{
  "$schema": "https://qft.org/data/schemas/qft-many-body-comparison-1.0.0.schema.json",
  "schema_version": "1.0.0",
  "artifact_id": "qft.artifact.many-body-quantum-matter.resonant-interactions-universal-quantum-gases-claim-validity-ledger",
  "table_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger",
  "canonical_source": {
    "route": "/many-body-quantum-matter/resonant-interactions-universal-quantum-gases/",
    "heading": "Comparison of claims and validity conditions",
    "anchor": "comparison-of-claims-and-validity-conditions"
  },
  "caption": "Inputs, checks, and limits of universal quantum-gas claims.",
  "common_scope": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "qualifications": {
    "markdown": "",
    "math": [],
    "links": []
  },
  "columns": [
    {
      "key": "claim",
      "label": "Claim"
    },
    {
      "key": "required_physical_input",
      "label": "Required physical input"
    },
    {
      "key": "declared_control_regime",
      "label": "Declared control regime"
    },
    {
      "key": "independent_check",
      "label": "Independent check"
    },
    {
      "key": "ceiling_on_the_conclusion",
      "label": "Ceiling on the conclusion"
    }
  ],
  "rows": [
    {
      "key": "a-contact-theory-predicts-two-body-scattering",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-contact-theory-predicts-two-body-scattering",
      "cells": {
        "claim": {
          "markdown": "A contact theory predicts two-body scattering",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "$a$ and a fixed regulator prescription",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "a"
            }
          ],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "$kR\\ll1$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "kR\\ll1"
            }
          ],
          "links": []
        },
        "independent_check": {
          "markdown": "cutoff independence after matching and pole position",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Leading zero-range amplitude only",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-shallow-pole-is-universal-beyond-leading-order",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-shallow-pole-is-universal-beyond-leading-order",
      "cells": {
        "claim": {
          "markdown": "A shallow pole is universal beyond leading order",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "$a$, $r_e$, and retained shape terms",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "a"
            },
            {
              "kind": "inlineMath",
              "tex": "r_e"
            }
          ],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "omitted terms small at $k\\sim\\kappa$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "k\\sim\\kappa"
            }
          ],
          "links": []
        },
        "independent_check": {
          "markdown": "pole and residue stable under next-order variation",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Stated effective-range order",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-resonance-is-broad-for-the-gas",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-resonance-is-broad-for-the-gas",
      "cells": {
        "claim": {
          "markdown": "A resonance is broad for the gas",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "width parameter such as $R^*$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "R^*"
            }
          ],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "$k_FR^*\\ll1$ and relevant collision energies small",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "k_FR^*\\ll1"
            }
          ],
          "links": []
        },
        "independent_check": {
          "markdown": "compare one- and two-channel predictions",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Broad-resonance observables in tested window",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-bosonic-observable-is-universal",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-bosonic-observable-is-universal",
      "cells": {
        "claim": {
          "markdown": "A bosonic observable is universal",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "$a$, $\\kappa_*$, and inelasticity information",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "a"
            },
            {
              "kind": "inlineMath",
              "tex": "\\kappa_*"
            }
          ],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "range separated and observation precedes loss",
          "math": [],
          "links": []
        },
        "independent_check": {
          "markdown": "three-body-parameter and lifetime variation",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Metastable, protocol-specific result",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-k-4-tail-measures-contact",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-k-4-tail-measures-contact",
      "cells": {
        "claim": {
          "markdown": "A $k^{-4}$ tail measures contact",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "k^{-4}"
            }
          ],
          "links": []
        },
        "required_physical_input": {
          "markdown": "contact convention and probe response",
          "math": [],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "$k_{\\rm many}\\ll k\\ll R^{-1}$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "k_{\\rm many}\\ll k\\ll R^{-1}"
            }
          ],
          "links": []
        },
        "independent_check": {
          "markdown": "adiabatic or pressure relation using same contact",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Leading short-distance coefficient",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-confined-gas-is-effectively-one-dimensional",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-confined-gas-is-effectively-one-dimensional",
      "cells": {
        "claim": {
          "markdown": "A confined gas is effectively one dimensional",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "3D scattering data and transverse spectrum",
          "math": [],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "energy and temperature below transverse gap",
          "math": [],
          "links": []
        },
        "independent_check": {
          "markdown": "include the first omitted transverse mode",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Low-energy effective coupling",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "a-virial-result-is-controlled",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--a-virial-result-is-controlled",
      "cells": {
        "claim": {
          "markdown": "A virial result is controlled",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "cluster data through order $z^n$",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "z^n"
            }
          ],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "$z\\ll1$ with decreasing terms",
          "math": [
            {
              "kind": "inlineMath",
              "tex": "z\\ll1"
            }
          ],
          "links": []
        },
        "independent_check": {
          "markdown": "next coefficient or fugacity-window stability",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Thermodynamics through the retained order",
          "math": [],
          "links": []
        }
      }
    },
    {
      "key": "few-body-data-suffice-for-an-equation-of-state",
      "row_id": "resonant-interactions-universal-quantum-gases-claim-validity-ledger--few-body-data-suffice-for-an-equation-of-state",
      "cells": {
        "claim": {
          "markdown": "Few-body data suffice for an equation of state",
          "math": [],
          "links": []
        },
        "required_physical_input": {
          "markdown": "all symmetry-allowed inputs at target order",
          "math": [],
          "links": []
        },
        "declared_control_regime": {
          "markdown": "every range, thermal, loss, and medium ratio bounded",
          "math": [],
          "links": []
        },
        "independent_check": {
          "markdown": "regulator/order/parameter covariance and benchmark",
          "math": [],
          "links": []
        },
        "ceiling_on_the_conclusion": {
          "markdown": "Observable and regime explicitly tested",
          "math": [],
          "links": []
        }
      }
    }
  ],
  "references": [
    {
      "id": "ref-40bb50aa6344ead2",
      "markdown": "Braaten, Eric, and Hans-Werner Hammer. “Universality in Few-Body Systems with Large Scattering Length.” *Physics Reports* **428** (2006): 259–390. [DOI](https://doi.org/10.1016/j.physrep.2006.03.001).",
      "urls": [
        "https://doi.org/10.1016/j.physrep.2006.03.001"
      ]
    }
  ],
  "reference_scope": "The reference catalog reproduces the canonical chapter bibliography. Cell links are associated with references only when their resolved URL matches; uncited chapter references are not asserted as claim-specific support."
}