cases/case25/grading_rubric.json

{
  "case_id": "Case25",
  "title": "High-Harmonic Generation in a Crystal Driven by Quantum Light",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.",
      "award_conditions": [
        "The supplied conduction band, ten occupied spin-momentum states and finite coherent-amplitude drive yield a consistently signed intraband current."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Judge the physical model and current identity directly.",
        "A localized convention defect does not erase independent state-weight or scope work that remains correct."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 3-4, Eqs. (13)-(15); p. 6, Sec. III B: The Fourier band and occupied spin-momentum states generate intraband current under the supplied finite pulse."
      ],
      "rc_relation": "RC0: absence or material invalidity of the central physical object. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "F2",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.",
      "award_conditions": [
        "Coherent, thermal, Fock and BSV weights are normalized at the common mean photon number, with exact versus APP status explicit."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Judge the quantum-state ensemble identity directly.",
        "Correct band/current work and honest scope remain independently creditable if a state weight is wrong."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 4-5, Eqs. (22)-(30): The four phase-space weights have the same stated mean photon number; coherent/thermal representation and approximate positive-P Fock/BSV branches must remain distinct."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Derive the finite-pulse state-conditioned spectrum with modulus square before phase-space averaging; do not replace it with a mean-current or Floquet-only spectrum.",
      "award_conditions": [
        "Finite-pulse spectral power is formed for each coherent amplitude before averaging over the state weight; mean-current and Floquet-only spectra are not substituted."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F1: Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.; F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in O2, O3, I2, I3, M3 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The spectrum must use the correct current family and state weights.",
        "Correct emitted-field moment work remains independently judgeable when the spectrum branch fails."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C: Finite-pulse spectra are averages of squared state-conditioned current transforms. Floquet markers and display offsets do not replace the finite-pulse output or common physical scale."
      ],
      "rc_relation": "RC0: absence or material invalidity of the central physical object. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Submit valid reconstructible finite-pulse spectra for all four correctly bound driving states, adequately exposing material harmonic structure and state-dependent extent.",
      "award_conditions": [
        "All four driving-state finite-pulse spectra exist with reconstructible state-bound data exposing material harmonic structure and extent.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F1: Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.; F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.; F3: Derive the finite-pulse state-conditioned spectrum with modulus square before phase-space averaging; do not replace it with a mean-current or Floquet-only spectrum.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in O2, O3, I2, I3, M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The spectra require the correct current, state weights and finite-pulse averaging construction.",
        "Correct emitted-field moment evidence and interpretation retain independent credit.",
        "Finite-pulse spectra are independently judgeable from field moments. A Floquet-only calculation can support mechanism but does not create the commissioned finite-pulse object."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C: Finite-pulse spectra are averages of squared state-conditioned current transforms. Floquet markers and display offsets do not replace the finite-pulse output or common physical scale."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Define or derive the generated-field first-moment relation from the current response, excluding the driving mode as specified, and submit its expectation for all four states with reconstructible data.",
      "award_conditions": [
        "The generated-field first moment follows the declared current relation with driver displacement excluded and is submitted for all four states.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F1: Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.; F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in O1, O3, I1, I3, M2, M4 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The state-conditioned generated-field mean requires the correct current and state weights.",
        "Spectral and field-uncertainty branches retain independent credit when the mean branch fails.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint p. 9, Eqs. (43)-(46), Fig. 4 and following discussion: The generated-field expectation excludes the driver-mode displacement and reflects the phase symmetry/anisotropy of the state weight."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Define or derive the generated-field second-moment/variance relation from current with specified zero-point subtraction, and submit the resulting nonnegative uncertainty for all four states with reconstructible data.",
      "award_conditions": [
        "The generated-field second moment/variance follows the declared zero-point convention and yields nonnegative reconstructed uncertainty for all four states.",
        "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F1: Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.; F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in O1, O2, I1, I2, M2, M4 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The state-conditioned uncertainty requires the correct current and state weights.",
        "Spectral and field-mean branches retain independent credit when uncertainty fails.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Eqs. (47)-(48), Fig. 4 and its caption: Generated-field uncertainty is the variance after the stated vacuum/driver exclusions, not the mean field or a claim of photon statistics."
