cases/case27/grading_rubric.json

{
  "case_id": "Case27",
  "title": "Numerical attoclock on atomic and molecular hydrogen",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Bind the finite circular pulse, intensity convention, hydrogen ionization potential and reference physical case.",
      "award_conditions": [
        "Finite circular-pulse support, intensity convention, hydrogen binding and reference-case inputs are consistently bound."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This obligation is independently judgeable; no upstream failure is propagated into it.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-2, Eq. (3) and following pulse/intensity paragraph: Finite circular-pulse support, intensity and ionization potential define the atomic reference case."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "F2",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Construct the coherent stationary-contribution amplitude and in-plane |D|^2 signal under a declared physically equivalent SFA convention.",
      "award_conditions": [
        "The atomic signal is a coherent stationary-contribution amplitude with a declared physically equivalent SFA convention and in-plane modulus-square observable."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F1: Bind the finite circular pulse, intensity convention, hydrogen ionization potential and reference physical case. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-2, Eqs. (4)-(7): Complex stationary contributions enter a coherent atomic amplitude; consistent action, prefactor and physical branch choices define the same SFA object."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Derive or correctly cite the atomic action and complex stationary equation.",
      "award_conditions": [
        "The action and complex stationary equation are correctly derived/cited and connected to the implemented saddle construction."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F1: Bind the finite circular pulse, intensity convention, hydrogen ionization potential and reference physical case. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-2, Eqs. (4)-(7): Complex stationary contributions enter a coherent atomic amplitude; consistent action, prefactor and physical branch choices define the same SFA object."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "F4",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Establish both H2 site-labeled stationary families with opposite-sign center-translation terms.",
      "award_conditions": [
        "Both molecular site families retain opposite-sign center-translation terms rather than duplicating the atomic equation."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This obligation is independently judgeable; no upstream failure is propagated into it.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, two-center LCAO paragraph and Eqs. (8)-(9): Translation phases produce opposite-sign site terms in the molecular stationary condition, not an unspecified molecular signal."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Provide a reconstructible atomic in-plane photoelectron distribution resolving major support, momentum scale, symmetry and Coulomb-free angular placement.",
      "award_conditions": [
        "A reconstructible atomic in-plane distribution has the correct main support, momentum scale, symmetry and Coulomb-free angular placement.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The physical PMD object is judged directly; formula or documentation defects affect it only when central-formula and output inspection establish actual object invalidity.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 1 lower SPM panel and caption; p. 063428-4, SFA/Coulomb comparison: Atomic Coulomb-free support is a finite crescent/semicircle on the vector-potential scale; the upper TDSE panel is not the required SFA output."
      ],
      "rc_relation": "RC0: absence or material invalidity of the central physical object. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Provide reconstructible coverage of both molecular site-family topologies for all three prescribed orientations.",
      "award_conditions": [
        "Both site-labeled molecular root families are supplied for each of the three orientations with judgeable topology.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F4: Establish both H2 site-labeled stationary families with opposite-sign center-translation terms. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion: Both site families split for in-plane orientations and coincide for the perpendicular orientation because E dot R vanishes."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify finite crescent/semicircular main support and characteristic momentum scale in the candidate's atomic distribution.",
      "award_conditions": [
        "The candidate identifies finite crescent/semicircular main support and its vector-potential momentum scale.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O1: Provide a reconstructible atomic in-plane photoelectron distribution resolving major support, momentum scale, symmetry and Coulomb-free angular placement. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 1 lower SPM panel and caption; p. 063428-4, SFA/Coulomb comparison: Atomic Coulomb-free support is a finite crescent/semicircle on the vector-potential scale; the upper TDSE panel is not the required SFA output."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Bind physically justified representative momenta and their complex-root families to the candidate's signal as reconstructible quantitative evidence.",
      "award_conditions": [
        "Representative momenta are physically bound to the candidate signal and have reconstructible complex-root families.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O1: Provide a reconstructible atomic in-plane photoelectron distribution resolving major support, momentum scale, symmetry and Coulomb-free angular placement. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in F3 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The root-to-signal evidence requires the candidate atomic signal; F3 documentation receives independent credit, while root identity is judged directly against the atomic complex stationary equation.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 1 c/f labels, Fig. 2 upper atomic panel and saddle-damping paragraphs: Representative momenta must be bound to the candidate signal; relative action/damping identifies a dominant main-lobe saddle, without requiring a root map at every momentum."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify nonzero opposite-sign separation of the two site families for both in-plane orientations.",
