{ "case_id": "Case24", "title": "Relativistic quantum kinetic theory: Higher order contributions in assisted Schwinger pair production", "atoms": [ { "criterion_id": "F1", "dimension": "F", "points": 0.5, "scientific_requirement": "Correctly bind both vector potentials, their electric-field relation, vacuum initial state, canonical/kinetic momentum convention, p_perp = 0 and both specified instances. Accept a self-consistent charged scalar or spinor model identified from actual equations/statistics; legal scalar identity is not a failure.", "award_conditions": [ "Both field instances, kinetic/canonical convention and vacuum initialization are bound to actual self-consistent scalar or spinor equations and counting." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint pp. 2-3, Eqs. (1)-(7); Supplemental p. 1, Eqs. (S1)-(S6), and Wigner derivation pp. 2-3: The source gives spinor homogeneous vacuum dynamics and the strong/weak field split. Scalar alternatives are accepted under the stated task/effective scientific rule, not claimed to be the source spinor numerical result." ], "rc_relation": "RC0: absence or material invalidity of the central physical object. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "F2", "dimension": "F", "points": 0.5, "scientific_requirement": "Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.", "award_conditions": [ "The executable weak-field recurrence treats the strong response nonperturbatively and retains distribution-level cross-order sources with a consistent truncation." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "F1: Correctly bind both vector potentials, their electric-field relation, vacuum initial state, canonical/kinetic momentum convention, p_perp = 0 and both specified instances. Accept a self-consistent charged scalar or spinor model identified from actual equations/statistics; legal scalar identity is not a failure. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 3, Eqs. (8)-(13), and footnote on the physical meaning of the total probability: A probability-level Taylor hierarchy preserves cross-order coherence; auxiliary-amplitude coefficients must be convolved before probability truncation." ], "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "F3", "dimension": "F", "points": 0.5, "scientific_requirement": "Define a reconstructible vacuum-subtracted asymptotic particle-number observable with one consistent declared normalization across cases/orders in the actual scalar or spinor model, preserving species/spin counting.", "award_conditions": [ "The asymptotic vacuum-subtracted occupation has one declared species/spin normalization across all cases and signed orders." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "F1: Correctly bind both vector potentials, their electric-field relation, vacuum initial state, canonical/kinetic momentum convention, p_perp = 0 and both specified instances. Accept a self-consistent charged scalar or spinor model identified from actual equations/statistics; legal scalar identity is not a failure. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in F2 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint pp. 2-3, Eqs. (1)-(7); Supplemental p. 1, Eqs. (S1)-(S6), and Wigner derivation pp. 2-3: The source gives spinor homogeneous vacuum dynamics and the strong/weak field split. Scalar alternatives are accepted under the stated task/effective scientific rule, not claimed to be the source spinor numerical result." ], "rc_relation": "RC0: absence or material invalidity of the central physical object. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "O1", "dimension": "O", "points": 0.5, "scientific_requirement": "Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence.", "award_conditions": [ "Instance I contains valid total, strong and leading signed-order spectra in the actual model; spinor source numbers are not imposed on scalar output.", "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [F2: Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.; F3: Define a reconstructible vacuum-subtracted asymptotic particle-number observable with one consistent declared normalization across cases/orders in the actual scalar or spinor model, preserving species/spin counting.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in O2, I2, I3, M4, M5, M6 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 4, Fig. 1 caption and discussion; Supplemental Fig. S3: Instance I illustrates strong-field dressing, assistance and signed-order reconstruction. Its printed curves are spinor examples; a legal scalar branch is checked in its own model, including a possibly negative central fourth-order coefficient." ], "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "O2", "dimension": "O", "points": 0.5, "scientific_requirement": "Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations.", "award_conditions": [ "Instance II contains a valid common-scale total and even/odd-family reconstruction with the material central/side relations preserved.", "The physical object is valid, its presentation adequate, and its claim adequate; record these three statuses separately. Reconstructible data alone do not create candidate identification or mechanism." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [F2: Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.; F3: Define a reconstructible vacuum-subtracted asymptotic particle-number observable with one consistent declared normalization across cases/orders in the actual scalar or spinor model, preserving species/spin counting.