{
  "case_id": "Case50",
  "title": "Spin-resolved relativistic three-photon Kapitza–Dirac transfer",
  "atoms": [
    {
      "criterion_id": "F1",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Bind the standing x-ray field, relativistic nr=2,nl=−1 three-photon Bragg branch, pure prepared-spin n=0 initial state and n=3 target; derive time-dependent Dirac mode equations with required energy/spin sectors and adjacent-mode coupling, treating rounded momentum as a resonance estimate.",
      "award_conditions": [
        "Bind the standing x-ray field, relativistic nr=2,nl=−1 three-photon Bragg branch, pure prepared-spin n=0 initial state and n=3 target; derive time-dependent Dirac mode equations with required energy/spin sectors and adjacent-mode coupling, treating rounded momentum as a resonance estimate."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for F2, F3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Ahrens et al. (2012), article pp. 043601-1–2, Eqs. (1)–(5), Fig. 1 and three-photon setup paragraph. Standing-wave Dirac sectors, adjacent momentum coupling, prepared spin and n=0→3 inelastic Bragg identity. Rounded incident momentum is a resonance estimate, not an exact hidden target."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "F2",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Define prepared/opposite n=3 occupations and their conditional spin-flip fraction with denominator equal to the sum of both final-spin n=3 probabilities.",
      "award_conditions": [
        "Define prepared/opposite n=3 occupations and their conditional spin-flip fraction with denominator equal to the sum of both final-spin n=3 probabilities."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: F1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for F3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3. Conditional fraction is opposite-spin occupation divided by both final-spin n=3 occupations. The analytic effective relation is an optional control, not a required generator."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "F3",
      "dimension": "F",
      "points": 0.5,
      "scientific_requirement": "Define independently extracted Dirac and matched time-dependent Klein–Gordon momentum-mode rates with common physical case, units and scale.",
      "award_conditions": [
        "Define independently extracted Dirac and matched time-dependent Klein–Gordon momentum-mode rates with common physical case, units and scale."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: F1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for F2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a); momentum-mode Klein–Gordon method note [23], p. 043601-5. Independently extracted matched Dirac/spinless rates on a common scale, not copied analytic points or a spin-resolved KG branch."
      ],
      "rc_relation": "Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award."
    },
    {
      "criterion_id": "O1",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present reconstructible incident n=0 survival and prepared-spin n=3 time exchange with auditable channel identity, bounded probabilities and accounting.",
      "award_conditions": [
        "Present reconstructible incident n=0 survival and prepared-spin n=3 time exchange with auditable channel identity, bounded probabilities and accounting.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F2). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O2, O3, O4, O5; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-2–3, Fig. 2, Eqs. (6a,b) and oscillation paragraph. Incident survival and both final-spin time branches. A local spin-changing inference can use the two final-spin branches independently of a separate missing incident-accounting conjunct."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "O2",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present opposite-spin n=3 occupation from the same dynamics with probability accounting sufficient to audit the spin-changing path.",
      "award_conditions": [
        "Present opposite-spin n=3 occupation from the same dynamics with probability accounting sufficient to audit the spin-changing path.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F2). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O3, O4, O5; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-2–3, Fig. 2, Eqs. (6a,b) and oscillation paragraph. Incident survival and both final-spin time branches. A local spin-changing inference can use the two final-spin branches independently of a separate missing incident-accounting conjunct."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "O3",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present justified-domain reconstructible spin-population momentum evidence: either the correct full conditional fraction and comparison/control, or explicitly bounded unnormalized/sparse flipped-branch evidence sufficient for a local trend.",
      "award_conditions": [
        "Either the full correct conditional spin fraction with a meaningful comparison/control is established, or explicitly bounded valid sparse/unnormalized flipped-branch evidence supports a local trend.",
        "The scope of credit is recorded: local mode does not establish the conditional denominator, full composition mechanism or indispensable full-object closure.",
        "Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: F1. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for F2, O1, O2, O4, O5; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "Only full conditional mode additionally requires F2's correct denominator. Local eligibility does not backfill F2, M2, M5 or full RC closure."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3 and momentum-adjustment paragraph. Momentum-dependent spin composition and supported local flipped-branch strength. Task local-mode credit is narrower than the full conditional object and cannot close that object by implication."
      ],
      "rc_relation": "Indispensable-component role: RC1 evaluates the full conditional-spin-fraction object independently. A half-point for eligible local unnormalized/sparse evidence does not close this role; display/claim failure alone still does not imply absence of a valid full object."
