Datasets:

Tauben
/

PhysClosureBench

Instantiate hydrogen, the two counterclockwise circular pulses and direct-SFA target/reference amplitudes; define all three physical cases with common gauge/continuum/normalization/final time and amplitude-before-square coherence.

Conditions for credit

Instantiate hydrogen, the two counterclockwise circular pulses and direct-SFA target/reference amplitudes; define all three physical cases with common gauge/continuum/normalization/final time and amplitude-before-square coherence.

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 F2; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Liu, Tian and Li (2026), author manuscript pp. 1–3, Eqs. (1)–(6); Appendix A, p. 7, Eqs. (A2)–(A10). Pulse, coherence, common final time and continuum-normalization identities. Direct-SFA production is the visible task model; the source TDSE solver and its pictures are not direct-SFA validation.

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

Derive computable three-spectrum complex recovery with correct analytic branch, full energy-dependent reference phase and sole global-phase freedom; define BTD/localized energy-time transforms retaining physical time, direction and case.

Conditions for credit

Derive computable three-spectrum complex recovery with correct analytic branch, full energy-dependent reference phase and sole global-phase freedom; define BTD/localized energy-time transforms retaining physical time, direction and case.

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. This identifies facts, not prerequisite half-point awards.
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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eqs. (3)–(4) and Hilbert-transform discussion; p. 3, Eqs. (6)–(7); Appendix B, pp. 7–8, Eqs. (B1)–(B2) and final translation paragraph. Three-spectrum complex recovery, analytic branch, known full reference phase and physical-time mapping. Exact reference-phase restoration is a task-required refinement of the source instant-reference approximation.

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

Achieve at least two-of-three valid, reconstructible, adequately presented and claim-adequate members of the fixed target/reference/coherent spectrum family under the common production, normalization and final-time identity.

Conditions for credit

Achieve at least two-of-three valid, reconstructible, adequately presented and claim-adequate members of the fixed target/reference/coherent spectrum family under the common production, normalization and final-time identity.
The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately.

Applicability and 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. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O2, O3, O4, O5, O6; 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eq. (2), Fig. 1(b); p. 3, Eq. (6), Fig. 2(b1–b3). Target/reference/coherent spectral identities and cross term. Two-of-three and three-of-three are task coverage milestones, not claims that source TDSE images must be reproduced.

Research Closure relation

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

O2 — O — 0.5 points

Scientific requirement

Achieve complete three-of-three spectrum coverage with each member valid and adequate and the coherent spectrum formed by common-normalization amplitude addition.

Conditions for credit

Achieve complete three-of-three spectrum coverage with each member valid and adequate and the coherent spectrum formed by common-normalization amplitude addition.
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 O1). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O3, O4, O5, O6; 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eq. (2), Fig. 1(b); p. 3, Eq. (6), Fig. 2(b1–b3). Target/reference/coherent spectral identities and cross term. Two-of-three and three-of-three are task coverage milestones, not claims that source TDSE images must be reproduced.

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 recovered/direct magnitude comparison against an independently retained noncircular target packet on a common energy domain.

Conditions for credit

Present reconstructible recovered/direct magnitude comparison against an independently retained noncircular target packet on a common energy domain.
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 O2). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O4, O5, O6; 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eqs. (3)–(4), Fig. 1(c) and global-phase paragraph; p. 3, Fig. 2(c1–c3). Magnitude and energy-dependent phase comparisons motivate recovery verification. Candidate direct-SFA reference must be independently retained, not copied from the recovered array; no source overlap threshold is imposed.

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 reconstructible recovered/direct energy-dependent phase comparison allowing only one global alignment, with the correct branch/reference phase and a domain capable of exposing node/branch errors.

Conditions for credit

Recovered phase is compared against an independently retained direct target on an adequate common energy domain.
Correct analytic branch and full reference phase are preserved, with only one constant global alignment; the domain can expose material node/branch error.
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 AND O2). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O3, O5, O6; 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eqs. (3)–(4), Fig. 1(c) and global-phase paragraph; p. 3, Fig. 2(c1–c3). Magnitude and energy-dependent phase comparisons motivate recovery verification. Candidate direct-SFA reference must be independently retained, not copied from the recovered array; no source overlap threshold is imposed.