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in O1, I1, M1, M2, M3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II: Instance II has central constructive and side destructive even/odd combination. Deep minima require the signed contributions and remainder to be retained on one scale, not copied spinor numbers in a scalar model." ], "rc_relation": "RC1: absence or material invalidity of an indispensable component, with the central object established. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "I1", "dimension": "I", "points": 0.5, "scientific_requirement": "Explicitly identify in the candidate's instance-I output its main total structure, relative strong/leading weak-order roles and material sign-changing contribution, without adding hidden peak-count or coordinate duties.", "award_conditions": [ "The candidate identifies the main instance-I total structure, relative strong/leading roles and a material sign-changing contribution in their actual-model evidence.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "O1: Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in I2, I3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 4, Fig. 1 caption and discussion; Supplemental Fig. S3: Instance I illustrates strong-field dressing, assistance and signed-order reconstruction. Its printed curves are spinor examples; a legal scalar branch is checked in its own model, including a possibly negative central fourth-order coefficient." ], "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "I2", "dimension": "I", "points": 0.5, "scientific_requirement": "Explicitly identify instance-II center enhancement and deep side minima as material structures in the candidate's actual-model evidence; arbitrary structures are not a substitute.", "award_conditions": [ "The candidate identifies instance-II center enhancement and deep side minima from their own model, not unrelated spectral extrema.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "O2: Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in I1, I3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II: Instance II has central constructive and side destructive even/odd combination. Deep minima require the signed contributions and remainder to be retained on one scale, not copied spinor numbers in a scalar model." ], "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "I3", "dimension": "I", "points": 0.5, "scientific_requirement": "Identify same/opposite signs of the leading even/odd families at the instance-II center and side minima, or equivalent constructive/destructive relations in the actual model.", "award_conditions": [ "The candidate identifies constructive/destructive signs or equivalent coherent relations of the leading even/odd families at center and side minima.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "O2: Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in I1, I2 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Total-spectrum identification and signed-family-relation identification are separately judgeable; failure of one does not automatically remove the other.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II: Instance II has central constructive and side destructive even/odd combination. Deep minima require the signed contributions and remainder to be retained on one scale, not copied spinor numbers in a scalar model." ], "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M1", "dimension": "M", "points": 0.5, "scientific_requirement": "Use instance-I signals to explain assistance by nonperturbative strong-field dressing and weak-field energy absorption. No fixed net photon number or spinor pure-weak-field selection rule is imposed on a scalar calculation.", "award_conditions": [ "The argument uses instance-I evidence to explain strong-field dressing and weak-field energy absorption without imposing spinor selection rules on a scalar model.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "I1: Explicitly identify in the candidate's instance-I output its main total structure, relative strong/leading weak-order roles and material sign-changing contribution, without adding hidden peak-count or coordinate duties. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M4, M5, M6 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 4, Fig. 1 caption and discussion; Supplemental Fig. S3: Instance I illustrates strong-field dressing, assistance and signed-order reconstruction. Its printed curves are spinor examples; a legal scalar branch is checked in its own model, including a possibly negative central fourth-order coefficient." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M2", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain that signed distribution-order contributions contain probability-level coherent cross terms, not independently positive channel probabilities.", "award_conditions": [ "Signed probability-order terms are correctly interpreted as coherent cross terms rather than independent nonnegative channel probabilities.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "I1: Explicitly identify in the candidate's instance-I output its main total structure, relative strong/leading weak-order roles and material sign-changing contribution, without adding hidden peak-count or coordinate duties. supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M4, M5, M6 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 3, footnote 2, and pp. 4-5 interference discussion: Distribution orders are signed coherent cross terms; only the reconstructed total is a physical nonnegative probability." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M3", "dimension": "M", "points": 0.5, "scientific_requirement": "Use actual-model order magnitudes and sign changes to explain reconstruction of the instance-I total. At the center, positive spinor fourth order and negative scalar fourth order are both valid model-conditioned possibilities; fourth order is not universally a positive two-photon probability.", "award_conditions": [ "Actual-model magnitudes and signs explain instance-I reconstruction, including legal negative scalar central fourth order; total occupation remains physical.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link.", "At the instance-I center accept a positive spinor or negative scalar fourth-order term when the actual equations, signed reconstruction and interpretation are self-consistent; no positive-channel interpretation is imposed." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [O1: Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence.; I1: Explicitly identify in the candidate's instance-I output its main total structure, relative strong/leading weak-order roles and material sign-changing contribution, without adding hidden peak-count or coordinate duties.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M4, M5, M6 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 4, Fig. 1 caption and discussion; Supplemental Fig. S3: Instance I illustrates strong-field dressing, assistance and signed-order reconstruction. Its printed curves are spinor examples; a legal scalar branch is checked in its own model, including a possibly negative central fourth-order coefficient." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M4", "dimension": "M", "points": 0.5, "scientific_requirement": "Use the candidate's instance-II central signal and even/odd relation to explain constructive combination physically, not merely state equal signs.", "award_conditions": [ "Instance-II central constructive interference is physically explained from the observed even/odd combination, not only its equal signs.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [I2: Explicitly identify instance-II center enhancement and deep side minima as material structures in the candidate's actual-model evidence; arbitrary structures are not a substitute.; I3: Identify same/opposite signs of the leading even/odd families at the instance-II center and side minima, or equivalent constructive/destructive relations in the actual model.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3, M5 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Preserve independently established instance-I and side-minimum mechanisms if the center explanation fails.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II: Instance II has central constructive and side destructive even/odd combination. Deep minima require the signed contributions and remainder to be retained on one scale, not copied spinor numbers in a scalar model." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M5", "dimension": "M", "points": 0.5, "scientific_requirement": "Use the candidate's instance-II side-minimum signal and even/odd relation to explain destructive near-cancellation; displaying a valley or examiner-supplied reasoning is insufficient.", "award_conditions": [ "The side minima are explained by evidence-bound destructive near-cancellation with the relative scales and remainder retained.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [I2: Explicitly identify instance-II center enhancement and deep side minima as material structures in the candidate's actual-model evidence; arbitrary structures are not a substitute.; I3: Identify same/opposite signs of the leading even/odd families at the instance-II center and side minima, or equivalent constructive/destructive relations in the actual model.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3, M4 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Preserve independently established instance-I and center mechanisms if the side-minimum explanation fails.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II: Instance II has central constructive and side destructive even/odd combination. Deep minima require the signed contributions and remainder to be retained on one scale, not copied spinor numbers in a scalar model." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M6", "dimension": "M", "points": 0.5, "scientific_requirement": "Connect momentum-dependent constructive/destructive competition in instance II to distinct weak-field energy-transfer routes in the dressed background. A justified bounded semiclassical explanation is accepted; mere signed addition or equating each probability order with one net-photon probability is not.", "award_conditions": [ "Momentum-dependent competition is connected to distinct dressed-background weak-field energy-transfer routes without assigning every order a unique net-photon probability.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [M4: Use the candidate's instance-II central signal and even/odd relation to explain constructive combination physically, not merely state equal signs.; M5: Use the candidate's instance-II side-minimum signal and even/odd relation to explain destructive near-cancellation; displaying a valley or examiner-supplied reasoning is insufficient.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint pp. 4-5, Figs. 1-3 and discussions of competing energy absorption in a dressed background: Different energy-transfer routes and their interference explain momentum-dependent competition. No probability order is universally a positive fixed-net-photon channel, and scalar/spinor results are not prefactor copies." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M7", "dimension": "M", "points": 0.5, "scientific_requirement": "Explain why the two instances exhibit different spectral forms and dominant competition within the same explicitly identified self-consistent probability-level model, without asserting universal scalar/spinor equivalence.", "award_conditions": [ "Differences between the two instances are explained within the same explicitly identified model without a universal scalar/spinor-equivalence claim.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [M1: Use instance-I signals to explain assistance by nonperturbative strong-field dressing and weak-field energy absorption. No fixed net photon number or spinor pure-weak-field selection rule is imposed on a scalar calculation.; M3: Use actual-model order magnitudes and sign changes to explain reconstruction of the instance-I total. At the center, positive spinor fourth order and negative scalar fourth order are both valid model-conditioned possibilities; fourth order is not universally a positive two-photon probability.; M4: Use the candidate's instance-II central signal and even/odd relation to explain constructive combination physically, not merely state equal signs.; M5: Use the candidate's instance-II side-minimum signal and even/odd relation to explain destructive near-cancellation; displaying a valley or examiner-supplied reasoning is insufficient.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M2, M6, V3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint pp. 4-5, Figs. 1-3 and discussions of competing energy absorption in a dressed background: Different energy-transfer routes and their interference explain momentum-dependent competition. No probability order is universally a positive fixed-net-photon channel, and scalar/spinor results are not prefactor copies." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M8", "dimension": "M", "points": 0.5, "scientific_requirement": "Correctly relate finite distribution-level hierarchy, signed coefficients and unexpanded full-field kinetic observable in the same model; distinguish amplitude truncation from probability truncation and do not treat scalar and spinor as one spectrum times a prefactor.", "award_conditions": [ "The relation between finite distribution hierarchy and unexpanded full-field occupation is correct; amplitude and probability truncations are not confused.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [F2: Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.; F3: Define a reconstructible vacuum-subtracted asymptotic particle-number observable with one consistent declared normalization across cases/orders in the actual scalar or spinor model, preserving species/spin counting.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3, M4, M5, M6 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 3, Eqs. (8)-(13), and footnote on the physical meaning of the total probability: A probability-level Taylor hierarchy preserves cross-order coherence; auxiliary-amplitude coefficients must be convolved before probability truncation." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "M9", "dimension": "M", "points": 0.5, "scientific_requirement": "Directly decide whether the finite hierarchy is faithful in both specified instances, consistently with both actual-model outputs, hierarchy identity and discriminating truncation evidence; the answer need not automatically be that it is faithful.", "award_conditions": [ "The candidate answers fidelity for both instances using actual-model spectra and discriminating truncation evidence, allowing a justified negative or bounded conclusion.", "Use the candidate's own valid and relevant evidence; a correct source narrative or examiner-created inference cannot replace the candidate's identification or causal link." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [O1: Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence.; O2: Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations.; M8: Correctly relate finite distribution-level hierarchy, signed coefficients and unexpanded full-field kinetic observable in the same model; distinguish amplitude truncation from probability truncation and do not treat scalar and spinor as one spectrum times a prefactor.; V1: Substantively test truncation fidelity for both instances within the same model using additional orders, unexpanded full-field evolution, analytic remainder or a functionally equivalent discriminating route; no fixed eighth order, probe or valley-error threshold is required.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3, M4, M5, M6, V3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "This judgment requires both outputs, correct hierarchy identity and material truncation evidence. A failed fidelity conclusion does not erase local mechanisms or honest scope; the same truncation gap is not charged twice.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 3, hierarchy truncation paragraph; Supplemental convergence discussion and Fig. S2; stated task fidelity question: Fidelity must be tested for both actual-model instances, including claim-material cancellation; additional orders or same-model full evolution are examples, not a fixed eighth-order duty." ], "rc_relation": "RC2: a missing or wrong indispensable principal mechanism/conclusion, with valid required signals. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "V1", "dimension": "V", "points": 0.5, "scientific_requirement": "Substantively test truncation fidelity for both instances within the same model using additional orders, unexpanded full-field evolution, analytic remainder or a functionally equivalent discriminating route; no fixed eighth order, probe or valley-error threshold is required.", "award_conditions": [ "Both instances receive a substantive same-model truncation test by extra orders, full evolution, remainder bounds or equivalent evidence relevant to material cancellation.", "The fidelity evidence must discriminate both specified instances in the same statistics model, including any cancellation accuracy actually claimed; one global error measure does not automatically establish a deep-minimum precision claim." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "All of [F2: Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.; O1: Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence.; O2: Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations.] supplies only a necessary physical/evidential premise, not an automatic atomic-score gate.", "Preserve independent correct work in M1, M2, M3, M4, M5, M6, V3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "Judge instance-local signs, spectra and reconstruction in the actual self-consistent scalar or spinor model. One instance failure does not erase an independently valid other instance, and legal scalar statistics are never an automatic failed identity or validation trigger." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 3, hierarchy truncation paragraph; Supplemental convergence discussion and Fig. S2; stated task fidelity question: Fidelity must be tested for both actual-model instances, including claim-material cancellation; additional orders or same-model full evolution are examples, not a fixed eighth-order duty." ], "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "V2", "dimension": "V", "points": 0.5, "scientific_requirement": "Only when candidate work triggers a material model/domain overextension, including spatial, backreaction, experimental or gamma-regime claims, or another risk altering order/sign/interference/fidelity conclusions, discriminate it or narrow scope. Legal scalar identity alone is not a risk.", "award_conditions": [ "An actual cross-model, weak-field-domain, spatial/backreaction or other material risk is discriminated or restricted; a legal scalar model alone is no trigger." ], "applicability_and_na": [ "Activate only for a concrete material risk arising from the candidate method, data, anomaly or claim; no fixed test is required in the absence of a trigger.", "Without a triggered risk record N/A and award 0.5 without positive validation evidence; this neither blocks RC4 nor triggers RC3.", "When triggered, accept functionally equivalent discriminating evidence or a justified restriction of the threatened claim." ], "dependencies": [ "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.", "Preserve independent correct work in V3 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty.", "A material overclaim is scored only here with its applicable risk consequence, not also by removing V3 for that same defect. An ordinary missing positive scope statement is V3-only and does not itself trigger V2/RC3." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 2, mean-field, no-backreaction and homogeneous assumptions; p. 5, Conclusion: Statistics identity and the demonstrated weak-field domain bound the conclusion. Legal scalar identity is not itself a validation threat." ], "rc_relation": "RC3: an unresolved applicable material risk, only after signals and principal mechanisms are substantially supported and all six risk-proof requirements are met. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." }, { "criterion_id": "V3", "dimension": "V", "points": 0.5, "scientific_requirement": "Positively identify the actual statistics/species model, common homogeneous prescribed vacuum-initialized setting, both instances and evidence-supported weak-field domain; ordinary scope omission is distinct from a material cross-model overclaim.", "award_conditions": [ "Actual statistics/species, homogeneous prescribed vacuum setting, both instances and evidence-supported weak-field domain are positively identified." ], "applicability_and_na": [ "Applicable to the stated scientific commission; N/A is not permitted." ], "dependencies": [ "Independently judgeable; no upstream atomic failure is mechanically propagated into this obligation.", "Preserve independent correct work in V2 and all branches not using the failed premise.", "Propagate the exact failed fact only; preserve independently established subobjects, candidate reasoning and scope. Do not charge a second generic incompleteness or numerical-reliability penalty." ], "equivalent_evidence": [ "Accept self-consistent charged scalar or spinor calculations, alternative Wigner bases and model-appropriate Bogoliubov/kinetic variables, declared consistent charge/vector-potential signs and a reconstructible common normalization. Scalar is not required to map to the same Dirac spectrum.", "Accept auxiliary mode evolution only when amplitude coefficients are convolved into the same probability-order coefficients and truncation meaning. Full-field evolution can validate but cannot replace the required order-resolved hierarchy.", "Accept equivalent signed-order grouping and bounded physically justified energy-route explanations preserving reconstruction and central/side interference. No exact source peaks, coefficient values, fixed hierarchy depth or unique fidelity test is required." ], "source_support": [ "Edwards et al. (2025), preprint