{
  "case_id": "Case40",
  "title": "Direct nondipole photon-momentum partition and its large-order limit",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Establish the source-equivalent direct length-gauge nondipole SFA amplitude with the same light-cone Volkov phase, leading-1/c momentum shift/prefactor and state-specific orbital-gradient matrix element.",
      "award_conditions": [
        "Establish the source-equivalent direct length-gauge nondipole SFA amplitude with the same light-cone Volkov phase, leading-1/c momentum shift/prefactor and state-specific orbital-gradient matrix element."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for F3, F4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "He, Lao and He (2017), article pp. 163203-1–2, Eqs. (1)–(5), especially Eq. (5). Direct length-gauge light-cone SFA amplitude and leading nondipole shift/prefactor. Velocity-gauge Table I is not the commissioned branch."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Bind 1s,2s,2p_0,2p_+1 and 2p_−1, their individual binding energies and momentum orbitals, right-circular polarization and propagation axis into a matched family.",
      "award_conditions": [
        "Bind 1s,2s,2p_0,2p_+1 and 2p_−1, their individual binding energies and momentum orbitals, right-circular polarization and propagation axis into a matched family."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for F1, F4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article pp. 163203-3–4, Fig. 2(a,b), caption and magnetic-sublevel paragraph. Five hydrogenic state identities, circular handedness and angular-momentum dependence. State-specific binding energies/orbitals follow standard hydrogenic quantum mechanics, not a fitted source ordering."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "F4",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Define normalized mean longitudinal momentum, paired mean kinetic energy, photon-order identity, a transparent comparison metric and reconstructible case-to-data mapping.",
      "award_conditions": [
        "Normalized mean longitudinal momentum and kinetic energy are paired from the same direct-branch distribution with the declared measure and units.",
        "Photon-order, comparison metric and underlying-data mapping permit independent reconstruction without substituting the asymptotic law for either computed moment."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for F1, F3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2. Normalized longitudinal mean and reconstructible one-member/full-five-state response roles. At-least-one valid member and five-of-five family completeness are separate task obligations."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present at least one physically valid, reconstructible and adequately displayed normalized-<pz> intensity-response member among the five specified states, with correct local state/case mapping.",
      "award_conditions": [
        "At least one specified-state intensity-response member is valid, reconstructible and locally adequate; state identity is checked for that member.",
        "Full five-state completeness is not a prerequisite to this local milestone.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F4). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O2, O3, O4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "Full five-state F3 credit is not a hard prerequisite for valid partial coverage; verify each member's physical identity locally."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2. Normalized longitudinal mean and reconstructible one-member/full-five-state response roles. At-least-one valid member and five-of-five family completeness are separate task obligations."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present a reconstructible same-direct-branch sequence of paired <pz> and <Ep> from few-photon to substantially larger absorbed-photon order.",
      "award_conditions": [
        "Present a reconstructible same-direct-branch sequence of paired <pz> and <Ep> from few-photon to substantially larger absorbed-photon order.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F4). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O3, O4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph. Finite-to-large-order partition structure. The task tests it with independently computed paired means and a transparent discrepancy; fitted coefficient panels are source evidence, not compulsory outputs."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Construct and present a transparent discrepancy or equivalent comparison from the paired sequence sufficient to discriminate finite-order and asymptotic behaviour and identify its domain.",
      "award_conditions": [
        "Construct and present a transparent discrepancy or equivalent comparison from the paired sequence sufficient to discriminate finite-order and asymptotic behaviour and identify its domain.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: O2. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph. Finite-to-large-order partition structure. The task tests it with independently computed paired means and a transparent discrepancy; fitted coefficient panels are source evidence, not compulsory outputs."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "O4",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present the full five-of-five homogeneous response family: every 1s,2s,2p_0,2p_+1,2p_−1 member is physically valid, reconstructible, adequately presented and mutually comparable under matched cases, normalization and coordinates.",
      "award_conditions": [
        "All five specified state members are physically valid, reconstructible and adequately presented.",
        "They form one mutually comparable family under matched cases, normalization and coordinates; fewer than five does not satisfy this full-coverage milestone.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F4). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O2, O3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2. Normalized longitudinal mean and reconstructible one-member/full-five-state response roles. At-least-one valid member and five-of-five family completeness are separate task obligations."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify sign, scale, intensity response and distinguishable relative l,m structure from the complete five-state comparable family.",
      "award_conditions": [
        "Identify sign, scale, intensity response and distinguishable relative l,m structure from the complete five-state comparable family."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: O4. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I2, I3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2. Normalized longitudinal mean and reconstructible one-member/full-five-state response roles. At-least-one valid member and five-of-five family completeness are separate task obligations."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify finite-to-larger-order discrepancy evolution from the candidate's paired sequence and comparison rather than taking the asymptotic relation as input.",
      "award_conditions": [
