{ "case_id": "Case02", "title": "Case02 — Calculation of multiphoton ionization amplitudes and cross sections of few-electron atoms", "atoms": [ { "criterion_id": "F1", "dimension": "F", "points": 0.5, "scientific_requirement": "Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.", "award_conditions": [ "The hydrogen-1s ordered hierarchy applies D=p_z exactly once with unit coefficient at each photon step; order and E1 channel parity are consistent." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.", "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-3-4, Eqs.(19),(20),(28)-(30), velocity-form paragraph following Eq.(21): ordered unit-coefficient dipole-response hierarchy." ], "rc_relation": "Role: central physical object. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "F2", "dimension": "F", "points": 0.5, "scientific_requirement": "For an open intermediate continuum, construct the multi-wave Coulomb asymptote containing every open intermediate wave and the selected final wave, and a determinate extraction uniquely isolating the final coefficient.", "award_conditions": [ "All open intermediate energies and the selected final energy occur in the outgoing Coulomb representation, and a determinate common complex system isolates the final coefficient." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.)", "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: central physical object. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "F3", "dimension": "F", "points": 0.5, "scientific_requirement": "Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.", "award_conditions": [ "The isolated final coefficient maps to absolute generalized partial cross sections with declared Coulomb normalization and K-dependent units; independently normalized shapes do not qualify." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: F2 (For an open intermediate continuum, construct the multi-wave Coulomb asymptote containing every open intermediate wave and the selected final wave, and a determinate extraction uniquely isolating the final coefficient.)", "Preserve independently valid work in I3, M3, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-2-5, Eqs.(11)-(15),(26),(27),(31) and SI-unit paragraph: Coulomb amplitude normalization and absolute order-dependent generalized cross sections." ], "rc_relation": "Role: central physical object. Trigger: the material physical identity is absent or invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "O4", "dimension": "O", "points": 0.5, "scientific_requirement": "Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.", "award_conditions": [ "All K=4 E1-allowed final waves are presented in absolute units over a domain sufficient for claimed openings and off-pole high-energy behavior.", "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); F3 (Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.)", "Preserve independently valid work in O1, O2, O3, I3, M3, M4, M5, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(b,c) and Sec.III.A threshold paragraphs: intermediate openings near 6.8/13.6 eV for K=3 and 4.5/6.8/13.6 eV for K=4, with off-pole high-energy decline." ], "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid." }, { "criterion_id": "O1", "dimension": "O", "points": 0.5, "scientific_requirement": "Present reconstructible absolute K=2 l=0 and l=2 partial cross sections and their arithmetic total, with correct case binding, stable large-scale structure and absolute normalization.", "award_conditions": [ "Absolute K=2 s/d partial cross sections and their arithmetic total are reconstructible with correct normalization and stable material structure.", "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); F3 (Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.)", "Preserve independently valid work in O2, O3, O4, I1, I3, M1, M3, M4, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(a), Eq.(31) and first Sec.III.A paragraph: hydrogen two-photon s/d partial cross sections and their arithmetic total; E1 channel parity extends to the higher orders in Fig.2(b,c)." ], "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid." }, { "criterion_id": "O2", "dimension": "O", "points": 0.5, "scientific_requirement": "Present reconstructible absolute K=3 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.", "award_conditions": [ "All K=3 E1-allowed final waves are presented in absolute units over a domain sufficient for claimed openings and off-pole high-energy behavior.", "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); F3 (Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.)", "Preserve independently valid work in O1, O3, O4, I3, M3, M4, M5, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(b,c) and Sec.III.A threshold paragraphs: intermediate openings near 6.8/13.6 eV for K=3 and 4.5/6.8/13.6 eV for K=4, with off-pole high-energy decline." ], "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid." }, { "criterion_id": "O3", "dimension": "O", "points": 0.5, "scientific_requirement": "Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.", "award_conditions": [ "A genuinely open-intermediate complex radial response and a common multi-wave reconstruction expose both components in the unscaled asymptotic region using one coefficient set.", "Record object_valid, presentation_adequate and claim_adequate separately; all three must hold for this 0.5-point Output award. Examiner reconstruction cannot supply omitted candidate presentation or reasoning." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); F2 (For an open intermediate continuum, construct the multi-wave Coulomb asymptote containing every open intermediate wave and the selected final wave, and a determinate extraction uniquely isolating the final coefficient.)", "Preserve independently valid work in O1, O2, O4, I1, I2, M1, M2, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: indispensable material component. Trigger: the required output object is absent or physically invalid. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure. Inadequate presentation or unsupported claim alone does not establish that an otherwise valid object is absent or physically invalid." }, { "criterion_id": "I1", "dimension": "I", "points": 0.5, "scientific_requirement": "Identify threshold/structure sequences organized by successive intermediate continuum openings, and off-pole high-energy decline, in the submitted K=3/4 absolute families.", "award_conditions": [ "The candidate identifies intermediate-opening sequences and high-energy off-pole decline in their K=3/4 absolute families." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: O4 (Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.)", "Preserve independently valid work in O1, O3, I3, M3, M4, M5, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(b,c) and Sec.III.A threshold paragraphs: intermediate openings near 6.8/13.6 eV for K=3 and 4.5/6.8/13.6 eV for K=4, with off-pole high-energy decline." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "I2", "dimension": "I", "points": 0.5, "scientific_requirement": "Identify the E1-allowed final partial-wave sets for K=2,3,4, the K=2 total, and correct order/channel binding across the submitted families.", "award_conditions": [ "The candidate identifies K=2 l=0,2; K=3 l=1,3; K=4 l=0,2,4 and the K=2 arithmetic total with correct order binding." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: O1 (Present reconstructible absolute K=2 l=0 and l=2 partial cross sections and their arithmetic total, with correct case binding, stable large-scale structure and absolute normalization.); O4 (Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.)", "Preserve independently valid work in O3, I3, M3, M4, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(a), Eq.(31) and first Sec.III.A paragraph: hydrogen two-photon s/d partial cross sections and their arithmetic total; E1 channel parity extends to the higher orders in Fig.2(b,c)." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "I3", "dimension": "I", "points": 0.5, "scientific_requirement": "Identify common multi-wave beating/asymptotic content caused by the open intermediate channel, the selected final coefficient, and the unscaled radial region actually supporting the identification.", "award_conditions": [ "The candidate identifies common complex beating, the selected final component and the actual unscaled region supporting that interpretation." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: O3 (Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.)", "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M1", "dimension": "M", "points": 0.5, "scientific_requirement": "Use energy conservation and the submitted K=3/4 structures to explain which intermediate absorption steps open and how they organize the observed sequences.", "award_conditions": [ "Energy conservation at successive absorption steps explains the observed K=3/4 threshold sequence and which intermediate continua open." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); O4 (Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.); I1 (Identify threshold/structure sequences organized by successive intermediate continuum openings, and off-pole high-energy decline, in the submitted K=3/4 absolute families.)", "Preserve independently valid work in O1, O3, I3, M3, M4, M5, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(b,c) and Sec.III.A threshold paragraphs: intermediate openings near 6.8/13.6 eV for K=3 and 4.5/6.8/13.6 eV for K=4, with off-pole high-energy decline." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M2", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain the allowed final waves, K=2 arithmetic total and retained off-pole high-energy behavior from absolute data rather than channel-free plot description.", "award_conditions": [ "The candidate explains E1 final-channel parity, K=2 summation and off-pole decline using their absolute data rather than channel-free descriptions." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F1 (Establish the ordered hydrogen-1s K-photon driven-response hierarchy with velocity-form D=p_z, one unit-coefficient application of D per step, and correct photon-order/E1-channel binding.); O1 (Present reconstructible absolute K=2 l=0 and l=2 partial cross sections and their arithmetic total, with correct case binding, stable large-scale structure and absolute normalization.); O4 (Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.); I2 (Identify the E1-allowed final partial-wave sets for K=2,3,4, the K=2 total, and correct order/channel binding across the submitted families.)", "Preserve independently valid work in O3, I3, M3, M4, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-5, Fig.2(a), Eq.(31) and first Sec.III.A paragraph: hydrogen two-photon s/d partial cross sections and their arithmetic total; E1 channel parity extends to the higher orders in Fig.2(b,c)." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M3", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain why a single outgoing final wave is insufficient with an open intermediate continuum and why open intermediate and selected final energy Coulomb waves must coexist.", "award_conditions": [ "The nonlocalized continuum driving term is connected to simultaneous intermediate- and final-energy outgoing waves; a lone final wave is shown insufficient." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F2 (For an open intermediate continuum, construct the multi-wave Coulomb asymptote containing every open intermediate wave and the selected final wave, and a determinate extraction uniquely isolating the final coefficient.); O3 (Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.); I3 (Identify common multi-wave beating/asymptotic content caused by the open intermediate channel, the selected final coefficient, and the unscaled radial region actually supporting the identification.)", "Preserve independently valid work in O1, O2, O4, M1, M2, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M4", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain how the common multi-wave coefficient system isolates the final outgoing coefficient and connects it to absolute cross sections; Wronskian, direct complex fit, surface relation and functionally equivalent extraction are accepted.", "award_conditions": [ "The candidate explains how the common coefficient system distinguishes the final outgoing coefficient and connects it to absolute cross sections." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F2 (For an open intermediate continuum, construct the multi-wave Coulomb asymptote containing every open intermediate wave and the selected final wave, and a determinate extraction uniquely isolating the final coefficient.); F3 (Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.); O3 (Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.); I3 (Identify common multi-wave beating/asymptotic content caused by the open intermediate channel, the selected final coefficient, and the unscaled radial region actually supporting the identification.)", "Preserve independently valid work in O1, O2, O4, M1, M2, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M5", "dimension": "M", "points": 0.5, "scientific_requirement": "Use one coefficient set to explain both real and imaginary response components and determine whether the fixed-wave-number Coulomb representation holds; separately fitting the components does not establish the common complex structure.", "award_conditions": [ "One complex coefficient set explains real and imaginary radial components and supports or bounds the fixed-wave-number representation." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: O3 (Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.); I3 (Identify common multi-wave beating/asymptotic content caused by the open intermediate channel, the selected final coefficient, and the unscaled radial region actually supporting the identification.)", "Preserve independently valid work in O1, O2, O4, M1, M2, M6, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article p.043108-4, Fig.1 and Eqs.(22)-(26): open-intermediate multi-wave asymptote and one complex coefficient system isolating the final wave. The two-wave omega=1 example is illustrative, not mandatory." ], "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M6", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain why coefficient recovery is supported only in the unscaled asymptotic region; distinguish ECS-contour response from real-axis continuation and state the boundary near exact bound poles or continuum resonances.", "award_conditions": [ "Coefficient recovery is justified only in an unscaled asymptotic region; ECS-contour damping, real-axis continuation and near-pole limitations are distinguished." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: O3 (Present a genuinely open-intermediate complex radial response and one common multi-wave outgoing reconstruction using the same coefficients for real and imaginary parts, with reconstructible data in the unscaled asymptotic region.); I3 (Identify common multi-wave beating/asymptotic content caused by the open intermediate channel, the selected final coefficient, and the unscaled radial region actually supporting the identification.)", "Preserve independently valid work in O1, O2, O4, M1, M2, M3, M4, M5, M7. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-2-5, Eq.(9), paragraph preceding Eq.(12), Fig.1 caption, Sec.III.A pole limitation: extraction inside unscaled asymptotic space; exact bound poles are outside the unmodified real-energy perturbative prescription." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M7", "dimension": "M", "points": 0.5, "scientific_requirement": "Close the dimensional and normalization chain from selected final coefficient through declared Coulomb normalization and order-dependent units to absolute K-photon cross sections; independently normalized curves are insufficient.", "award_conditions": [ "The coefficient, Coulomb normalization, photon order and units form one absolute dimensional chain without arbitrary channel-by-channel rescaling." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: F3 (Convert the isolated outgoing final coefficient to absolute partial generalized cross sections under declared Coulomb normalization, cross-section convention and photon-order units. The K=2 arithmetic total belongs to O1 alone.); O1 (Present reconstructible absolute K=2 l=0 and l=2 partial cross sections and their arithmetic total, with correct case binding, stable large-scale structure and absolute normalization.); O4 (Present reconstructible absolute K=4 cross sections for all E1-allowed partial waves over a chosen energy domain sufficient for the claimed channel openings and off-pole high-energy behavior.)", "Preserve independently valid work in O3, I3, M3, M5, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-2-5, Eqs.(11)-(15),(26),(27),(31) and SI-unit paragraph: Coulomb amplitude normalization and absolute order-dependent generalized cross sections." ], "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the principal physical mechanism is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "M8", "dimension": "M", "points": 0.5, "scientific_requirement": "Synthesize the absolute partial-wave families and open-intermediate complex response into a supported off-pole, perturbative single-electron-ejection extraction conclusion, without assuming every outgoing fit is meaningful.", "award_conditions": [ "The absolute partial families and open-intermediate reconstruction jointly support a bounded off-pole, perturbative single-electron-ejection extraction conclusion." ], "applicability_and_na": [ "This scientific obligation applies to an assessable submission; it is not a risk-triggered N/A item." ], "dependencies": [ "Necessary scientific premise: all of: M3 (Explain why a single outgoing final wave is insufficient with an open intermediate continuum and why open intermediate and selected final energy Coulomb waves must coexist.); M4 (Explain how the common multi-wave coefficient system isolates the final outgoing coefficient and connects it to absolute cross sections; Wronskian, direct complex fit, surface relation and functionally equivalent extraction are accepted.); M5 (Use one coefficient set to explain both real and imaginary response components and determine whether the fixed-wave-number Coulomb representation holds; separately fitting the components does not establish the common complex structure.); M7 (Close the dimensional and normalization chain from selected final coefficient through declared Coulomb normalization and order-dependent units to absolute K-photon cross sections; independently normalized curves are insufficient.)", "Preserve independently valid work in O1, O2, O3, O4, I1, I2, I3, M1, M2, M6. Propagate only a failure of the actual necessary physical premise, not a composite atom's unrelated missing branch or display defect." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-4-5, Eqs.(23)-(31), Fig.1 and Sec.III.A discussion of radial extent/collocation resolution; p.043108-9, Sec.IV limits the extraction to single-electron ejection. Evidence synthesis and triggered alternative tests are task-defined, not prescribed source solver settings." ], "rc_relation": "Role: principal mechanism or commissioned conclusion. Trigger: the commissioned scientific conclusion is absent or wrong. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "V1", "dimension": "V", "points": 0.5, "scientific_requirement": "Positively restrict the conclusion to the specified hydrogen single-electron ionization, ordered perturbation theory, real-energy off-pole response, sampled energy domain and selected unscaled asymptotic region.", "award_conditions": [ "The candidate positively states hydrogen single-electron ionization, ordered perturbation, real-energy off-pole domain and actual unscaled recovery interval." ], "applicability_and_na": [ "This positive scientific scope/evidence duty is unconditional; N/A is not permitted." ], "dependencies": [ "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.", "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-2-5, Eq.