      ],
      "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify material four-state odd-harmonic structure and relative extent/ordering, including similar coherent/Fock extents and extended thermal/bright-squeezed-vacuum response, from submitted spectra.",
      "award_conditions": [
        "The candidate identifies odd harmonics, similar coherent/Fock extent and extended thermal/BSV response from finite-pulse spectra.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O1: Submit valid reconstructible finite-pulse spectra for all four correctly bound driving states, adequately exposing material harmonic structure and state-dependent extent. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I2, I3, M1, M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The identification must come from a valid candidate-presented spectral object.",
        "The distribution-sampling relation and generated-field signal/mechanism branches remain independently judgeable when spectral identification fails.",
        "Finite-pulse spectra are independently judgeable from field moments. A Floquet-only calculation can support mechanism but does not create the commissioned finite-pulse object."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C: Finite-pulse spectra are averages of squared state-conditioned current transforms. Floquet markers and display offsets do not replace the finite-pulse output or common physical scale."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify from generated-field expectation data which state-conditioned means are materially nonzero, vanishing or burst-like, allowing small sampling residuals.",
      "award_conditions": [
        "The candidate identifies which generated means vanish and which exhibit coherent burst-like structure, allowing small numerical residuals.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link.",
        "The coherent driver produces a nonzero coherent/burst-like generated mean; Fock, thermal and BSV means vanish up to declared numerical residuals. A vanishing mean is not absence of radiation."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O2: Define or derive the generated-field first-moment relation from the current response, excluding the driving mode as specified, and submit its expectation for all four states with reconstructible data. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I1, I3, M2, M4 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The mean-field signature must come from a valid candidate-presented first-moment output.",
        "Spectral and uncertainty identification remain independent.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint p. 9, Eqs. (43)-(46), Fig. 4 and following discussion: The generated-field expectation excludes the driver-mode displacement and reflects the phase symmetry/anisotropy of the state weight."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify stationary versus time-dependent state-conditioned uncertainty from candidate-presented data, without examiner-created observation.",
      "award_conditions": [
        "The candidate identifies stationary versus time-dependent generated uncertainty from the four-state moment data.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link.",
        "Identify constant Fock/thermal uncertainty versus time-dependent BSV uncertainty from the generated-field data, with the stated zero-point subtraction; coherent-field mean structure is a distinct observable."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "O3: Define or derive the generated-field second-moment/variance relation from current with specified zero-point subtraction, and submit the resulting nonnegative uncertainty for all four states with reconstructible data. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I1, I2, M2, M4 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The uncertainty signature must come from a valid candidate-presented second-moment output.",
        "Spectral and mean-field identification remain independent.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Eqs. (47)-(48), Fig. 4 and its caption: Generated-field uncertainty is the variance after the stated vacuum/driver exclusions, not the mean field or a claim of photon statistics."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Relate radial/angular width and shape of each phase-space distribution to the range and symmetry of coherent amplitudes sampled at common mean photon number.",
      "award_conditions": [
        "The explanation links each phase-space radial/angular shape to sampled coherent amplitudes at equal mean photon number.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The distribution-to-sampled-amplitude relation requires correctly bound state weights but does not require the candidate's spectral signal to be present.",
        "Spectral identification/linkage is charged through I1/M2; the independently correct distribution relation and both field-moment mechanisms retain credit."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 4-5, Eqs. (22)-(30): The four phase-space weights have the same stated mean photon number; coherent/thermal representation and approximate positive-P Fock/BSV branches must remain distinct."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain spectral extent through overlap between the state phase-space distribution and nonlinear band-dependent harmonic response, not mean photon number alone.",
      "award_conditions": [
        "The candidate explains spectral extent through distribution overlap with the nonlinear band response, not a change in mean photon number alone.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [M1: Relate radial/angular width and shape of each phase-space distribution to the range and symmetry of coherent amplitudes sampled at common mean photon number.; I1: Identify material four-state odd-harmonic structure and relative extent/ordering, including similar coherent/Fock extents and extended thermal/bright-squeezed-vacuum response, from submitted spectra.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M3, M5, M7 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The overlap explanation requires the distribution-sampling relation and candidate-identified spectral signal.",
        "Field-moment and cross-representation branches remain independent.",