      "award_conditions": [
        "The candidate identifies nonzero opposite-sign site-family splitting for both in-plane orientations.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O2: Provide reconstructible coverage of both molecular site-family topologies for all three prescribed orientations. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion: Both site families split for in-plane orientations and coincide for the perpendicular orientation because E dot R vanishes."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify coincidence of the two stationary families for R parallel e_z, the perpendicular degeneracy.",
      "award_conditions": [
        "The candidate identifies coincidence of the two families for the perpendicular molecular orientation.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O2: Provide reconstructible coverage of both molecular site-family topologies for all three prescribed orientations. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion: Both site families split for in-plane orientations and coincide for the perpendicular orientation because E dot R vanishes."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use roots to distinguish the dominant contribution from more strongly damped alternatives through relative weights.",
      "award_conditions": [
        "Candidate root/action information establishes relative damping or weights distinguishing the dominant contribution from alternatives.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O3: Bind physically justified representative momenta and their complex-root families to the candidate's signal as reconstructible quantitative evidence. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 1 c/f labels, Fig. 2 upper atomic panel and saddle-damping paragraphs: Representative momenta must be bound to the candidate signal; relative action/damping identifies a dominant main-lobe saddle, without requiring a root map at every momentum."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Connect relevant roots to saddle contributions and their coherent sum, declaring physical branch selection and amplitude conventions.",
      "award_conditions": [
        "Physical root selection, saddle amplitudes and coherent summation are connected under one consistent convention.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F2: Construct the coherent stationary-contribution amplitude and in-plane |D|^2 signal under a declared physically equivalent SFA convention.; F3: Derive or correctly cite the atomic action and complex stationary equation.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "All named physical premises are necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-2, Eqs. (4)-(7): Complex stationary contributions enter a coherent atomic amplitude; consistent action, prefactor and physical branch choices define the same SFA object."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Obtain the opposite-sign site term from the center-translated linear-combination-of-atomic-orbitals phase.",
      "award_conditions": [
        "The opposite site term is obtained from the translated two-center LCAO phase with consistent center labels.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F4: Establish both H2 site-labeled stationary families with opposite-sign center-translation terms. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, two-center LCAO paragraph and Eqs. (8)-(9): Translation phases produce opposite-sign site terms in the molecular stationary condition, not an unspecified molecular signal."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Correctly bind the site equation to the three target orientations e_x, e_y and e_z.",
      "award_conditions": [
        "The site equation is correctly specialized to the two in-plane and one perpendicular target orientations.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F4: Establish both H2 site-labeled stationary families with opposite-sign center-translation terms. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Only the named physical premise is necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion: Both site families split for in-plane orientations and coincide for the perpendicular orientation because E dot R vanishes."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M5",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use the candidate's atomic distribution and root/action-damping evidence at a representative main-peak momentum to explain how the least-damped contribution dominates main support; no momentum-by-momentum root map of the whole support is required.",
      "award_conditions": [
        "The main-lobe explanation connects the candidate distribution to least-damped action/root evidence at a representative peak momentum.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [I1: Identify finite crescent/semicircular main support and characteristic momentum scale in the candidate's atomic distribution.; M1: Use roots to distinguish the dominant contribution from more strongly damped alternatives through relative weights.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "All named physical premises are necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 1 c/f labels, Fig. 2 upper atomic panel and saddle-damping paragraphs: Representative momenta must be bound to the candidate signal; relative action/damping identifies a dominant main-lobe saddle, without requiring a root map at every momentum."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "For weaker flank/fringe features actually claimed, explain their origin through coherent multipath contributions and relative damping; no unclaimed fringe structure is imposed.",
      "award_conditions": [
