cases/case16/grading_rubric.json

{
  "case_id": "Case16",
  "title": "Case16 — Spectral confinement of phase-matched slow electrons",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Establish the full phase-matched Floquet–Bloch production equation, retaining quadratic dispersion, recoil-dependent nearest-neighbour $Ap$ coupling, and $A^2$ next-nearest-neighbour coupling as one model identity.",
      "award_conditions": [
        "The executable formulation is physically consistent with the starting Hamiltonian and retains these terms; a linear nonrecoil walk is not the production model.",
        "A localized coefficient or convention issue affects this atom only if it materially changes model identity or the actual physics."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "No upstream prerequisite; judge the full model on its own evidence."
      ],
      "equivalent_evidence": [
        "Consistent charge, optical-phase, sideband-index and global-phase conventions are accepted when the resulting physical model is unchanged.",
        "Consistently retained Lorentz-factor corrections are accepted; they are negligible for the specified slow-electron cases."
      ],
      "source_support": [
        "Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9)."
      ],
      "rc_relation": "Role: central object. Trigger: material model identity absent or invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "F2",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Establish a quantitative discriminator for the competition between sideband hopping and the quadratic on-site term without replacing the full production model.",
      "award_conditions": [
        "Derive a competition measure or a physically equivalent discriminator with the correct limits and ordering; field-strength labels alone are insufficient."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid full-model foundation in F1 for a discriminator of the commissioned model. Failure of the discriminator does not erase independently correct propagation or Case-I/Case-II mechanisms."
      ],
      "equivalent_evidence": [
        "The ratio $\\rho=\\beta_d/|\\kappa|$, an explicitly related reciprocal or other dimensionless ratio, direct spectral analysis, or an equivalent asymptotic reduction is accepted when physical equivalence and ordering are shown."
      ],
      "source_support": [
        "Eldar et al. (2024), Eq. (2), definition of $\\rho$ and regime discussion, pp. 3–4; Supplemental Material Eqs. (S25)–(S26)."
      ],
      "rc_relation": "Role: indispensable component. Trigger: material discriminator identity absent or invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Define the shared observable: sideband probability $P_n(t)=|a_n(t)|^2$, a declared sideband-to-energy mapping, and common absolute state normalization across both cases.",
      "award_conditions": [
        "Probability, energy mapping, and absolute normalization form a reconstructible observable identity. Individual case/initial-state bindings are assessed locally under O1, O3 and O4, not folded into this shared atom."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "No upstream prerequisite. A failure of the shared probability identity does not erase an independently correct regime discriminator."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization."
      ],
      "source_support": [
        "Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9).",
        "Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4.",
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: central object. Trigger: material observable identity absent or invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present valid, reconstructible full-model evidence sufficient to judge Case-I spectral confinement for the 100 eV reference, $n=0$ initial state, 1.54 eV photon, and 0.5 V nm$^{-1}$ field.",
      "award_conditions": [
        "The physical object is valid and reconstructible; its presentation exposes the claim-relevant structure; the associated claim stays within what the object supports. Assess these three conditions separately.",
        "Bounded turning plus recurrent reconcentration is sufficient. Functionally equivalent confined-evolution evidence is also accepted; early broadening alone with no confinement or return evidence is insufficient."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the full production model in F1 and the shared probability/mapping/normalization identity in F3. A Case-I local binding or interpretation failure does not erase independent regime or Case-II work."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization.",
        "Unitary propagation, direct ODE integration, Krylov methods, split propagation, or another converged realization of the stated Hamiltonian are accepted. No particular solver, basis size, sampling grid, plotting range, or panel layout is required.",
        "A time-energy map, sideband-probability evolution, or another physically equivalent confined-evolution representation is accepted. Exact source times, colours, crop and displayed energy range are not targets."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4."
      ],
      "rc_relation": "Role: indispensable component. Trigger: output object absent or physically invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC. Presentation or claim inadequacy alone is not an automatic component-invalidity decision."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present quantitative regime evidence for both cases with correct ordering, approximate scale, and support for the crossover.",
      "award_conditions": [
        "The discriminator is valid and reconstructible, adequately presented, and supports the claim made from it.",
        "Correct two-case ordering, approximate scale, and crossover support must all be established. Two unexplained Raman–Nath/Bragg labels are insufficient."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid quantitative discriminator in F2. Failure here does not erase independently correct raw propagation evidence in either case."
