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"case_id": "Case37",
"title": "Recoil-resolved free-electron photon statistics across low and high coupling",
"atoms": [
{
"criterion_id": "F1",
"dimension": "F",
"points": 0.5,
"scientific_requirement": "Instantiate the initially empty single-mode joint electron–photon ladder, first-link phase matching, exact relativistic link mismatch, bosonic square-root coupling and consistent phase/index conventions.",
"award_conditions": [
"Instantiate the initially empty single-mode joint electron–photon ladder, first-link phase matching, exact relativistic link mismatch, bosonic square-root coupling and consistent phase/index conventions."
],
"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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Sirotin et al. (2026), main text p. 6, joint-state paragraph; SI pp. 20–22, Eqs. (S34)–(S44). Vacuum joint ladder, link-dependent recoil, bosonic factors and exact direct/autonomous evolution. Populations are invariant under exact autonomization; fidelity-sensitive phases need not be."
],
"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": "Provide executable direct propagation or exact autonomous construction and derive normalized populations, moments and factorial second-order correlation from the final amplitudes with reconstructible conventions.",
"award_conditions": [
"Provide executable direct propagation or exact autonomous construction and derive normalized populations, moments and factorial second-order correlation from the final amplitudes with reconstructible conventions."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Judge this scientific obligation directly; no fixed prerequisite award.",
"Preserve independently valid evidence for F1, F3; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
"Independently award the observable construction; upstream model defects propagate only where they actually change the scientific object."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Sirotin et al. (2026), main text p. 6, joint-state paragraph; SI pp. 20–22, Eqs. (S34)–(S44). Vacuum joint ladder, link-dependent recoil, bosonic factors and exact direct/autonomous evolution. Populations are invariant under exact autonomization; fidelity-sensitive phases need not be."
],
"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": "Bind exact mismatch or a declared sigma estimate, both low/high-coupling domains, weak-recoil reference and at least one self-consistent electron/photon/length realization, distinguishing approximations from exact production.",
"award_conditions": [
"Bind exact mismatch or a declared sigma estimate, both low/high-coupling domains, weak-recoil reference and at least one self-consistent electron/photon/length realization, distinguishing approximations from exact production."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Judge this scientific obligation directly; no fixed prerequisite award.",
"Preserve independently valid evidence for F1, F2; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text p. 6, Eqs. (5)–(6); SI pp. 30–31, Eqs. (S64)–(S65), Fig. S4(a). Physical energy/photon/length mapping and the approximate phase-matching width; the physical example is not a compulsory parameter tuple."
],
"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 valid reconstructible exact-ladder low-coupling weak-recoil and finite-recoil populations with suitable moments/correlation and a common representation adequate for the stated claims.",
"award_conditions": [
"Exact-ladder low-coupling weak- and finite-recoil populations are reconstructible and correctly normalized.",
"Same-population means and factorial correlation, or statistically equivalent variance/Fano/Mandel evidence, support the claimed statistical classification; a qualitative label alone is insufficient.",
"Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (F1 AND F2). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for O2, O3, I2, I3, M3, M4; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.",
"Physical-mapping or domain-presentation weaknesses are localized, not automatic upstream gates on a valid low-coupling object."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs. Weak-recoil Poisson reference, restricted ladder and antibunching/sub-Poissonian statistics. Joint populations or equivalent electron sidebands with reconstructible statistics are accepted."
],
"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 exact-ladder higher-coupling state-resolved evidence exposing finite-ladder coherent response families through populations or equivalent electron sidebands with suitable statistics.",
"award_conditions": [
"Higher-coupling production retains the exact recoil-dependent ladder and state-resolved population or equivalent electron-sideband evidence.",
"The presentation and same-object statistics distinguish coherent finite-ladder occupation dynamics from a large normalized correlation at very low mean.",
"Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (F1 AND F2). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for O1, O3, I1, I3, M1, M2; 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b). Coherent redistribution, returns, few-level oscillations and large factorial correlation at low mean. A capped correlation map alone cannot establish occupation dynamics or high yield."