        "Finite-pulse spectra are independently judgeable from field moments. A Floquet-only calculation can support mechanism but does not create the commissioned finite-pulse object."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 6-7, Eq. (36), Fig. 2 and Sec. III C: Broad thermal/BSV radial distributions overlap nonlinear harmonic response at larger amplitudes; equal mean photon number alone does not fix spectral extent."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain generated-field mean using phase symmetry or anisotropy of the state weight carried through nonlinear current, delimiting what the identified first-moment signature supports.",
      "award_conditions": [
        "State-weight phase symmetry/anisotropy explains the identified generated-field means and their interpretation boundary.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.; I2: Identify from generated-field expectation data which state-conditioned means are materially nonzero, vanishing or burst-like, allowing small sampling residuals.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I3, M1, M2, M4, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The mean-field symmetry mechanism requires the correct state weight and the candidate-identified first-moment signature only.",
        "The uncertainty mechanism in M5 and spectral mechanism/conclusion work remain independently creditable when the mean branch fails.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint p. 9, Eqs. (43)-(46), Fig. 4 and following discussion: The generated-field expectation excludes the driver-mode displacement and reflects the phase symmetry/anisotropy of the state weight."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use identified spectral structures to determine how and where the four equal-mean-photon states differ; display floors, arbitrary scaling and supporting markers are not finite-pulse evidence.",
      "award_conditions": [
        "The spectral conclusion follows the actual four-state finite-pulse structures rather than a plotting floor, state offset or analytic marker.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "I1: Identify material four-state odd-harmonic structure and relative extent/ordering, including similar coherent/Fock extents and extended thermal/bright-squeezed-vacuum response, from submitted spectra. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The state inference must rest on the candidate-identified finite-pulse spectral signal.",
        "Field-moment inference remains independent.",
        "Finite-pulse spectra are independently judgeable from field moments. A Floquet-only calculation can support mechanism but does not create the commissioned finite-pulse object."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C: Finite-pulse spectra are averages of squared state-conditioned current transforms. Floquet markers and display offsets do not replace the finite-pulse output or common physical scale."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M5",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain generated-field uncertainty using phase symmetry or anisotropy carried through nonlinear current, delimiting what the identified second-moment signature supports.",
      "award_conditions": [
        "The uncertainty explanation carries phase symmetry/anisotropy through the nonlinear current and distinguishes second moment from mean.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.; I3: Identify stationary versus time-dependent state-conditioned uncertainty from candidate-presented data, without examiner-created observation.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in I2, M2, M3, M4 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The uncertainty symmetry mechanism requires the correct state weight and the candidate-identified second-moment signature only.",
        "The mean mechanism in M3 and spectral mechanism/conclusion work remain independently creditable when the uncertainty branch fails.",
        "First moment and zero-point-subtracted second moment are independently scoreable; a zero mean does not imply absent radiation or zero uncertainty, and failure of one moment does not erase the other."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Eqs. (47)-(48), Fig. 4 and its caption: Generated-field uncertainty is the variance after the stated vacuum/driver exclusions, not the mean field or a claim of photon statistics."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain why common mean photon number does not fix the response and why phase-space averaging at the correct mathematical level controls state dependence.",
      "award_conditions": [
        "The candidate explains why equal mean photon number does not fix the response and why averaging at the correct mathematical level matters.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [M1: Relate radial/angular width and shape of each phase-space distribution to the range and symmetry of coherent amplitudes sampled at common mean photon number.; M2: Explain spectral extent through overlap between the state phase-space distribution and nonlinear band-dependent harmonic response, not mean photon number alone.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "This explanation depends on the distribution-sampling and response-overlap links.",
        "Field-moment mechanism and inference remain independent."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 6-7, Eq. (36), Fig. 2 and Sec. III C: Broad thermal/BSV radial distributions overlap nonlinear harmonic response at larger amplitudes; equal mean photon number alone does not fix spectral extent."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Bind spectral and time-domain observables to the same inputs and distinguish robust state structure from normalization, finite-pulse broadening and the limited role of APP/Floquet analysis.",
      "award_conditions": [
        "Spectral and temporal evidence share the same physical inputs and distinguish robust state structure from finite-pulse broadening, normalization and APP scope.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [O1: Submit valid reconstructible finite-pulse spectra for all four correctly bound driving states, adequately exposing material harmonic structure and state-dependent extent.; O2: Define or derive the generated-field first-moment relation from the current response, excluding the driving mode as specified, and submit its expectation for all four states with reconstructible data.; O3: Define or derive the generated-field second-moment/variance relation from current with specified zero-point subtraction, and submit the resulting nonnegative uncertainty for all four states with reconstructible data.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M1, M2, M3, M4, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Cross-representation reasoning requires valid spectra and both field-moment outputs.",
