{
  "case_id": "Case05",
  "title": "Case05 — High-Harmonic Generation in Solids with and without Topological Edge States",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Establish the specified N=100, soft-core, spin-neutral continuous-space x-only LSD model and fixed M/D+/D- geometries; an SSH/tight-binding replacement is not equivalent.",
      "award_conditions": [
        "The model is the N=100 spin-neutral continuous-space soft-core x-only LSD chain at the fixed M/D+/D- geometries, not SSH/tight binding."
      ],
      "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 independently valid work in F2, F3, F4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-2, Eqs.(1)-(6): soft-core continuous-space x-only LSD chain, spin-neutral occupation and fixed dimerized geometries; not a tight-binding replacement."
      ],
      "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": "Construct finite-chain stationary analysis capable of distinguishing metal, insulating manifolds and edge-localized states from the continuous-space calculation rather than assigning labels from the sign of delta.",
      "award_conditions": [
        "The finite-chain stationary calculation can distinguish metallic/insulating manifolds and localize edge states without assigning physics solely from the sign of distortion."
      ],
      "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 independently valid work in F1, O1, I1, M1. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets."
      ],
      "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": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Construct the specified driven dynamics from the corresponding occupied stationary orbitals and the spin-summed total-dipole spectrum. A current/acceleration representation requires its relation and frequency weighting to that object.",
      "award_conditions": [
        "Driven dynamics starts from the appropriate occupied stationary orbitals and yields the coherent spin-summed total-dipole spectrum; derivative forms explicitly account for frequency weights."
      ],
      "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 independently valid work in F1, F4, O2, O3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-3, Eqs.(7),(8): driven KS orbitals, spin-summed coherent total dipole and squared Fourier modulus; current/acceleration variants require omega^2/omega^4 weighting."
      ],
      "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": "Consistently use units, specified field, Fourier convention, orbital occupation and one common constant spectral scale across M/D+/D- and dynamical comparisons.",
      "award_conditions": [
        "Field, units, Fourier convention, occupation and one common constant spectral scale remain consistent across all phase and dynamical comparisons."
      ],
      "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 independently valid work in F3, O2, O3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-3, Eqs.(7),(8): driven KS orbitals, spin-summed coherent total dipole and squared Fourier modulus; current/acceleration variants require omega^2/omega^4 weighting."
      ],
      "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": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present reconstructible stationary evidence distinguishing M from both insulators and establishing D-'s in-gap pair, occupation and edge localization relative to D+.",
      "award_conditions": [
        "Reconstructible stationary evidence distinguishes metal and insulators and establishes the D- in-gap pair, its occupation and edge localization relative to D+.",
        "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: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve independently valid work in F1, F2, I1, M1. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets."
      ],
      "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 common-scale spin-summed total-dipole HHG spectra for M, D+ and D-, making sub-gap relative strength and case ordering assessable.",
      "award_conditions": [
        "Common-scale total-dipole spectra for M/D+/D- expose correct material sub-gap ordering and the large insulating contrast.",
        "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: F3 (Construct the specified driven dynamics from the corresponding occupied stationary orbitals and the spin-summed total-dipole spectrum. A current/acceleration representation requires its relation and frequency weighting to that object.)",
        "Preserve independently valid work in F4, O3, I2, M2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and sub-gap discussion: large D- (source phase B) / D+ (source phase A) contrast on common scale; exact 14-order magnitude or colors are not targets."
      ],
      "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 a valid comparison/intervention showing that evolving internal Kohn-Sham potential does not generate the D-/D+ sub-gap contrast and that M is more sensitive to that dynamics; full/frozen propagation is accepted but not mandatory.",
      "award_conditions": [
        "A valid causal comparison establishes robust insulating contrast and stronger metal sensitivity to evolving internal KS potential; a named full/frozen plot is not mandatory.",
        "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: F3 (Construct the specified driven dynamics from the corresponding occupied stationary orbitals and the spin-summed total-dipole spectrum. A current/acceleration representation requires its relation and frequency weighting to that object.)",
        "Preserve independently valid work in O2, O4, I3, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and frozen-KS/screening paragraphs: insulating contrast survives frozen internal potential, while metal is much more sensitive. A functionally equivalent intervention is accepted."
