{ "case_id": "Case46", "title": "CEP-controlled directional ATI from coherent direct and rescattered amplitudes", "atoms": [ { "criterion_id": "F1", "dimension": "F", "points": 0.5, "scientific_requirement": "Bind electric field, CEP, E=−dA/dt, zero endpoint A, pulse/frequency/intensity/binding/ponderomotive inputs and declared longitudinal left/right convention; do not treat the field waveform as A.", "award_conditions": [ "Bind electric field, CEP, E=−dA/dt, zero endpoint A, pulse/frequency/intensity/binding/ponderomotive inputs and declared longitudinal left/right convention; do not treat the field waveform as A." ], "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 F2, 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Suárez et al. (2015), author manuscript pp. 5, 8–9, Eqs. (1), (11)–(13) and endpoint/envelope paragraph. Electric-field CEP identity and integrated vector potential. The waveform is E, not A; a direction convention fixes the asymmetry sign." ], "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": "F2", "dimension": "F", "points": 0.5, "scientific_requirement": "Implement the stated d(q), B(q), g(q1,q2), independent Gamma/gamma/epsilon inputs and source-equivalent action, saddle momentum, complex direct b0 and single-rescattering b1, preserving branch signs/phases/normalization and coherent addition.", "award_conditions": [ "The stated d, B and g with independent fixed inputs feed a phase/sign-consistent complex direct and saddle-reduced single-return construction.", "Both actions, return momentum, normalization and coherent amplitude sum are retained; source transcription ambiguities do not override the visible formula policy." ], "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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 8–15, Eqs. (12)–(18), (26), (28), (31)–(37). Complex direct/single-rescattered engine, action, saddle, dipole and continuum element. The visible task fixes the source ambiguities: negative B, 2 Gamma-squared numerators and independent Gamma/gamma inputs; source Eq. (25) does not override them." ], "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": "Define reconstructible longitudinal probabilities/asymmetry and two-dimensional direct, rescattered, signed-interference and coherent-total quantities with shared normalization, units and component/case mapping.", "award_conditions": [ "Define reconstructible longitudinal probabilities/asymmetry and two-dimensional direct, rescattered, signed-interference and coherent-total quantities with shared normalization, units and component/case mapping." ], "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, F2; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript p. 10, Eq. (15); p. 19, Eq. (38); p. 23, Fig. 6(a–d). Signed amplitude interference, common-normalization total and longitudinal left/right definition; reconstructible component reporting is task-defined." ], "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 a reconstructible CEP-resolved longitudinal P_minus/P_plus and asymmetry family with sufficient phase/energy coverage for symmetry/asymmetry, broad cutoff ordering and convention-correct high-energy sign structure.", "award_conditions": [ "Present a reconstructible CEP-resolved longitudinal P_minus/P_plus and asymmetry family with sufficient phase/energy coverage for symmetry/asymmetry, broad cutoff ordering and convention-correct high-energy sign structure.", "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 F2 AND F3). 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." ], "equivalent_evidence": [ "Accept mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 19–21, Eq. (38), Fig. 3 and Fig. 4(a,b). CEP-dependent directional asymmetry and high-energy sign under the stated left/right convention, not source pixels or a mandatory phase grid." ], "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": "O2", "dimension": "O", "points": 0.5, "scientific_requirement": "Present same-family longitudinal direct, rescattered, signed-interference and total spectra exposing low/high-energy hierarchy and the claim-relevant coherent relation.", "award_conditions": [ "Present same-family longitudinal direct, rescattered, signed-interference and total spectra exposing low/high-energy hierarchy and the claim-relevant coherent relation.", "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 F2 AND F3). 