        "Identify finite-to-larger-order discrepancy evolution from the candidate's paired sequence and comparison rather than taking the asymptotic relation as input."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (O2 AND O3). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I1, I3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph. Finite-to-large-order partition structure. The task tests it with independently computed paired means and a transparent discrepancy; fitted coefficient panels are source evidence, not compulsory outputs."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Distinguish an evidence-supported asymptotic domain from still-finite-order samples, with comparison scope and uncertainty.",
      "award_conditions": [
        "Distinguish an evidence-supported asymptotic domain from still-finite-order samples, with comparison scope and uncertainty."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: O3. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I1, I2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph. Finite-to-large-order partition structure. The task tests it with independently computed paired means and a transparent discrepancy; fitted coefficient panels are source evidence, not compulsory outputs."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain initial-angular-momentum dependence of longitudinal partition from the candidate's multistate evidence.",
      "award_conditions": [
        "Explain initial-angular-momentum dependence of longitudinal partition from the candidate's multistate evidence."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M2, M3, M4, M5, M6, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article pp. 163203-3–4, Fig. 2(a,b), state-dependent partition paragraphs; Eq. (5), p. 163203-2. Orbital-gradient momentum structure and initial angular momentum explain state dependence; opposite magnetic substates need not have identical longitudinal means."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Connect that dependence to bound-state momentum structure without treating source ordering as an imposed prior.",
      "award_conditions": [
        "Connect that dependence to bound-state momentum structure without treating source ordering as an imposed prior."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M3, M4, M5, M6, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article pp. 163203-3–4, Fig. 2(a,b), state-dependent partition paragraphs; Eq. (5), p. 163203-2. Orbital-gradient momentum structure and initial angular momentum explain state dependence; opposite magnetic substates need not have identical longitudinal means."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain the under-barrier contribution to the observed finite-intensity state dependence.",
      "award_conditions": [
        "Explain the under-barrier contribution to the observed finite-intensity state dependence."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M4, M5, M6, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (6), tunnelling/under-barrier paragraph, Eq. (7), Fig. 2(b). Continuum energy and under-barrier/bound-state contributions are distinct. The velocity-gauge limiting Eq. (6) does not replace the length-gauge result."
      ],
      "rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Distinguish and connect continuum-energy and bound/under-barrier contributions to longitudinal partition.",
      "award_conditions": [
        "Distinguish and connect continuum-energy and bound/under-barrier contributions to longitudinal partition."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M5, M6, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (6), tunnelling/under-barrier paragraph, Eq. (7), Fig. 2(b). Continuum energy and under-barrier/bound-state contributions are distinct. The velocity-gauge limiting Eq. (6) does not replace the length-gauge result."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M5",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Infer the large-order partition relation from the candidate's paired moments, not mere repetition of a known formula.",
      "award_conditions": [
        "Infer the large-order partition relation from the candidate's paired moments, not mere repetition of a known formula."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I2. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M4, M6, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and large-order limit paragraph; length-gauge qualification on p. 163203-2. Source limits alpha→1, beta→1/3, gamma→0 motivate the tested relation. A same-direct-branch paired-moment discrepancy is the task-defined numerical test, not a printed separate source figure."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use the explicit discrepancy to test <pz>→(<Ep>+Ip/3)/c and explain finite-order deviations and the chosen metric.",
      "award_conditions": [
        "Use the explicit discrepancy to test <pz>→(<Ep>+Ip/3)/c and explain finite-order deviations and the chosen metric."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (O3 AND I2). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M4, M5, M7, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and large-order limit paragraph; length-gauge qualification on p. 163203-2. Source limits alpha→1, beta→1/3, gamma→0 motivate the tested relation. A same-direct-branch paired-moment discrepancy is the task-defined numerical test, not a printed separate source figure."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Establish that the relation follows from independently computed same-direct-branch observables, not separate fitted coefficient panels or formula substitution.",
      "award_conditions": [
        "Establish that the relation follows from independently computed same-direct-branch observables, not separate fitted coefficient panels or formula substitution."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (O2 AND O3). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M4, M5, M6, M8; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and large-order limit paragraph; length-gauge qualification on p. 163203-2. Source limits alpha→1, beta→1/3, gamma→0 motivate the tested relation. A same-direct-branch paired-moment discrepancy is the task-defined numerical test, not a printed separate source figure."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M8",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "State evidence-supported direct circular-polarization asymptotic scope and its finite-to-asymptotic mechanism without extrapolation to rescattering, linear polarization, spin or absolute yield.",
      "award_conditions": [
        "State evidence-supported direct circular-polarization asymptotic scope and its finite-to-asymptotic mechanism without extrapolation to rescattering, linear polarization, spin or absolute yield."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (I2 AND I3). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M4, M5, M6, M7; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-2, length-gauge paragraph; pp. 163203-3–4, circular versus linear/rescattering branch discussion. Direct circular asymptotic scope and separation from rescattering, spin and absolute-yield claims."
