{
  "case_id": "Case09",
  "title": "Case09 — High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Construct a coordinate-space B-spline TDSE for the specified one-dimensional periodic potential and both drives with consistent executable and auditable field coupling.",
      "award_conditions": [
        "The primary coordinate-space B-spline TDSE preserves the specified periodic potential and both physical drives with auditable field coupling."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-1-2, Eqs.(1)-(7), Fig.1(a-d), paragraph choosing k=0 in band 2: B-spline generalized metric and physical initial-state identity, not finite-box ordinal."
      ],
      "rc_relation": "Role: central physical object. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "F2",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.",
      "award_conditions": [
        "The initial state is physically band 2 at k=0 and is demonstrably preserved in every representation or auxiliary calculation."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-1-2, Eqs.(1)-(7), Fig.1(a-d), paragraph choosing k=0 in band 2: B-spline generalized metric and physical initial-state identity, not finite-box ordinal."
      ],
      "rc_relation": "Role: indispensable material component. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.",
      "award_conditions": [
        "A consistent radiation/current observable and its gauge, field amplitude, frequency, carrier time/phase and units define the HHG comparison."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article p.033852-2, Eq.(8), and p.033852-3 Fourier/window paragraph: laser-induced current and spectrum. Gauge/derivative equivalence requires actual observable mapping, not appearance alone."
      ],
      "rc_relation": "Role: central physical object. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "F4",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.",
      "award_conditions": [
        "The actual theoretical/control framework has reproducible physical observables and can test interpretation/reliability; a relabelled duplicate or static-only driven check is insufficient."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present correct, reconstructible, claim-sufficient Case-A harmonic evidence spanning the full material response, including the first plateau and weaker second plateau beyond the first cutoff.",
      "award_conditions": [
        "Case-A spectrum resolves the full material response, including the first plateau and much weaker second plateau beyond its first cutoff, with correct state/object identity.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: F1 (Construct a coordinate-space B-spline TDSE for the specified one-dimensional periodic potential and both drives with consistent executable and auditable field coupling.); F2 (Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.); F3 (Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.)",
        "Preserve independently valid work in O2, O3, O4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article p.033852-3, Fig.2 and two-plateau paragraph: first plateau and roughly five-orders-weaker second plateau, with cutoff scales near harmonic orders 40 and 120; bands 2/3 and higher-band pathways differ."
      ],
      "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present correct, reconstructible, claim-sufficient Case-B full material harmonic response with proper case, initial-state, observable and representation identity.",
      "award_conditions": [
        "Case-B full material response is reconstructible and correctly bound to its drive, initial state, observable and representation.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: F1 (Construct a coordinate-space B-spline TDSE for the specified one-dimensional periodic potential and both drives with consistent executable and auditable field coupling.); F2 (Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.); F3 (Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.)",
        "Preserve independently valid work in O1, O3, O4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present reconstructible Case-B causal decomposition or genuinely equivalent evidence distinguishing low-order and higher-first-plateau contributions; static labels, duplicate currents, swapped components or false spectral-power addition are insufficient.",
      "award_conditions": [
        "A causal current-level or truly equivalent distinction separates low-order and higher-first-plateau mechanisms without duplicate/swapped components or false power addition.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: F2 (Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.); F3 (Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.); F4 (Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.)",
        "Preserve independently valid work in O1, O2, O4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid."