    },
    {
      "criterion_id": "O4",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present independently extracted Dirac rate versus |pE|/k from reconstructible dynamics, declaring estimator, physical units/normalization and case identity.",
      "award_conditions": [
        "Present independently extracted Dirac rate versus |pE|/k from reconstructible dynamics, declaring estimator, physical units/normalization and case identity.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (F1 AND F3). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O2, O3, O5; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a), numerical comparison paragraph. Separate Dirac and KG rate outputs; both must be independently auditable before identifying a matched-rate comparison. Either valid branch retains local credit if the other fails."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "O5",
      "dimension": "O",
      "points": 0.5,
      "scientific_requirement": "Present independently extracted matched Klein–Gordon spinless rates versus |pE|/k from reconstructible dynamics with comparable parameters, units and extraction.",
      "award_conditions": [
        "Present independently extracted matched Klein–Gordon spinless rates versus |pE|/k from reconstructible dynamics with comparable parameters, units and extraction.",
        "The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: F3. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for O1, O2, O3, O4; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a), numerical comparison paragraph. Separate Dirac and KG rate outputs; both must be independently auditable before identifying a matched-rate comparison. Either valid branch retains local credit if the other fails."
      ],
      "rc_relation": "Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object."
    },
    {
      "criterion_id": "I1",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify prepared/opposite diffracted pathways and bounded exchange from the candidate's time evidence, not total diffraction alone.",
      "award_conditions": [
        "Identify prepared/opposite diffracted pathways and bounded exchange from the candidate's time evidence, not total diffraction alone."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (O1 AND O2). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I2, I3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-2–3, Fig. 2, Eqs. (6a,b) and oscillation paragraph. Incident survival and both final-spin time branches. A local spin-changing inference can use the two final-spin branches independently of a separate missing incident-accounting conjunct."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "I2",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify material momentum dependence from eligible O3 evidence at its supported strength: full conditional/control statements only in full mode, local flipped-strength trends only in bounded local mode.",
      "award_conditions": [
        "Identify material momentum dependence from eligible O3 evidence at its supported strength: full conditional/control statements only in full mode, local flipped-strength trends only in bounded local mode."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: O3. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I1, I3; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3 and momentum-adjustment paragraph. Momentum-dependent spin composition and supported local flipped-branch strength. Task local-mode credit is narrower than the full conditional object and cannot close that object by implication."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "I3",
      "dimension": "I",
      "points": 0.5,
      "scientific_requirement": "Identify matched Dirac/spinless rate relations and momentum dependence from both independently auditable rate branches, separating composition from rate effects.",
      "award_conditions": [
        "Identify matched Dirac/spinless rate relations and momentum dependence from both independently auditable rate branches, separating composition from rate effects."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (O4 AND O5). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for I1, I2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a), numerical comparison paragraph. Separate Dirac and KG rate outputs; both must be independently auditable before identifying a matched-rate comparison. Either valid branch retains local credit if the other fails."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M1",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use both presented final-spin n=3 time branches to establish or exclude a spin-changing path; missing incident-survival/accounting remains local to its owning criteria, while total diffraction alone is insufficient.",
      "award_conditions": [
        "Both candidate-presented final-spin n=3 time branches support establishing or excluding a spin-changing path.",
        "Missing incident-survival/accounting work is judged locally elsewhere, while total diffraction alone cannot establish a spin-changing branch."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for M2, M3, M4, M5, M6; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "The two presented final-spin time branches, not the complete incident-survival/accounting conjunct of I1, supply this local pathway inference."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-2–3, Fig. 2, Eqs. (6a,b) and oscillation paragraph. Incident survival and both final-spin time branches. A local spin-changing inference can use the two final-spin branches independently of a separate missing incident-accounting conjunct."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M2",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Connect both final-spin amplitudes, the true conditional denominator and observed spin composition, preserving prepared/opposite invariants under basis-arrow reversal.",
      "award_conditions": [
        "Both final-spin components and their true summed n=3 conditional denominator support the composition explanation.",
        "Prepared/opposite spin invariants remain mapped under arrow reversal; local unnormalized O3 eligibility is insufficient for this full conditional obligation."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: O3. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M3, M4, M5, M6; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "O3 local-mode credit is not sufficient: independently verify both final-spin components and the full correct conditional denominator. Incident-survival accounting is not a necessary premise of this conditional composition explanation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3 and momentum-adjustment paragraph. Momentum-dependent spin composition and supported local flipped-branch strength. Task local-mode credit is narrower than the full conditional object and cannot close that object by implication."