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.

O5 — O — 0.5 points

Scientific requirement

Present at least one independently auditable direct/recovered time-domain comparator, BTD or claim-equivalent localized representation, under identical direction, case and physical-time identity with supported periodic feature scale.

Conditions for credit

At least one direct/recovered BTD or claim-equivalent time/localized comparator is independently auditable.
Direction, case and pulse-defined physical time agree and the claimed periodic feature scale is supported; unrelated peak recentering is not accepted.
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 AND O3 AND O4). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O6; 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 3–4, Eq. (7), Fig. 2(d1–d3), BTD interpretation paragraphs. Birth-time transform and periodic ionization content. Source angular delays and cross-regime maxima are not direct-SFA targets; physical time may not be recentered to align peaks.

Research Closure relation

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

O6 — O — 0.5 points

Scientific requirement

Present claim-adequate localized energy-time evidence under the same case/time mapping, exposing energy support, periodicity and declared localization/resolution.

Conditions for credit

Present claim-adequate localized energy-time evidence under the same case/time mapping, exposing energy support, periodicity and declared localization/resolution.
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 O3 AND O4). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O5; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.
An additional direct/recovered comparator is scored only in O5, not imposed twice.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 4, 6–8, Fig. 3(a1–a3), Fig. 5, Appendix B Eqs. (B1)–(B2) and final time-translation paragraph. Localized energy-time support and resolution tradeoff. Source Gabor examples provide transform identity only; SST Fig. 6 is excluded and no source window width is mandatory.

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 magnitude and energy-dependent phase recovery fidelity from the candidate's comparisons, distinguishing a free global phase from material phase-domain error.

Conditions for credit

Identify magnitude and energy-dependent phase recovery fidelity from the candidate's comparisons, distinguishing a free global phase from material phase-domain error.

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 AND O4). 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eqs. (3)–(4), Fig. 1(c) and global-phase paragraph; p. 3, Fig. 2(c1–c3). Magnitude and energy-dependent phase comparisons motivate recovery verification. Candidate direct-SFA reference must be independently retained, not copied from the recovered array; no source overlap threshold is imposed.

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 three-spectrum physical consistency, target ATI-scale structure and coherent cross-term evidence from complete coverage, not rescaling or incoherent addition.

Conditions for credit

Identify three-spectrum physical consistency, target ATI-scale structure and coherent cross-term evidence from complete coverage, not rescaling or incoherent addition.

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: O2. 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 2, Eq. (2), Fig. 1(b); p. 3, Eq. (6), Fig. 2(b1–b3). Target/reference/coherent spectral identities and cross term. Two-of-three and three-of-three are task coverage milestones, not claims that source TDSE images must be reproduced.

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 periodic structure, physical-time consistency and supported feature scale from valid submitted same-case time/energy-time evidence.

Conditions for credit

Identify periodic structure, physical-time consistency and supported feature scale from valid submitted same-case time/energy-time 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: O6. 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.
The absence of O5's independent comparator does not automatically invalidate this identification.
Judge valid candidate-presented periodicity and physical time directly; O5 comparator absence alone is not missing candidate identification, and an independently auditable equivalent time representation remains eligible.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 3–4, Eq. (7), Fig. 2(d1–d3), BTD interpretation paragraphs. Birth-time transform and periodic ionization content. Source angular delays and cross-regime maxima are not direct-SFA targets; physical time may not be recentered to align peaks.

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

Use three spectra and independently retained direct/recovered complex evidence to decide whether spectral phase and birth-time content were recovered.

Conditions for credit

Use three spectra and independently retained direct/recovered complex evidence to decide whether spectral phase and birth-time content were recovered.

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 I2). 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 2–3, Eqs. (2)–(6), analytic-signal and global-phase discussion. Task-defined internal-recovery decision from candidate spectra and independent complex comparison. The cross term retains relative phase; a wrong branch or energy-dependent reference phase cannot be repaired by global alignment.

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 how the coherent cross term encodes relative spectral phase and why analytic branch/full reference phase cannot be replaced by global alignment.

Conditions for credit

Explain how the coherent cross term encodes relative spectral phase and why analytic branch/full reference phase cannot be replaced by global alignment.