    },
    {
      "criterion_id": "O4",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present a reconstructible comparison between the primary B-spline result and a control exposing a distinct material failure mode, stating the physical range in which agreement or conflict is established.",
      "award_conditions": [
        "The candidate presents a reconstructible control comparison that can expose a distinct material failure mode and states the actual physical range of agreement/conflict.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: F4 (Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.)",
        "Preserve independently valid work in O1, O2, O3, V2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify both Case-A plateaus, approximate cutoff scales near orders 40 and 120, and the much weaker second plateau in the submitted evidence.",
      "award_conditions": [
        "The candidate identifies Case-A plateaus, cutoff scales near 40/120 and much weaker second branch in their presented evidence."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: O1 (Present correct, reconstructible, claim-sufficient Case-A harmonic evidence spanning the full material response, including the first plateau and weaker second plateau beyond the first cutoff.)",
        "Preserve independently valid work in I2, I3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article p.033852-3, Fig.2 and two-plateau paragraph: first plateau and roughly five-orders-weaker second plateau, with cutoff scales near harmonic orders 40 and 120; bands 2/3 and higher-band pathways differ."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify the important same-band/intraband contribution among the first few Case-B harmonics, or a genuinely physically equivalent low-order contribution, from presented causal evidence.",
      "award_conditions": [
        "The candidate identifies an important same-band/intraband contribution at low Case-B orders or a genuinely equivalent causal contribution."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: O3 (Present reconstructible Case-B causal decomposition or genuinely equivalent evidence distinguishing low-order and higher-first-plateau contributions; static labels, duplicate currents, swapped components or false spectral-power addition are insufficient.)",
        "Preserve independently valid work in I1, I3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify different-band/interband dominance at higher orders of the Case-B first plateau, or a genuinely physically equivalent high-order contribution, from presented causal evidence.",
      "award_conditions": [
        "The candidate identifies different-band/interband dominance at higher first-plateau orders or a genuinely equivalent causal contribution."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: O3 (Present reconstructible Case-B causal decomposition or genuinely equivalent evidence distinguishing low-order and higher-first-plateau contributions; static labels, duplicate currents, swapped components or false spectral-power addition are insufficient.)",
        "Preserve independently valid work in I1, I2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Link calculated band structure and driven dynamics to Case A's two plateaus, cutoff scales and weak second branch in an evidence-grounded causal account.",
      "award_conditions": [
        "Calculated bands and driven evidence explain Case-A plateau/cutoff ordering and weak higher branch rather than naming two plateaus alone."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: I1 (Identify both Case-A plateaus, approximate cutoff scales near orders 40 and 120, and the much weaker second plateau in the submitted evidence.); F4 (Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.)",
        "Preserve independently valid work in M2, M3, M4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article p.033852-3, Fig.2 and two-plateau paragraph: first plateau and roughly five-orders-weaker second plateau, with cutoff scales near harmonic orders 40 and 120; bands 2/3 and higher-band pathways differ."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain from submitted evidence the important low-order Case-B intraband/same-band contribution or a physically equivalent low-order causal account.",
      "award_conditions": [
        "Presented causal evidence grounds the important low-order Case-B contribution and its physical origin."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: I2 (Identify the important same-band/intraband contribution among the first few Case-B harmonics, or a genuinely physically equivalent low-order contribution, from presented causal evidence.); F4 (Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.)",
        "Preserve independently valid work in M1, M3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain from submitted evidence the higher-first-plateau Case-B interband/different-band dominance or a physically equivalent high-order causal account.",
      "award_conditions": [
        "Presented causal evidence grounds higher-first-plateau Case-B dominance and its physical origin."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: I3 (Identify different-band/interband dominance at higher orders of the Case-B first plateau, or a genuinely physically equivalent high-order contribution, from presented causal evidence.); F4 (Give the actual theoretical/auxiliary framework for interpreting radiation and testing reliability with reproducible observables; a renamed duplicate, static-only driven-validation claim or undefined mechanism object is insufficient.)",
        "Preserve independently valid work in M1, M2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "State where the Case-B physical account applies or changes between low and higher-first-plateau orders, without extending a local mechanism to the full response without evidence.",
      "award_conditions": [
        "The candidate locates where the Case-B mechanism applies or changes and does not extend a local decomposition globally without evidence."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: I2 (Identify the important same-band/intraband contribution among the first few Case-B harmonics, or a genuinely physically equivalent low-order contribution, from presented causal evidence.); I3 (Identify different-band/interband dominance at higher orders of the Case-B first plateau, or a genuinely physically equivalent high-order contribution, from presented causal evidence.)",