      ],
      "rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
    },
    {
      "criterion_id": "M3",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Explain momentum-dependent spin coupling at the level supported by I2: full conditional data permit relative channel redistribution; eligible local data support only explicitly restricted flipped-branch strength reasoning.",
      "award_conditions": [
        "Explain momentum-dependent spin coupling at the level supported by I2: full conditional data permit relative channel redistribution; eligible local data support only explicitly restricted flipped-branch strength reasoning."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I2. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M4, M5, M6; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "Local I2/M3 credit does not close full channel-composition mechanism or upgrade to M5."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3 and momentum-adjustment paragraph. Momentum-dependent spin composition and supported local flipped-branch strength. Task local-mode credit is narrower than the full conditional object and cannot close that object by implication."
      ],
      "rc_relation": "Principal-mechanism role: RC2 concerns the full conditional channel-composition mechanism independently of local atomic credit. Eligible restricted flipped-branch reasoning retains its half-point without closing the full required mechanism."
    },
    {
      "criterion_id": "M4",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Use the matched Klein–Gordon control to distinguish spin-dependent channel/rate effects from shared spinless standing-wave dynamics.",
      "award_conditions": [
        "Use the matched Klein–Gordon control to distinguish spin-dependent channel/rate effects from shared spinless standing-wave dynamics."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: I3. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M5, M6, M7; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, role-of-spin paragraphs, Fig. 3, Fig. 4(a,b), Eqs. (7)–(9), (12). Positive spin-coupling mechanism, conditional redistribution and matched spinless rate effect. The full composition conclusion needs the actual full conditional denominator, not only sparse flipped population."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M5",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Connect |pE|/k, spin-dependent channel composition and coherent oscillation rate to determine whether spin changes composition, rate or both; the composition part requires the full correct conditional object, not local-mode backfill.",
      "award_conditions": [
        "Matched independently identified rates and the correctly constructed full conditional composition support deciding whether spin changes composition, rate or both.",
        "Bounded local O3/I2/M3 credit does not backfill the full conditional part of this integrated mechanism."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (I2 AND I3). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M1, M2, M3, M4, M6, M7; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
        "Independently verify the full conditional-object gate for the composition part; local O3/I2/M3 evidence cannot close it, even if both independent numerical rate branches are valid."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, role-of-spin paragraphs, Fig. 3, Fig. 4(a,b), Eqs. (7)–(9), (12). Positive spin-coupling mechanism, conditional redistribution and matched spinless rate effect. The full composition conclusion needs the actual full conditional denominator, not only sparse flipped population."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "M6",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Integrate Dirac spin branches and matched spinless control to determine spin's distinguishable effect on resonant three-photon transfer as a model-specific, not universal, conclusion.",
      "award_conditions": [
        "Integrate Dirac spin branches and matched spinless control to determine spin's distinguishable effect on resonant three-photon transfer as a model-specific, not universal, conclusion."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: (M1 AND M3 AND M4 AND M5). This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for M2, M7; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article pp. 043601-3–4, role-of-spin paragraphs, Fig. 3, Fig. 4(a,b), Eqs. (7)–(9), (12). Positive spin-coupling mechanism, conditional redistribution and matched spinless rate effect. The full composition conclusion needs the actual full conditional denominator, not only sparse flipped population."
      ],
      "rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
    },
    {
      "criterion_id": "M7",
      "dimension": "M",
      "points": 0.5,
      "scientific_requirement": "Distinguish the model-specific spin conclusion established by Dirac/Klein–Gordon evidence from universal/nonrelativistic claims; treat the source-effective analytic relation as optional, not a missing duty.",
      "award_conditions": [
        "Distinguish the model-specific spin conclusion established by Dirac/Klein–Gordon evidence from universal/nonrelativistic claims; treat the source-effective analytic relation as optional, not a missing duty."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Necessary scientific premises: M6. This identifies facts, not prerequisite half-point awards.",
        "Preserve independently valid evidence for V1, V2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-2, inelastic Bragg paragraph; pp. 043601-3–4, nonrelativistic comparison and conclusion. Model-specific relativistic three-photon domain; taking a universal nonrelativistic limit is not licensed by qualitative resemblance."
