Datasets:

Tauben
/

PhysClosureBench

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for F2, F3, F4; 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.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Milošević (2013), article pp. 023417-3–5, Eqs. (20)–(22), (27)–(28), (42)–(44). Monochromatic field/order, forward target, p-state dipole and short-range Yukawa kernel, with common conventions.

Research Closure relation

Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award.

F2 — F — 0.5 points

Scientific requirement

Implement the complete complex energy/intermediate-channel rescattering amplitude with −k dOmega measure, channel sum, PV-plus-pole Cauchy branch and off-shell content.

Conditions for credit

Implement the complete complex energy/intermediate-channel rescattering amplitude with −k dOmega measure, channel sum, PV-plus-pole Cauchy branch and off-shell content.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for F1, F3, F4; 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.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-3–4, Eqs. (29)–(35), (37)–(41). Complete complex energy/channel amplitude, negative k angular measure and principal-value plus pole propagation; closed channels remain in off-shell amplitude.

Research Closure relation

Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award.

F3 — F — 0.5 points

Scientific requirement

Construct the delta-pole-only on-shell reduction and phase-compatible direct amplitude sharing the complete rescattered branch's phase identity.

Conditions for credit

Construct the delta-pole-only on-shell reduction and phase-compatible direct amplitude sharing the complete rescattered branch's phase identity.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for F1, F2, F4; 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.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-3–5, Eqs. (28), (30), (32)–(36), (45); p. 023417-2, Eq. (8). Phase-compatible direct and pole-only comparator, differential rate and coherent addition. The energy-variable denominator changes the apparent Cauchy sign together with the kernel measure.

Research Closure relation

Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award.

F4 — F — 0.5 points

Scientific requirement

Define the differential rate, one common display normalization, reconstructible complex/component mapping and |TD+TR|² total.

Conditions for credit

Define the differential rate, one common display normalization, reconstructible complex/component mapping and |TD+TR|² total.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for F1, 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.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-3–5, Eqs. (28), (30), (32)–(36), (45); p. 023417-2, Eq. (8). Phase-compatible direct and pole-only comparator, differential rate and coherent addition. The energy-variable denominator changes the apparent Cauchy sign together with the kernel measure.

Research Closure relation

Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award.

O1 — O — 0.5 points

Scientific requirement

Present a reconstructible complete off-shell-plus-on-shell forward rescattering spectrum retaining complex information needed for the complete-branch claim.

Conditions for credit

Present a reconstructible complete off-shell-plus-on-shell forward rescattering spectrum retaining complex information needed for the complete-branch claim.
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 N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (F1 AND F2 AND F4). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O2, O3, O4; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-5, Fig. 2; p. 023417-6, Figs. 3–5 and principal-value discussion. Complete versus pole-only broad spectral difference; PV/channel component figures are canonical, not compulsory layouts or hidden peak targets.

Research Closure 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.

O2 — O — 0.5 points

Scientific requirement

Present the same-case, common-normalization on-shell comparator with enough evidence for material comparison against the complete spectrum.

Conditions for credit

Present the same-case, common-normalization on-shell comparator with enough evidence for material comparison against the complete spectrum.
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 N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: F3. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O1, O3, O4; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.
A local common rate-prefactor weakness propagates only if it actually changes this same-order comparison.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-5, Fig. 2; p. 023417-6, Figs. 3–5 and principal-value discussion. Complete versus pole-only broad spectral difference; PV/channel component figures are canonical, not compulsory layouts or hidden peak targets.

Research Closure 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.

O3 — O — 0.5 points

Scientific requirement

Present reconstructible phase-compatible direct, complete rescattered and coherent-total evidence making the cross term and rescattering consequence auditable.

Conditions for credit

Present reconstructible phase-compatible direct, complete rescattered and coherent-total evidence making the cross term and rescattering consequence auditable.
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 N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (F2 AND F3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O1, O2, O4; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.
Do not propagate an unrelated cross-order prefactor defect into an unaffected same-energy coherence relation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-7, Fig. 7 and direct/rescattered comparison paragraph; Eqs. (8), (28)–(29), (45). Coherent total can differ materially despite a smaller separate rescattering rate. Source 5D results support physics but do not establish local execution of that route.