],
"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 claim-relevant exact/low-coupling-approximation, tail, phase or representation discrimination, or equivalent evidence, with data/run identity sufficient to expose an artefactual central response and delimit approximation applicability.",
"award_conditions": [
"Present claim-relevant exact/low-coupling-approximation, tail, phase or representation discrimination, or equivalent evidence, with data/run identity sufficient to expose an artefactual central response and delimit approximation applicability.",
"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. This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for O1, O2, I1, I2, M1, M3; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c). Exact production versus first-order sinc/Magnus approximation; alternative exact representations and informative tail/phase discrimination remain acceptable without a prescribed solver."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
},
{
"criterion_id": "I1",
"dimension": "I",
"points": 0.5,
"scientific_requirement": "Identify how recoil changes low-coupling ladder support and statistical classification from the candidate's own populations and statistics.",
"award_conditions": [
"Identify how recoil changes low-coupling ladder support and statistical classification from the candidate's own populations and statistics."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: O1. This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for I2, I3, M3, M4; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs. Weak-recoil Poisson reference, restricted ladder and antibunching/sub-Poissonian statistics. Joint populations or equivalent electron sidebands with reconstructible statistics are accepted."
],
"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 recoil/coupling-dependent coherent redistribution, return or few-level structures from the candidate's higher-coupling state-resolved evidence, not source ridge coordinates.",
"award_conditions": [
"Identify material recoil/coupling-dependent coherent redistribution, return or few-level structures from the candidate's higher-coupling state-resolved evidence, not source ridge coordinates."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: O2. This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for I1, I3, M1, M2; 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b). Coherent redistribution, returns, few-level oscillations and large factorial correlation at low mean. A capped correlation map alone cannot establish occupation dynamics or high yield."
],
"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 where the low-coupling approximation agrees with exact production and where approximation, tail, phase or representation sensitivity becomes material.",
"award_conditions": [
"Identify where the low-coupling approximation agrees with exact production and where approximation, tail, phase or representation sensitivity becomes material."
],
"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, I2, M1, M3; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c). Exact production versus first-order sinc/Magnus approximation; alternative exact representations and informative tail/phase discrimination remain acceptable without a prescribed solver."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
},
{
"criterion_id": "M1",
"dimension": "M",
"points": 0.5,
"scientific_requirement": "Explain the identified low-coupling response through successive-link mismatch restricting accessible ladder support, not a detached statistical label.",
"award_conditions": [
"Explain the identified low-coupling response through successive-link mismatch restricting accessible ladder support, not a detached statistical label."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (F1 AND I1). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for M3, M4, M5; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs. Weak-recoil Poisson reference, restricted ladder and antibunching/sub-Poissonian statistics. Joint populations or equivalent electron sidebands with reconstructible statistics are accepted."
],
"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": "Ground the low-coupling statistical/nonclassical claim in the candidate's populations, mean and factorial correlation within the studied domain; one summary statistic alone is not complete state proof.",
"award_conditions": [
"Ground the low-coupling statistical/nonclassical claim in the candidate's populations, mean and factorial correlation within the studied domain; one summary statistic alone is not complete state proof."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for M3, M4, M5, 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs. Weak-recoil Poisson reference, restricted ladder and antibunching/sub-Poissonian statistics. Joint populations or equivalent electron sidebands with reconstructible statistics are accepted."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
},
{
"criterion_id": "M3",
"dimension": "M",
"points": 0.5,
"scientific_requirement": "Explain identified higher-coupling structures as coherent occupation redistribution within the recoil-limited multilevel ladder, including supported returns or few-level behaviour.",
"award_conditions": [
"Explain identified higher-coupling structures as coherent occupation redistribution within the recoil-limited multilevel ladder, including supported returns or few-level behaviour."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (F1 AND I2). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for M1, M2, M4, M5; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b). Coherent redistribution, returns, few-level oscillations and large factorial correlation at low mean. A capped correlation map alone cannot establish occupation dynamics or high yield."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
},
{
"criterion_id": "M4",
"dimension": "M",
"points": 0.5,
"scientific_requirement": "Distinguish genuine high-coupling occupation dynamics from low-mean large-correlation features and display artefacts using state-resolved distributions or equivalent underlying evidence.",
"award_conditions": [
"Distinguish genuine high-coupling occupation dynamics from low-mean large-correlation features and display artefacts using state-resolved distributions or equivalent underlying evidence."
