Grading Rubric — Case48
Observable-dependent approximation hierarchy for quantum HHG from a correlated chain
Atomic Scoring Criteria
Each criterion awards 0 or 0.5 points, for a maximum of 10. Explicit N/A rules apply where stated.
F1 — F — 0.5 points
Scientific requirement
Instantiate the specified eight-site periodic half-filled Hubbard ring, spin-symmetric K=0 ground state, parameters and matched Nc=10/18 pulses, deriving diagonal/transition currents from common propagation for all routes.
Conditions for credit
- Instantiate the specified eight-site periodic half-filled Hubbard ring, spin-symmetric K=0 ground state, parameters and matched Nc=10/18 pulses, deriving diagonal/transition currents from common propagation for all routes.
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, 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Lange and Madsen (2025), author manuscript pp. 2–3, Eqs. (9)–(12); pp. 7–8, Eqs. (42)–(44). Half-filled eight-site Hubbard dynamics, common propagated diagonal/transition currents and two pulse durations. Source Eq. (20) full decoupled branch is background, not a fourth required route.
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 ground-transition branches, H1,H3,…,H59 physical mode product, same-branch other-mode norm factors and one global normalization, without independently normalizing modes.
Conditions for credit
- Implement the ground-transition branches, H1,H3,…,H59 physical mode product, same-branch other-mode norm factors and one global normalization, without independently normalizing modes.
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 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.
This criterion scores executable state identity, not the additional explanation of why the global trace holds.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 4–5, Eq. (19), Eqs. (22a,b). Ground-transition branch product and initial vacuum. Same-branch other-mode norm product and one global trace are physical entailments of the task-adopted multimode state; fixed odd H1–H59 modes are task-defined, not a source display grid.
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
Derive and implement S, Q and optimized-squeezing moments from the defined local Markov-type Fock equation, preserving its eliminated-branch/state identity.
Conditions for credit
- Derive and implement S, Q and optimized-squeezing moments from the defined local Markov-type Fock equation, preserving its eliminated-branch/state identity.
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 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript p. 5, Eqs. (23)–(26); p. 3, Eqs. (13), (15)–(16). Local-time elimination and Fock-state moments preserve the Markov-type route identity rather than silently using an independent branch.
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
Derive moments from the defined displaced-correlation MSA state, retaining W, beta, electronic expectation, higher-commutator neglect and second-order g0 identity.
Conditions for credit
- Derive moments from the defined displaced-correlation MSA state, retaining W, beta, electronic expectation, higher-commutator neglect and second-order g0 identity.
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 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 5–7, Eqs. (28)–(41), especially (30)–(34), (38). MSA displaced current-correlation state, electronic expectation, discarded higher commutator and second-order coupling content; moment derivation must retain the declared order.
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 first coverage: at least one of the fixed six pulse×observable members is valid, reconstructible, adequately presented and locally claim-adequate for all three routes.
Conditions for credit
At least one fixed pulse×observable member is valid, reconstructible, adequately presented and locally claim-adequate across all three retained routes.
Count members only after checking matched model, case, route and normalization; one successful route alone is not a covered member.
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: (F1 AND F2 AND F3 AND F4). 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.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript p. 8, Fig. 3(a–c); p. 9, Fig. 4(a–c); observable definitions p. 3, Eqs. (13), (15)–(16). Exactly six pulse×observable members, each compared across the three retained routes. First, broad and full coverage are task-defined milestones, not a source figure-count requirement or proof of named-member inference.
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 broad coverage: at least three valid presented members covering both pulses and at least two observable types, including Q or optimized squeezing, after the O1 milestone.
Conditions for credit
O1 is established and at least three members are valid and adequately presented across all three retained routes.
Coverage includes both pulses, at least two observable types and Q or optimized squeezing; this milestone does not backfill missing members or candidate inference.
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: O1. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for 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.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript p. 8, Fig. 3(a–c); p. 9, Fig. 4(a–c); observable definitions p. 3, Eqs. (13), (15)–(16). Exactly six pulse×observable members, each compared across the three retained routes. First, broad and full coverage are task-defined milestones, not a source figure-count requirement or proof of named-member inference.
Research Closure relation
Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.
O3 — O — 0.5 points
Scientific requirement
Achieve full coverage: all six pulse×observable members are valid, reconstructible, adequately presented and mutually comparable under matched model, route, normalization, domain and resolution.
Conditions for credit
All six pulse×observable members are valid, reconstructible and adequately presented across the three retained routes.
Matched case, normalization, domain and resolution make the complete family mutually comparable; partial coverage retains its already established lower milestones.
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: O2. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for O1; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript p. 8, Fig. 3(a–c); p. 9, Fig. 4(a–c); observable definitions p. 3, Eqs. (13), (15)–(16). Exactly six pulse×observable members, each compared across the three retained routes. First, broad and full coverage are task-defined milestones, not a source figure-count requirement or proof of named-member inference.
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 principal harmonic/higher-frequency structure and route agreement/disagreement from both valid, reconstructible and presented spectrum pulse members, without inferring full state fidelity.