Research Closure 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.

O4 — O — 0.5 points

Scientific requirement

Present claim-adequate channel interrogation through grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence, without mandatory n'=n or closed/open panels.

Conditions for credit

A candidate-chosen grouping, removal/reconstruction, complex sensitivity or equivalent diagnostic interrogates the complete intermediate-channel amplitude.
It has enough force for the actual channel claim; neither prior full-spectrum presentation nor a particular selected-channel/closed-open panel is required.
Record object_valid, presentation_adequate and claim_adequate separately; all must hold for the applicable full or explicitly bounded local award.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: F2. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O2, O3; 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.
Requires the correct complete-complex construction, not prior full spectrum-presentation credit.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-6, Fig. 6 and selected n-prime=n paragraph. Bounded no-net-photon channel role in very-low-energy structure. Other channels/interference remain material; another discriminating grouping/removal analysis is equally acceptable.

Research Closure 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.

I1 — I — 0.5 points

Scientific requirement

Identify the complete/on-shell spectral relation, affected domain and significance of off-shell content from the candidate's own comparison.

Conditions for credit

Identify the complete/on-shell spectral relation, affected domain and significance of off-shell content from the candidate's own comparison.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (O1 AND O2). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for I2, I3; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-5, Fig. 2; p. 023417-6, Figs. 3–5 and principal-value discussion. Complete versus pole-only broad spectral difference; PV/channel component figures are canonical, not compulsory layouts or hidden peak targets.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

I2 — I — 0.5 points

Scientific requirement

Identify a bounded intermediate-channel account from the candidate's interrogation; do not generalize n'=n to the whole response without evidence.

Conditions for credit

Identify a bounded intermediate-channel account from the candidate's interrogation; do not generalize n'=n to the whole response without evidence.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: O4. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for I1, I3; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-6, Fig. 6 and selected n-prime=n paragraph. Bounded no-net-photon channel role in very-low-energy structure. Other channels/interference remain material; another discriminating grouping/removal analysis is equally acceptable.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

I3 — I — 0.5 points

Scientific requirement

Identify the coherent total's material difference from separate direct/rescattered responses; smaller separate rescattering rate alone cannot establish negligibility.

Conditions for credit

Identify the coherent total's material difference from separate direct/rescattered responses; smaller separate rescattering rate alone cannot establish negligibility.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: O3. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for I1, I2; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-7, Fig. 7 and direct/rescattered comparison paragraph; Eqs. (8), (28)–(29), (45). Coherent total can differ materially despite a smaller separate rescattering rate. Source 5D results support physics but do not establish local execution of that route.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

M1 — M — 0.5 points

Scientific requirement

Explain that on-shell reduction retains the pole while the complete branch additionally contains PV/off-shell complex amplitude.

Conditions for credit

Explain that on-shell reduction retains the pole while the complete branch additionally contains PV/off-shell complex amplitude.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M2, M3, M4, M5, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-3–4, Eqs. (29)–(41); p. 023417-6, Figs. 3–5 and amplitude-interference paragraph. On-shell retains the pole; complete propagation adds complex PV/off-shell content. Squared-sector rate addition is physically inequivalent.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

M2 — M — 0.5 points

Scientific requirement

Explain why PV, pole and channel sectors combine as complex amplitudes and why separately squared sector rates are physically inequivalent.

Conditions for credit

Explain why PV, pole and channel sectors combine as complex amplitudes and why separately squared sector rates are physically inequivalent.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (O1 AND O2). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M3, M4, M5, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-3–4, Eqs. (29)–(41); p. 023417-6, Figs. 3–5 and amplitude-interference paragraph. On-shell retains the pole; complete propagation adds complex PV/off-shell content. Squared-sector rate addition is physically inequivalent.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

M3 — M — 0.5 points

Scientific requirement

Use genuinely discriminating candidate-chosen complex/decomposition evidence to explain the intermediate-propagation origin of the material spectral difference.