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 I2). 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 2–3, Eqs. (2)–(6), analytic-signal and global-phase discussion. Task-defined internal-recovery decision from candidate spectra and independent complex comparison. The cross term retains relative phase; a wrong branch or energy-dependent reference phase cannot be repaired by global alignment.

Research Closure relation

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

M3 — M — 0.5 points

Scientific requirement

Connect the target ATI-scale energy comb to periodic direct ionization without hidden source angular-delay or cross-regime mechanisms.

Conditions for credit

Connect the target ATI-scale energy comb to periodic direct ionization without hidden source angular-delay or cross-regime mechanisms.

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, 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript p. 3, Eqs. (5)–(7), periodic burst discussion; Fig. 2(b1–b3). Fourier entailment links ATI-scale energy spacing to cycle-scale direct emission. It does not import source Coulomb/angular-delay or multiple-regime mechanisms.

Research Closure relation

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

M4 — M — 0.5 points

Scientific requirement

Use submitted BTD or equivalent time-domain evidence to explain cycle-scale bursts and the pulse-defined physical birth-time origin, rejecting nonphysical peak recentering.

Conditions for credit

Use submitted BTD or equivalent time-domain evidence to explain cycle-scale bursts and the pulse-defined physical birth-time origin, rejecting nonphysical peak recentering.

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, 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 3–4, Eq. (7), Fig. 2(d1–d3), BTD interpretation paragraphs. Birth-time transform and periodic ionization content. Source angular delays and cross-regime maxima are not direct-SFA targets; physical time may not be recentered to align peaks.

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

Explain compatibility of localized energy-time support with the energy comb/time bursts and the physical localization–resolution tradeoff.

Conditions for credit

Explain compatibility of localized energy-time support with the energy comb/time bursts and the physical localization–resolution tradeoff.

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: (O6 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M3, M4, M6, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 4, 6–8, Fig. 3(a1–a3), Fig. 5, Appendix B Eqs. (B1)–(B2) and final time-translation paragraph. Localized energy-time support and resolution tradeoff. Source Gabor examples provide transform identity only; SST Fig. 6 is excluded and no source window width is mandatory.

Research Closure relation

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

M6 — M — 0.5 points

Scientific requirement

Combine complex recovery, submitted time periodicity and energy-time consistency into a bounded direct-SFA internal-reconstruction conclusion without duplicating O5's comparator requirement.

Conditions for credit

The candidate integrates complex recovery with submitted periodic time content and localized energy-time consistency.
O5 independent-comparator credit is not charged again as a hard gate where time inference is independently established; genuine phase/time identity damage still limits dependent claims.

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: (M1 AND M3 AND M4 AND M5). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M2, M7; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 3–4, Eqs. (5)–(7); Appendix B, pp. 7–8. Task-defined synthesis of complex recovery, periodic time content and localized support. O5 independent-comparator credit is not duplicated as another hard gate on independently established time inference.

Research Closure relation

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

M7 — M — 0.5 points

Scientific requirement

Distinguish interferometric reconstruction within direct SFA from TDSE/Coulomb/SST or multi-regime validation, limiting the model claim to submitted evidence.

Conditions for credit

Distinguish interferometric reconstruction within direct SFA from TDSE/Coulomb/SST or multi-regime validation, limiting the model claim to submitted 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: (M2 AND M6). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for V1, V2; failure here does not automatically erase those criteria.
Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation.

Accepted equivalent evidence

Accept sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript Appendix A, pp. 6–7, Eqs. (A1), (A6)–(A10); Appendix B, pp. 7–8. Explicit model distinction: source uses Coulomb TDSE/scattering projection and may show SST; the commissioned result is direct-SFA internal reconstruction, not equivalence to those methods.

Research Closure relation

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

V1 — V — 0.5 points

Scientific requirement

State the hydrogen circular-pulse direct-SFA and actual direction/energy/time domain without TDSE, Coulomb, rescattering, t-SURFF, SST, absolute-yield or multi-regime claims.

Conditions for credit

State the hydrogen circular-pulse direct-SFA and actual direction/energy/time domain without TDSE, Coulomb, rescattering, t-SURFF, SST, absolute-yield or multi-regime 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript Appendix A, pp. 6–7, Eqs. (A1), (A6)–(A10); Appendix B, pp. 7–8. Explicit model distinction: source uses Coulomb TDSE/scattering projection and may show SST; the commissioned result is direct-SFA internal reconstruction, not equivalence to those methods.