      ],
      "rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Limit conclusions to the calculated direct length-gauge-equivalent right-circular branch, specified states and supported order/intensity domain.",
      "award_conditions": [
        "Limit conclusions to the calculated direct length-gauge-equivalent right-circular branch, specified states and supported order/intensity domain."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for V2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-2, length-gauge paragraph; pp. 163203-3–4, circular versus linear/rescattering branch discussion. Direct circular asymptotic scope and separation from rescattering, spin and absolute-yield claims."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "If actual ATI resolution, subtractive-moment cancellation, pulse convention, order sampling, state/case binding or claims trigger a concrete threat to a central conclusion, provide discriminating evidence or proportionate restriction; otherwise record N/A.",
      "award_conditions": [
        "If actual ATI resolution, subtractive-moment cancellation, pulse convention, order sampling, state/case binding or claims trigger a concrete threat to a central conclusion, provide discriminating evidence or proportionate restriction; otherwise record N/A.",
        "When triggered, award only if claim-relevant discriminating evidence resolves the threat or an evidence-based restriction makes it immaterial; merely acknowledging an unresolved material threat is not positive validation credit. With no trigger, record N/A and retain the fixed 0.5 without claiming performed validation."
      ],
      "applicability_and_na": [
        "Applies only when the submitted method, data, anomaly or conclusion creates the concrete material risk described here. If untriggered, record not_applicable and retain the fixed 0.5 contribution; this is not evidence that validation was performed, and does not block RC4."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for V1; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept algebraically equivalent leading-1/c direct length-gauge amplitudes with the same light-cone phase, momentum shift and orbital-gradient matrix element; consistent Cartesian/cylindrical integration and analytic hydrogenic or independently validated numerical Fourier orbitals are allowed. Dipole-only, velocity-gauge table targets and rescattering-only calculations are not equivalent.",
        "Accept transparent alternative discrepancy metrics, photon-order sequences and pulse conventions that discriminate finite from asymptotic behaviour and state their domain. No fixed intensity/frequency grid, N=4/8/16 sequence, coefficient fit, delta-pz curve, source panel or absolute yield is required.",
        "Presented energy-resolved probabilities or an absorbed-energy ledger that uniquely reconstruct paired mean kinetic energy are equivalent to separately listed <Ep>, provided direct branch, pulse convention, normalization and finite-order domain agree. Nominal order/frequency labels, substitution of the asymptotic formula or examiner reconstruction from unpresented data are not equivalent. The comparison/residual must still be candidate-presented."
      ],
      "source_support": [
        "Article p. 163203-3, Eq. (7) and paragraph on unreliable extraction of the small ponderomotive coefficient; Eq. (5), p. 163203-2. Concrete moment-resolution and finite-order inference threats. Conditional validation is task-defined and imposes no particular ATI mesh or residual metric."
      ],
      "rc_relation": "Material-validation role: RC3 only for a triggered, unresolved threat to a central conclusion, with all six burden-of-proof elements established. N/A does not block RC4."