        "Correct branch-local mechanism and conclusions retain credit if cross-representation binding is incomplete."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Secs. IV-V and Conclusion; Appendix A: Spectral and temporal claims share the same intraband input and have observable-dependent APP limits; APP does not establish emitted photon statistics or quantitative squeezing."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "M8",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Integrate spectra and field moments with the distribution-response mechanism to state what driver-state information is and is not defensibly encoded in generated radiation.",
      "award_conditions": [
        "The final answer integrates spectra and field moments to state which driver-state information is supported and which photon-statistical conclusions are not.",
        "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [M2: Explain spectral extent through overlap between the state phase-space distribution and nonlinear band-dependent harmonic response, not mean photon number alone.; M3: Explain generated-field mean using phase symmetry or anisotropy of the state weight carried through nonlinear current, delimiting what the identified first-moment signature supports.; M4: Use identified spectral structures to determine how and where the four equal-mean-photon states differ; display floors, arbitrary scaling and supporting markers are not finite-pulse evidence.; M5: Explain generated-field uncertainty using phase symmetry or anisotropy carried through nonlinear current, delimiting what the identified second-moment signature supports.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in M1, M6, M7 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "The final conclusion requires both principal mechanism branches and both candidate-grounded inference branches.",
        "Correct independent mechanism and cross-representation work retains atomic credit when the integrated conclusion does not close."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Secs. IV-V and Conclusion; Appendix A: Spectral and temporal claims share the same intraband input and have observable-dependent APP limits; APP does not establish emitted photon statistics or quantitative squeezing."
      ],
      "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Establish that the spectral claim is supported by finite-pulse calculation, not solely a Floquet marker, arbitrary normalization or an APP result beyond its stated role.",
      "award_conditions": [
        "Finite-pulse computation actually supports the spectral claim; Floquet-only markers or an unjustified APP extrapolation do not establish it."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "All of [F3: Derive the finite-pulse state-conditioned spectrum with modulus square before phase-space averaging; do not replace it with a mean-current or Floquet-only spectrum.; O1: Submit valid reconstructible finite-pulse spectra for all four correctly bound driving states, adequately exposing material harmonic structure and state-dependent extent.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.",
        "Preserve independent correct work in V2, V3, M3, M5 and all branches not using the failed premise.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Branch validation requires the finite-pulse construction and candidate-presented spectrum.",
        "Risk response, honest scope and field-moment reasoning remain independent.",
        "Finite-pulse spectra are independently judgeable from field moments. A Floquet-only calculation can support mechanism but does not create the commissioned finite-pulse object."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C: Finite-pulse spectra are averages of squared state-conditioned current transforms. Floquet markers and display offsets do not replace the finite-pulse output or common physical scale."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "When quadrature, sampling, Fourier/differentiation method, anomaly or claim raises a concrete material alternative, supply discriminating evidence capable of exposing it or leave the outcome unresolved for adjudication.",
      "award_conditions": [
        "A real quadrature, derivative, transform, sampling or representation threat is discriminated or retained explicitly for adjudication under the existing applicability rule."
      ],
      "applicability_and_na": [
        "Activate only for a concrete material risk arising from the candidate method, data, anomaly or claim; no fixed test is required in the absence of a trigger.",
        "Without a triggered risk record N/A and award 0.5 without positive validation evidence; this neither blocks RC4 nor triggers RC3.",
        "When triggered, accept functionally equivalent discriminating evidence or a justified restriction of the threatened claim."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Evaluate the risk trigger and evidence independently of the atomic award.",
        "Untriggered preferred checks cannot reduce other credit, and one numerical defect is charged only to the evidence or inference it damages."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 5-7, Eqs. (32)-(37), and pp. 9-10, Eqs. (46)-(48), Sec. V: Averaging level, phase-space sampling, finite-pulse treatment and APP scope can threaten specific claims. Equivalent discrimination is task-defined only for material triggered risks."
      ],
      "rc_relation": "RC3: an unresolved applicable material risk, only after signals and principal mechanisms are substantially supported and all six risk-proof requirements are met. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    },
    {
      "criterion_id": "V3",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Limit conclusions to the intraband single-conduction-band independent-electron model, stated equal-mean-photon comparison and APP-supported observables; do not claim unsupported emitted photon statistics or quantitative squeezing.",
      "award_conditions": [
        "The conclusion is bounded to the single-conduction-band intraband model, equal-mean-photon comparison and APP-supported observables; quantitative squeezing is not inferred."