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 16–17, Fig. 1(a,b); p. 21, Fig. 4(a,b); Eq. (15), p. 10. Direct-low/rescattered-high hierarchy and coherent contribution. Classical 2 Up/10 Up scales are broad anchors, not exact numerical cutoff thresholds." ], "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": "Present reconstructible polarization-plane direct/rescattered SFA components exposing central low-momentum ATI support and outer larger-momentum ring/arc support.", "award_conditions": [ "Present reconstructible polarization-plane direct/rescattered SFA components exposing central low-momentum ATI support and outer larger-momentum ring/arc support.", "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 F2 AND F3). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for O1, O2, 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 23–24, Fig. 6(a,b) and component interpretation paragraph. Central direct ATI and outer rescattering support in the polarization plane; these are SFA components, not the TDSE panel in Fig. 5(b)." ], "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 same-family two-dimensional signed interference and coherent total with both interference signs, shared normalization, py reflection, CEP-compatible pz asymmetry and amplitude-level identity.", "award_conditions": [ "Same-family 2D signed interference and coherent total are reconstructible on a shared normalization.", "Both interference signs, transverse reflection, CEP-compatible longitudinal asymmetry and amplitude-level total identity remain physically valid.", "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 F2 AND F3). 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 23–24, Fig. 6(c,d), Eq. (15), p. 10. Signed texture and coherent total preserve transverse reflection and CEP-compatible longitudinal imbalance. The task explicitly requires both interference signs; the source logarithmic display cannot erase this identity." ], "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 phase-dependent longitudinal symmetry/asymmetry and material sign structure from the candidate's own valid left/right evidence under the declared convention.", "award_conditions": [ "Identify phase-dependent longitudinal symmetry/asymmetry and material sign structure from the candidate's own valid left/right evidence under the declared convention." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: O1. 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 19–21, Eq. (38), Fig. 3 and Fig. 4(a,b). CEP-dependent directional asymmetry and high-energy sign under the stated left/right convention, not source pixels or a mandatory phase grid." ], "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 direct-low/rescattered-high support, broad cutoff hierarchy and the material signed-interference role from the candidate's longitudinal components.", "award_conditions": [ "Identify direct-low/rescattered-high support, broad cutoff hierarchy and the material signed-interference role from the candidate's longitudinal components." ], "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 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 16–17, Fig. 1(a,b); p. 21, Fig. 4(a,b); Eq. (15), p. 10. Direct-low/rescattered-high hierarchy and coherent contribution. Classical 2 Up/10 Up scales are broad anchors, not exact numerical cutoff thresholds." ], "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": "Identify central/outer component support, signed texture, py symmetry, CEP-compatible pz asymmetry and coherent-total relation from the candidate's two-dimensional data.", "award_conditions": [ "Identify central/outer component support, signed texture, py symmetry, CEP-compatible pz asymmetry and coherent-total relation from the candidate's two-dimensional data." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (O3 AND O4). 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 23–24, Fig. 6(c,d), Eq. (15), p. 10. Signed texture and coherent total preserve transverse reflection and CEP-compatible longitudinal imbalance. The task explicitly requires both interference signs; the source logarithmic display cannot erase this identity." ], "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 how few-cycle electric-field CEP changes waveform symmetry, birth/return timing and accumulated semiclassical phases, not vector-potential phase mislabelling.", "award_conditions": [ "Explain how few-cycle electric-field CEP changes waveform symmetry, birth/return timing and accumulated semiclassical phases, not vector-potential phase mislabelling." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (F1 AND I1). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M2, M5; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 20–21, Fig. 3, Fig. 4(a,b), waveform/phase and birth-return paragraphs; Eqs. (12)–(17), pp. 8–11. Waveform CEP alters event timing and phases, leading to directional coherent response; source vector-potential phase mislabelling is not an equivalent explanation." ], "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": "Explain how bound–free and continuum–continuum factors weight direct/rescattered paths and connect them to central low-energy and higher-energy/outer support.", "award_conditions": [ "Explain how bound–free and continuum–continuum factors weight direct/rescattered paths and connect them to central low-energy and higher-energy/outer support." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (F2 AND I2). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M1, M5; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 8–15, Eqs. (12), (14), (17), (28), (37); pp. 16–17, Fig. 1(a,b). Bound-free and continuum-continuum weighting of direct/return pathways and their energy-domain support." ], "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": "Connect CEP-conditioned amplitude phases and signed interference causally to total left/right imbalance, rather than merely noting coexistence.", "award_conditions": [ "Connect CEP-conditioned amplitude phases and signed interference causally to total left/right imbalance, rather than merely noting coexistence." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (I1 AND I2 AND M1 AND M2). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M5, M6; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 20–21, Fig. 3, Fig. 4(a,b), waveform/phase and birth-return paragraphs; Eqs. (12)–(17), pp. 8–11. Waveform CEP alters event timing and phases, leading to directional coherent response; source vector-potential phase mislabelling is not an equivalent explanation." ], "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim." }, { "criterion_id": "M4", "dimension": "M", "points": 0.5, "scientific_requirement": "Use the candidate's longitudinal asymmetry, component hierarchy and coherence mechanism to conclude how CEP controls directional ATI in this model.", "award_conditions": [ "Use the candidate's longitudinal asymmetry, component hierarchy and coherence mechanism to conclude how CEP controls directional ATI in this model." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (I1 AND I2 AND M3). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M6; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 20–21, Fig. 3, Fig. 4(a,b), waveform/phase and birth-return paragraphs; Eqs. (12)–(17), pp. 8–11. Waveform CEP alters event timing and phases, leading to directional coherent response; source vector-potential phase mislabelling is not an equivalent explanation." ], "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": "M5", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain how the same b0/b1 engine assembles central direct, outer rescattered and signed interference into the two-dimensional total while retaining py symmetry and CEP-dependent pz directionality.", "award_conditions": [ "Explain how the same b0/b1 engine assembles central direct, outer rescattered and signed interference into the two-dimensional total while retaining py symmetry and CEP-dependent pz directionality." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (F2 AND I3). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M1, M2, M3; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 23–24, Fig. 6(a–d), same-engine component paragraphs; Eq. (15), p. 10. One engine assembles the 2D total; signed/common-scale relations distinguish it from incoherent or independently normalized panels." ], "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 two-dimensional support, signs, symmetry and coherent identity to discriminate the shared pathway account from incoherent sums, independent normalization or model conflation.", "award_conditions": [ "Use two-dimensional support, signs, symmetry and coherent identity to discriminate the shared pathway account from incoherent sums, independent normalization or model conflation." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (I3 AND M5). This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for M4; 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 23–24, Fig. 6(a–d), same-engine component paragraphs; Eq. (15), p. 10. One engine assembles the 2D total; signed/common-scale relations distinguish it from incoherent or independently normalized panels." ], "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": "M7", "dimension": "M", "points": 0.5, "scientific_requirement": "Establish compatibility of both domains with one field-conditioned pathway mechanism involving timing, matrix-element weights and amplitude interference.", "award_conditions": [ "Establish compatibility of both domains with one field-conditioned pathway mechanism involving timing, matrix-element weights and amplitude interference." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: (M4 AND M6). This identifies facts, not prerequisite half-point awards.", "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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 19–24, Figs. 3, 4(a,b), 6(a–d); action/amplitude chain Eqs. (12)–(18). Task-defined cross-domain compatibility and bounded causal synthesis, not proof of exact Coulomb or TDSE behavior." ], "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": "M8", "dimension": "M", "points": 0.5, "scientific_requirement": "Answer whether one coherent direct/rescattered account explains longitudinal asymmetry and polarization-plane structure, distinguishing demonstrated compatibility from model-conditional explanation.", "award_conditions": [ "Answer whether one coherent direct/rescattered account explains longitudinal asymmetry and polarization-plane structure, distinguishing demonstrated compatibility from model-conditional explanation." ], "applicability_and_na": [ "Applies to every assessable submission; absence or failure is scored 0, not N/A." ], "dependencies": [ "Necessary scientific premises: M7. This identifies facts, not prerequisite half-point awards.", "Preserve independently valid evidence for V1, 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 19–24, Figs. 3, 4(a,b), 6(a–d); action/amplitude chain Eqs. (12)–(18). Task-defined cross-domain compatibility and bounded causal synthesis, not proof of exact Coulomb or TDSE behavior." ], "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim." }, { "criterion_id": "V1", "dimension": "V", "points": 0.5, "scientific_requirement": "State the fixed-parameter nonlocal short-range first-rescattering SFA, few-cycle, low-depletion and actual one-/two-dimensional observation domain, distinct from TDSE/Coulomb/inversion/experimental/universal claims.", "award_conditions": [ "State the fixed-parameter nonlocal short-range first-rescattering SFA, few-cycle, low-depletion and actual one-/two-dimensional observation domain, distinct from TDSE/Coulomb/inversion/experimental/universal claims." ], "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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript p. 5, low-ionization SFA paragraph; pp. 22–24, potential/low-energy limitations and conclusion. Fixed short-range first-rescattering SFA scope; source Fig. 5(b) TDSE comparison is excluded and no target inversion is established." ], "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 field/CEP/direction binding, B/g transcription, incoherent addition, lost interference sign, normalization, denominator masks, quadrature aliasing, anomalies or claims threaten the central conclusion, provide discrimination or proportionate restriction; otherwise record N/A.", "award_conditions": [ "If actual field/CEP/direction binding, B/g transcription, incoherent addition, lost interference sign, normalization, denominator masks, quadrature aliasing, anomalies or claims threaten the central conclusion, provide discrimination 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 mapped gauges/phases, units, reversed coordinates, common normalization, quadrature/sampling/interpolation and output transforms preserving electric-field CEP, momentum direction, component identity, interference sign and coherent total.", "Accept reconstructible component spectra, momentum maps, tables and equivalent discriminating evidence without fixed phase mesh, energy/momentum range, source panels, weak-tail details or absolute log intensity. TDSE is not a substitute for the specified SFA components or a requirement for full credit." ], "source_support": [ "Author manuscript pp. 8–15, Eqs. (12)–(18), (28), (33), (37); p. 19, Eq. (38). Task-defined conditional phase, matrix-element, aliasing, denominator-mask and normalization risks; no unique mesh or preferred check is required." ], "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 stated waveform is E(t)=E0 sin²(omega t/(2Nc))sin(omega t+phi) zhat with Nc=4, omega=0.057, I=10^14 W cm^−2 and Ip=0.5. Obtain A(t)=−integral_0^t E(s)ds