Conditions for credit

Use genuinely discriminating candidate-chosen complex/decomposition evidence to explain the intermediate-propagation origin of the material spectral difference.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M4, M5, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-5–6, Figs. 2–6, closed/open-channel and forward-scattering discussion. Candidate-grounded intermediate-propagation mechanism; no mandatory closed/open or selected-channel panels.

Research Closure relation

Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award.

M4 — M — 0.5 points

Scientific requirement

Give a bounded channel mechanism: if n'=n is used, assign only its supported local role and retain other channels/interference where material.

Conditions for credit

Give a bounded channel mechanism: if n'=n is used, assign only its supported local role and retain other channels/interference where material.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: I2. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M3, M5, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-6, Fig. 6 and selected n-prime=n paragraph. Bounded no-net-photon channel role in very-low-energy structure. Other channels/interference remain material; another discriminating grouping/removal analysis is equally acceptable.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

M5 — M — 0.5 points

Scientific requirement

Connect relative direct/rescattered phase and signed cross contribution to the total, explaining why separate-rate magnitudes cannot determine coherent consequence.

Conditions for credit

Connect relative direct/rescattered phase and signed cross contribution to the total, explaining why separate-rate magnitudes cannot determine coherent consequence.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: I3. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M3, M4, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-7, Fig. 7 and direct/rescattered comparison paragraph; Eqs. (8), (28)–(29), (45). Coherent total can differ materially despite a smaller separate rescattering rate. Source 5D results support physics but do not establish local execution of that route.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

M6 — M — 0.5 points

Scientific requirement

Combine complete/on-shell and direct/rescattered coherence evidence to answer how complete intermediate propagation changes the spectrum and whether rescattering is consequential.

Conditions for credit

The candidate combines identified complete/on-shell spectral consequences with identified direct/rescattered coherent consequences to answer the commissioned question.
A detailed local selected-channel attribution is scored separately and is not a second hard gate on this supported synthesis.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (I1 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, 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.
Detailed bounded channel attribution is owned by M4 and is not a second hard gate on this integrated conclusion.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-5–7, Figs. 2, 5, 7 and conclusion. Task-defined synthesis of complete/on-shell response and direct/rescattered coherence. A local selected-channel attribution gap is not another automatic hard gate on independently supported synthesis.

Research Closure relation

Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award.

M7 — M — 0.5 points

Scientific requirement

Restrict the causal conclusion to the established monochromatic short-range forward first-Born ISFA domain without claiming a locally unexecuted five-dimensional route.

Conditions for credit

Restrict the causal conclusion to the established monochromatic short-range forward first-Born ISFA domain without claiming a locally unexecuted five-dimensional route.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Necessary scientific premises: (I1 AND 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.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-4–5, numerical target setting; pp. 023417-6–7, short-range/Coulomb and finite-pulse discussion. Established forward monochromatic first-Born scope; source 5D and UA are optional representations, not required claimed executions.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

V1 — V — 0.5 points

Scientific requirement

State the actual monochromatic-field, short-range-target, forward first-Born and spectral domain, distinct from Coulomb, finite-pulse, experimental, other-species and five-dimensional-execution claims.

Conditions for credit

State the actual monochromatic-field, short-range-target, forward first-Born and spectral domain, distinct from Coulomb, finite-pulse, experimental, other-species and five-dimensional-execution claims.

Applicability and N/A

Applies to every assessable submission; absence or failure is scored 0, not N/A.

Scientific dependencies and independent credit

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for V2; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article pp. 023417-4–5, numerical target setting; pp. 023417-6–7, short-range/Coulomb and finite-pulse discussion. Established forward monochromatic first-Born scope; source 5D and UA are optional representations, not required claimed executions.

Research Closure relation

Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.

V2 — V — 0.5 points

Scientific requirement

If actual Cauchy/phase conventions, channel completeness, pole interpolation, truncation, incoherent addition, route claims or other submitted facts create a central material risk, provide discrimination or proportionate restriction; otherwise record N/A.