      ],
      "applicability_and_na": [
        "Applicable to the stated scientific commission; N/A is not permitted."
      ],
      "dependencies": [
        "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.",
        "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.",
        "Honest scope is independently judgeable and cannot depend on upstream success.",
        "A correct evidence-based limitation retains credit even if an observable or mechanism fails."
      ],
      "equivalent_evidence": [
        "Accept algebraically consistent current/Fourier signs, time origins, units and common radiation/spectral prefactors with retained raw data and reconstructible state binding.",
        "Accept alternative phase-space quadrature/sampling, valid derivative and transform procedures, displays and defensible extent definitions; no source floor, time crop, burst width, analytic marker set or fixed photon-number scan is required.",
        "Floquet/Bessel analysis and overlap cutoff estimates may support mechanism and spectral extent but cannot replace finite-pulse evidence. First and second moments must remain consistently defined under any radiation normalization."
      ],
      "source_support": [
        "Gothelf et al. (2025), preprint pp. 9-10, Secs. IV-V and Conclusion; Appendix A: Spectral and temporal claims share the same intraband input and have observable-dependent APP limits; APP does not establish emitted photon statistics or quantitative squeezing."
      ],
      "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure."
    }
  ],
  "case_scientific_rules": [
    "The coherent-amplitude current is j_alpha(t) = sum_{q,mu} dE(k)/dk evaluated at k = q + A_alpha(t). The finite-pulse spectrum is S(omega) proportional to omega^2 integral d^2alpha P(alpha) |integral j_alpha(t) exp(i omega t) dt|^2. Squaring before averaging is essential; averaging current first can erase noncoherent spectral signals by phase symmetry.",
    "With driving mode and zero-point contribution excluded, generated mean field is -4/(3c^3) integral P(alpha) j_alpha_double_dot(t) d^2alpha. Variance is 16/(9c^6) times {integral P(alpha) [j_alpha_double_dot]^2 d^2alpha - [integral P(alpha) j_alpha_double_dot d^2alpha]^2}; uncertainty is its nonnegative square root. Driver traces or current itself are not this radiation observable.",
    "At equal mean photon number spectra show odd harmonics. Coherent and large-n Fock states have similar sharp extents, while thermal and bright squeezed vacuum have substantially broader higher-order tails. Approximate source extent landmarks 25,25,59,69 indicate structure classes, not required endpoints or a unique cutoff estimator.",
    "The distribution-overlap mechanism combines radial weight with nonlinear band harmonic response. Narrow radial distributions sample limited coherent amplitudes; broad thermal/squeezed weights sample larger amplitudes at the same mean photon number. Generic photon-statistics language without this causal link is incomplete.",
    "Generated coherent mean is nonzero and burst-like, with negligible zero-point-subtracted uncertainty in this uncorrelated model. Fock and thermal means vanish and uncertainties are approximately stationary; bright-squeezed-vacuum mean vanishes with strongly time-dependent uncertainty. Phase symmetry/anisotropy explains these relations; small numerical residual means compatible with sampling are allowed.",
    "APP is accepted for Fock and squeezed-vacuum spectra and the first two generated-field moments at the stated large photon number. It does not establish emitted photon-number statistics or a quantitatively correct squeezing parameter.",
    "Periodic band-Fourier/Bessel expressions, with odd amplitudes given by weighted sums of J_n(l g_tilde0 |alpha|), g_tilde0 = 2 a g0/sqrt(omega0), and distribution-overlap estimates are interpretation or cross-checks, not substitutes for finite-pulse spectra. A constant band offset b0 is immaterial.",
    "Concrete risks include periodic-only substitution, averaging before squaring, state/mean-photon misbinding, quadrature erasing tails/anisotropy, derivative/FFT artifacts, driver/radiation confusion or APP overextension. The source and finite-pulse data adjudicate fine structure; a supporting Floquet reference does not establish finite-pulse leakage floors.",
    "The score is the sum of twenty material-core criteria, each worth either 0 or 0.5, for a maximum of 10.0. Research Closure is descriptive and never caps, floors, rescales or repairs the score.",
    "Judge physical object identity and correctness before display similarity. For each output record object_valid, presentation_adequate and claim_adequate separately; all three must hold for its atomic award. A presentation or claim defect alone is not evidence that an existing physical object is absent or invalid.",