(9), paragraph preceding Eq.(12), Fig.1 caption, Sec.III.A pole limitation: extraction inside unscaled asymptotic space; exact bound poles are outside the unmodified real-energy perturbative prescription." ], "rc_relation": "Role: no independent closure role. Trigger: no independent closure trigger. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." }, { "criterion_id": "V2", "dimension": "V", "points": 0.5, "scientific_requirement": "When the fit, data, region, anomaly or conclusion exposes a material risk, challenge omitted open waves, coefficient non-identifiability, recovery-region contamination or normalization freedom with discriminating evidence, or narrow the claim; a missing prescribed convergence sequence alone is insufficient.", "award_conditions": [ "When a material risk is exposed, capable evidence challenges omitted waves, coefficient degeneracy, region contamination or normalization freedom, or the claim is narrowed; otherwise N/A." ], "applicability_and_na": [ "Applicable only when a concrete material risk arises from the submitted method, data, anomaly or claim and threatens a retained conclusion.", "If untriggered in an assessable attempt, record N/A with fixed 0.5-point contribution, exclude it from applicable functional-completion counts, make no positive validation-evidence claim and do not block RC4." ], "dependencies": [ "Necessary scientific premise: No fixed upstream prerequisite; judge this scientific obligation on its own evidence.", "Preserve all other independently valid work; no additional unlisted hard dependency or duplicate deduction is implied." ], "equivalent_evidence": [ "Globally equivalent Coulomb phases, radial signs and plotting conventions are accepted after alignment.", "Wronskians, direct complex fits and equivalent surface-flux relations are accepted only when they recover the same outgoing final coefficient and absolute Coulomb normalization.", "The open-intermediate radial case is student-selected: K=2, omega=1 a.u., l=2 is canonical but not compulsory. The real and imaginary parts must use one coefficient set.", "A broadened response can provide secondary evidence but cannot replace the primary real-energy off-pole object. Exact pole heights, source plot bounds, basis and layout are not targets." ], "source_support": [ "Mihelic and Horvat (2021), article pp.043108-4-5, Eqs.(23)-(31), Fig.1 and Sec.III.A discussion of radial extent/collocation resolution; p.043108-9, Sec.IV limits the extraction to single-electron ejection. Evidence synthesis and triggered alternative tests are task-defined, not prescribed source solver settings." ], "rc_relation": "Role: material validation/scope. Trigger: a triggered, conclusion-threatening material validation/scope risk remains unresolved. Assess the substantive role condition independently of the atomic award; an atom receiving zero does not automatically establish a closure failure." } ], "case_scientific_rules": [ "Each of the exact twenty criteria is worth 0 or 0.5 points; the sum is Atomic Score on 0-10. Judge the essential scientific core: an immaterial notation or display imperfection alone does not remove credit. Do not interpolate or inherit numerical partial-credit ranges.", "Object existence, candidate presentation, candidate identification and physical inference are distinct evidence acts. Examiner-generated plots, slices, fits or diagnostics may verify an object but cannot supply omitted candidate presentation, identification, mechanism or validation.", "For every Output atom separately establish object_valid, presentation_adequate and claim_adequate. Credit requires all three. Assess physical identity, global pattern and important secondary structure, not a single superficial resemblance.", "Apply only scientifically necessary dependencies, branch-locally. A composite prerequisite failing for an unrelated part does not erase independently established evidence. Identify the exact failed premise and why the downstream argument cannot stand independently; do not turn a half-point result into a mechanical gate.", "Risk-triggered validation is applicable only to a concrete threat arising from the submitted method, data, anomaly or claim. In an assessable attempt, an untriggered criterion is N/A and contributes its fixed 0.5 points, but supplies no positive validation evidence and does not block RC4.", "Positive model/scope duties are unconditional where specified. Merely omitting an examiner-preferred convergence procedure, control or plot does not create a material risk. A valid run with no assessable scientific work is 0 and RC0 before atom summation; N/A cannot create credit for it.", "Research