p. 2, mean-field, no-backreaction and homogeneous assumptions; p. 5, Conclusion: Statistics identity and the demonstrated weak-field domain bound the conclusion. Legal scalar identity is not itself a validation threat." ], "rc_relation": "No direct closure trigger; affect a dependent required scientific role only through a demonstrated failed necessary premise.. Evaluate actual trigger facts separately from the atomic award; the most severe failed required role determines closure." } ], "case_scientific_rules": [ "Both charged scalar and spinor formulations are accepted when self-consistent in equations, initial state, statistics, counting and normalization. They are not generally related by a single spin/prefactor conversion. Wrong fields, mixed statistics, a scalar solution mislabeled Dirac, missing signed mechanisms and unsupported extrapolation remain failures.", "Let kinetic q = p - e A and qdot = e E with E = -Adot. At p_perp = 0 and m = 1 an equivalent model-conditioned kinetic system is ndot = W u/2, udot = W(1 + 2 s n) - 2 omega v, vdot = 2 omega u, omega^2 = 1 + q^2, with vacuum n = u = v = 0. Spinor uses s = -1, W = qdot/omega^2; scalar uses s = +1, W = q qdot/omega^2.", "Spinor normalization follows |alpha|^2 + |beta|^2 = 1 and scalar follows |alpha|^2 - |beta|^2 = 1. A kinetic f = 2n source convention must not be confused with another formula using f = |beta|^2. A continuum density g n/(2 pi)^3 is allowed with declared counting, e.g. scalar g = 1 and spinor g = 2; no hidden common plotted factor is required.", "For the spinor Wigner representation, sdot = 2(q_parallel-eA) v_perp - 2 q_perp v_parallel, v_parallel_dot = 2 q_perp s - 2 m v_perp, v_perp_dot = -2(q_parallel-eA)s + 2m v_parallel. The free vacuum components are (-m,-q_parallel,-q_perp)/omega_q. An induced sector projects asymptotically as N (m s_r + q_parallel v_parallel_r + q_perp v_perp_r)/omega_q. This is a spinor representation, not a compulsory scalar equation.", "Write A = Astrong + lambda Aweak(actual), keep the strong response unexpanded and define n_k = (partial_lambda^k n)(0)/k!. At lambda = 1 the probability truncation is sum_{k<=K} n_k, not |sum_{k<=K} beta_k|^2. Auxiliary amplitude coefficients must yield n_k = sum_{i+j=k} Re(beta_i beta_j*), including background/high-order cross terms.", "In particular n4 includes |beta2|^2, 2 Re(beta1 beta3*) and 2 Re(beta0 beta4*) rather than an isolated positive two-photon probability. Instance-I scalar center n4 can be negative while spinor n4 is small positive. Judge the candidate's actual-model magnitudes/signs, total reconstruction and coherent interpretation; no fixed coefficient value is a new duty.", "The spinor source anchors show a central instance-I peak with structured outer peaks and a sign-changing first-order contribution, while instance II has central constructive even/odd combination and deep side near-cancellation. The accepted scalar branch must independently establish the shared material output/identification/mechanism requirements in its own model, not reproduce the spinor numerical curves.", "In instance II the leading grouping n2+n4 and n3 is distinct from all retained even and all retained odd orders. Do not disguise the remainder as normalization error or omit it from reconstruction. Equivalent grouping is accepted if it preserves the probability-level relations and total.", "Truncation evidence must discriminate both instances, including claim-material cancellation. Additional orders, same-model full evolution, analytic remainder or equivalent evidence are accepted; no K = 8, momentum probe, global/local tolerance or source coordinate is mandated. A small global error alone does not automatically establish a precision claim about a deep minimum.", "Supported scope is homogeneous prescribed external fields, vacuum initialization, the two specified instances and demonstrated weak-field region gamma = 1/xi > 1. Do not generalize to gamma <= 1, spatial/backreacting dynamics or experiment without evidence. Local instance failures propagate only through necessary premises, never automatically across the other instance.", "Concrete risks include canonical/kinetic or vacuum misbinding, inconsistent relative scales, clipping signed orders, omission of a locally canceling term, partial sums mistaken for full observables and unsupported cross-model prefactor equivalence. Treat ordinary scope documentation separately from triggered material overclaims and avoid duplicate ownership.", "The score is the sum of twenty material-core criteria, each worth either 0 or 0.5, for a maximum of 10.0. Research Closure is descriptive and never caps, floors, rescales or repairs the score.", "Judge physical object identity and correctness before display similarity. For each output record object_valid, presentation_adequate and claim_adequate separately; all three must hold for its atomic award. A presentation or claim defect alone is not evidence that an existing physical object is absent or invalid.", "Candidate-created data, candidate presentation, candidate identification, physical inference and validation are distinct evidential acts. Examiner inspection or reconstruction may verify an object but cannot create missing candidate identification, reasoning or risk resolution.", "A dependency propagates only the failed necessary physical fact, not the whole upstream atomic verdict or unrelated missing detail. Name the failed premise, the downstream obligation needing it, and why independent candidate-presented evidence does not establish it. Preserve unaffected branches, independent work