    "Candidate-created data, candidate presentation, candidate identification, physical inference and validation are distinct evidential acts. Examiner inspection or reconstruction may verify an object but cannot create missing candidate identification, reasoning or risk resolution.",
    "A dependency propagates only the failed necessary physical fact, not the whole upstream atomic verdict or unrelated missing detail. Name the failed premise, the downstream obligation needing it, and why independent candidate-presented evidence does not establish it. Preserve unaffected branches, independent work and honest scope.",
    "Numerical reliability has no separate generic penalty. Charge a defect where it materially damages a physical object or evidence, and a dependent conclusion only when that same defect makes the conclusion false or unsupported. Missing a preferred convergence study, control, solver or named self-check is not itself a failure.",
    "Risk-triggered validation applies only to a concrete material risk arising from the candidate method, data, anomaly or claim. With no such risk record N/A and award the fixed atomic credit without treating it as positive validation evidence; N/A does not block RC4. Unconditional scope obligations cannot be N/A.",
    "Before assigning RC3 establish all six: the concrete failure mode; its origin in candidate work; the fair task or standard-knowledge basis; the threatened central conclusion; why submitted evidence cannot expose or exclude it; and examples of functionally equivalent discriminating evidence or an honest scope restriction that would resolve it. A generic possibility of further testing is insufficient.",
    "Use bounded examiner-side source/reference readback only to resolve a material scientific uncertainty, not to demand pixels, an undeclared tolerance or additional candidate deliverables. A provisional score of at least 9.0 or provisional RC4 triggers one bounded central-output-anchor check and one principal-mechanism-link check. Reference conflict or insufficiency remains an explicit unresolved decision, not an invented target.",
    "Evaluate closure-role trigger facts independently of atomic awards and select the most severe failed required role across all indispensable and co-primary claims. Supporting claims do not independently set overall closure. A point loss alone is not a closure-gate fact.",
    "A valid candidate-caused incomplete or timeout attempt with no scientifically assessable required submission has score 0.0 and RC0 before N/A bookkeeping. Assessable incomplete work is graded from the submitted evidence; unsubmitted scratch or logs do not become scientific answers.",
    "For state weight $P(\\alpha)$ and current $j_\\alpha(t)$, $S(\\omega)\\propto\\int d^2\\alpha\\,P(\\alpha)|\\int dt\\,w(t)j_\\alpha(t)e^{i\\omega t}|^2$ (Eqs. (32)-(33), finite-pulse form Eq. (37)). This is not $|\\int P\\,\\widetilde j_\\alpha|^2$. In the periodic limit odd-order weights involve $\\int P(\\alpha)[\\sum_lC_lJ_n(l\\widetilde g_0|\\alpha|)]^2d^2\\alpha$ (Eq. (36)), explaining distribution-response overlap but not replacing finite-pulse evidence.",
    "Let $X_\\alpha(t)=d^2j_\\alpha/dt^2$ and $\\langle\\cdot\\rangle_P=\\int d^2\\alpha P(\\alpha)(\\cdot)$. The generated field obeys $\\langle E\\rangle_{\\rm HHG}=-4\\langle X\\rangle_P/(3c^3)$ and $\\operatorname{Var}(E)_{\\rm HHG}=16[\\langle X^2\\rangle_P-\\langle X\\rangle_P^2]/(9c^6)$ (Eqs. (46)-(48)), with driver and zero-point terms removed as specified. The plotted uncertainty is the nonnegative square root of that variance.",
    "The coherent weight is concentrated at one complex amplitude; the thermal weight is circular Gaussian. The APP Fock weight is $P_n=e^{-|\\alpha|^2}|\\alpha|^{2n}/(\\pi n!)$, while APP BSV has the anisotropic Gaussian in Eq. (30). The APP is observable dependent, not an exact density-operator replacement for all output statistics."
  ],
  "closure_requirements": {
    "RC0": [
      "Required claim CL1, central object: All of [F1: Bind the implementation to the visible one-dimensional conduction band, ten occupied spin-momentum states, coherent-amplitude vector potential and finite pulse, constructing intraband current with consistent units and signs.; F3: Derive the finite-pulse state-conditioned spectrum with modulus square before phase-space averaging; do not replace it with a mean-current or Floquet-only spectrum.]. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Apply for a missing or materially invalid required central object; a presentation-only or claim-only defect does not establish object invalidity.",
      "No scientifically assessable required submission gives the terminal 0.0/RC0 disposition."