      ],
      "equivalent_evidence": [
        "An analytic regime boundary, parameter map, targeted quantitative comparisons, direct spectral analysis, or another equivalent discriminator is accepted. Exact decimals, the source map layout, and the authors' sampling grid are not requirements."
      ],
      "source_support": [
        "Eldar et al. (2024), Eq. (2), definition of $\\rho$ and regime discussion, pp. 3–4; Supplemental Material Eqs. (S25)–(S26)."
      ],
      "rc_relation": "Role: indispensable component. Trigger: output object absent or physically invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present full-ladder Case-II evolution at 0.1 V nm$^{-1}$ from the $n=-1$ (12 eV) initial state, resolving coherent exchange between the 12 eV and 20 eV states.",
      "award_conditions": [
        "The full-ladder object and its local case binding are valid and reconstructible, the exchange is adequately presented, and the claim is supported.",
        "Exchange with the 16 eV reference is not the required exchanged-state identity. A separately imposed two-state model without full-ladder justification is insufficient."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the full production model in F1 and the shared probability/mapping/normalization identity in F3. Exchanged-state identity and remaining-ladder suppression are distinct aspects: failure of one does not mechanically erase valid independent evidence for the other."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization.",
        "Unitary propagation, direct ODE integration, Krylov methods, split propagation, or another converged realization of the stated Hamiltonian are accepted. No particular solver, basis size, sampling grid, plotting range, or panel layout is required.",
        "Exact transfer time, source colours and panel arrangement are not requirements."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: indispensable component. Trigger: output object absent or physically invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "O4",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present the small remaining-ladder probability outside the 12/20 eV pair in the same correctly bound Case-II full-ladder evolution, using common absolute normalization.",
      "award_conditions": [
        "The full-ladder suppression object is valid and reconstructible, adequately presented, and supports the claim that the other sidebands are a small correction.",
        "An independently renormalized two-state subset cannot establish suppression of the excluded ladder."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the full production model in F1 and the shared probability/mapping/normalization identity in F3. Assess suppression independently of the distinct exchanged-state output aspect in O3."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization.",
        "Unitary propagation, direct ODE integration, Krylov methods, split propagation, or another converged realization of the stated Hamiltonian are accepted. No particular solver, basis size, sampling grid, plotting range, or panel layout is required.",
        "Full-ladder probability maps, absolute component populations, or equivalent evidence of the remaining-state weight are accepted. The reference pair-probability value is not an exact cutoff."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: indispensable component. Trigger: output object absent or physically invalid. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Explicitly identify physically located turning and recurrent reconcentration in the candidate's own Case-I evidence, rather than reporting early broadening alone.",
      "award_conditions": [
        "The candidate identifies these structures from adequately presented valid evidence. An examiner's discovery of recurrence cannot supply candidate identification."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the relevant valid and presented Case-I output in O1. An unrelated unmet presentation or claim condition does not erase identification independently established from a valid presented sub-object."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization.",
        "Descriptions or quantitative extraction of turning and reconcentration are accepted without an exact recurrence time."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify Case I as hopping-dominated and Case II as on-site-dominated from the candidate's own quantitative evidence, with correct scale ordering.",
      "award_conditions": [
        "The candidate identifies the ordering and regime interpretation; correct values merely present in data do not supply identification."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the relevant valid and presented quantitative evidence in O2; propagate only the failed fact actually needed for this identification."
      ],
      "equivalent_evidence": [
        "Equivalent dimensionless conventions, including a declared reciprocal ratio with reversed numerical inequalities, are accepted when the physical ordering is correct."
      ],
      "source_support": [
        "Eldar et al. (2024), Eq. (2), definition of $\\rho$ and regime discussion, pp. 3–4; Supplemental Material Eqs. (S25)–(S26)."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Explicitly identify the $n=-1$ (12 eV) and $n=+1$ (20 eV) exchanged states from the candidate's own Case-II evidence.",
      "award_conditions": [
        "The identification is candidate-authored and supported by the correctly mapped and presented exchange signal."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the relevant valid and presented exchanged-state evidence in O3. Missing remaining-ladder identification does not erase this independent identification."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    },
    {
      "criterion_id": "I4",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Explicitly identify strong suppression of the other sidebands from the candidate's own Case-II full-ladder evidence under common absolute normalization.",
      "award_conditions": [
        "The candidate identifies the small remaining-ladder correction; the suppression is not created by subset renormalization."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the relevant valid and presented remaining-ladder evidence in O4. Exchanged-state identification and suppression identification remain separately assessable."
