2015), Gapped and gapless phases of frustration-free spin-1/2 chains. Theorem 1 supplies the complete qubit predecessor; it does not classify distinct intersecting qutrit supports.",
      "url": "https://arxiv.org/abs/1503.04035",
      "doi": "10.48550/arXiv.1503.04035"
    },
    {
      "citation": "Movassagh et al. (2010), Unfrustrated qudit chains and their ground states. Section II, equation (11), is the antecedent for the ground-space recurrence; the candidate proves persistence along its entire specified fibre.",
      "url": "https://arxiv.org/abs/1001.1006",
      "doi": "10.48550/arXiv.1001.1006"
    },
    {
      "citation": "Bravyi et al. (2012), Criticality without frustration for quantum spin-1 chains. Step 1 uses the known unbiased hopping gap and separates hopping from pair creation.",
      "url": "https://arxiv.org/abs/1203.5801",
      "doi": "10.48550/arXiv.1203.5801"
    },
    {
      "citation": "Lemm (2019), Gaplessness is not generic for translation-invariant spin chains. Established adjacent-overlap and finite-size methods precede the saturation analysis here.",
      "url": "https://arxiv.org/abs/1903.00108",
      "doi": "10.48550/arXiv.1903.00108"
    },
    {
      "citation": "Rai et al. (2026), A hierarchy of spectral gap certificates for frustration-free spin systems. A broader certificate framework, not a classification of this specified support geometry.",
      "url": "https://doi.org/10.22331/q-2026-04-13-2065",
      "doi": "10.22331/q-2026-04-13-2065"
    }
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