      ],
      "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 candidate-designed calculation/control distinguishing edge occupation/incomplete coherent cancellation from gap-only, distortion-only or generic edge-state-presence explanations; no named canonical control is mandatory.",
      "award_conditions": [
        "A candidate-designed intervention distinguishes occupation/incomplete cancellation from at least one gap-only, distortion-only or generic edge-presence alternative.",
        "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: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve independently valid work in O2, O3, I4, M3, M4, M5. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples."
      ],
      "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": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify from the submitted stationary evidence metallic M, insulating D+/D-, and the occupied in-gap edge-state pair and end localization in D-.",
      "award_conditions": [
        "The candidate identifies metallic M, insulating D+/D- and D- occupied edge-pair/end localization in their stationary 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 reconstructible stationary evidence distinguishing M from both insulators and establishing D-'s in-gap pair, occupation and edge localization relative to D+.)",
        "Preserve independently valid work in F2, M1. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets."
      ],
      "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 correct common-scale sub-gap HHG case ordering and large D-/D+ contrast, without relying on absolute scale.",
      "award_conditions": [
        "The candidate identifies common-scale sub-gap case ordering and large D-/D+ contrast without relying on absolute yield."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: O2 (Present correct common-scale spin-summed total-dipole HHG spectra for M, D+ and D-, making sub-gap relative strength and case ordering assessable.)",
        "Preserve independently valid work in F4, M2, I3. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and sub-gap discussion: large D- (source phase B) / D+ (source phase A) contrast on common scale; exact 14-order magnitude or colors are not targets."
      ],
      "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 from the causal comparison the robustness of the insulating contrast and stronger M sensitivity to the evolving self-consistent potential.",
      "award_conditions": [
        "The candidate identifies the robust insulating contrast and larger metal response to the evolving-potential intervention."
      ],
      "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 a valid comparison/intervention showing that evolving internal Kohn-Sham potential does not generate the D-/D+ sub-gap contrast and that M is more sensitive to that dynamics; full/frozen propagation is accepted but not mandatory.)",
        "Preserve independently valid work in I2, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and frozen-KS/screening paragraphs: insulating contrast survives frozen internal potential, while metal is much more sensitive. A functionally equivalent intervention is accepted."
      ],
      "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": "I4",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify a concrete discriminating signal consistent with edge occupation and incomplete coherent cancellation and capable of distinguishing at least one alternative.",
      "award_conditions": [
        "The candidate identifies the actual discriminating control signal and the alternative it distinguishes."
      ],
      "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 candidate-designed calculation/control distinguishing edge occupation/incomplete coherent cancellation from gap-only, distortion-only or generic edge-state-presence explanations; no named canonical control is mandatory.)",
        "Preserve independently valid work in M3, M4, M5. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples."
      ],
      "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": "Explain finite-chain phase identities and the finite-chain/topological character of D- edge states using continuous-space evidence, not generic detached SSH language.",
      "award_conditions": [
        "Continuous-space finite-chain evidence supports phase/topological edge assignments; detached SSH terminology 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: all of: F2 (Construct finite-chain stationary analysis capable of distinguishing metal, insulating manifolds and edge-localized states from the continuous-space calculation rather than assigning labels from the sign of delta.); I1 (Identify from the submitted stationary evidence metallic M, insulating D+/D-, and the occupied in-gap edge-state pair and end localization in D-.)",
        "Preserve independently valid work in F1, O1. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets."
      ],
      "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": "Infer how topology is encoded in the low-energy HHG response from the identified common-scale sub-gap ordering, limiting the conclusion to supported relative rather than absolute strength.",
      "award_conditions": [
        "The identified relative sub-gap HHG ordering supports the stated topology-sensitive inference without claiming absolute strength."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: I2 (Identify the correct common-scale sub-gap HHG case ordering and large D-/D+ contrast, without relying on absolute scale.)",
        "Preserve independently valid work in O2, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and sub-gap discussion: large D- (source phase B) / D+ (source phase A) contrast on common scale; exact 14-order magnitude or colors are not targets."
      ],
      "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": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use discriminating evidence to explain near-cancellation of strong individual-orbital radiation in the coherent fully occupied D+ valence total and incomplete cancellation in D-.",
      "award_conditions": [
        "Causal evidence connects strong individual orbital emission to near-cancellation in the filled D+ valence total and incomplete cancellation in D-."
      ],
      "applicability_and_na": [
        "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Necessary scientific premise: I4 (Identify a concrete discriminating signal consistent with edge occupation and incomplete coherent cancellation and capable of distinguishing at least one alternative.)",
        "Preserve independently valid work in M4, M5, O4. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples."