        "Preserve independently valid work in M2, M3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M5",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "For Case A, decide coordinate-space reliability from evidence covering every material response range in the claim and challenged by a distinct-failure-mode control; explicitly localize the conclusion when evidence is local.",
      "award_conditions": [
        "Case-A reliability is supported over every material response range included in the claim and challenged by a distinct-failure-mode control; a local claim is explicitly localized."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: O1 (Present correct, reconstructible, claim-sufficient Case-A harmonic evidence spanning the full material response, including the first plateau and weaker second plateau beyond the first cutoff.); O4 (Present a reconstructible comparison between the primary B-spline result and a control exposing a distinct material failure mode, stating the physical range in which agreement or conflict is established.)",
        "Preserve independently valid work in M6, M2, M3, V2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the commissioned scientific conclusion is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "For Case B, reach an evidence-bounded coordinate-space reliability decision using that case's material response and a control exposing a distinct failure mode.",
      "award_conditions": [
        "Case-B reliability follows that case's actual response and a control capable of exposing a distinct material failure mode."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: all of: O2 (Present correct, reconstructible, claim-sufficient Case-B full material harmonic response with proper case, initial-state, observable and representation identity.); O4 (Present a reconstructible comparison between the primary B-spline result and a control exposing a distinct material failure mode, stating the physical range in which agreement or conflict is established.)",
        "Preserve independently valid work in M5, M1, V2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the commissioned scientific conclusion is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explicitly reconcile material agreement/conflict among primary, auxiliary/control and mechanism evidence and explain its effect on retained reliability and physical conclusions; do not suppress a conclusion-changing conflict.",
      "award_conditions": [
        "The candidate reconciles conclusion-changing agreement/conflict among primary, control and causal evidence instead of suppressing it."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: O4 (Present a reconstructible comparison between the primary B-spline result and a control exposing a distinct material failure mode, stating the physical range in which agreement or conflict is established.)",
        "Preserve independently valid work in V2, M1, M2, M3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-1-4, SAE model boundary, Sec.III.A numerical-stability comparison and last p.033852-4 model-potential qualification. Reconciling material disagreements and bounded reliability claims are task-defined, not universal experimental equivalence."
      ],
      "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the commissioned scientific conclusion is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "When the method, data, anomaly or retained claim exposes a concrete material failure risk, challenge it with functionally capable evidence and state its effective physical range. A same-integrator basis comparison sharing the error class is not sufficient by itself.",
      "award_conditions": [
        "A concrete triggered failure risk is challenged by functionally capable evidence over its actual range; shared-integrator basis convergence alone does not establish distinct sensitivity."
      ],
      "applicability_and_na": [
        "Applicable only when a concrete material risk arises from the submitted method, data, anomaly or claim and threatens a retained conclusion.",
        "If untriggered in an assessable attempt, record N/A with fixed 0.5-point contribution, exclude it from applicable functional-completion counts, make no positive validation-evidence claim and do not block RC4."
      ],
      "dependencies": [
        "Necessary scientific premise: O4 (Present a reconstructible comparison between the primary B-spline result and a control exposing a distinct material failure mode, stating the physical range in which agreement or conflict is established.)",
        "Preserve independently valid work in O1, O2, M1, M2, M3, V2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations."
      ],
      "rc_relation": "Role: material validation/scope. Trigger: a triggered, conclusion-threatening material validation/scope risk remains unresolved. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Restrict conclusions to the established one-dimensional single-active-electron model and physical domain, without equating numerical smoothness, precision or apparent convergence with physical certainty.",
      "award_conditions": [
        "The conclusion remains within the 1D SAE model and established physical range, without treating smoothness or precision as certainty."
      ],
      "applicability_and_na": [
        "This positive scientific scope/evidence duty is unconditional; N/A is not permitted."
      ],
      "dependencies": [
        "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied."
      ],
      "equivalent_evidence": [
        "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target, especially band-2 k=0 identity, and produce converged evidence.",
        "Physically equivalent current-level causal distinctions are accepted in place of the canonical same-band/different-band decomposition; no named Houston or Bloch auxiliary route is compulsory.",
        "Independent periodic formulations, controlled analytic limits or comparably strong falsification routes are accepted by the error classes they expose, not their filenames or basis labels."