and honest scope.", "Numerical reliability has no separate generic penalty. Charge a defect where it materially damages a physical object or evidence, and a dependent conclusion only when that same defect makes the conclusion false or unsupported. Missing a preferred convergence study, control, solver or named self-check is not itself a failure.", "Risk-triggered validation applies only to a concrete material risk arising from the candidate method, data, anomaly or claim. With no such risk record N/A and award the fixed atomic credit without treating it as positive validation evidence; N/A does not block RC4. Unconditional scope obligations cannot be N/A.", "Before assigning RC3 establish all six: the concrete failure mode; its origin in candidate work; the fair task or standard-knowledge basis; the threatened central conclusion; why submitted evidence cannot expose or exclude it; and examples of functionally equivalent discriminating evidence or an honest scope restriction that would resolve it. A generic possibility of further testing is insufficient.", "Use bounded examiner-side source/reference readback only to resolve a material scientific uncertainty, not to demand pixels, an undeclared tolerance or additional candidate deliverables. A provisional score of at least 9.0 or provisional RC4 triggers one bounded central-output-anchor check and one principal-mechanism-link check. Reference conflict or insufficiency remains an explicit unresolved decision, not an invented target.", "Evaluate closure-role trigger facts independently of atomic awards and select the most severe failed required role across all indispensable and co-primary claims. Supporting claims do not independently set overall closure. A point loss alone is not a closure-gate fact.", "A valid candidate-caused incomplete or timeout attempt with no scientifically assessable required submission has score 0.0 and RC0 before N/A bookkeeping. Assessable incomplete work is graded from the submitted evidence; unsubmitted scratch or logs do not become scientific answers.", "Write $A=A_s+\\lambda A_w$ with the actual weak pulse at $\\lambda=1$, $E=-\\dot A$, and $n_k=\\partial_\\lambda^kn|_0/k!$. The distribution-level truncation is $n^{[K]}=\\sum_{k=0}^Kn_k$. If $\\beta=\\sum_k\\lambda^k\\beta_k$, then $n_k=\\sum_{i+j=k}\\operatorname{Re}(\\beta_i\\beta_j^*)$; notably $n_4=|\\beta_2|^2+2\\operatorname{Re}(\\beta_1\\beta_3^*+\\beta_0\\beta_4^*)$. Squaring a truncated amplitude retains different probability orders.", "For an explicitly declared same-model normalization at $p_\\perp=0$, let $q=p-eA$, $\\omega=\\sqrt{m^2+q^2}$ and vacuum $n=u=v=0$. A model-conditioned kinetic representation is $\\dot n=Wu/2$, $\\dot u=W(1+2sn)-2\\omega v$, $\\dot v=2\\omega u$. Spinor has $s=-1$, $W=m\\dot q/\\omega^2$; scalar has $s=+1$, $W=q\\dot q/\\omega^2$. These accepted equivalent formulations are a model-conditioned derivation; the cited article explicitly derives the spinor hierarchy.", "Spinor and scalar satisfy respectively $|\\alpha|^2+|\\beta|^2=1$ and $|\\alpha|^2-|\\beta|^2=1$. Track whether the distribution is $n=|\\beta|^2$, $f=2n$ or a continuum density with declared degeneracy. Negative scalar $n_4(0)$ is not a failure, nor is it equivalent to negative total occupation. The same-model total and interference, not a universal positive fourth order, determine correctness." ], "closure_requirements": { "RC0": [ "Required claim CL1, central object: All of [F1: Correctly bind both vector potentials, their electric-field relation, vacuum initial state, canonical/kinetic momentum convention, p_perp = 0 and both specified instances. Accept a self-consistent charged scalar or spinor model identified from actual equations/statistics; legal scalar identity is not a failure.; F3: Define a reconstructible vacuum-subtracted asymptotic particle-number observable with one consistent declared normalization across cases/orders in the actual scalar or spinor model, preserving species/spin counting.]. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Apply for a missing or materially invalid required central object; a presentation-only or claim-only defect does not establish object invalidity.", "No scientifically assessable required submission gives the terminal 0.0/RC0 disposition." ], "RC1": [ "Required claim CL1, indispensable component: F2: Construct an executable Wigner/particle-distribution probability-level Taylor recurrence in the weak-field parameter xi with the strong field nonperturbative and correct cross-order sources. Auxiliary amplitudes must be convolved into same-order distribution coefficients with the same truncation meaning.. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Required claim CL2, indispensable component: O1: Provide physically valid reconstructible adequately presented instance-I total, strong and leading signed-order spectra in the actual self-consistent model; do not impose the spinor numerical spectrum on scalar evidence.. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Required claim CL3, indispensable component: O2: Provide physically valid reconstructible adequately presented instance-II total/even/odd-family spectra on a common scale in the actual self-consistent model, retaining total reconstruction and material family relations.. Judge role-trigger facts, not the sum or pattern of atomic scores.", "The central object is established but a required physical component is absent or materially invalid. Preserve independent valid branches; do not invent a hidden canonical plot, sampling grid or implementation duty." ], "RC2": [ "Required claim CL2, principal mechanism: All of [M1: Use instance-I signals to explain assistance by nonperturbative strong-field dressing and weak-field energy absorption. No fixed net photon number or spinor pure-weak-field selection rule is imposed on a scalar calculation.; M2: Explain that signed distribution-order contributions contain probability-level coherent cross terms, not independently positive channel probabilities.; M3: Use actual-model order magnitudes and sign changes to explain reconstruction of the instance-I total. At the center, positive spinor fourth order and negative scalar fourth order are both valid model-conditioned possibilities; fourth order is not universally a positive two-photon probability.]. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Required claim CL3, principal mechanism: All of [M4: Use the candidate's instance-II central signal and even/odd relation to explain constructive combination physically, not merely state equal signs.; M5: Use the candidate's instance-II side-minimum signal and even/odd relation to explain destructive near-cancellation; displaying a valley or examiner-supplied reasoning is insufficient.; M6: Connect momentum-dependent constructive/destructive competition in instance II to distinct weak-field energy-transfer routes in the dressed background. A justified bounded semiclassical explanation is accepted; mere signed addition or equating each probability order with one net-photon probability is not.]. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Required claim CL4, principal mechanism: All of [M7: Explain why the two instances exhibit different spectral forms and dominant competition within the same explicitly identified self-consistent probability-level model, without asserting universal scalar/spinor equivalence.; M8: Correctly relate finite distribution-level hierarchy, signed coefficients and unexpanded full-field kinetic observable in the same model; distinguish amplitude truncation from probability truncation and do not treat scalar and spinor as one spectrum times a prefactor.; M9: Directly decide whether the finite hierarchy is faithful in both specified instances, consistently with both actual-model outputs, hierarchy identity and discriminating truncation evidence; the answer need not automatically be that it is faithful.]. Judge role-trigger facts, not the sum or pattern of atomic scores.", "Required signal objects/components exist, but an indispensable principal mechanism or co-primary scientific conclusion is missing, wrong or unsupported by candidate-presented and identified evidence. Correct data do not let an examiner supply missing reasoning." ], "RC3": [ "Required claim CL4, material validation: V2: Only when candidate work triggers a material model/domain overextension, including spatial, backreaction, experimental or gamma-regime claims, or another risk altering order/sign/interference/fidelity conclusions, discriminate it or narrow scope. Legal scalar identity alone is not a risk.. Judge role-trigger facts, not the sum or pattern of atomic scores.", "All indispensable signal and principal-mechanism roles are substantially supported, but a concrete triggered material risk remains unresolved and inadequately restricted.", "Establish the failure mode, candidate origin, fair task/knowledge basis, threatened central conclusion, insufficiency of submitted evidence, and equivalent resolution or scope restriction. Missing an examiner-preferred check alone never suffices." ], "RC4": [ "Every nonempty required role in every indispensable and co-primary claim above is scientifically established from candidate evidence, with the correct case/model/observable identities and evidence-bounded conclusion.", "All applicable material risks are discriminated or honestly restricted; no unresolved material ambiguity remains. Risk N/A does not block closure or supply positive validation evidence. Honest limitations cannot repair a false signal or mechanism.", "Select the most severe failed required role if any; otherwise RC4. No score threshold, score ceiling or required display/solver checklist determines this classification." ] }, "references": [ { "citation": "James P. Edwards, Naser Ahmadiniaz, Sebastian M. Schmidt, and Christian Kohlfürst. 2025. Relativistic quantum kinetic theory: Higher order contributions in assisted Schwinger pair production. Physical Review D, 112, L031901.", "url": "https://doi.org/10.1103/mgst-vn3c", "scientific_scope": "Probability-level weak-field hierarchy in a nonperturbative strong background, signed coherent contributions and case-specific truncation fidelity. Published numerical figures are spinor anchors; accepted scalar evidence is judged using self-consistent Klein-Gordon/kinetic identities and its own model-supported scientific relations, not a hidden spinor curve.", "anchors": [ "preprint pp. 2-3, Eqs. (1)-(7); Supplemental p. 1, Eqs. (S1)-(S6), and Wigner derivation pp. 2-3", "preprint p. 3, Eqs. (8)-(13), and footnote on the physical meaning of the total probability", "preprint p. 4, Fig. 1 caption and discussion; Supplemental Fig. S3", "preprint p. 5, Fig. 3 and interference paragraphs; Supplemental Table II", "preprint p. 3, footnote 2, and pp. 4-5 interference discussion", "preprint pp. 4-5, Figs. 1-3 and discussions of competing energy absorption in a dressed background", "preprint p. 3, hierarchy truncation paragraph; Supplemental convergence discussion and Fig. S2; stated task fidelity question", "preprint p. 2, mean-field, no-backreaction and homogeneous assumptions; p. 5, Conclusion" ] } ] }