Conditions for credit

If actual Cauchy/phase conventions, channel completeness, pole interpolation, truncation, incoherent addition, route claims or other submitted facts create a central material risk, provide discrimination or proportionate restriction; otherwise record N/A.
When triggered, award only if claim-relevant discriminating evidence resolves the threat or an evidence-based restriction makes it immaterial; merely acknowledging an unresolved material threat is not positive validation credit. With no trigger, record N/A and retain the fixed 0.5 without claiming performed validation.

Applicability and N/A

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.

Scientific dependencies and independent credit

Judge this scientific obligation directly; no fixed prerequisite award.
Preserve independently valid evidence for V1; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept algebraically equivalent gauges, generalized-Bessel expansions, quadrature, truncation, common normalization and diagnostics only if the kernel, −k dOmega channel measure, Cauchy sign, complex off-shell content and relative direct phase are preserved.
Pole/PV, closed/open PV and isolated n'=n decompositions are sufficient canonical examples, not required panels. Channel grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence is accepted if it exposes the same complete/on-shell and coherence physics.
Five-dimensional and uniform-approximation implementations are optional, not hidden duties. Do not claim unexecuted routes. No fixed spectral interval, source display, weak oscillation, exact extremum, normalization height or preferred decomposition.

Scientific source locations

Article p. 023417-4, Eqs. (32)–(41); p. 023417-6, channel interference; p. 023417-7, Fig. 7. Task-defined triggered Cauchy, phase, channel, pole interpolation and coherent-normalization risks; no universal fixed quadrature or decomposition.

Research Closure 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.

Research 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 AND F4): Bind the stated monochromatic field, Up/Ip, p-state dipole, Yukawa matrix element, photon-order energy, forward direction and common conventions. Implement the complete complex energy/intermediate-channel rescattering amplitude with −k dOmega measure, channel sum, PV-plus-pole Cauchy branch and off-shell content. Construct the delta-pole-only on-shell reduction and phase-compatible direct amplitude sharing the complete rescattered branch's phase identity. Define the differential rate, one common display normalization, reconstructible complex/component mapping and |TD+TR|² total.

RC1

The central object exists, but a required indispensable component is absent or materially invalid. A failed presentation or overextended claim alone does not establish object absence or invalidity.
(O1 AND O2): Present a reconstructible complete off-shell-plus-on-shell forward rescattering spectrum retaining complex information needed for the complete-branch claim. Present the same-case, common-normalization on-shell comparator with enough evidence for material comparison against the complete spectrum.
(O3 AND O4): Present reconstructible phase-compatible direct, complete rescattered and coherent-total evidence making the cross term and rescattering consequence auditable. Present claim-adequate channel interrogation through grouping, removal/reconstruction, complex sensitivity or equally discriminating evidence, without mandatory n'=n or closed/open panels.

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.
M3: Use genuinely discriminating candidate-chosen complex/decomposition evidence to explain the intermediate-propagation origin of the material spectral difference.
M6: Combine complete/on-shell and direct/rescattered coherence evidence to answer how complete intermediate propagation changes the spectrum and whether rescattering is consequential.

RC3

Required objects and principal mechanisms are established, but a concrete material validation or scope threat to a central conclusion remains unresolved.
V2: If actual Cauchy/phase conventions, channel completeness, pole interpolation, truncation, incoherent addition, route claims or other submitted facts create a central material risk, provide discrimination or proportionate restriction; otherwise record N/A.
Before assigning RC3, establish all six elements: the concrete failure mode; its origin in the submitted method/data/conclusion; a fair recognizable scientific risk; the affected central conclusion; why the submitted evidence cannot discriminate or exclude it; and functionally equivalent remedies. Evidence-based restriction may remove unsupported breadth, but cannot repair an incorrect signal, absent component or missing mechanism.

RC4

Every indispensable and co-primary claim has all its nonempty required role slots closed; supporting claims do not independently determine overall RC. Preserve AND/OR branch logic shown above and use scientific gate outcomes, not atomic scores.
No triggered material risk remains unresolved. Untriggered risk criteria recorded N/A do not block RC4 and do not establish validation evidence.
Classify descriptively by the most severe applicable failed required role (lowest RC), without score ceilings or target distributions. A score/RC tension prompts scientific review, not forced adjustment.