      ],
      "source_support": [
        "Guan et al. (2016), article pp.033852-1-4, SAE model boundary, Sec.III.A numerical-stability comparison and last p.033852-4 model-potential qualification. Reconciling material disagreements and bounded reliability claims are task-defined, not universal experimental equivalence."
      ],
      "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure."
    }
  ],
  "case_scientific_rules": [
    "Each of the exact twenty criteria is worth 0 or 0.5 points; the sum is Atomic Score on 0-10. Judge the essential scientific core: an immaterial notation or display imperfection alone does not remove credit. Do not interpolate or inherit numerical partial-credit ranges.",
    "Object existence, candidate presentation, candidate identification and physical inference are distinct evidence acts. Examiner-generated plots, slices, fits or diagnostics may verify an object but cannot supply omitted candidate presentation, identification, mechanism or validation.",
    "For every Output atom separately establish object_valid, presentation_adequate and claim_adequate. Credit requires all three. Assess physical identity, global pattern and important secondary structure, not a single superficial resemblance.",
    "Apply only scientifically necessary dependencies, branch-locally. A composite prerequisite failing for an unrelated part does not erase independently established evidence. Identify the exact failed premise and why the downstream argument cannot stand independently; do not turn a half-point result into a mechanical gate.",
    "Risk-triggered validation is applicable only to a concrete threat arising from the submitted method, data, anomaly or claim. In an assessable attempt, an untriggered criterion is N/A and contributes its fixed 0.5 points, but supplies no positive validation evidence and does not block RC4.",
    "Positive model/scope duties are unconditional where specified. Merely omitting an examiner-preferred convergence procedure, control or plot does not create a material risk. A valid run with no assessable scientific work is 0 and RC0 before atom summation; N/A cannot create credit for it.",
    "Research Closure is descriptive and independent of Atomic Score and criterion awards. Evaluate scientific role conditions against their specified facts, not zero-valued atoms; choose the lowest applicable level across required roles of all indispensable/co-primary claims. Supporting claims alone do not determine overall RC.",
    "Numerical defects matter where they damage a physical object or make a scientific conclusion false/unsupported; do not deduct the same defect again as generic incompleteness, missing convergence or validation. A fluent interpretation does not rescue a materially incorrect signal.",
    "Accept physically valid equivalent methods and representations after aligning identity, units, gauge/basis, normalization, signs, support and component relations as applicable. Do not invent a tolerance when the evidence cannot distinguish equivalence from error; retain a bounded issue for review.",
    "Reference verification is required when a material decision depends on resolution, fine structure, component completeness, formula identity or representational equivalence, including nonstandard projections, cropping or reconstruction used for full Output credit. Consult the applicable scientific source and appropriately scoped validated references after establishing observable, case/component and convention mapping. It cannot complete missing candidate reasoning.",
    "A provisional Atomic Score of at least 9.0 or provisional RC4 requires examiner-side focused verification of one central output and one principal-mechanism link against applicable scientific references, checking identity, sign, ordering and overall pattern. This is not an additional candidate deliverable or pixel-matching requirement.",
    "Approximate band-family ranges are -14.31 to -14.13, -4.86 to -2.66, 1.52 to 9.19 and 9.99 to 23.88 eV. These support physical identity, not fixed plot bounds.",
    "A survey ending near the first Case-A cutoff cannot establish the full response. Correct explicitly local results retain their own credit, but cannot supply the omitted weak high-energy branch.",
    "Component spectral powers must not be arithmetically forced to equal the coherent total. A correct total retains independent credit when causal decomposition fails.",
    "Two bases sharing a one-cell sector, integrator and observable reduction may verify basis convergence but do not automatically provide independent failure-mode sensitivity.",
    "No source basis size, finite-box ordinal, box, absorber, window, projector, fixed grid or plot layout is a target; physical equivalence must be demonstrated rather than inferred from code sharing or visual similarity.",