    ],
    "RC1": [
      "Required claim CL1, indispensable component: All of [F2: Instantiate and normalize coherent, thermal, Fock and bright-squeezed-vacuum phase-space weights at the common mean photon number, preserving exact versus APP branch identity.; O1: Submit valid reconstructible finite-pulse spectra for all four correctly bound driving states, adequately exposing material harmonic structure and state-dependent extent.; O2: Define or derive the generated-field first-moment relation from the current response, excluding the driving mode as specified, and submit its expectation for all four states with reconstructible data.; O3: Define or derive the generated-field second-moment/variance relation from current with specified zero-point subtraction, and submit the resulting nonnegative uncertainty for all four states with reconstructible data.]. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "The central object is established but a required physical component is absent or materially invalid. Preserve independent valid branches; do not invent a hidden canonical plot, sampling grid or implementation duty."
    ],
    "RC2": [
      "Required claim CL2, principal mechanism: All of [M2: Explain spectral extent through overlap between the state phase-space distribution and nonlinear band-dependent harmonic response, not mean photon number alone.; M3: Explain generated-field mean using phase symmetry or anisotropy of the state weight carried through nonlinear current, delimiting what the identified first-moment signature supports.; M5: Explain generated-field uncertainty using phase symmetry or anisotropy carried through nonlinear current, delimiting what the identified second-moment signature supports.; M8: Integrate spectra and field moments with the distribution-response mechanism to state what driver-state information is and is not defensibly encoded in generated radiation.]. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "Required signal objects/components exist, but an indispensable principal mechanism or co-primary scientific conclusion is missing, wrong or unsupported by candidate-presented and identified evidence. Correct data do not let an examiner supply missing reasoning."
    ],
    "RC3": [
      "Required claim CL3, material validation: V2: When quadrature, sampling, Fourier/differentiation method, anomaly or claim raises a concrete material alternative, supply discriminating evidence capable of exposing it or leave the outcome unresolved for adjudication.. Judge role-trigger facts, not the sum or pattern of atomic scores.",
      "All indispensable signal and principal-mechanism roles are substantially supported, but a concrete triggered material risk remains unresolved and inadequately restricted.",
      "Establish the failure mode, candidate origin, fair task/knowledge basis, threatened central conclusion, insufficiency of submitted evidence, and equivalent resolution or scope restriction. Missing an examiner-preferred check alone never suffices."
    ],
    "RC4": [
      "Every nonempty required role in every indispensable and co-primary claim above is scientifically established from candidate evidence, with the correct case/model/observable identities and evidence-bounded conclusion.",
      "All applicable material risks are discriminated or honestly restricted; no unresolved material ambiguity remains. Risk N/A does not block closure or supply positive validation evidence. Honest limitations cannot repair a false signal or mechanism.",
      "Select the most severe failed required role if any; otherwise RC4. No score threshold, score ceiling or required display/solver checklist determines this classification."
    ]
  },
  "references": [
    {
      "citation": "Rasmus Vesterager Gothelf, Christian Saugbjerg Lange, and Lars Bojer Madsen. 2025. High-Harmonic Generation in a Crystal Driven by Quantum Light. arXiv, 2502.11803.",
      "url": "https://arxiv.org/abs/2502.11803v2",
      "scientific_scope": "Four-state finite-pulse intraband HHG and generated-field first/second moments, phase-space distribution-response mechanism and APP scope. Floquet structural support does not replace finite-pulse spectra or justify emitted photon statistics.",
      "anchors": [
        "preprint pp. 3-4, Eqs. (13)-(15); p. 6, Sec. III B",
        "preprint pp. 4-5, Eqs. (22)-(30)",
        "preprint pp. 5-7, Eqs. (32)-(37), Fig. 1 and Sec. III B-C",
        "preprint p. 9, Eqs. (43)-(46), Fig. 4 and following discussion",
        "preprint pp. 9-10, Eqs. (47)-(48), Fig. 4 and its caption",
        "preprint pp. 6-7, Eq. (36), Fig. 2 and Sec. III C",
        "preprint pp. 9-10, Secs. IV-V and Conclusion; Appendix A",
        "preprint pp. 5-7, Eqs. (32)-(37), and pp. 9-10, Eqs. (46)-(48), Sec. V"
      ]
    }
  ]
}