      ],
      "rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
    },
    {
      "criterion_id": "V1",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "State the commissioned relativistic three-photon standing-wave finite-pulse model and actual momentum/resonance domain without universal nonrelativistic or hidden-source extrapolation.",
      "award_conditions": [
        "State the commissioned relativistic three-photon standing-wave finite-pulse model and actual momentum/resonance domain without universal nonrelativistic or hidden-source extrapolation."
      ],
      "applicability_and_na": [
        "Applies to every assessable submission; absence or failure is scored 0, not N/A."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for V2; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-2, inelastic Bragg paragraph; pp. 043601-3–4, nonrelativistic comparison and conclusion. Model-specific relativistic three-photon domain; taking a universal nonrelativistic limit is not licensed by qualitative resemblance."
      ],
      "rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
    },
    {
      "criterion_id": "V2",
      "dimension": "V",
      "points": 0.5,
      "scientific_requirement": "If actual photon/momentum branch, energy/spin sectors, detuning/resonance, rate resolution, case/normalization matching, conditional denominator or other submitted facts threaten the central conclusion, provide discrimination or restriction; otherwise record N/A.",
      "award_conditions": [
        "If actual photon/momentum branch, energy/spin sectors, detuning/resonance, rate resolution, case/normalization matching, conditional denominator or other submitted facts threaten the central conclusion, provide discrimination or restriction; otherwise record N/A.",
        "When triggered, award only if claim-relevant discriminating evidence resolves the threat or an evidence-based restriction makes it immaterial; merely acknowledging an unresolved material threat is not positive validation credit. With no trigger, record N/A and retain the fixed 0.5 without claiming performed validation."
      ],
      "applicability_and_na": [
        "Applies only when the submitted method, data, anomaly or conclusion creates the concrete material risk described here. If untriggered, record not_applicable and retain the fixed 0.5 contribution; this is not evidence that validation was performed, and does not block RC4."
      ],
      "dependencies": [
        "Judge this scientific obligation directly; no fixed prerequisite award.",
        "Preserve independently valid evidence for V1; failure here does not automatically erase those criteria.",
        "Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
      ],
      "equivalent_evidence": [
        "Accept consistent spinor phases, gauges, bases and numerical propagation preserving the prepared positive-energy n=0 state, n=3 momentum order, positive/negative energy sectors as needed, spin projection along E and the physical probabilities. A basis-arrow reversal is allowed when prepared/opposite identity is mapped.",
        "Source-effective analytic formulas are optional perturbative controls, not full-credit duties. Converged reconstructible Dirac/Klein–Gordon dynamics, numerical/asymptotic/symmetry/ratio or equivalent controls may establish the same channel/rate conclusion.",
        "No fixed exact resonance coordinate, hidden momentum offset, mode span, tolerance, cycle stride, peak window, scan density, period, source layout or pointwise target. Declared common absolute/normalized rate scales are accepted when matched and auditable."
      ],
      "source_support": [
        "Article p. 043601-2, Eqs. (4)–(5); p. 043601-4, resonance-resolution paragraph; Ahrens et al. (2013), author manuscript Sec. 5.1, pp. 9–11, Fig. 8. Triggered sector/detuning/rate/denominator threats. Follow-up clarifies adopted original data and rounded momentum, not a hidden exact coordinate or extra candidate method."
      ],
      "rc_relation": "Material-validation role: RC3 only for a triggered, unresolved threat to a central conclusion, with all six burden-of-proof elements established. N/A does not block RC4."