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 pulse/case mismatch, missing reference, branch selection, reference nodes/phase, common final time, birth-time translation, circular comparison or another ambiguity threatens recovery, provide discrimination or evidence-based restriction; otherwise record N/A.

Conditions for credit

If actual pulse/case mismatch, missing reference, branch selection, reference nodes/phase, common final time, birth-time translation, circular comparison or another ambiguity threatens recovery, provide discrimination or evidence-based 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 sign-consistent Fourier/Hilbert conventions, equivalent direct-SFA gauges, explicitly mapped energy/momentum normalization and adequate localized energy-time transforms preserving the same physical case, direction and pulse-defined time.
One common global packet phase is free. Energy-dependent reference phase, analytic branch and physical time origin are not removable by a global alignment or peak recentering. Direction, localization, masks, sampling and display are choices, not hidden fixed targets.
BTD or a claim-equivalent time representation may supply the independent direct/recovered comparator. Independently valid time-signal identification and synthesis do not automatically fail because that additional comparator is absent.

Scientific source locations

Author manuscript pp. 2–3, Eq. (3), Eq. (6) and reference-support paragraph; Appendix B, pp. 7–8, physical-time translation. Task-defined conditional risk from reference nodes/phase, carrier sampling, common-time binding or circular verification. No fixed reference mask, overlap threshold or transform window is prescribed.

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): Instantiate hydrogen, the two counterclockwise circular pulses and direct-SFA target/reference amplitudes; define all three physical cases with common gauge/continuum/normalization/final time and amplitude-before-square coherence. Derive computable three-spectrum complex recovery with correct analytic branch, full energy-dependent reference phase and sole global-phase freedom; define BTD/localized energy-time transforms retaining physical time, direction and case.

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.
O2: Achieve complete three-of-three spectrum coverage with each member valid and adequate and the coherent spectrum formed by common-normalization amplitude addition.
(O3 AND O4 AND O6): Present reconstructible recovered/direct magnitude comparison against an independently retained noncircular target packet on a common energy domain. Present reconstructible recovered/direct energy-dependent phase comparison allowing only one global alignment, with the correct branch/reference phase and a domain capable of exposing node/branch errors. Present claim-adequate localized energy-time evidence under the same case/time mapping, exposing energy support, periodicity and declared localization/resolution.

RC2

Required objects are scientifically valid, but a required principal mechanism or state-dependence inference is absent or incorrect. Reference evidence cannot provide omitted candidate reasoning.
M2: Explain how the coherent cross term encodes relative spectral phase and why analytic branch/full reference phase cannot be replaced by global alignment.
M5: Explain compatibility of localized energy-time support with the energy comb/time bursts and the physical localization–resolution tradeoff.
M7: Distinguish interferometric reconstruction within direct SFA from TDSE/Coulomb/SST or multi-regime validation, limiting the model claim to submitted evidence.

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 pulse/case mismatch, missing reference, branch selection, reference nodes/phase, common final time, birth-time translation, circular comparison or another ambiguity threatens recovery, provide discrimination or evidence-based 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