        "Any weaker flank/fringe features actually claimed are explained through coherent multiple contributions and relative damping, without imposing unclaimed fringes.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [O3: Bind physically justified representative momenta and their complex-root families to the candidate's signal as reconstructible quantitative evidence.; M1: Use roots to distinguish the dominant contribution from more strongly damped alternatives through relative weights.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "All named physical premises are necessary for this downstream claim.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Atomic PMD and representative-root binding are separate evidence facts. A correct main-lobe dominance argument need not map every momentum; explain weaker features only when the candidate claims them."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article p. 063428-3, Fig. 2 upper atomic panel/insets and fringe-damping discussion: Weaker features actually claimed require a coherent multipath/damping explanation; no hidden unclaimed fringe family is imposed."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use the same E(t_s) dot R geometry to explain opposite-sign in-plane splitting and disappearance of the perpendicular term/degeneracy.",
      "award_conditions": [
        "The same E dot R geometry explains in-plane splitting and its disappearance for the perpendicular case.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "M3: Obtain the opposite-sign site term from the center-translated linear-combination-of-atomic-orbitals phase. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I2, I3 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The common geometric explanation requires the centre-translated opposite-sign term; I2 and I3 retain independent signal-identification credit and are not hard prerequisites for judging a candidate-presented geometric explanation.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "A candidate-presented common geometric explanation is judged from its center-phase premise; I2/I3 signal identifications remain separately creditable and are not mechanical prerequisites for an otherwise valid geometry derivation.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion: Both site families split for in-plane orientations and coincide for the perpendicular orientation because E dot R vanishes."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Limit atomic conclusions to the Coulomb-free SFA signal and saddle organization.",
      "award_conditions": [
        "Atomic conclusions are positively limited to the Coulomb-free SFA signal and its saddle organization."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This obligation is independently judgeable; no upstream failure is propagated into it.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-2-4, Eqs. (4)-(9), SFA/TDSE comparison and molecular-stationarity discussion; stated task output boundary: The assigned molecular conclusion ends at root topology. Molecular PMD, widths, Coulomb offsets and tunneling times do not follow from the commissioned saddle-only evidence. Risk checks remain conditional."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Limit molecular conclusions to root topology, not uncomputed photoelectron distributions, widths, offsets or ordering.",
      "award_conditions": [
        "Molecular conclusions stop at stationary-family topology, not uncomputed widths, offsets or molecular distributions."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This obligation is independently judgeable; no upstream failure is propagated into it.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice.",
        "Molecular root topology needs both site families and the stated orientations. Correct perpendicular degeneracy remains independently creditable from an in-plane failure; no molecular PMD or width is a dependency or substitute."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-2-4, Eqs. (4)-(9), SFA/TDSE comparison and molecular-stationarity discussion; stated task output boundary: The assigned molecular conclusion ends at root topology. Molecular PMD, widths, Coulomb offsets and tunneling times do not follow from the commissioned saddle-only evidence. Risk checks remain conditional."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V3",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "If candidate results or atomic/molecular scope claims trigger a material branch, identity or overextension risk, discriminate it or honestly restrict scope.",
      "award_conditions": [
        "A material complex-branch, root-binding or extrapolation risk from actual work is discriminated or restricted; otherwise use N/A."
      ],
      "applicability_and_na": [
        "Activate only for a concrete material risk arising from the candidate method, data, anomaly or claim; no fixed test is required in the absence of a trigger.",
        "Without a triggered risk record N/A and award 0.5 without positive validation evidence; this neither blocks RC4 nor triggers RC3.",
        "When triggered, accept functionally equivalent discriminating evidence or a justified restriction of the threatened claim."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This obligation is independently judgeable; no upstream failure is propagated into it.",
        "Credit for branches not using the failed premise is preserved; the same defect is not charged twice."
      ],
      "equivalent_evidence": [
        "Accept declared rotations/reflections, units and common normalizations preserving physical identity; complex-time, invertible vector-potential coordinates, quantitative tables or equivalent root representations can replace source marker plots.",
        "Accept self-chosen physically justified representative momenta and physically equivalent SFA conventions with declared coherent amplitude and branch mapping; no hidden center/flank direction or exact marker size is prescribed.",
        "Source panel layouts, root solver, grid, fitting window, color, crop and pointwise tolerances are not duties. Additional exploratory molecular distributions/widths cannot substitute for the required topology or establish assigned-observable claims."
      ],
      "source_support": [
        "Serov et al. (2019), article pp. 063428-2-4, Eqs. (4)-(9), SFA/TDSE comparison and molecular-stationarity discussion; stated task output boundary: The assigned molecular conclusion ends at root topology. Molecular PMD, widths, Coulomb offsets and tunneling times do not follow from the commissioned saddle-only evidence. Risk checks remain conditional."