      ],
      "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": "Connect half occupation of the D- in-gap edge pair to incomplete coherent cancellation and explain why an in-gap eigenvalue alone does not establish the mechanism.",
      "award_conditions": [
        "The half-occupied D- in-gap pair explains incomplete coherent cancellation; an in-gap eigenvalue alone 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: I4 (Identify a concrete discriminating signal consistent with edge occupation and incomplete coherent cancellation and capable of distinguishing at least one alternative.)",
        "Preserve independently valid work in M3, M5, I1. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples."
      ],
      "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": "Explicitly reject at least one plausible gap-only, distortion-only or edge-state-presence-only alternative using the causal discriminating evidence.",
      "award_conditions": [
        "The candidate explicitly rejects at least one plausible alternative using their discriminating calculation rather than association 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: I4 (Identify a concrete discriminating signal consistent with edge occupation and incomplete coherent cancellation and capable of distinguishing at least one alternative.)",
        "Preserve independently valid work in 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": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples."
      ],
      "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": "Use the identified ordering and intervention sensitivity to distinguish robust D-/D+ insulating contrast from stronger M sensitivity to the evolving self-consistent potential.",
      "award_conditions": [
        "The causal comparison separates the robust insulating contrast from stronger metallic sensitivity to time-dependent screening/internal potential."
      ],
      "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 correct common-scale sub-gap HHG case ordering and large D-/D+ contrast, without relying on absolute scale.); I3 (Identify from the causal comparison the robustness of the insulating contrast and stronger M sensitivity to the evolving self-consistent potential.)",
        "Preserve independently valid work in O2, O3, M2. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and frozen-KS/screening paragraphs: insulating contrast survives frozen internal potential, while metal is much more sensitive. A functionally equivalent intervention is accepted."
      ],
      "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": "Bound the conclusion to the specified finite-chain continuous-space x-only LSD model, honestly handling known energy-curve differences or unresolved numerical features without manufacturing certainty or demanding exact source-curve reproduction.",
      "award_conditions": [
        "The conclusion remains within the specified finite-chain x-only LSD model and handles energy-curve uncertainty honestly without demanding exact source minima."
      ],
      "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 independently valid work in M1, M2, M3, M4, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-4, model equations and conclusion delimit finite-chain mean-field scope. Exact Fig.1(a) energy-curve reproduction is excluded by the task/effective rubric because its printed-functional reproduction remains unresolved; risk validation is task-defined."
      ],
      "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": "Only when the model, transform/rescaling, occupation control or claim creates a concrete material risk to the central conclusion, provide targeted discriminating evidence; otherwise record N/A with fixed credit and no evidence claim.",
      "award_conditions": [
        "Only a concrete material risk in the model, transform, occupation intervention or claim requires discriminating validation; otherwise N/A."
      ],
      "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: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.",
        "Preserve independently valid work in F1, F2, F3, F4, O1, O2, O3, O4, I1, I2, I3, I4, M1, M2, M3, M4, M5, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect."
      ],
      "equivalent_evidence": [
        "Global orbital phases, momentum signs, Fourier conventions and one common constant spectrum scale are accepted after alignment. Current/acceleration spectra require a demonstrated relation and treatment of frequency weighting.",
        "Stationary momentum maps, real-space localization, projectors, spectral functions or equivalent evidence are accepted by their ability to establish state and occupation identity.",
        "A full/frozen comparison or any genuinely isolating intervention may test evolving internal potential. Incoherent-orbital-sum, controlled occupation, impurity, orbital-removal or equivalent interventions may test cancellation only when they distinguish the competing explanations."
      ],
      "source_support": [
        "Bauer and Hansen (2018), article pp.177401-2-4, model equations and conclusion delimit finite-chain mean-field scope. Exact Fig.1(a) energy-curve reproduction is excluded by the task/effective rubric because its printed-functional reproduction remains unresolved; risk validation is task-defined."
      ],
      "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."