      ],
      "equivalent_evidence": [
        "Use sideband index, electron energy, or a correctly transformed equivalent representation with declared mapping and the same absolute state normalization."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Connect arrest of Case-I expansion to full-model quadratic curvature and the on-site detuning that increases with sideband order.",
      "award_conditions": [
        "The candidate makes the physical connection between expansion arrest and growing curvature-induced detuning; generic confinement wording detached from evidence is insufficient.",
        "Failure to identify recurrent reconcentration does not erase an independently established expansion-arrest mechanism."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid full-model foundation in F1 only. I1's recurrence identification and M2's recurrence mechanism are distinct obligations and are not prerequisites of M1."
      ],
      "equivalent_evidence": [
        "An equivalent curvature/energy–momentum-mismatch argument grounded in the full model is accepted; finite-basis or normalization explanations are not physical substitutes."
      ],
      "source_support": [
        "Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9).",
        "Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: principal mechanism absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use the candidate's recurrent signal to explain reconcentration after turning, completing the local curvature-detuning-to-reconcentration mechanism chain.",
      "award_conditions": [
        "The candidate connects the identified recurrence to the established curvature-induced arrest mechanism rather than merely naming a recurring feature."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the arrest–curvature mechanism in M1 and the candidate's own turning/reconcentration identification in I1. This local recurrence chain is not a prerequisite of the integrated conclusion in M8."
      ],
      "equivalent_evidence": [
        "A physically equivalent explanation of the recurrent full-model signal is accepted without exact timing or a prescribed diagnostic."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 1(d), p. 2, and plane-wave confinement discussion, pp. 3–4."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain how the quantitative competition separates Raman–Nath, Bragg and crossover behaviour, rather than assigning regimes from field-strength labels.",
      "award_conditions": [
        "The candidate links the valid quantitative evidence and its identified case ordering to hopping-dominated, on-site-dominated and crossover physics."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the relevant quantitative evidence in O2 and its candidate identification in I2. Independently correct propagation and local case mechanisms retain credit."
      ],
      "equivalent_evidence": [
        "Equivalent ratios, analytic limits, or spectral arguments are accepted if they explain the same physical competition and crossover."
      ],
      "source_support": [
        "Eldar et al. (2024), Eq. (2), definition of $\\rho$ and regime discussion, pp. 3–4; Supplemental Material Eqs. (S25)–(S26)."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: principal mechanism absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain Case-II selectivity of the $n=\\pm1$ pair using the full-ladder quadratic energy structure and the correctly identified exchanged-state signal, including the pair's relative near-degeneracy and other-order detuning.",
      "award_conditions": [
        "The candidate establishes the full-model origin of the pair's selectivity rather than reciting a fitted two-level model.",
        "Exchange-channel descriptions consistent with the full equation are accepted; a local-pathway omission alone is diagnostic and does not invalidate the principal selectivity mechanism."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid full model in F1 and the candidate's correct exchanged-state identification in I3. Missing independent remaining-ladder identification does not automatically erase this selectivity argument."
      ],
      "equivalent_evidence": [
        "An exchange pathway through the intermediate state, direct $A^2$ next-nearest-neighbour coupling, effective elimination, or any other pathway consistent with the full equation is accepted. Local pathway detail is diagnostic, not an independently scored requirement; incompleteness alone does not imply a principal-mechanism failure."
      ],
      "source_support": [
        "Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9).",
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: principal mechanism absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC. A pathway claim matters only if its inconsistency materially changes selectivity or the reduced-manifold mechanism."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain the small remaining-sideband correction through off-resonant detuning in the full-model quadratic energy structure.",
      "award_conditions": [
        "The candidate connects the presented and identified remaining-ladder suppression to the physical detuning mechanism, not a numerical deletion of states."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid full model in F1, the valid suppression evidence in O4, and the candidate's suppression identification in I4."
      ],
      "equivalent_evidence": [
        "Equivalent full-model detuning or effective-elimination reasoning is accepted when it preserves the physically small remaining-ladder contribution."