      ],
      "rc_relation": "RC3: an unresolved applicable material risk, only after signals and principal mechanisms are substantially supported and all six risk-proof requirements are met. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    }
  ],
  "case_scientific_rules": [
    "The assigned molecular calculation ends at site-resolved complex saddle topology for three orientations. Molecular distributions, angular widths, offsets, TDSE/Coulomb offsets and tunneling times are not required results and cannot replace the topology.",
    "Atomic action is S_k(t) = integral^t {[k + A(t')]^2/2 + Ip} dt', with stationary equation [k + A(t_s)]^2/2 + Ip = 0. A canonical hydrogenic amplitude is D(k) proportional to sum_s [S_k_double_prime(t_s)]^(-1) exp[i S_k(t_s)], under declared consistent amplitude and branch conventions; the in-plane probability is proportional to |D(kx,ky,kz=0)|^2.",
    "At the reference case the atomic signal has finite crescent/semicircular main support with momentum scale set by the pulse vector potential and no substantial SFA offset from that direction. One physical saddle overwhelmingly dominates the principal lobe; other roots may be strongly suppressed.",
    "Representative momenta must be physically associated with the candidate's signal. Relative complex action/damping supports dominance. Larger imaginary time can characterize damped alternatives, but quoted root equations alone do not bind roots to the coherent signal. No full-support root map or exact source selector angle is required.",
    "Two-center translation phases give [k + A(t_sj)]^2/2 + Ip - (-1)^j E(t_sj) dot R/2 = 0. For either in-plane orientation the center term is nonzero with opposite signs; for R parallel e_z both in-plane field and detected momentum are orthogonal to R, the site term vanishes and the families coincide.",
    "Stationary topology alone does not determine a molecular distribution, fitted width or offset. A molecular amplitude/prefactor is not uniquely fixed by the assigned source equations; exploratory molecular calculations must remain clearly labeled and cannot become a grading reference.",
    "Concrete risks include an unphysical complex branch, omission of a materially contributing family, roots unrelated to submitted momenta, numerical branch jumps mistaken for molecular splitting or uncomputed molecular observable claims. Root residuals and generic convergence statements have no detachable score.",
    "The score is the sum of twenty material-core criteria, each worth either 0 or 0.5, for a maximum of 10.0. Research Closure is descriptive and never caps, floors, rescales or repairs the score.",
    "Judge physical object identity and correctness before display similarity. For each output record object_valid, presentation_adequate and claim_adequate separately; all three must hold for its atomic award. A presentation or claim defect alone is not evidence that an existing physical object is absent or invalid.",
    "Candidate-created data, candidate presentation, candidate identification, physical inference and validation are distinct evidential acts. Examiner inspection or reconstruction may verify an object but cannot create missing candidate identification, reasoning or risk resolution.",
    "A dependency propagates only the failed necessary physical fact, not the whole upstream atomic verdict or unrelated missing detail. Name the failed premise, the downstream obligation needing it, and why independent candidate-presented evidence does not establish it. Preserve unaffected branches, independent work and honest scope.",
    "Numerical reliability has no separate generic penalty. Charge a defect where it materially damages a physical object or evidence, and a dependent conclusion only when that same defect makes the conclusion false or unsupported. Missing a preferred convergence study, control, solver or named self-check is not itself a failure.",
    "Risk-triggered validation applies only to a concrete material risk arising from the candidate method, data, anomaly or claim. With no such risk record N/A and award the fixed atomic credit without treating it as positive validation evidence; N/A does not block RC4. Unconditional scope obligations cannot be N/A.",
    "Before assigning RC3 establish all six: the concrete failure mode; its origin in candidate work; the fair task or standard-knowledge basis; the threatened central conclusion; why submitted evidence cannot expose or exclude it; and examples of functionally equivalent discriminating evidence or an honest scope restriction that would resolve it. A generic possibility of further testing is insufficient.",
    "Use bounded examiner-side source/reference readback only to resolve a material scientific uncertainty, not to demand pixels, an undeclared tolerance or additional candidate deliverables. A provisional score of at least 9.0 or provisional RC4 triggers one bounded central-output-anchor check and one principal-mechanism-link check. Reference conflict or insufficiency remains an explicit unresolved decision, not an invented target.",
    "Evaluate closure-role trigger facts independently of atomic awards and select the most severe failed required role across all indispensable and co-primary claims. Supporting claims do not independently set overall closure. A point loss alone is not a closure-gate fact.",
    "A valid candidate-caused incomplete or timeout attempt with no scientifically assessable required submission has score 0.0 and RC0 before N/A bookkeeping. Assessable incomplete work is graded from the submitted evidence; unsubmitted scratch or logs do not become scientific answers.",
    "The atomic action is $S_{\\mathbf k}(t)=\\int^t\\{[\\mathbf k+\\mathbf A(t^{\\prime})]^2/2+I_p\\}dt^{\\prime}$, with $[\\mathbf k+\\mathbf A(t_s)]^2/2+I_p=0$ (Eq. (6)). A hydrogenic convention gives $D(\\mathbf k)\\propto\\sum_s[S_{\\mathbf k}^{\\prime\\prime}(t_s)]^{-1}e^{iS_{\\mathbf k}(t_s)}$ (Eq. (7)), and $P(k_x,k_y,0)\\propto|D|^2$. The action damping and prefactors, not root count alone, establish relative importance.",
    "For two translated centers, $[\\mathbf k+\\mathbf A(t_{sj})]^2/2+I_p-(-1)^j\\mathbf E(t_{sj})\\cdot\\mathbf R/2=0$ (Eq. (9)). Opposite site labels interchange signs; for $\\mathbf R\\parallel\\mathbf e_z$ with in-plane field the site term vanishes. This establishes stationary-family topology, not an uncomputed molecular PMD or attoclock width."