Applicability and Equivalent Evidence

The complete amplitude is T_n^R=sum_(n') integral_0^infinity f_(n')(x)/(x−x_(n')−i0) dx, x_(n')=n'omega−Ip−Up, with the stated kernel and −k dOmega measure. The Cauchy identity is 1/(x−x_n'−i0)=PV[1/(x−x_n')]+i pi delta(x−x_n'). The on-shell branch retains only the pole; the complete branch also retains off-shell PV amplitude.
All intermediate sectors combine as complex amplitudes, not separately squared rates. Pole-only propagation misses material low-energy structure and can differ over part of the plateau; assess the established domain rather than exact peak depths or weak oscillations.
The no-net-photon n'=n channel can have a bounded very-low-energy role supported by forward short-range scattering, but does not alone explain the complete low-energy family. Other channels and complex interference remain potentially material. An alternative discriminating channel account is allowed.
The physical total is W_n^coh=K_n|T_n^D+T_n^R|² with common phase and normalization. A smaller separate rescattering rate does not imply negligible coherent consequence because the signed cross term can be material.
O4 channel interrogation depends on correct complete-complex construction, not mandatory O1 spectrum presentation. Local cross-order rate-prefactor weakness does not automatically invalidate an unaffected same-order pole comparison or same-energy coherent identity.
M6's combined complete/on-shell/coherent conclusion does not impose I2/M4 local channel attribution as a second hard gate; M4 alone owns that detailed bounded attribution. Independent correct causal scope remains creditable.
Scope is the stated long monochromatic field, short-range p-state/Yukawa target, forward emission and first-Born rescattering ISFA, not Coulomb accuracy, finite-pulse depletion, experimental agreement or quantitative other-species transfer. Numerical/channel/pole interpolation and phase defects affect only actually damaged scientific facts.
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.
$E_p=n\omega-I_p-U_p$, $x=k^2/2$, $x_{n\prime}=n\prime\omega-I_p-U_p$ and $T_n^R=\sum_{n\prime}\int_0^\infty f_{n\prime}(x)/(x-x_{n\prime}-i0)dx$ (Eqs. (31)–(32)). The kernel is $f_{n\prime}=-k\int d\Omega_k,T_{pk}(n-n\prime)T_{ki}^{(0)}(n\prime)$, retaining the negative sign and the $k,d\Omega_k$ energy measure (Eq. (33)); $T_{pk}$ includes the short-range scattering element and Bessel coefficient (Eq. (25)).
The required Cauchy identity is $(x-x_{n\prime}-i0)^{-1}=\mathcal P(x-x_{n\prime})^{-1}+i\pi\delta(x-x_{n\prime})$. Therefore $T_{n,\rm pole}^R=i\pi\sum_{x_{n\prime}\ge0}f_{n\prime}(x_{n\prime})$ (Eq. (36)); negative $x_{n\prime}$ channels have no pole but retain the complete off-shell integral. The alternative source $1/(y+i0)$ convention is equivalent only after transforming both denominator and kernel sign.
The direct Fourier coefficient is $T_{ki}^{(0)}(n)=(1/T)\int_0^Tdt,\langle\mathbf k+\mathbf A(t)|\mathbf r\cdot\mathbf E(t)|\psi_i\rangle e^{i[\mathbf k\cdot\boldsymbol\alpha(t)+U_1(t)+n\omega t]}$ (Eqs. (27)–(28)), with $\dot{\boldsymbol\alpha}=\mathbf A$, $\dot U_1=A^2/2-U_p$ and $T=2\pi/\omega$. It must share the rescattering phase convention.
At fixed forward order, $w_n^R=2\pi p|T_n^R|^2$ and $w_n^{\rm coh}=2\pi p|T_n^D+T_n^R|^2$ (Eq. (45), amplitude identity Eq. (8)). The signed cross term is $4\pi p\operatorname{Re}(T_n^DT_n^{R*})$. Small $|T_n^R|^2$ alone does not bound this term or establish negligible rescattering.