      ],
      "source_support": [
        "Eldar et al., Phys. Rev. Lett. 132, 035001 (2024), Eq. (1), pp. 2–3; Supplemental Material derivation through Eq. (S9).",
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: principal mechanism absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Assess whether the reduced Case-II manifold supports model-level coherent control, using correct exchange identity, remaining-ladder suppression and full-ladder dynamics.",
      "award_conditions": [
        "The candidate connects the full-ladder exchange and suppression mechanisms to the requested coherent-control conclusion; a two-level recital alone is insufficient."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the valid exchange output in O3, valid suppression output in O4, the full-model selectivity mechanism in M4, and remaining-ladder detuning mechanism in M6."
      ],
      "equivalent_evidence": [
        "A full-model-supported reduced-manifold argument is accepted without a unique fitted two-level parametrization or preferred exchange route."
      ],
      "source_support": [
        "Eldar et al. (2024), Fig. 2(a)–(b), p. 4, and Bragg-regime discussion."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: commissioned coherent-control conclusion absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "M8",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Unify Case-I confinement and Case-II selective exchange through the shared competition of quadratic curvature and hopping, directly answering whether and how the field achieves these forms of model-level spectral control.",
      "award_conditions": [
        "The candidate integrates the two cases through supported physics rather than presenting disconnected descriptions.",
        "Functionally equivalent Case-I confinement evidence is accepted; one particular late-time recurrence signature is not the only integrated-closure route."
      ],
      "applicability_and_na": [
        "This obligation applies to a scientifically assessable submission; it is not a risk-triggered N/A item."
      ],
      "dependencies": [
        "Requires the independent Case-I arrest/confinement mechanism in M1, the quantitative regime mechanism in M3, and the full-model reduced-manifold coherent-control conclusion in M7. I1 and M2 are local scored recurrence duties, not prerequisites of this integrated conclusion."
      ],
      "equivalent_evidence": [
        "A source-consistent integrated conclusion built from physically equivalent evidence and the stated model is accepted."
      ],
      "source_support": [
        "Eldar et al. (2024), regime and confinement discussion/conclusions, pp. 4–5."
      ],
      "rc_relation": "Role: principal mechanism. Trigger: commissioned integrated conclusion absent or wrong. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "When the candidate's own method, data, anomaly or claim exposes a concrete material risk, exclude it with discriminating evidence or honestly restrict the conclusion.",
      "award_conditions": [
        "When applicable, evidence must be capable of exposing the identified threat: finite-basis reflection, an invalid nonrecoil reduction, wrong case/unit/normalization, an unsupported two-level reduction, or an unresolved alternative between physical selectivity and numerical suppression.",
        "Generic norm conservation, refinement, residuals or caveats count only if they address the particular material risk; no preferred test is compulsory."
      ],
      "applicability_and_na": [
        "Risk-triggered only; the examiner must identify a concrete answer-specific material threat with a fair basis in the visible problem or standard reasoning.",
        "If no such risk applies, record N/A and award the fixed 0.5 point without claiming validation evidence; the ten-point denominator is unchanged.",
        "If the sealed submission contains no scientifically assessable required submission, the terminal 0/RC0 disposition takes precedence and N/A credit is not awarded."
      ],
      "dependencies": [
        "No blanket upstream prerequisite. Judge the risk and its resolution from the evidence and claim actually submitted; do not add a second penalty for a defect already owned by another atom."
      ],
      "equivalent_evidence": [
        "Analytic checks, interventions, alternative models or estimators, cross-representation tests, held-out evidence, relevant convergence controls, or evidence-based scope limitation are accepted when they have the necessary discriminating force."
      ],
      "source_support": [
        "Eldar et al. (2024), regime and confinement discussion/conclusions, pp. 4–5."
      ],
      "rc_relation": "Role: material validation. Trigger: triggered material risk remains unresolved. Assess the substantive role condition independently of the atomic award; a zero score does not automatically lower RC. N/A does not block closure and does not constitute validation evidence. Honest restriction removes only the unsupported broader claim, not a wrong signal or mechanism."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "Limit the conclusion to the one-dimensional phase-matched quadratic Floquet–Bloch model and to the candidate's actual evidence.",
      "award_conditions": [
        "Distinguish model-level spectral control from multidimensional device efficiency, completed experimental performance or universal electron trapping."
      ],
      "applicability_and_na": [
        "This visible scope duty is unconditional; N/A is not permitted."