],
"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, M3, M5; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b). Coherent redistribution, returns, few-level oscillations and large factorial correlation at low mean. A capped correlation map alone cannot establish occupation dynamics or high yield."
],
"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": "Relate the studied recoil/coupling domain to at least one self-consistent electron-energy, photon-energy and interaction-length realization, respecting exact/estimated mismatch applicability.",
"award_conditions": [
"Relate the studied recoil/coupling domain to at least one self-consistent electron-energy, photon-energy and interaction-length realization, respecting exact/estimated mismatch applicability."
],
"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 M1, M2, M3, M4; failure here does not automatically erase those criteria.",
"Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text p. 6, Eqs. (5)–(6); SI pp. 30–31, Eqs. (S64)–(S65), Fig. S4(a). Physical energy/photon/length mapping and the approximate phase-matching width; the physical example is not a compulsory parameter tuple."
],
"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": "Unify both branches: relativistic successive-link mismatch limits the active ladder while interaction strength governs coherent occupation dynamics within it.",
"award_conditions": [
"Unify both branches: relativistic successive-link mismatch limits the active ladder while interaction strength governs coherent occupation dynamics within it."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (M1 AND M3). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for M2, M4, M5, 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 6–8, Eqs. (5)–(7), Fig. 4(c–f); SI p. 22, Eqs. (S41)–(S44). Physical synthesis: mismatch limits accessible levels, coupling governs coherent dynamics. Integrated state-generation conclusions must come from the candidate rather than source labels."
],
"rc_relation": "Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award."
},
{
"criterion_id": "M7",
"dimension": "M",
"points": 0.5,
"scientific_requirement": "Establish from the candidate's discriminating signal where first-order phase-matching or weak-coherent descriptions are adequate and where exact finite-recoil dynamics is required.",
"award_conditions": [
"Establish from the candidate's discriminating signal where first-order phase-matching or weak-coherent descriptions are adequate and where exact finite-recoil dynamics is required."
],
"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, 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."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c). Exact production versus first-order sinc/Magnus approximation; alternative exact representations and informative tail/phase discrimination remain acceptable without a prescribed solver."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
},
{
"criterion_id": "M8",
"dimension": "M",
"points": 0.5,
"scientific_requirement": "Integrate low/high-coupling signals, the unified finite-ladder mechanism, approximation boundary and physical realization into the strongest evidence-supported state-generation conclusion.",
"award_conditions": [
"Integrate low/high-coupling signals, the unified finite-ladder mechanism, approximation boundary and physical realization into the strongest evidence-supported state-generation conclusion."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Necessary scientific premises: (M2 AND M5 AND M6 AND M7). This identifies facts, not prerequisite half-point awards.",
"Preserve independently valid evidence for M1, M3, V1, V3; 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 separately assessed low-mean/high-correlation caveat or proportionate restriction belongs to M4 and is not a second hard gate on this integrated conclusion."
],
"equivalent_evidence": [
"Accept direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 6–8, Eqs. (5)–(7), Fig. 4(c–f); SI p. 22, Eqs. (S41)–(S44). Physical synthesis: mismatch limits accessible levels, coupling governs coherent dynamics. Integrated state-generation conclusions must come from the candidate rather than source labels."