    "A standard B-spline Galerkin construction produces a nonorthogonal field-free problem and driven TDSE for the supplied periodic potential and field. The physical initial state is band 2 at $k=0$; it must remain the same state under any representation change. Source-used values such as 3600 splines, the $[-240,240]$ box, midpoint Crank-Nicolson, outer absorber and Hanning window are reproduction evidence, not grading requirements.",
    "Gauge-, basis-, boundary-, propagator- and transform-equivalent realizations are accepted when they preserve the physical target and produce converged evidence. A finite-box ordinal, shared code path or visually similar spectrum does not by itself establish physical equivalence.",
    "Canonical field-free evidence contains four corresponding band families with approximate ranges $-14.31$ to $-14.13$ eV, $-4.86$ to $-2.66$ eV, $1.52$ to $9.19$ eV and $9.99$ to $23.88$ eV.",
    "For Case A, the physical harmonic response has two plateau regions. Their approximate cutoffs are near orders 40 and 120, with the second plateau much weaker than the first. A calculation or displayed survey that stops around the first cutoff has not established the full response even if that range is smooth and converged.",
    "For Case B, source-backed causal evidence shows an important same-band or intraband contribution among the first few harmonics and different-band or interband dominance over higher orders of the first plateau. Equivalent current-level causal distinctions are accepted. Component spectral powers must not be arithmetically forced to equal the coherent total.",
    "The source canonical route uses periodic Bloch and moving-band/Houston descriptions. They are not secretly required by name. Full credit requires evidence capable of exposing a material failure mode not shared with the primary calculation. This may be an independently formulated periodic method, an analytically controlled limit, or another comparably strong falsification route.",
    "Two bases that converge to the same one-cell sector with the same integrator and observable reduction can provide useful basis verification, but they do not automatically receive full independence credit. Grade the error classes the comparison can expose, not whether two filenames or basis labels differ.",
    "For F1-F3/O3/M2-M4, article p.033852-2, Eqs.(4),(6)-(12), a nonorthogonal B-spline representation obeys $iS\\dot C=H(t)C$, with norm $C^\\dagger SC$. For physical band projectors $P_b$,\n$$j_{\\rm intra}=-\\frac em\\sum_b\\langle\\psi|P_b\\hat pP_b|\\psi\\rangle,\\quad\nj_{\\rm inter}=-\\frac em\\sum_{b\\ne b'}\\langle\\psi|P_b\\hat pP_{b'}|\\psi\\rangle,\\quad j=j_{\\rm intra}+j_{\\rm inter}.$$\nThus $|\\widetilde j|^2=|\\widetilde j_{\\rm intra}|^2+|\\widetilde j_{\\rm inter}|^2+2\\Re(\\widetilde j_{\\rm intra}\\widetilde j_{\\rm inter}^*)$. In a different gauge the current and projectors must be transformed consistently. Equivalent methods must preserve the same physical k=0 band-2 state and causal content; matching a finite-box ordinal, swapping component names, or comparing two implementations sharing the relevant error class does not establish equivalence or independent reliability."
  ],
  "closure_requirements": {
    "RC0": [
      "The central physical object is absent or materially invalid, or a valid run contains no assessable scientific work. This is a scientific fact determination, not a low-score rule.",
      "Required indispensable claim CL1, central physical object: all of: F1 (Construct a coordinate-space B-spline TDSE for the specified one-dimensional periodic potential and both drives with consistent executable and auditable field coupling.); F3 (Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.). Judge the role-specific failure facts, not the corresponding atom scores."
    ],
    "RC1": [
      "With a valid central object, an indispensable material component needed for the commissioned research question is absent or invalid. Presentation or claim inadequacy alone does not establish object absence/invalidity.",
      "Required indispensable claim CL1, indispensable material component: all of: F2 (Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.); O1 (Present correct, reconstructible, claim-sufficient Case-A harmonic evidence spanning the full material response, including the first plateau and weaker second plateau beyond the first cutoff.); O2 (Present correct, reconstructible, claim-sufficient Case-B full material harmonic response with proper case, initial-state, observable and representation identity.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required co primary claim CL2, indispensable material component: O3 (Present reconstructible Case-B causal decomposition or genuinely equivalent evidence distinguishing low-order and higher-first-plateau contributions; static labels, duplicate currents, swapped components or false spectral-power addition are insufficient.). Judge the role-specific failure facts, not the corresponding atom scores."