      ],
      "dependencies": [
        "No upstream prerequisite; honest scope restrictions retain independent credit but cannot repair a wrong signal or mechanism."
      ],
      "equivalent_evidence": [
        "A scientifically specific limitation to the established model and evidence is accepted; generic caveats or an asserted completed device are not substitutes."
      ],
      "source_support": [
        "Eldar et al. (2024), regime and confinement discussion/conclusions, pp. 4–5."
      ],
      "rc_relation": "No direct RC gate. Failure of this atom does not automatically lower RC; assess only any separately established material consequence for a required closure role."
    }
  ],
  "case_scientific_rules": [
    "The score is the sum of twenty binary awards of 0 or 0.5 point, for a maximum of 10. F=1.5, O=2.0, I=2.0, M=3.5 and V=1.0. There is no interpolation, RC ceiling, floor, multiplier or post-hoc score adjustment. The criterion labels are identifiers; no additional mechanism criterion is implied by the gap in their numbering.",
    "An assessable incomplete submission is graded from the scientific evidence actually sealed. A valid candidate-caused timeout/incomplete attempt with no scientifically assessable required submission receives 0 and RC0; unsealed scratch and run logs cannot supply scientific evidence or N/A credit.",
    "One accepted nonrelativistic convention for the full sideband equation is $$\n\\begin{aligned}i\\hbar\\dot a_n={}&-\\left[n\\hbar(\\omega-v_0k_z)-n^2\\frac{(\\hbar k_z)^2}{2m}+U_p\\right]a_n\\\\&+\\delta\\left(k_{n+1}-\\frac{k_z}{2}\\right)a_{n+1}e^{i\\phi}+\\delta\\left(k_{n-1}+\\frac{k_z}{2}\\right)a_{n-1}e^{-i\\phi}\\\\&-\\frac{U_p}{2}\\left(a_{n+2}e^{2i\\phi_0}+a_{n-2}e^{-2i\\phi_0}\\right),\\end{aligned}\n$$ with $k_n=k_0+n k_z$, $U_p=e^2E_f^2/(4m\\omega^2)$, $\\delta=e\\hbar E_f/(2m\\omega)$ and $\\phi=\\phi_0+\\pi/2$. Phase matching $k_z=\\omega/v_0$ removes the linear diagonal term. A removable global phase, consistent convention transformation, or consistently retained relativistic factors does not change the model identity.",
    "For interpretation, neglecting recoil variation in the nearest-neighbour coefficient and small ponderomotive coupling gives $$\ni\\hbar\\dot a_n=\\beta_d n^2a_n+\\kappa a_{n+1}+\\kappa^*a_{n-1},\\qquad\\beta_d=\\frac{(\\hbar k_z)^2}{2m},\\quad\\kappa=\\frac{eE_f\\hbar k_0}{2m\\omega}e^{i(\\phi_0+\\pi/2)},\\quad\\rho=\\frac{\\beta_d}{|\\kappa|}.\n$$ This is an analytical interpretation, not a replacement production model. $\\rho\\ll1$ is hopping-dominated Raman–Nath behaviour; $\\rho\\gg1$ is on-site-dominated Bragg behaviour; crossover is near $\\rho\\sim1$. Equivalent discriminators must show the same physics and ordering.",
    "The observable is $P_n(t)=|a_n(t)|^2$ with absolute state normalization and a declared energy mapping. Under the stated sideband-energy convention, $E_n=E_{\\rm ref}+n\\hbar\\omega$; a correctly transformed equivalent representation is accepted. Independent normalization of a selected state subset cannot establish full-ladder probability or suppression.",
    "Case I uses the 100 eV reference, initial $n=0$, 1.54 eV photon, 0.5 V nm$^{-1}$ field and $\\phi_0=0$. The characteristic full-model evolution is initial multi-order transport followed by slowing, bounded turning and recurrent reconcentration near the initial energy. Other functionally equivalent confinement evidence is accepted for the confinement output and integrated conclusion; early broadening alone or a smooth finite-basis distribution is insufficient.",
    "Case II uses the 16 eV reference, initial 12 eV state $n=-1$, 4 eV photon, 0.1 V nm$^{-1}$ field and $\\phi_0=0$. The task's unit convention is $10^8$ V m$^{-1}=0.1$ V nm$^{-1}$; the inconsistent source-caption unit must not replace it. Exchange is concentrated in $n=-1$ (12 eV) and $n=+1$ (20 eV), not the 16 eV reference, while the rest of the full ladder remains a small correction.",
    "Broad quantitative anchors are Case-I $\\rho$ of order $10^{-2}$ and Case-II $\\rho$ of order $10^1$. Source times near 9 and 19 fs, a reference transfer near 680 fs and reference pair probability near 0.975 are comparison anchors, not exact timing, decimal, probability-threshold or display obligations.",