],
"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": "Establish with claim-relevant evidence that production is the exact finite-recoil ladder rather than a sinc, weak-coherent, Poisson or fixed-level surrogate; any exact equivalent representation is acceptable.",
"award_conditions": [
"Establish with claim-relevant evidence that production is the exact finite-recoil ladder rather than a sinc, weak-coherent, Poisson or fixed-level surrogate; any exact equivalent representation is acceptable."
],
"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, V3; 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c). Exact production versus first-order sinc/Magnus approximation; alternative exact representations and informative tail/phase discrimination remain acceptable without a prescribed solver."
],
"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": "When actual branch, tail, phase indexing, truncation, representation, anomaly or claim raises a concrete material alternative, supply evidence capable of exposing it or leave the conclusion unresolved for adjudication; otherwise record N/A.",
"award_conditions": [
"When actual branch, tail, phase indexing, truncation, representation, anomaly or claim raises a concrete material alternative, supply evidence capable of exposing it or leave the conclusion unresolved for adjudication; 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, V3; 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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"SI pp. 21–22, Eqs. (S39)–(S44); main text p. 7, low-mean correlation paragraph; Fig. S5(b), SI p. 31. Concrete representation, truncation and low-mean interpretation threats; conditional risk handling is task-defined, not a universal required check."
],
"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."
},
{
"criterion_id": "V3",
"dimension": "V",
"points": 0.5,
"scientific_requirement": "Limit conclusions to the initially empty single-mode, first-link single-photon phase-matched branch and actual recoil/coupling domain; do not infer excluded multiphoton, driven-field, horizon or universal state-engineering claims.",
"award_conditions": [
"Limit conclusions to the initially empty single-mode, first-link single-photon phase-matched branch and actual recoil/coupling domain; do not infer excluded multiphoton, driven-field, horizon or universal state-engineering claims."
],
"applicability_and_na": [
"Applies to every assessable submission; absence or failure is scored 0, not N/A."
],
"dependencies": [
"Judge this scientific obligation directly; no fixed prerequisite award.",
"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 direct nonautonomous propagation, exact autonomization, unitary/Krylov or any mathematically equivalent solver preserving the exact finite-detuning ladder. Approximate sinc/Magnus, Poisson, weak-coherent or fixed-two-level models may be comparisons, not production substitutes.",
"Accept adaptive/irregular sampling, maps, cuts, trajectories, distributions, electron-loss sidebands and functionally equivalent representations. No fixed sigma/coupling mesh, ridge coordinate/count, return coordinate, display cap, truncation, solver or source layout.",
"For positive mean occupation, a full normalized P(n), or the mean with variance, Fano factor or Mandel Q sufficient to reconstruct factorial g^(2), is equivalent to separately tabulated g^(2), when all statistics refer to the same final P(n). A qualitative sub-Poissonian label or unreconstructible summary is not equivalent. Representation equivalence supplies no missing candidate identification, mechanism or scope."
],
"source_support": [
"Main text pp. 6–7, joint vacuum state and Sec. III A; Sec. III B, pp. 7–9, as an excluded contrast. Scope is the initially empty single-mode first-link single-photon branch; two-photon/QPM, driven PINEM and horizon cases are not commissioned outputs."
],
"rc_relation": "Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim."