    ],
    "RC2": [
      "With the central object and indispensable components present, a required principal mechanism, state-dependence account or commissioned conclusion remains absent or wrong.",
      "Required indispensable claim CL1, principal mechanism or commissioned conclusion: all of: M5 (For Case A, decide coordinate-space reliability from evidence covering every material response range in the claim and challenged by a distinct-failure-mode control; explicitly localize the conclusion when evidence is local.); M6 (For Case B, reach an evidence-bounded coordinate-space reliability decision using that case's material response and a control exposing a distinct failure mode.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required co primary claim CL2, principal mechanism or commissioned conclusion: all of: M1 (Link calculated band structure and driven dynamics to Case A's two plateaus, cutoff scales and weak second branch in an evidence-grounded causal account.); M2 (Explain from submitted evidence the important low-order Case-B intraband/same-band contribution or a physically equivalent low-order causal account.); M3 (Explain from submitted evidence the higher-first-plateau Case-B interband/different-band dominance or a physically equivalent high-order causal account.); M4 (State where the Case-B physical account applies or changes between low and higher-first-plateau orders, without extending a local mechanism to the full response without evidence.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required indispensable claim CL3, principal mechanism or commissioned conclusion: M7 (Explicitly reconcile material agreement/conflict among primary, auxiliary/control and mechanism evidence and explain its effect on retained reliability and physical conclusions; do not suppress a conclusion-changing conflict.). Judge the role-specific failure facts, not the corresponding atom scores."
    ],
    "RC3": [
      "With the central object, indispensable components and principal mechanism established, a concrete material validation/scope risk remains unresolved and threatens a central conclusion. It is not triggered by omission of a preferred named check.",
      "Before assigning RC3, establish all six elements: 1. Identify the concrete unexcluded failure mode. 2. Explain its origin in the submitted method, data or conclusion. 3. Identify a fair basis in the task or standard scientific reasoning for recognizing this risk. 4. Identify the threatened central conclusion. 5. Explain why submitted evidence cannot expose/exclude the failure; naming it alone does not resolve it. 6. Give examples of functionally equivalent remedies without prescribing one method.",
      "Required indispensable claim CL3, material validation/scope: V1 (When the method, data, anomaly or retained claim exposes a concrete material failure risk, challenge it with functionally capable evidence and state its effective physical range. A same-integrator basis comparison sharing the error class is not sufficient by itself.). Judge the role-specific failure facts, not the corresponding atom scores."
    ],
    "RC4": [
      "All required scientific roles of every indispensable and co-primary claim close, including applicable material-risk treatment and evidence-bounded scope. Supporting claims do not independently determine RC; untriggered risk N/A does not block RC4.",
      "Choose the lowest applicable level when required role failures coexist. RC has no score cap, floor or multiplier and cannot be inferred mechanically from criterion awards. A scope restriction removes unsupported breadth but cannot repair a wrong signal, absent component or missing principal mechanism.",
      "Close CL1 central physical object: all of: F1 (Construct a coordinate-space B-spline TDSE for the specified one-dimensional periodic potential and both drives with consistent executable and auditable field coupling.); F3 (Define a physically consistent HHG radiation observable in the chosen gauge/representation and consistently state field amplitude, frequency, carrier phase/time origin and units.).",
      "Close CL1 indispensable material component: all of: F2 (Prepare band 2 at k=0 as the same physical initial state and preserve that identity under every representation change and auxiliary calculation.); O1 (Present correct, reconstructible, claim-sufficient Case-A harmonic evidence spanning the full material response, including the first plateau and weaker second plateau beyond the first cutoff.); O2 (Present correct, reconstructible, claim-sufficient Case-B full material harmonic response with proper case, initial-state, observable and representation identity.).",