Closure is descriptive and independent of Atomic Score and criterion awards. Evaluate scientific role conditions against their specified facts, not zero-valued atoms; choose the lowest applicable level across required roles of all indispensable/co-primary claims. Supporting claims alone do not determine overall RC.", "Numerical defects matter where they damage a physical object or make a scientific conclusion false/unsupported; do not deduct the same defect again as generic incompleteness, missing convergence or validation. A fluent interpretation does not rescue a materially incorrect signal.", "Accept physically valid equivalent methods and representations after aligning identity, units, gauge/basis, normalization, signs, support and component relations as applicable. Do not invent a tolerance when the evidence cannot distinguish equivalence from error; retain a bounded issue for review.", "Reference verification is required when a material decision depends on resolution, fine structure, component completeness, formula identity or representational equivalence, including nonstandard projections, cropping or reconstruction used for full Output credit. Consult the applicable scientific source and appropriately scoped validated references after establishing observable, case/component and convention mapping. It cannot complete missing candidate reasoning.", "A provisional Atomic Score of at least 9.0 or provisional RC4 requires examiner-side focused verification of one central output and one principal-mechanism link against applicable scientific references, checking identity, sign, ordering and overall pattern. This is not an additional candidate deliverable or pixel-matching requirement.", "The E1-allowed families are K=2: l=0,2; K=3: l=1,3; K=4: l=0,2,4. The K=2 arithmetic total has only one scoring and indispensable-component owner, O1.", "A local presentation or low-energy resolution gap in K=3 does not erase independent channel identity, absolute normalization or a correctly bounded inference supported by the valid remaining evidence.", "The overall extraction synthesis requires the common multi-wave representation, final-coefficient isolation, common complex reconstruction and absolute normalization. Supporting fine opening sequences, family interpretation and recovery-region explanation retain independent scoring; they are not silently promoted to additional all-or-nothing central gates.", "The canonical construction solves the ordered hierarchy", "$$\n(E_0+j\\omega-H)\\widehat\\Psi_j=D\\widehat\\Psi_{j-1},\n\\qquad j=1,\\ldots,K,\n$$", "with one unit-coefficient application of $D=p_z$ at each step. In an open partial wave, the large-radius radial response is a linear combination of the outgoing Coulomb waves associated with every open intermediate energy and the selected final energy. Wronskians sampled at several radii in the unscaled asymptotic region determine the coefficients; the coefficient of the final wave gives the ionization amplitude and hence the conventional generalized cross section. Equivalent direct linear fits or surface-flux formulations are acceptable only when they recover the same outgoing coefficient and absolute normalization.", "the two-photon family contains $\\ell=0,2$ partial cross sections and their arithmetic total;", "the three-photon family contains $\\ell=1,3$ and resolves sequences approaching about 6.8 and 13.6 eV, associated with the successive opening of intermediate ATI steps;", "the four-photon family contains $\\ell=0,2,4$ and resolves corresponding sequences approaching about 4.5, 6.8 and 13.6 eV;", "away from resonant poles, the selected ATI branches decrease at high photon energy over the retained range; exact pole heights are not finite physical anchors;", "the source's canonical $K=2$, $\\omega=1$ a.u., $(\\ell,m)=(2,0)$ example has a beat-like unscaled complex radial response described by one common pair of outgoing Coulomb waves with $k=1$ and $k=\\sqrt3$; this is a reference anchor, not the only acceptable student-selected open-intermediate response;", "coefficient recovery must use the unscaled region. The damped computed ECS response beyond $R_0$ and a real-axis analytic continuation are different objects and their post-$R_0$ difference is not fit evidence.", "For F2-F3/M4-M7, article pp.043108-3-5, Eqs.(14),(23)-(31), define\n$$H_\\ell(k;r)=e^{-i\\eta_\\ell(k)}[F_\\ell(k;r)+iG_\\ell(k;r)],\\quad\n\\widehat P_\\ell(r)\\sim\\sum_q A_{\\ell q}H_\\ell(k_q;r),\\quad k_q=\\sqrt{2(E_0+j_q\\omega)}>0.$$\nWith the article's outgoing Coulomb convention and final index $n$,\n$$B_\\ell=i^{-\\ell+1}\\sqrt{\\frac{k_n}{2\\pi}}A_{\\ell n},\\qquad\n\\sigma^{(K)}_\\ell=2\\pi\\left(\\frac{2\\pi}{c\\omega}\\right)^K|B_\\ell|^2.$$\nThe SI conversion factor is $a_0^{2K}t_0^{K-1}$, giving area$^K$ time$^{K-1}$. The canonical extraction uses $M_{pq}=W\\{F_\\ell(k_n;r),H_\\ell(k_q;r)\\}_{r_p}$ and $b_p=W\\{F_\\ell(k_n;r),\\widehat P_\\ell(r)\\}_{r_p}$, with $W\\{f,g\\}=fg'-f'g$ and $r_p