    }
  ],
  "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.",
    "Use the source-stated +0.265 and -0.165 geometries for driven/stationary comparisons. An optional energy scan may establish converged dimerization-lowered two-basin structure and undistorted instability, but exact minima, well depths, barrier, curvature and pixels are excluded anchors.",
    "The unresolved published energy-curve detail does not invalidate the source-fixed-geometry driven analysis and does not determine RC. Do not tune unconverged conventions to imitate it.",
    "Changing occupation can redefine the physical target. Mere edge-state presence, a gap or a stationary association does not establish the causal occupation/interference explanation.",
    "For the stated continuous-space x-only LSD model, the undimerized finite chain is a half-filled metal. Both nonzero shifts dimerize the chain and open an insulating gap. D- has an almost-degenerate in-gap edge-state pair with only the lower state occupied; the occupied edge state is localized near the chain ends and broad in momentum. D+ lacks this half-occupied in-gap pair.",
    "The Hanning-windowed total-dipole spectra support these central anchors:",
    "below the D+ valence-to-first-conduction gap, D- emits many orders of magnitude more strongly than D+; D+ retains chiefly low odd harmonics;",
    "the large D-/D+ contrast persists when the internal KS potential is frozen, so it is not created by the time dependence of the mean-field potential;",
    "the metallic M case changes much more between full and frozen propagation, consistent with screening by the time-dependent metallic density;",
    "individual occupied orbitals emit strongly in the sub-gap range, but their coherent total nearly cancels for the fully occupied D+ valence manifold. The half-occupied edge-state pair in D- makes the cancellation incomplete.",
    "The source's canonical discriminating evidence is the comparison between the true coherent total spectrum and an incoherent sum of individual-orbital spectra. Supplemental controls using nontopological impurity edge states and a virtual electron occupying the second D- edge state support the same occupation/interference mechanism. Students need not reproduce a named control, but high mechanism credit requires evidence with equivalent causal discrimination; the presence of an edge-state eigenvalue alone is insufficient.",
    "the task fixes $+0.265$ and $-0.165$ only as source-stated geometries for the stationary and driven comparison;",
    "an energy scan is optional supporting evidence, not a hidden required output;",
    "if submitted, grade only a converged dimerization-lowered two-basin structure and the instability of the undistorted case;",
    "never grade the exact source minima, relative well depth, central maximum, curvature or pixel similarity;",
    "do not tune an unconverged numerical convention to imitate the published curve.",
    "The task fixes the physical chain, geometries and total-dipole spectral class, not full/frozen branches, a particular window, a ground-state plot list or one mechanism diagnostic. Accept momentum-resolved stationary maps, real-space edge-state localization, projectors, spectral functions or equivalent stationary evidence by function. Accept the canonical full/frozen comparison or another intervention only when it genuinely tests the role of the evolving internal potential. Accept an incoherent-sum comparison, controlled edge occupation, impurity control, orbital-removal test or another intervention for the cancellation evidence/mechanism only when it genuinely distinguishes the competing physical explanations.",
    "Do not accept an SSH result as equivalent to the specified interacting continuous-space model. Do not grade source colours, crops, arrows, exact energy-curve shape, absolute HHG magnitude or a fixed number of figures. Charge an upstream numerical defect first to the physical output evidence it damages; A dependent scientific inference loses credit only when the same defect makes the inference false or unsupported.",
    "For F1/F3-F4/M3-M4, article pp.177401-2-4, Eqs.(3),(4),(8) and Fig.3, the spin densities and internal potential obey\n$$n_\\sigma(x,t)=\\sum_{i\\in\\mathrm{occ}}|\\varphi_{\\sigma i}(x,t)|^2,\\quad\nu[n](x,t)=\\int\\frac{n(x',t)dx'}{\\sqrt{(x-x')^2+1}},\\quad\nv_{x,\\sigma}=-[6n_\\sigma/\\pi]^{1/3}.$$\nThe coherent observable is\n$$X(t)=\\sum_{i,\\sigma}\\int x|\\varphi_{\\sigma i}(x,t)|^2dx,\\qquad\nS_D(\\Omega)\\propto|\\mathcal F[wX](\\Omega)|^2,$$\nnot $\\sum_{i,\\sigma}|\\mathcal F[wx_{\\sigma i}]|^2$. The latter is an optional cancellation diagnostic. Derivative-form spectra differ by $\\Omega^2$ or $\\Omega^4$ when transform/window boundary terms are controlled. Frozen-KS comparison freezes the internal Hartree/exchange potential at its stationary value, not the external laser coupling.\n\nThe continuous-space orbital equation is $i\\partial_t\\varphi_{\\sigma i}=[-\\partial_x^2/2+v_{\\rm ion}(x)-iA(t)\\partial_x+u[n]+v_{x,\\sigma}]\\varphi_{\\sigma i}$, with $v_{\\rm ion}(x)=-\\sum_j[(x-x_j)^2+1]^{-1/2}$ and $x_j=(j-(N+1)/2)a-(-1)^j\\delta$. This is the x-only LSD velocity-gauge convention of source Eqs.(1)-(6); a common time-dependent scalar A-squared phase does not change the graded density/dipole."