An accepted direct amplitude is M(p)=−i integral d[p+A(t)]·E(t) exp[−i S(p,t)]dt, S=integral_t^tf {[p+A(t')]²/2+Ip}dt'. Both counterclockwise circular pulse amplitudes must share hydrogen case, direction, energy-normalized continuum and final time tf.
The observables are P0=|psi0|², Px=|psix|², P=|psi0+psix|², never separately rescaled or incoherently added probabilities. The three-member homogeneous family is target-only, reference-only and coherent; O1 requires at least two valid/adequate/claim-adequate members, O2 requires all three.
Form W=(P−P0−Px)/(2sqrt(Px))=Re[psi0 exp(−i arg psix)]. Recover with the appropriate analytic-signal branch and restore the full energy-dependent reference phase. For positive delay the stated convention has negative energy-carrier slope; positive-Fourier-support analytic signals therefore require conjugation.
Assess magnitude and energy-dependent phase against an independently retained direct target packet, allowing only one global phase. Comparing the same stored target array to itself is circular and cannot establish recovery. Handle unresolved reference nodes with a declared uncertainty-based valid domain, not hidden thresholds.
The physical BTD is B_phi(tau0)=|(1/sqrt(2pi))integral_0^infinity psi(E,phi;tf)exp[−i(tau0−tf)E]dE|². A localized representation can use G(E,tau)=|integral psi(E';tf)eta(E'−E)exp[−i tau(E'−E)]dE'|² with tau=tau0−tf. Restore the pulse-defined physical time, not a recentered peak axis.
Robust direct-SFA content is an ATI-scale energy comb, approximately cycle-separated birth bursts and compatible periodic energy-time support. Exact source angles/delays, envelope, overlap value, display floor and raster are not targets.
O5 owns the independent direct/recovered time comparator. O6 owns claim-adequate localized energy-time evidence; I3, M4 and M6 may use valid submitted same-case time evidence without imposing O5's local comparator a second time as a hard gate.
Source-TDSE imagery verifies formula/pulse identity only and is not direct-SFA image gold. Scope excludes full 3D TDSE, Coulomb/scattering-state projections, t-SURFF equivalence, SST, absolute yield, source angular delays and multiple regimes. Optional broader work cannot compensate for missing specified direct-SFA work.
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.
One admissible direct-SFA convention is $M(\mathbf p)=-i\int dt,\mathbf d[\mathbf p+\mathbf A(t)]\cdot\mathbf E(t)e^{-iS(\mathbf p,t)}$, $S=\int_t^{t_f}{[\mathbf p+\mathbf A(t\prime)]^2/2+I_p}dt\prime$. This is the task-defined production model, not a formula claimed to be the source TDSE solver. The energy-normalized packet is $\psi(E,\Omega;t_f)=(2E)^{1/4}\psi(\mathbf p=\sqrt{2E},\Omega;t_f)$ (source Eq. (A8)); direction $\Omega$, normalization and common $t_f$ are preserved for both pulses.
$P_0=|\psi_0|^2$, $P_x=|\psi_x|^2$, $P=|\psi_0+\psi_x|^2$ and $W=(P-P_0-P_x)/(2\sqrt{P_x})=\operatorname{Re}[\psi_0e^{-i\arg\psi_x}]$ on the reference-supported domain (Eqs. (2)–(6), exact algebra). The correct analytic-signal branch recovers the complex carrier; multiply by the full known $e^{i\arg\psi_x(E)}$, not a constant reference phase. For the stated positive delay and source Fourier convention a positive-support analytic signal requires the conjugate branch. Only one overall constant phase is free.
$B_\Omega(\tau_0)=|(2\pi)^{-1/2}\int_0^\infty\psi(E,\Omega;t_f)e^{-i(\tau_0-t_f)E}dE|^2$ (Eq. (7), common-final-time generalization). A suitable local representation is $G(E,\tau)=|\int_0^\infty dE\prime,\psi(E\prime;t_f)\eta(E\prime-E)e^{-i\tau(E\prime-E)}|^2$, with $\tau=\tau_0-t_f$ and declared localization window $\eta$ (Appendix B). The pulse-defined physical birth time is $\tau_0=\tau+t_f$, not a peak-recentered axis.
For cycle-separated coherent direct bursts, the energy/time Fourier relation gives ATI spacing $\Delta E\simeq\omega$ and burst spacing $T=2\pi/\omega$ in atomic units. This is a source-supported physical entailment, not an exact envelope/angle-delay prediction. Localization broadens conjugate time/energy resolution; it must be adequate for the features claimed, without a prescribed source width or SST algorithm.
All source figures in this crosswalk are conceptual/formula and broad-structure support only. None is direct-SFA image gold or evidence of candidate TDSE/Coulomb/SST validation. Correct two-spectrum local coverage retains O1 when the third is missing, but it cannot establish the full three-spectrum I2 claim or manufacture a missing complex recovery.
An independent direct/recovered time comparator is the local O5 award condition. Its absence does not erase independently auditable localized O6 evidence or candidate time-periodicity reasoning; however, genuine analytic-branch/reference-phase/time-origin errors propagate to the outputs they actually invalidate.