    }
  ],
  "case_scientific_rules": [
    "The direct amplitude is W_L^(1)(p)=−i integral d eta exp{i[S_p(eta)+Ip eta]}(1−pz/c) E(eta)·i grad_k phi_tilde_0(k), evaluated at k=(p_perp+A(eta),pz−[Ep+Ip]/c), with Ep=p²/2 and the corresponding nondipole light-cone Volkov phase.",
    "Set P(p)=|W_L^(1)(p)|² and form <pz>=integral pz P d³p/integral P d³p and <Ep>=integral Ep P d³p/integral P d³p from the same distribution. Overall orbital normalization cancels; peak displacement, standard deviation or an unnormalized integral is not the required mean.",
    "The matched 1400-nm right-circular family contains 1s,2s,2p_0,2p_+1,2p_−1 with their correct Ip and momentum orbitals, propagation along z and consistent helicity. Judge positive source-scale means, systematic intensity response and distinguishable angular-momentum-dependent relative structure, not exact source samples.",
    "The large-order direct relation is <pz>→(<Ep>+Ip/3)/c. It must be tested with independently computed paired observables and a transparent discrepancy; imposing the relation or separate alpha/beta/gamma fits does not verify it. Finite-order deviations and actual asymptotic domain must be distinguished.",
    "O1 is an at-least-one-valid-member milestone; O4 is the five-of-five physically valid, reconstructible, adequately presented and mutually comparable milestone. Preserve O1 when full coverage fails. Cross-state identification I1 requires five comparable members, not a lone member.",
    "Independent coefficient panels, Table-I velocity-gauge values, rescattering, spin, absolute yield and a complete momentum-distribution display are outside required scope. ATI under-resolution, cancellation of close moments, pulse conventions and finite-order-as-asymptotic claims matter only when actually material.",
    "Score exactly twenty fixed 0/0.5 material-core criteria. Do not import numerical score ranges or RC score caps. A valid run with no assessable scientific work receives score 0 and RC0 before criterion summation; assessable partial work and timeouts are graded on available evidence.",
    "Object existence, candidate presentation, candidate identification, physical inference and validation are different acts. Inspect every candidate-authored claim-bearing figure. A reference, rerender or examiner reconstruction can verify an object but cannot create missing candidate reasoning.",
    "When a material decision depends on feature resolution, fine structure, component relations, formula identity, representation equivalence or suspicious morphology, perform a focused applicable source/reference check. Establish model/observable/case/component, units, basis/gauge, normalization and transformations first; do not turn source-only display choices into new duties. A provisional score at least 9.0 or provisional RC4 requires a focused check of one central output and one principal-mechanism link. Reference verification is examiner-side, not a new candidate deliverable.",
    "In atomic units, $W_L^{(1)}(\\mathbf p)=-i\\int d\\eta\\,e^{i[S_{\\mathbf p}(\\eta)+I_p\\eta]}(1-p_z/c)\\mathbf E(\\eta)\\cdot i\\nabla_{\\mathbf k}\\widetilde\\phi_0(\\mathbf k)|_{\\mathbf k=(\\mathbf p_\\perp+\\mathbf A,\\,p_z-[p^2/2+I_p]/c)}$ (Eq. (5)). Here $\\eta=t-z/c$, $\\mathbf E=-\\partial_\\eta\\mathbf A$ and $S_{\\mathbf p}=\\int^\\eta[p^2/2+(\\mathbf A\\cdot\\mathbf p+A^2/2)/(1-p_z/c)]d\\eta\\prime$. Algebraically equivalent leading-$1/c$ forms are accepted in the nonrelativistic nondipole domain; an arbitrary gauge swap in SFA is not automatically equivalent.",
    "For $P(\\mathbf p)=|W_L^{(1)}|^2$, $\\langle p_z\\rangle=\\int p_zP\\,d^3p/\\int P\\,d^3p$ and $\\langle E_p\\rangle=\\int(p^2/2)P\\,d^3p/\\int P\\,d^3p$. Both moments must use the same direct-branch state, case, measure and distribution. A peak shift or unnormalized numerator is not the longitudinal mean; overall orbital normalization cancels.",
    "The independently tested asymptotic relation is $\\langle p_z\\rangle\\to[\\langle E_p\\rangle+I_p/3]/c$. An admissible residual is $D=c\\langle p_z\\rangle-\\langle E_p\\rangle-I_p/3$ or a transparently scaled equivalent; it is computed from paired moments rather than used to generate either moment. $N=\\lfloor I_p/\\omega\\rfloor+1$ is the source threshold-order convention, with pulse/channel conventions declared. The five hydrogenic states have $I_p=1/(2n^2)$, so $1s$ and $n=2$ states cannot share an incorrectly fixed binding energy.",
    "O1 requires at least one locally valid specified-state member; O4 requires all five mutually comparable members. A missing member damages the complete family and conclusions genuinely requiring it, but cannot erase valid local-member output or independently valid foundations. The large-order paired family is independently assessable; neither a missing Delta-pz curve nor unsubmitted coefficient-fit panels creates another component failure."