  ],
  "closure_requirements": {
    "RC0": [
      "Required claim CL1, central object: O1: Provide a reconstructible atomic in-plane photoelectron distribution resolving major support, momentum scale, symmetry and Coulomb-free angular placement.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Apply for a missing or materially invalid required central object; a presentation-only or claim-only defect does not establish object invalidity.",
      "No scientifically assessable required submission gives the terminal 0.0/RC0 disposition."
    ],
    "RC1": [
      "Required claim CL1, indispensable component: O3: Bind physically justified representative momenta and their complex-root families to the candidate's signal as reconstructible quantitative evidence.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Required claim CL2, indispensable component: All of [F4: Establish both H2 site-labeled stationary families with opposite-sign center-translation terms.; O2: Provide reconstructible coverage of both molecular site-family topologies for all three prescribed orientations.]. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "The central object is established but a required physical component is absent or materially invalid. Preserve independent valid branches; do not invent a hidden canonical plot, sampling grid or implementation duty."
    ],
    "RC2": [
      "Required claim CL1, principal mechanism: M5: Use the candidate's atomic distribution and root/action-damping evidence at a representative main-peak momentum to explain how the least-damped contribution dominates main support; no momentum-by-momentum root map of the whole support is required.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Required claim CL2, principal mechanism: M7: Use the same E(t_s) dot R geometry to explain opposite-sign in-plane splitting and disappearance of the perpendicular term/degeneracy.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Required signal objects/components exist, but an indispensable principal mechanism or co-primary scientific conclusion is missing, wrong or unsupported by candidate-presented and identified evidence. Correct data do not let an examiner supply missing reasoning."
    ],
    "RC3": [
      "Required claim CL4, material validation: V3: If candidate results or atomic/molecular scope claims trigger a material branch, identity or overextension risk, discriminate it or honestly restrict scope.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "All indispensable signal and principal-mechanism roles are substantially supported, but a concrete triggered material risk remains unresolved and inadequately restricted.",
      "Establish the failure mode, candidate origin, fair task/knowledge basis, threatened central conclusion, insufficiency of submitted evidence, and equivalent resolution or scope restriction. Missing an examiner-preferred check alone never suffices."
    ],
    "RC4": [
      "Every nonempty required role in every indispensable and co-primary claim above is scientifically established from candidate evidence, with the correct case/model/observable identities and evidence-bounded conclusion.",
      "All applicable material risks are discriminated or honestly restricted; no unresolved material ambiguity remains. Risk N/A does not block closure or supply positive validation evidence. Honest limitations cannot repair a false signal or mechanism.",
      "Select the most severe failed required role if any; otherwise RC4. No score threshold, score ceiling or required display/solver checklist determines this classification."
    ]
  },
  "references": [
    {
      "citation": "Vladislav V. Serov, Alexander W. Bray, and Anatoli S. Kheifets. 2019. Numerical attoclock on atomic and molecular hydrogen. Physical Review A, 99, 063428.",
      "url": "https://doi.org/10.1103/PhysRevA.99.063428",
      "scientific_scope": "Atomic Coulomb-free SFA distribution and signal-to-saddle relation, plus two-site molecular root topology for three orientations. Molecular widths/distributions and Coulomb/TDSE/timing claims lie outside the assigned observable scope.",
      "anchors": [
        "article p. 063428-2, Eq. (3) and following pulse/intensity paragraph",
        "article p. 063428-2, Eqs. (4)-(7)",
        "article pp. 063428-3-4, two-center LCAO paragraph and Eqs. (8)-(9)",
        "article p. 063428-3, Fig. 1 lower SPM panel and caption; p. 063428-4, SFA/Coulomb comparison",
        "article pp. 063428-3-4, Fig. 2 lower molecular panel, Eq. (9) and following orientation discussion",
        "article p. 063428-3, Fig. 1 c/f labels, Fig. 2 upper atomic panel and saddle-damping paragraphs",
        "article p. 063428-3, Fig. 2 upper atomic panel/insets and fringe-damping discussion",
        "article pp. 063428-2-4, Eqs. (4)-(9), SFA/TDSE comparison and molecular-stationarity discussion; stated task output boundary"
      ]
    }
  ]
}