    }
  ],
  "case_scientific_rules": [
    "An accepted standing field is A(x,t)=−E/k cos(k·x)sin(ckt)w(t), with i partial_t psi=[c(−i grad+eA/c)·alpha+beta mc²]psi. Expand positive/negative free Dirac spinors at p+n k, keeping adjacent-mode coupling and required spin/energy sectors. The Bragg process has nr=2,nl=−1 and target n=3 from pure prepared-spin positive-energy n=0.",
    "Define final prepared/opposite probabilities at n=3 and P_flip=P_3,opposite/(P_3,prepared+P_3,opposite). This is conditional on diffraction, not unnormalized opposite population or total initial norm. Klein–Gordon dynamics has no spin-resolved branch.",
    "For x=|p_E|/k, optional source-effective relations are P0=cos²(Omega_R T/2), P3=sin²(Omega_R T/2), P3,prepared≈(Omega0 T/2)²(25/2)x², P3,opposite≈(Omega0 T/2)², P_flip=1/[1+(25/2)x²], Omega_D=Omega0 sqrt[1+(25/2)x²], Omega_KG=Omega0(5/sqrt(2))x, Omega0=e³|E|³/(24m³c⁵k²). Their omission does not lose credit; analytic curves relabelled independent numerical data are invalid evidence.",
    "Treat the rounded source momentum and stated envelope as resonance starting information, not a hidden exact coordinate. Establish the commissioned finite-intensity resonant dynamics using declared claim-adequate resonance localization. The same-author clarification may support examiner identity interpretation but cannot impose an undisclosed momentum offset.",
    "O3 admits full conditional evidence with control, or explicitly bounded eligible unnormalized/sparse flipped-branch evidence. Local mode can earn its half-point and proportionate I2/M3 credit only for local strength/dependence. Wrong branch, wrong denominator or copied analytic values are not eligible local evidence.",
    "O3 local credit does not establish conditional denominator, complete channel composition, M2, M5, RC1 full-object closure or RC2 full-mechanism closure. Evaluate scientific role outcomes separately from the local atomic award.",
    "M1 can be established directly by the two candidate-presented final-spin time branches even when incident-survival/accounting conjuncts of O1/O2/I1 are incomplete. M2's conditional composition likewise does not hard-gate on complete incident accounting, but does require the true conditional object and both components.",
    "Extract Dirac and Klein–Gordon rates independently from reconstructible oscillatory dynamics with matched field, momentum/resonance, envelope and normalization. One valid solver's rate retains local credit when the other is absent, but cannot establish the cross-model comparison.",
    "Bound claims to the demonstrated relativistic three-photon finite-pulse Dirac/Klein–Gordon and momentum/resonance domain. A nonrelativistic two-photon replacement, unmatched controls or missing material spin/energy sectors is not equivalent. Numerical reliability has no separate points; threats must actually change a branch, ratio, rate or conclusion.",
    "Score exactly twenty fixed 0/0.5 material-core criteria. Do not import numerical score ranges or RC score caps. A valid run with no assessable scientific work receives score 0 and RC0 before criterion summation; assessable partial work and timeouts are graded on available evidence.",
    "Object existence, candidate presentation, candidate identification, physical inference and validation are different acts. Inspect every candidate-authored claim-bearing figure. A reference, rerender or examiner reconstruction can verify an object but cannot create missing candidate reasoning.",
    "When a material decision depends on feature resolution, fine structure, component relations, formula identity, representation equivalence or suspicious morphology, perform a focused applicable source/reference check. Establish model/observable/case/component, units, basis/gauge, normalization and transformations first; do not turn source-only display choices into new duties. A provisional score at least 9.0 or provisional RC4 requires a focused check of one central output and one principal-mechanism link. Reference verification is examiner-side, not a new candidate deliverable.",
    "With $\\hbar=1$ and the source Gaussian-unit convention, $\\mathbf A(\\mathbf x,t)=-(\\mathbf E/k)\\cos(\\mathbf k\\cdot\\mathbf x)\\sin(ckt)w(t)$ and $i\\partial_t\\psi=[c(-i\\nabla+e\\mathbf A/c)\\cdot\\boldsymbol\\alpha+\\beta mc^2]\\psi$ (Eqs. (1)–(2)). The electron charge is $-e$. Free modes have momentum $\\mathbf p+n\\mathbf k$, energies $\\pm\\sqrt{m^2c^4+c^2|\\mathbf p+n\\mathbf k|^2}$ and prepared/opposite spin projections along $\\mathbf E$; the cosine field couples adjacent n with the spinor matrix elements of $\\mathbf E\\cdot\\boldsymbol\\alpha$ (Eqs. (3)–(4)).",