  ],
  "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 (Establish the specified N=100, soft-core, spin-neutral continuous-space x-only LSD model and fixed M/D+/D- geometries; an SSH/tight-binding replacement is not equivalent.); F2 (Construct finite-chain stationary analysis capable of distinguishing metal, insulating manifolds and edge-localized states from the continuous-space calculation rather than assigning labels from the sign of delta.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required indispensable claim CL2, central physical object: all of: F3 (Construct the specified driven dynamics from the corresponding occupied stationary orbitals and the spin-summed total-dipole spectrum. A current/acceleration representation requires its relation and frequency weighting to that object.); F4 (Consistently use units, specified field, Fourier convention, orbital occupation and one common constant spectral scale across M/D+/D- and dynamical comparisons.). 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: O1 (Present reconstructible stationary evidence distinguishing M from both insulators and establishing D-'s in-gap pair, occupation and edge localization relative to D+.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required indispensable claim CL2, indispensable material component: all of: O2 (Present correct common-scale spin-summed total-dipole HHG spectra for M, D+ and D-, making sub-gap relative strength and case ordering assessable.); O3 (Present a valid comparison/intervention showing that evolving internal Kohn-Sham potential does not generate the D-/D+ sub-gap contrast and that M is more sensitive to that dynamics; full/frozen propagation is accepted but not mandatory.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required co primary claim CL3, indispensable material component: O4 (Present a candidate-designed calculation/control distinguishing edge occupation/incomplete coherent cancellation from gap-only, distortion-only or generic edge-state-presence explanations; no named canonical control is mandatory.). 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: M1 (Explain finite-chain phase identities and the finite-chain/topological character of D- edge states using continuous-space evidence, not generic detached SSH language.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required indispensable claim CL2, principal mechanism or commissioned conclusion: all of: M2 (Infer how topology is encoded in the low-energy HHG response from the identified common-scale sub-gap ordering, limiting the conclusion to supported relative rather than absolute strength.); M6 (Use the identified ordering and intervention sensitivity to distinguish robust D-/D+ insulating contrast from stronger M sensitivity to the evolving self-consistent potential.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required co primary claim CL3, principal mechanism or commissioned conclusion: all of: M3 (Use discriminating evidence to explain near-cancellation of strong individual-orbital radiation in the coherent fully occupied D+ valence total and incomplete cancellation in D-.); M4 (Connect half occupation of the D- in-gap edge pair to incomplete coherent cancellation and explain why an in-gap eigenvalue alone does not establish the mechanism.); M5 (Explicitly reject at least one plausible gap-only, distortion-only or edge-state-presence-only alternative using the causal discriminating evidence.). Judge the role-specific failure facts, not the corresponding atom scores.",
      "Required co primary claim CL4, principal mechanism or commissioned conclusion: M7 (Bound the conclusion to the specified finite-chain continuous-space x-only LSD model, honestly handling known energy-curve differences or unresolved numerical features without manufacturing certainty or demanding exact source-curve reproduction.). 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 co primary claim CL3, material validation/scope: V1 (Only when the model, transform/rescaling, occupation control or claim creates a concrete material risk to the central conclusion, provide targeted discriminating evidence; otherwise record N/A with fixed credit and no evidence claim.). 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 (Establish the specified N=100, soft-core, spin-neutral continuous-space x-only LSD model and fixed M/D+/D- geometries; an SSH/tight-binding replacement is not equivalent.); F2 (Construct finite-chain stationary analysis capable of distinguishing metal, insulating manifolds and edge-localized states from the continuous-space calculation rather than assigning labels from the sign of delta.).",
      "Close CL1 indispensable material component: O1 (Present reconstructible stationary evidence distinguishing M from both insulators and establishing D-'s in-gap pair, occupation and edge localization relative to D+.).",
      "Close CL1 principal mechanism or commissioned conclusion: M1 (Explain finite-chain phase identities and the finite-chain/topological character of D- edge states using continuous-space evidence, not generic detached SSH language.).",