    "A basis edge masquerading as trapping, a linear nonrecoil walk used as the production model, or a fitted two-state model lacking full-ladder justification does not establish the commissioned physical object. A materially wrong but internally coherent signal cannot establish a dependent mechanism by narrative consistency.",
    "For every output atom distinguish physical object validity, presentation adequacy and claim adequacy. All three are needed for the award, but output-atom failure alone is not an RC1 decision. Data existence, candidate presentation, candidate identification and physical inference are different acts. Examiner verification may check the object and presentation but cannot create candidate identification, mechanism or validation.",
    "Dependency propagation follows the exact failed physical fact, not the numerical zero of a composite atom. Identify the failed fact, the downstream obligation needing it, and why the candidate's own presented evidence cannot independently establish that obligation. Independent branches retain credit. Assign the initial deduction where the defect first has a physical consequence; downstream unsupported claims may fail on their explicit necessary premises, but no generic completeness, convergence or validation deduction is added.",
    "Local coefficient, phase, charge, index, solver, sampling or basis issues are diagnostic unless they materially change the model, case/observable identity, discriminator or output correctness. Local exchange-channel details likewise are diagnostic unless they materially damage selectivity, remaining-ladder suppression or the reduced-manifold mechanism. No preferred intermediate-state route is compulsory.",
    "Output comparison tests claim-relevant physical identity, support, symmetry, ordering, approximate scale, time ordering and component relations after a legitimate representation bridge. Principal structure, global morphology and material secondary structure must support the visible claim; agreement in one conspicuous feature is insufficient when other claim-relevant structures differ materially. Exact pixels, colour, smoothing, source crop, layout and author-specific numerical implementation are not targets.",
    "A material uncertainty about resolution, physical fine structure, branch identity or a noncanonical representation calls for a bounded examiner reference check, not an added candidate duty. A provisional score of at least 9.0 or provisional RC4 triggers one central-output and one principal-mechanism verification. An unresolved scientifically material equivalence dispute requires bounded review rather than an invented tolerance.",
    "The conclusion concerns spectral control in the stated one-dimensional model. A localized Gaussian-packet study, relative-phase trap-width scan, Maxwell-field simulation, multidimensional propagation and experimental-device efficiency are outside the required task."
  ],
  "closure_requirements": {
    "RC0": [
      "The central object is absent or materially invalid. The indispensable central-object claim requires the full phase-matched quadratic Floquet–Bloch production model with recoil-dependent $Ap$ and $A^2$ couplings (F1) and the shared sideband probability, declared energy mapping and common absolute normalization (F3). Each role is judged on material identity, not on the numerical atom award.",
      "A scientifically empty sealed required submission also has the terminal 0/RC0 disposition; N/A bookkeeping cannot create credit."
    ],
    "RC1": [
      "The central object is valid, but an indispensable component is absent or physically invalid: Case-I confinement evidence under the correct branch (O1), a valid quantitative hopping/on-site discriminator (F2) and two-case ordering/scale/crossover evidence (O2), or Case-II correctly bound coherent 12/20 eV exchange (O3) and small remaining-ladder probability under common absolute normalization (O4).",
      "For F2 the trigger is material discriminator identity absent or invalid; for O1–O4 it is output object absent or physically invalid. A presentation or claim failure that gives an O atom zero does not by itself prove this trigger. Case-I confinement may be established by functionally equivalent evidence rather than one unique late-time signature."