}
],
"case_scientific_rules": [
"The joint basis is |n>_(eγ)=|E0−n hbar omega>_e tensor |n>_γ, with E_n=E0−n hbar omega, k_n=sqrt(E_n²−m_e²c⁴)/(hbar c) and Delta k_n=k_n−k_(n+1)−kappa; choose Delta k_0=0. Preserve link-dependent relativistic mismatch and sqrt(n+1) bosonic coupling.",
"The exact production evolution is dc_n/dz=g_tilde exp(−i Delta k_n z)sqrt(n+1)c_(n+1)−g_tilde* exp(+i Delta k_(n−1)z)sqrt(n)c_(n−1), omitting the lower term at n=0, with c_n(0)=delta_n0 and g_Qu=g_tilde L. Exact transformations must leave final P(n)=|c_n(L)|² invariant.",
"From the same normalized final populations obtain mean n=sum nP(n) and g^(2)=sum n(n−1)P(n)/(mean n)². At zero mean the ratio is undefined, not evidence of a statistical class. For positive mean, g^(2)=1+[variance−mean]/mean²=1+Q/mean=1+(F−1)/mean.",
"Define sigma through Delta k_sigma L/2=pi at the first phase-matching zero. For electron kinetic energies below about 150 keV, the declared estimate sigma≈155[E_kin(keV)]^(3/2)/([E_ph(eV)]² L(micrometres)) is permitted; identify estimate versus exact mismatch and give at least one self-consistent physical realization.",
"Both low coupling |g_Qu| approximately at most 0.5, including a weak-recoil reference, and nonperturbative higher coupling are required. Weak recoil/low coupling approaches weak-coherent/Poisson behaviour; stronger recoil mismatches successive links, suppresses higher occupation and can yield sub-Poissonian response.",
"Higher coupling produces coherent redistribution within a recoil-limited ladder, including occupation accumulation near its boundary, return toward zero loss, or few-level oscillations where supported. A large g^(2) near tiny mean is not high-yield emission or sufficient state proof; a capped correlation map alone cannot establish state-generation claims.",
"Physical mapping and F3 domain/presentation weaknesses are localized rather than automatic gates on otherwise valid F2/O1 evidence. Independently correct branch work survives failure of another branch. M4's low-mean/statistical caveat is not imposed a second time as a hard gate on M8.",
"The scope excludes multiphoton phase matching, two-photon/QPM parity, NOON fidelity, driven classical-field operation, analogue horizons, arbitrary initial cavity states and experimental collection efficiency. Generic missing refinement or an examiner-preferred cross-check is not automatically RC3.",
"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.",
"Let $E_n=mc^2+K_0-n\\hbar\\omega$ be total electron energy after $n$ emitted photons, $k_n=\\sqrt{E_n^2-m^2c^4}/(\\hbar c)$, and $\\Delta_n=k_n-k_{n+1}-\\kappa$, with $\\kappa=k_0-k_1$ for first-link phase matching. This relativistic dispersion is exact in the stated ladder model; a nonrelativistic $\\sigma$ estimate does not replace exact propagation.",
"For $|\\psi(z)\\rangle=\\sum_{n\\ge0}c_n(z)|E_n,n\\rangle$, a convention equivalent to SI Eq. (S38) is $\\partial_zc_n=\\widetilde g\\sqrt{n+1}e^{-i\\Delta_n z}c_{n+1}-\\widetilde g^*\\sqrt n e^{i\\Delta_{n-1}z}c_{n-1}$, $c_n(0)=\\delta_{n0}$, with the negative-index term absent. Here $z\\in[0,L]$, $g_{\\rm Qu}=\\widetilde gL$; the generator is anti-Hermitian. Consistent conjugation/sign redefinitions are equivalent, but deleting successive-link mismatch is not.",
"$P_n=|c_n(L)|^2$, $\\sum_nP_n=1$, $\\bar n=\\sum_nnP_n$, $g^{(2)}(0)=\\sum_nn(n-1)P_n/\\bar n^2$. The factorial second moment is not $\\langle n^2\\rangle$. For $\\bar n>0$, the Fano factor is $F=\\operatorname{Var}(n)/\\bar n=1+\\bar n[g^{(2)}-1]$; this is a derived statistically equivalent classification when the underlying populations/mean are available. At zero mean the normalized correlation is undefined, and large $g^{(2)}$ at tiny mean does not imply high yield.",
"For the stated small fractional energy-loss approximation, $\\sigma\\simeq155[K_0(\\mathrm{keV})]^{3/2}/\\{[\\hbar\\omega(\\mathrm{eV})]^2L(\\mu\\mathrm m)\\}$ for electron kinetic energy below about 150 keV (SI Eq. (S65)); Eq. (S64) covers the broader relativistic estimate. $\\sigma$ estimates the first mismatch zero, not a sharp exact population cutoff; $g_{\\rm Qu}$ and $\\sigma$ jointly govern higher-coupling dynamics.",
"Exact low- and higher-coupling families are both required; a correct low branch retains its own credit when the high branch is absent or approximate, and vice versa. Electron energy-loss sidebands are statistically equivalent to photon populations in this vacuum single-mode joint ladder, not a new observable obligation. Fano/Mandel or other invertibly related statistics are accepted if they preserve the factorial-moment classification and the population/mean evidence needed by the claim."