with zero endpoints; Up=E0²/(4omega²). Treating the waveform as A changes the physical CEP and adds an envelope-derivative error.", "Use the stated nonlocal dipole and continuum–continuum elements with independent fixed Gamma=1, gamma=38, epsilon=0.2. B has the specified negative sign, the continuum numerators contain 2Gamma², and the prescribed epsilon signs/complex conjugations must be retained. Do not recompute Gamma/gamma from a hidden bound-state relation.", "Direct b0 uses the bound–free matrix element; b1 additionally retains the saddle return momentum, diffusion factor, continuum–continuum element and complex action. Declare phase, action, saddle and normalization conventions. Construct P_total=|b0+b1|²=|b0|²+|b1|²+2Re(b0 b1*), not added rates.", "For P_minus at pz<0 and P_plus at pz>0 at matched energy, A=(P_minus−P_plus)/(P_minus+P_plus). Judge only adequately supported denominators; masks cannot manufacture the asymmetry.", "Direct weight is primarily low energy (classical scale about 2Up); rescattering reaches a higher plateau (about 10Up). Interference materially affects low-energy structure and generally weakens in the high-energy domain. Near phi=90 degrees direct longitudinal response is approximately reflection symmetric while rescattering can remain asymmetric; phi=0 may be asymmetric. In the stated convention the broad phi=0 high-energy source family is predominantly negative, not every weak bin.", "In the polarization plane, direct response has central low-momentum ATI texture, rescattering reaches outer rings/arcs, signed interference contributes coherent texture, and the total must match amplitude addition. Linear polarization preserves py→−py reflection while CEP can allow pz asymmetry.", "Retain interference sign and shared component normalization; |interference| alone or per-panel normalization cannot close relative-weight relations. Both observation domains must come from the same amplitude engine, not two conflated physical models.", "Scope is the stated fixed-parameter short-range first-rescattering SFA, few-cycle pulse and low-depletion regime; no quantitative TDSE agreement, Coulomb low-energy accuracy, target inversion or experimental/universal CEP conclusion. Optional one-dimensional TDSE context is bounded; unresolved two-dimensional TDSE is not a scoring reference.", "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.", "$\\mathbf E(t)=E_0\\sin^2[\\omega t/(2N_c)]\\sin(\\omega t+\\phi)\\hat{\\mathbf z}$, $\\mathbf A(t)=-\\int_0^t\\mathbf E(s)ds$ and $U_p=E_0^2/(4\\omega^2)$ (source Eq. (1) and pulse paragraph; stated $N_c=4$). $A(0)=A(t_f)=0$ binds the canonical/final kinetic momentum. Directly substituting the supplied E waveform as A changes the physical CEP case.", "$S(\\mathbf p,t,t\\prime)=\\int_{t\\prime}^t[(\\mathbf p+\\mathbf A(s))^2/2+I_p]ds$, $\\mathbf p_s=-(t\\prime-t\\prime\\prime)^{-1}\\int_{t\\prime\\prime}^{t\\prime}\\mathbf A(s)ds$ (Eqs. (13), (16)); $b_0=i\\int_0^{t_f}dt\\prime\\,\\mathbf E(t\\prime)\\cdot\\mathbf d[\\mathbf p+\\mathbf A(t\\prime)]e^{-iS(\\mathbf p,t_f,t\\prime)}$. The source single-return Eq. (17) retains both excursion actions, the matrix element $\\mathbf g$ and complex diffusion factor $[\\pi/(\\varepsilon+i(t\\prime-t\\prime\\prime)/2)]^{3/2}$; omitting their relative phase is not equivalent.", "The visible benchmark resolves the printed source branches by $B(\\mathbf q)=-2\\gamma(q^2+\\Gamma^2)^{-1/2}[1-4\\pi^2i\\gamma/(|\\mathbf q|+i\\Gamma)]^{-1}$ and the stated $2\\Gamma^2$ numerators of $\\mathbf g$, with $\\Gamma=1$, $\\gamma=38$, $\\epsilon=0.2$ independent fixed inputs. These are task-defined formula identities, not a demand to reproduce the conflicting signs/powers in source Eqs. (25), (33), (37).", "$P_{\\rm tot}=|b_0+b_1|^2=|b_0|^2+|b_1|^2+2\\operatorname{Re}(b_0b_1^*)$ and $\\mathcal A=(P_--P_+)/(P_-+P_+)$ (Eqs. (15), (38)), where minus/plus mean negative/positive longitudinal momentum at the same energy. The cross term is signed; its absolute value or independently normalized components are different objects. At a vanishing denominator the asymmetry is undefined, requiring a declared claim-adequate domain rather than artificial extreme values.", "The bound-free identity is $\\mathbf d(\\mathbf q)=i\\mathcal N\\mathbf q[(q^2+\\Gamma^2)+(q^2/2+I_p)]/[(q^2+\\Gamma^2)^{3/2}(q^2/2+I_p)^2]$, $\\mathcal N^2=\\sqrt{2I_p}(\\Gamma+\\sqrt{2I_p})^2/(4\\pi^2)$ (Eqs. (26), (28)). The fixed continuum element is $\\mathbf g_{12}=iB_2\\mathbf q_1(3q_1^2-q_2^2+2\\Gamma^2+i\\epsilon)/[(q_1^2+\\Gamma^2)^{3/2}(q_2^2-q_1^2-i\\epsilon)^2]-iB_1^*\\mathbf