    ],
    "RC2": [
      "Required physical signals/components are substantially valid, but a required principal mechanism is absent or wrong, or the commissioned conclusion is absent or wrong.",
      "The Case-I co-primary claim requires the curvature-induced growth of on-site detuning to explain expansion arrest/confinement (M1). Its trigger is a missing/wrong principal mechanism. Local recurrence identification (I1) and explanation (M2) remain scored but are not separate closure gates.",
      "The regime/reduced-manifold co-primary claim requires the quantitative Raman–Nath/Bragg/crossover mechanism (M3), full-ladder near-degeneracy/selectivity of the correct exchanged pair (M4), off-resonant detuning of the remaining ladder (M6), and the supported model-level coherent-control assessment (M7). M3/M4/M6 trigger on a missing/wrong principal mechanism; M7 triggers on a missing/wrong commissioned conclusion. No unique local exchange pathway is a closure gate.",
      "The integrated co-primary claim requires a candidate-authored cross-case answer through the shared curvature/hopping competition (M8), supported by Case-I confinement mechanism, quantitative regime mechanism and Case-II reduced-manifold control. Its trigger is a missing/wrong commissioned conclusion; it does not require I1 or M2 as its sole closure route."
    ],
    "RC3": [
      "Signals and principal mechanisms are substantially correct, but a concrete answer-specific material risk remains unresolved and the conclusion is not adequately restricted. The integrated claim's material-validation role (V1) triggers only when such a material risk is unresolved; no prescribed validation method is compulsory.",
      "Before using RC3, record all six elements: (1) concrete unexcluded failure mode; (2) origin in the candidate's own method/data/claim; (3) fair visible evidence or standard-reasoning basis for recognizing it; (4) central conclusion threatened; (5) why submitted evidence cannot expose/exclude it; (6) examples of functionally equivalent resolving evidence. If any element is missing, RC3 does not apply by default.",
      "Candidate-specific threats can include basis reflection, an invalid nonrecoil reduction, wrong case/unit/initial-state binding, subset renormalization, unsupported two-state reduction, or inability to distinguish true selectivity from numerical suppression. These are possible risks, not a required checklist.",
      "Discriminating evidence or an honest restriction to the established domain can resolve the risk. Generic refinement, norm conservation, residuals or caveats neither automatically resolve nor automatically trigger it. Honest limitation does not repair a wrong signal, mechanism or an omitted commissioned component."
    ],
    "RC4": [
      "All nonempty required roles of the indispensable and co-primary claims close: central model/observable identity (F1/F3); Case-I confinement component (O1) and arrest/confinement mechanism (M1); quantitative regime and Case-II exchange/suppression components (F2/O2/O3/O4) with regime, selectivity, remaining-ladder and coherent-control mechanisms/conclusion (M3/M4/M6/M7); integrated cross-case conclusion (M8) with any applicable material risk resolved or honestly restricted (V1).",
      "Role conditions are substantive gate outcomes, not requirements that every F/O/I/M/V atom earn its point. In particular I1/M2 do not force one recurrence signature as the only complete research route, and V1 N/A does not block RC4 or claim validation evidence.",
      "The shared conclusion must remain model- and evidence-bounded. Unresolved material scientific ambiguity blocks an RC4 decision pending bounded adjudication. When several required roles fail, report the lowest corresponding RC; supporting work does not override a failed indispensable/co-primary role. RC classification never changes the independent atomic score."
    ]
  },
  "references": [
    {
      "citation": "M. Eldar, Z. Chen, Y. Pan, and M. Krüger, “Self-Trapping of Slow Electrons in the Energy Domain,” Physical Review Letters 132, 035001 (2024), doi:10.1103/PhysRevLett.132.035001.",
      "url": "https://doi.org/10.1103/PhysRevLett.132.035001",
      "anchors": [
        "Eq. (1), pp. 2–3: full sideband dynamics.",
        "Fig. 1(d), p. 2; discussion pp. 3–4: plane-wave confinement.",
        "Eq. (2), definition of rho and regime discussion pp. 3–4.",
        "Fig. 2(a)–(b), p. 4: regime map and Bragg exchange.",
        "Discussion/conclusions pp. 4–5: tunability and model interpretation."
      ],
      "scientific_scope": "Quadratic electron dispersion, phase-matched sideband dynamics, spectral confinement, and the Raman–Nath/Bragg regime distinction."
    },
    {
      "citation": "Supplemental Material to M. Eldar et al., Physical Review Letters 132, 035001 (2024).",
      "url": "https://link.aps.org/supplemental/10.1103/PhysRevLett.132.035001",
      "anchors": [
        "Derivation through Eq. (S9): full sideband model.",
        "Eqs. (S25)–(S26): regime discrimination."
      ],
      "scientific_scope": "Derivation of the sideband equation and the regime parameter, including its effective-interaction-length interpretation."
    }
  ]
}