      "Close CL1 principal mechanism or commissioned conclusion: all of: M5 (For Case A, decide coordinate-space reliability from evidence covering every material response range in the claim and challenged by a distinct-failure-mode control; explicitly localize the conclusion when evidence is local.); M6 (For Case B, reach an evidence-bounded coordinate-space reliability decision using that case's material response and a control exposing a distinct failure mode.).",
      "Close CL2 indispensable material component: O3 (Present reconstructible Case-B causal decomposition or genuinely equivalent evidence distinguishing low-order and higher-first-plateau contributions; static labels, duplicate currents, swapped components or false spectral-power addition are insufficient.).",
      "Close CL2 principal mechanism or commissioned conclusion: all of: M1 (Link calculated band structure and driven dynamics to Case A's two plateaus, cutoff scales and weak second branch in an evidence-grounded causal account.); M2 (Explain from submitted evidence the important low-order Case-B intraband/same-band contribution or a physically equivalent low-order causal account.); M3 (Explain from submitted evidence the higher-first-plateau Case-B interband/different-band dominance or a physically equivalent high-order causal account.); M4 (State where the Case-B physical account applies or changes between low and higher-first-plateau orders, without extending a local mechanism to the full response without evidence.).",
      "Close CL3 principal mechanism or commissioned conclusion: M7 (Explicitly reconcile material agreement/conflict among primary, auxiliary/control and mechanism evidence and explain its effect on retained reliability and physical conclusions; do not suppress a conclusion-changing conflict.).",
      "Close CL3 material validation/scope: V1 (When the method, data, anomaly or retained claim exposes a concrete material failure risk, challenge it with functionally capable evidence and state its effective physical range. A same-integrator basis comparison sharing the error class is not sufficient by itself.)."
    ]
  },
  "references": [
    {
      "citation": "Zhong Guan, Xiao-Xin Zhou, and Xue-Bin Bian. 2016. *High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses*. *Physical Review A*, 93, 033852.",
      "url": "https://doi.org/10.1103/PhysRevA.93.033852",
      "scientific_scope": "High-order-harmonic generation from periodic potentials driven by few-cycle laser pulses",
      "anchors": [
        "Guan et al. (2016), article pp.033852-1-2, Eqs.(1)-(7), Fig.1(a-d), paragraph choosing k=0 in band 2: B-spline generalized metric and physical initial-state identity, not finite-box ordinal.",
        "Guan et al. (2016), article p.033852-2, Eq.(8), and p.033852-3 Fourier/window paragraph: laser-induced current and spectrum. Gauge/derivative equivalence requires actual observable mapping, not appearance alone.",
        "Guan et al. (2016), article pp.033852-2-4, Fig.1(a,b), Fig.2 and Fig.4(a,b), Sec.III.A comparison to Bloch/Houston expansions: static bands plus driven response comparisons. Distinct failure-mode sensitivity and case/range-specific reliability conclusions are task-defined; no named auxiliary method is compulsory. Secondary standard-method support: Wu et al. (2015), article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23) and Fig.5: matched physical k=0 valence initial state, Houston/Bloch equivalence and band-truncation limitations.",
        "Guan et al. (2016), article p.033852-3, Fig.2 and two-plateau paragraph: first plateau and roughly five-orders-weaker second plateau, with cutoff scales near harmonic orders 40 and 120; bands 2/3 and higher-band pathways differ.",
        "Guan et al. (2016), article pp.033852-2-4, Eqs.(9)-(12), Fig.4(a,b): same-band/intraband low-order contribution and different-band/interband higher-first-plateau dominance at Case-B intensity. Current components combine coherently; their powers do not simply add.",
        "Guan et al. (2016), article pp.033852-1-4, SAE model boundary, Sec.III.A numerical-stability comparison and last p.033852-4 model-potential qualification. Reconciling material disagreements and bounded reliability claims are task-defined, not universal experimental equivalence."
      ]
    },
    {
      "citation": "Mengxi Wu, Shambhu Ghimire, David A. Reis, Kenneth J. Schafer, and Mette B. Gaarde. 2015. *High-harmonic generation from Bloch electrons in solids*. *Physical Review A*, 91, 043839.",
      "url": "https://doi.org/10.1103/PhysRevA.91.043839",
      "scientific_scope": "Secondary periodic Bloch/Houston method and physical-equivalence boundary, not a required auxiliary implementation.",
      "anchors": [
        "Article pp.043839-2,6, Secs.II-III, Fig.4(a-c), Eqs.(22),(23), Fig.5."
      ]
    }
  ]
}