      "Close CL2 central physical object: all of: F3 (Construct the specified driven dynamics from the corresponding occupied stationary orbitals and the spin-summed total-dipole spectrum. A current/acceleration representation requires its relation and frequency weighting to that object.); F4 (Consistently use units, specified field, Fourier convention, orbital occupation and one common constant spectral scale across M/D+/D- and dynamical comparisons.).",
      "Close CL2 indispensable material component: all of: O2 (Present correct common-scale spin-summed total-dipole HHG spectra for M, D+ and D-, making sub-gap relative strength and case ordering assessable.); O3 (Present a valid comparison/intervention showing that evolving internal Kohn-Sham potential does not generate the D-/D+ sub-gap contrast and that M is more sensitive to that dynamics; full/frozen propagation is accepted but not mandatory.).",
      "Close CL2 principal mechanism or commissioned conclusion: all of: M2 (Infer how topology is encoded in the low-energy HHG response from the identified common-scale sub-gap ordering, limiting the conclusion to supported relative rather than absolute strength.); M6 (Use the identified ordering and intervention sensitivity to distinguish robust D-/D+ insulating contrast from stronger M sensitivity to the evolving self-consistent potential.).",
      "Close CL3 indispensable material component: O4 (Present a candidate-designed calculation/control distinguishing edge occupation/incomplete coherent cancellation from gap-only, distortion-only or generic edge-state-presence explanations; no named canonical control is mandatory.).",
      "Close CL3 principal mechanism or commissioned conclusion: all of: M3 (Use discriminating evidence to explain near-cancellation of strong individual-orbital radiation in the coherent fully occupied D+ valence total and incomplete cancellation in D-.); M4 (Connect half occupation of the D- in-gap edge pair to incomplete coherent cancellation and explain why an in-gap eigenvalue alone does not establish the mechanism.); M5 (Explicitly reject at least one plausible gap-only, distortion-only or edge-state-presence-only alternative using the causal discriminating evidence.).",
      "Close CL3 material validation/scope: V1 (Only when the model, transform/rescaling, occupation control or claim creates a concrete material risk to the central conclusion, provide targeted discriminating evidence; otherwise record N/A with fixed credit and no evidence claim.).",
      "Close CL4 principal mechanism or commissioned conclusion: M7 (Bound the conclusion to the specified finite-chain continuous-space x-only LSD model, honestly handling known energy-curve differences or unresolved numerical features without manufacturing certainty or demanding exact source-curve reproduction.)."
    ]
  },
  "references": [
    {
      "citation": "Dieter Bauer and Kenneth K. Hansen. 2018. *High-Harmonic Generation in Solids with and without Topological Edge States*. *Physical Review Letters*, 120, 177401.",
      "url": "https://doi.org/10.1103/PhysRevLett.120.177401",
      "scientific_scope": "High-Harmonic Generation in Solids with and without Topological Edge States",
      "anchors": [
        "Bauer and Hansen (2018), article p.177401-2, Eqs.(1)-(6): soft-core continuous-space x-only LSD chain, spin-neutral occupation and fixed dimerized geometries; not a tight-binding replacement.",
        "Bauer and Hansen (2018), article pp.177401-2-3, Fig.1(b-d), Eq.(6) and edge-state paragraph: half-filled metal, dimerized insulators and an in-gap edge pair with only its lower member occupied. Fig.1(a) exact energy minima are not compulsory targets.",
        "Bauer and Hansen (2018), article p.177401-3, Eqs.(7),(8): driven KS orbitals, spin-summed coherent total dipole and squared Fourier modulus; current/acceleration variants require omega^2/omega^4 weighting.",
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and sub-gap discussion: large D- (source phase B) / D+ (source phase A) contrast on common scale; exact 14-order magnitude or colors are not targets.",
        "Bauer and Hansen (2018), article pp.177401-3-4, Fig.2(a,b) and frozen-KS/screening paragraphs: insulating contrast survives frozen internal potential, while metal is much more sensitive. A functionally equivalent intervention is accepted.",
        "Bauer and Hansen (2018), article p.177401-4, Fig.3 and paragraph beginning \"The key question\": incoherent orbital powers versus coherent total distinguish cancellation. The same page, \"Topological vs other edge states\", describes impurity/occupation controls and a virtual extra electron filling the vacant edge state; these are optional canonical causal examples.",
        "Bauer and Hansen (2018), article pp.177401-2-4, model equations and conclusion delimit finite-chain mean-field scope. Exact Fig.1(a) energy-curve reproduction is excluded by the task/effective rubric because its printed-functional reproduction remains unresolved; risk validation is task-defined."
      ]
    }
  ]
}