  ],
  "closure_requirements": {
    "RC0": [
      "The central physical object is absent or materially invalid. Establish this scientific fact independently of atomic credit; a local notation or presentation weakness is not itself RC0.",
      "(F1 AND F3 AND F4): Establish the source-equivalent direct length-gauge nondipole SFA amplitude with the same light-cone Volkov phase, leading-1/c momentum shift/prefactor and state-specific orbital-gradient matrix element. Bind 1s,2s,2p_0,2p_+1 and 2p_−1, their individual binding energies and momentum orbitals, right-circular polarization and propagation axis into a matched family. Define normalized mean longitudinal momentum, paired mean kinetic energy, photon-order identity, a transparent comparison metric and reconstructible case-to-data mapping."
    ],
    "RC1": [
      "The central object exists, but a required indispensable component is absent or materially invalid. A failed presentation or overextended claim alone does not establish object absence or invalidity.",
      "O4: Present the full five-of-five homogeneous response family: every 1s,2s,2p_0,2p_+1,2p_−1 member is physically valid, reconstructible, adequately presented and mutually comparable under matched cases, normalization and coordinates.",
      "(O2 AND O3): Present a reconstructible same-direct-branch sequence of paired <pz> and <Ep> from few-photon to substantially larger absorbed-photon order. Construct and present a transparent discrepancy or equivalent comparison from the paired sequence sufficient to discriminate finite-order and asymptotic behaviour and identify its domain."
    ],
    "RC2": [
      "Required objects are scientifically valid, but a required principal mechanism or state-dependence inference is absent or incorrect. Reference evidence cannot provide omitted candidate reasoning.",
      "M3: Explain the under-barrier contribution to the observed finite-intensity state dependence.",
      "M8: State evidence-supported direct circular-polarization asymptotic scope and its finite-to-asymptotic mechanism without extrapolation to rescattering, linear polarization, spin or absolute yield."
    ],
    "RC3": [
      "Required objects and principal mechanisms are established, but a concrete material validation or scope threat to a central conclusion remains unresolved.",
      "V2: If actual ATI resolution, subtractive-moment cancellation, pulse convention, order sampling, state/case binding or claims trigger a concrete threat to a central conclusion, provide discriminating evidence or proportionate restriction; otherwise record N/A.",
      "Before assigning RC3, establish all six elements: the concrete failure mode; its origin in the submitted method/data/conclusion; a fair recognizable scientific risk; the affected central conclusion; why the submitted evidence cannot discriminate or exclude it; and functionally equivalent remedies. Evidence-based restriction may remove unsupported breadth, but cannot repair an incorrect signal, absent component or missing mechanism."
    ],
    "RC4": [
      "Every indispensable and co-primary claim has all its nonempty required role slots closed; supporting claims do not independently determine overall RC. Preserve AND/OR branch logic shown above and use scientific gate outcomes, not atomic scores.",
      "No triggered material risk remains unresolved. Untriggered risk criteria recorded N/A do not block RC4 and do not establish validation evidence.",
      "Classify descriptively by the most severe applicable failed required role (lowest RC), without score ceilings or target distributions. A score/RC tension prompts scientific review, not forced adjustment."
    ]
  },
  "references": [
    {
      "citation": "Pei-Lun He, Di Lao, and Feng He. 2017. *Strong Field Theories beyond Dipole Approximations in Nonrelativistic Regimes*. *Physical Review Letters*, 118, 163203.",
      "url": "https://doi.org/10.1103/PhysRevLett.118.163203",
      "scientific_scope": "Direct length-gauge nondipole circular-polarization SFA for five hydrogen states and independent same-branch paired momentum/energy tests. Source fitted alpha/beta/gamma panels support the asymptotic interpretation but are not required reproductions.",
      "anchors": [
        "He, Lao and He (2017), article pp. 163203-1–2, Eqs. (1)–(5), especially Eq. (5).",
        "Article pp. 163203-3–4, Fig. 2(a,b), caption and magnetic-sublevel paragraph.",
        "Article p. 163203-3, Fig. 2(a), Eq. (7); Eq. (5), p. 163203-2.",
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and adjoining photon-order paragraph.",
        "Article pp. 163203-3–4, Fig. 2(a,b), state-dependent partition paragraphs; Eq. (5), p. 163203-2.",
        "Article p. 163203-3, Eq. (6), tunnelling/under-barrier paragraph, Eq. (7), Fig. 2(b).",
        "Article p. 163203-3, Eq. (7), Fig. 1(a–c) and large-order limit paragraph; length-gauge qualification on p. 163203-2.",
        "Article p. 163203-2, length-gauge paragraph; pp. 163203-3–4, circular versus linear/rescattering branch discussion.",
        "Article p. 163203-3, Eq. (7) and paragraph on unreliable extraction of the small ponderomotive coefficient; Eq. (5), p. 163203-2."
      ]
    }
  ]
}