    "Three-photon conservation is $\\mathbf p\\prime=\\mathbf p+(n_r-n_l)\\mathbf k=\\mathbf p+3\\mathbf k$, $E(\\mathbf p\\prime)-E(\\mathbf p)=(n_r+n_l)ck=ck$ for $n_r=2$, $n_l=-1$ (Bragg discussion, Eq. (5)). The initial positive-energy prepared-spin n=0 amplitude is one and all other amplitudes zero. Matching the resonant driven case may require justified localization around the rounded momentum; no hidden exact offset is compulsory.",
    "Let $P_{3,\\rm prep}=|c_3^{+\\uparrow}|^2$, $P_{3,\\rm opp}=|c_3^{+\\downarrow}|^2$. Then $F_{\\rm flip}=P_{3,\\rm opp}/(P_{3,\\rm prep}+P_{3,\\rm opp})$ where the denominator is resolved and nonzero (Eq. (8)). It is conditional on diffraction into n=3, not on the total electron population or incident survival. Consistent reversal of basis arrows preserves prepared/opposite invariants.",
    "Optional perturbative comparator only: with $x=|p_E|/k$ and $\\Omega_0=e^3|E|^3/(24m^3c^5k^2)$, $F_{\\rm flip}\\simeq[1+(25/2)x^2]^{-1}$, $\\Omega_D\\simeq\\Omega_0\\sqrt{1+(25/2)x^2}$, $\\Omega_{\\rm KG}\\simeq\\Omega_0(5/\\sqrt2)x$ (Eqs. (7)–(9), (12)). These are Bragg/short-time effective relations, not substitutes for independently propagated Dirac/KG rate evidence. For the two-level-envelope interpretation $P_3(T)\\simeq\\sin^2(\\Omega_RT/2)$ (Eq. (6b)), state the estimator, time unit and matched pulse convention.",
    "O3/I2/M3 permit explicitly bounded local credit from valid unnormalized or sparse flipped-branch momentum evidence. That credit does not establish F2, M2, the full composition part of M5, or the indispensable full conditional-object role under RC1; full Research Closure requires the correct conditional object and both matched numerical rate branches.",
    "M1 is judged from the two candidate-presented final-spin n=3 time branches. Missing incident survival or probability accounting remains local to its owning obligations and is not a second all-or-nothing gate on the supported spin-changing path. Conversely, total diffraction alone cannot establish either final-spin path.",
    "Figure 8 of Ahrens et al. (2013) explicitly adopts the original three-photon data and Sec. 5.1 distinguishes the rounded momentum representation and driven resonance behavior. This is citation/identity clarification only; it does not require an exact 0.347 mc value, hidden resonance offset or the source analytic comparator."
  ],
  "closure_requirements": {
    "RC0": [
      "The central physical object is absent or materially invalid. Establish this scientific fact independently of atomic credit; a local notation or presentation weakness is not itself RC0.",
      "(F1 AND F2 AND F3): Bind the standing x-ray field, relativistic nr=2,nl=−1 three-photon Bragg branch, pure prepared-spin n=0 initial state and n=3 target; derive time-dependent Dirac mode equations with required energy/spin sectors and adjacent-mode coupling, treating rounded momentum as a resonance estimate. Define prepared/opposite n=3 occupations and their conditional spin-flip fraction with denominator equal to the sum of both final-spin n=3 probabilities. Define independently extracted Dirac and matched time-dependent Klein–Gordon momentum-mode rates with common physical case, units and scale."
    ],
    "RC1": [
      "The central object exists, but a required indispensable component is absent or materially invalid. A failed presentation or overextended claim alone does not establish object absence or invalidity.",
      "(O1 AND O2): Present reconstructible incident n=0 survival and prepared-spin n=3 time exchange with auditable channel identity, bounded probabilities and accounting. Present opposite-spin n=3 occupation from the same dynamics with probability accounting sufficient to audit the spin-changing path.",
      "O3: Present justified-domain reconstructible spin-population momentum evidence: either the correct full conditional fraction and comparison/control, or explicitly bounded unnormalized/sparse flipped-branch evidence sufficient for a local trend.",
      "(O4 AND O5): Present independently extracted Dirac rate versus |pE|/k from reconstructible dynamics, declaring estimator, physical units/normalization and case identity. Present independently extracted matched Klein–Gordon spinless rates versus |pE|/k from reconstructible dynamics with comparable parameters, units and extraction.",
      "For O3 the indispensable closure object is the full correct conditional spin-flip observable. Eligible unnormalized/sparse local evidence can earn the O3 half-point but cannot close this full conditional-object role or repair the denominator."