],
"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): Instantiate the initially empty single-mode joint electron–photon ladder, first-link phase matching, exact relativistic link mismatch, bosonic square-root coupling and consistent phase/index conventions. Provide executable direct propagation or exact autonomous construction and derive normalized populations, moments and factorial second-order correlation from the final amplitudes with reconstructible conventions. Bind exact mismatch or a declared sigma estimate, both low/high-coupling domains, weak-recoil reference and at least one self-consistent electron/photon/length realization, distinguishing approximations from exact production."
],
"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: Present valid reconstructible exact-ladder low-coupling weak-recoil and finite-recoil populations with suitable moments/correlation and a common representation adequate for the stated claims.",
"O2: Present exact-ladder higher-coupling state-resolved evidence exposing finite-ladder coherent response families through populations or equivalent electron sidebands with suitable statistics."
],
"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.",
"(M6 AND M8): Unify both branches: relativistic successive-link mismatch limits the active ladder while interaction strength governs coherent occupation dynamics within it. Integrate low/high-coupling signals, the unified finite-ladder mechanism, approximation boundary and physical realization into the strongest evidence-supported state-generation conclusion."
],
"RC3": [
"Required objects and principal mechanisms are established, but a concrete material validation or scope threat to a central conclusion remains unresolved.",
"V2: When actual branch, tail, phase indexing, truncation, representation, anomaly or claim raises a concrete material alternative, supply evidence capable of exposing it or leave the conclusion unresolved for adjudication; 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": "Maxim Sirotin, Andrei Rasputnyi, Tomáš Chlouba, Roy Shiloh, and Peter Hommelhoff. 2026. *Quantum computing and quantum optics with recoiled free electrons*. *arXiv*, 2405.06560. Version 2; first submitted in 2024, revised in 2026.",
"url": "https://arxiv.org/abs/2405.06560v2",
"scientific_scope": "Single-photon vacuum-cavity exact recoil ladder across low and nonperturbative higher coupling. Source version is arXiv:2405.06560v2 (19 May 2026); driven control, horizon and two-photon sections are excluded except to identify the scope boundary.",
"anchors": [
"Sirotin et al. (2026), main text p. 6, joint-state paragraph; SI pp. 20–22, Eqs. (S34)–(S44).",
"Main text p. 6, Eqs. (5)–(6); SI pp. 30–31, Eqs. (S64)–(S65), Fig. S4(a).",
"Main text pp. 7–8, Eq. (7), Fig. 4(a–c), low-coupling single-photon paragraphs.",
"Main text pp. 7–8, Fig. 4(d–f), high-coupling paragraphs; SI p. 31, Fig. S5(a,b).",
"SI pp. 21–22, Eqs. (S38)–(S44), limitation paragraph after Eq. (S41); main text p. 8, Fig. 4(c).",
"Main text pp. 6–8, Eqs. (5)–(7), Fig. 4(c–f); SI p. 22, Eqs. (S41)–(S44).",
"SI pp. 21–22, Eqs. (S39)–(S44); main text p. 7, low-mean correlation paragraph; Fig. S5(b), SI p. 31.",
"Main text pp. 6–7, joint vacuum state and Sec. III A; Sec. III B, pp. 7–9, as an excluded contrast."
]
}
]
}
|