q_2(3q_2^2-q_1^2+2\\Gamma^2-i\\epsilon)/[(q_2^2+\\Gamma^2)^{3/2}(q_1^2-q_2^2+i\\epsilon)^2]$, $B_j=B(\\mathbf q_j)$ (task-fixed reading of Eq. (37)). Here $\\Gamma$ is the potential shape scale, $\\gamma$ its independent coupling input and $\\epsilon$ the declared pole regularization; none is the Keldysh parameter." ], "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 F2 AND F3): Bind electric field, CEP, E=−dA/dt, zero endpoint A, pulse/frequency/intensity/binding/ponderomotive inputs and declared longitudinal left/right convention; do not treat the field waveform as A. Implement the stated d(q), B(q), g(q1,q2), independent Gamma/gamma/epsilon inputs and source-equivalent action, saddle momentum, complex direct b0 and single-rescattering b1, preserving branch signs/phases/normalization and coherent addition. Define reconstructible longitudinal probabilities/asymmetry and two-dimensional direct, rescattered, signed-interference and coherent-total quantities with shared normalization, units and component/case 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.", "(O1 AND O2): Present a reconstructible CEP-resolved longitudinal P_minus/P_plus and asymmetry family with sufficient phase/energy coverage for symmetry/asymmetry, broad cutoff ordering and convention-correct high-energy sign structure. Present same-family longitudinal direct, rescattered, signed-interference and total spectra exposing low/high-energy hierarchy and the claim-relevant coherent relation.", "(O3 AND O4): Present reconstructible polarization-plane direct/rescattered SFA components exposing central low-momentum ATI support and outer larger-momentum ring/arc support. Present same-family two-dimensional signed interference and coherent total with both interference signs, shared normalization, py reflection, CEP-compatible pz asymmetry and amplitude-level identity." ], "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.", "M4: Use the candidate's longitudinal asymmetry, component hierarchy and coherence mechanism to conclude how CEP controls directional ATI in this model.", "M6: Use two-dimensional support, signs, symmetry and coherent identity to discriminate the shared pathway account from incoherent sums, independent normalization or model conflation.", "M7: Establish compatibility of both domains with one field-conditioned pathway mechanism involving timing, matrix-element weights and amplitude interference." ], "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 field/CEP/direction binding, B/g transcription, incoherent addition, lost interference sign, normalization, denominator masks, quadrature aliasing, anomalies or claims threaten the central conclusion, provide discrimination 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": "Noslen Suárez, Alexis Chacón, Marcelo F. Ciappina, Jens Biegert, and Maciej Lewenstein. 2015. *Above-threshold ionization and photoelectron spectra in atomic systems driven by strong laser fields*. *Physical Review A*, 92, 063421.", "url": "https://doi.org/10.1103/PhysRevA.92.063421", "scientific_scope": "Fixed-parameter short-range SFA complex b0/b1 engine shared by longitudinal CEP and polarization-plane components. Source manuscript page numbers are used explicitly; task-visible B/g/parameter corrections govern, and TDSE/Coulomb quantitative comparisons are excluded.", "anchors": [ "Suárez et al. (2015), author manuscript pp. 5, 8–9, Eqs. (1), (11)–(13) and endpoint/envelope paragraph.", "Author manuscript pp. 8–15, Eqs. (12)–(18), (26), (28), (31)–(37).", "Author manuscript p. 10, Eq. (15); p. 19, Eq. (38); p. 23, Fig. 6(a–d).", "Author manuscript pp. 19–21, Eq. (38), Fig. 3 and Fig. 4(a,b).", "Author manuscript pp. 16–17, Fig. 1(a,b); p. 21, Fig. 4(a,b); Eq. (15), p. 10.", "Author manuscript pp. 23–24, Fig. 6(a,b) and component interpretation paragraph.", "Author manuscript pp. 23–24, Fig. 6(c,d), Eq. (15), p. 10.", "Author manuscript pp. 20–21, Fig. 3, Fig. 4(a,b), waveform/phase and birth-return paragraphs; Eqs. (12)–(17), pp. 8–11.", "Author manuscript pp. 8–15, Eqs. (12), (14), (17), (28), (37); pp. 16–17, Fig. 1(a,b).", "Author manuscript pp. 23–24, Fig. 6(a–d), same-engine component paragraphs; Eq. (15), p. 10.", "Author manuscript pp. 19–24, Figs. 3, 4(a,b), 6(a–d); action/amplitude chain Eqs. (12)–(18).", "Author manuscript p. 5, low-ionization SFA paragraph; pp. 22–24, potential/low-energy limitations and conclusion.", "Author manuscript pp. 8–15, Eqs. (12)–(18), (28), (33), (37); p. 19, Eq. (38)." ] } ] }