    ],
    "RC2": [
      "Required objects are scientifically valid, but a required principal mechanism or state-dependence inference is absent or incorrect. Reference evidence cannot provide omitted candidate reasoning.",
      "M2: Connect both final-spin amplitudes, the true conditional denominator and observed spin composition, preserving prepared/opposite invariants under basis-arrow reversal.",
      "M3: Explain momentum-dependent spin coupling at the level supported by I2: full conditional data permit relative channel redistribution; eligible local data support only explicitly restricted flipped-branch strength reasoning.",
      "M6: Integrate Dirac spin branches and matched spinless control to determine spin's distinguishable effect on resonant three-photon transfer as a model-specific, not universal, conclusion.",
      "M7: Distinguish the model-specific spin conclusion established by Dirac/Klein–Gordon evidence from universal/nonrelativistic claims; treat the source-effective analytic relation as optional, not a missing duty.",
      "For M3 the required closure fact is the full channel-composition mechanism supported by correct conditional evidence. Eligible branch-local flipped-strength reasoning can earn the M3 half-point but cannot close that full mechanism role. M2 and M5 likewise require the full conditional object; branch-local credit does not backfill them."
    ],
    "RC3": [
      "Required objects and principal mechanisms are established, but a concrete material validation or scope threat to a central conclusion remains unresolved.",
      "V2: If actual photon/momentum branch, energy/spin sectors, detuning/resonance, rate resolution, case/normalization matching, conditional denominator or other submitted facts threaten the central conclusion, provide discrimination or restriction; otherwise record N/A.",
      "Before assigning RC3, establish all six elements: the concrete failure mode; its origin in the submitted method/data/conclusion; a fair recognizable scientific risk; the affected central conclusion; why the submitted evidence cannot discriminate or exclude it; and functionally equivalent remedies. Evidence-based restriction may remove unsupported breadth, but cannot repair an incorrect signal, absent component or missing mechanism."
    ],
    "RC4": [
      "Every indispensable and co-primary claim has all its nonempty required role slots closed; supporting claims do not independently determine overall RC. Preserve AND/OR branch logic shown above and use scientific gate outcomes, not atomic scores.",
      "No triggered material risk remains unresolved. Untriggered risk criteria recorded N/A do not block RC4 and do not establish validation evidence.",
      "Classify descriptively by the most severe applicable failed required role (lowest RC), without score ceilings or target distributions. A score/RC tension prompts scientific review, not forced adjustment."
    ]
  },
  "references": [
    {
      "citation": "Sven Ahrens, Heiko Bauke, Christoph H. Keitel, and Carsten Müller. 2012. *Spin Dynamics in the Kapitza-Dirac Effect*. *Physical Review Letters*, 109, 043601.",
      "url": "https://doi.org/10.1103/PhysRevLett.109.043601",
      "scientific_scope": "Relativistic three-photon spin-resolved standing-wave Dirac transfer, its true conditional final-spin fraction and independently propagated matched spinless rates. Local flipped-branch credit is explicitly weaker than full conditional closure; source-effective curves are optional controls.",
      "anchors": [
        "Ahrens et al. (2012), article pp. 043601-1–2, Eqs. (1)–(5), Fig. 1 and three-photon setup paragraph.",
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3.",
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a); momentum-mode Klein–Gordon method note [23], p. 043601-5.",
        "Article pp. 043601-2–3, Fig. 2, Eqs. (6a,b) and oscillation paragraph.",
        "Article p. 043601-3, Eqs. (7a,b)–(8), Fig. 3 and momentum-adjustment paragraph.",
        "Article pp. 043601-3–4, Eqs. (9), (12), Fig. 4(a), numerical comparison paragraph.",
        "Article pp. 043601-3–4, role-of-spin paragraphs, Fig. 3, Fig. 4(a,b), Eqs. (7)–(9), (12).",
        "Article p. 043601-2, inelastic Bragg paragraph; pp. 043601-3–4, nonrelativistic comparison and conclusion.",
        "Article p. 043601-2, Eqs. (4)–(5); p. 043601-4, resonance-resolution paragraph; Ahrens et al. (2013), author manuscript Sec. 5.1, pp. 9–11, Fig. 8."
      ]
    },
    {
      "citation": "Sven Ahrens, Heiko Bauke, Christoph H. Keitel, and Carsten Müller. 2013. Kapitza-Dirac effect in the relativistic regime. Physical Review A 88, 012115.",
      "url": "https://doi.org/10.1103/PhysRevA.88.012115",
      "scientific_scope": "This follow-up clarifies the identity of the adopted data and the resonance setting. It does not prescribe an analytic comparator or an exact incident momentum, and is not supplied to candidates as an offline reference.",
      "anchors": [
        "Author manuscript Sec. 5.1, pp. 9–11, Fig. 8: adopted original data, rounded momentum representation and driven-resonance qualification."
      ]
    }
  ]
}