Conditions for credit
- Identify principal harmonic/higher-frequency structure and route agreement/disagreement from both valid, reconstructible and presented spectrum pulse members, without inferring full state fidelity.
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 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.
Named-member eligibility is direct; generic coverage credit neither formally gates nor supplies this identification.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 8–10, Fig. 3(a), Fig. 4(a), spectrum paragraphs. Both pulse spectra: main harmonic train, higher-frequency structure and route relation. Spectrum agreement alone does not identify full emitted-state fidelity.
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 Q route differences and duration/frequency-region changes from both valid presented Q pulse members satisfying denominator/resolution limits.
Conditions for credit
- Identify Q route differences and duration/frequency-region changes from both valid presented Q pulse members satisfying denominator/resolution limits.
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 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.
Named-member eligibility is direct; generic coverage cannot fill a missing or invalid Q member.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 8–10, Fig. 3(b), Fig. 4(b), denominator explanation after Eq. (40). Both pulse Mandel-Q members, with photon-number validity and defensible resolution. Source raw spikes or clipped plotting range are not targets.
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 squeezing route/region differences and duration-dependent validity from both valid presented optimized-squeezing pulse members.
Conditions for credit
- Identify squeezing route/region differences and duration-dependent validity from both valid presented optimized-squeezing pulse members.
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 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.
Named-member eligibility is direct; generic coverage cannot fill a missing or invalid squeezing member.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 8–11, Fig. 3(c), Fig. 4(c), low-harmonic squeezing discussion; Eq. (41), p. 7. Both pulse optimized-squeezing members and observable/region-dependent duration relation; no fixed inset/peak coordinate.
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 all three observable families to establish an approximation hierarchy, not full emitted-state fidelity from spectral agreement alone.
Conditions for credit
- Use all three observable families to establish an approximation hierarchy, not full emitted-state fidelity from spectral agreement alone.
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 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M2, M3, M4, M5, M6, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 6–7, Eqs. (32)–(41); pp. 9–11, Figs. 3–4 and spectral/nonclassical comparison paragraphs. Positive mechanism for why main spectral agreement need not preserve photon-number fluctuations or quadrature noise; no universal winning observable is imposed.
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 main spectral agreement does not guarantee photon-number-fluctuation or quadrature-noise fidelity, without an imposed universal winning observable.
Conditions for credit
- Explain why main spectral agreement does not guarantee photon-number-fluctuation or quadrature-noise fidelity, without an imposed universal winning observable.
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 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M3, M4, M5, M6, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 6–7, Eqs. (32)–(41); pp. 9–11, Figs. 3–4 and spectral/nonclassical comparison paragraphs. Positive mechanism for why main spectral agreement need not preserve photon-number fluctuations or quadrature noise; no universal winning observable is imposed.
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 matched Nc=10/18 evidence to determine how approximation error changes with duration and frequency region.
Conditions for credit
- Use matched Nc=10/18 evidence to determine how approximation error changes with duration and frequency region.
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 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M4, M5, M6, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 9–11, Figs. 3–5, longer-pulse and averaged-error limitations paragraphs. Matched duration/frequency-region validity synthesis. Fig. 5 additional pulse-length scan is contextual, not a required new sweep.
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
Synthesize observables and both pulses to give an observable-dependent duration validity boundary rather than unrelated plots.
Conditions for credit
- Synthesize observables and both pulses to give an observable-dependent duration validity boundary rather than unrelated plots.
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). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M2, M5, M6, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 9–11, Figs. 3–5, longer-pulse and averaged-error limitations paragraphs. Matched duration/frequency-region validity synthesis. Fig. 5 additional pulse-length scan is contextual, not a required new sweep.
Research Closure relation
Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award.
M5 — M — 0.5 points
Scientific requirement
Connect diagonal-current control of spectra with transition-current-correlation control of Q and squeezing.
Conditions for credit
- Connect diagonal-current control of spectra with transition-current-correlation control of Q and squeezing.
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 AND I3). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M3, M4, M6, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 5–7, Eqs. (25)–(41); pp. 10–11, discarded-commutator explanation. Diagonal current controls leading spectra while transition-current correlations control nonclassical moments; local-time and higher-commutator reductions discard different information.
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
Explain how local-time elimination and higher-commutator/second-order reductions discard correlation content and produce observable-dependent agreement/failure.
Conditions for credit
- Explain how local-time elimination and higher-commutator/second-order reductions discard correlation content and produce observable-dependent agreement/failure.
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: M5. This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M1, M2, M3, M4, M7, M8; 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 5–7, Eqs. (25)–(41); pp. 10–11, discarded-commutator explanation. Diagonal current controls leading spectra while transition-current correlations control nonclassical moments; local-time and higher-commutator reductions discard different information.
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
Give an evidence-bounded conclusion and explicit equivalent-representation mapping to branch/state/order/global trace, explaining the same-branch other-mode product and single global normalization.
Conditions for credit
The conclusion is bounded by the established hierarchy, duration evidence and reduction mechanism.
An explicit equivalent-representation mapping explains branch/state/order/global trace, including same-branch other-mode norms and single global normalization; a correct executable F2 alone does not supply that explanation.
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 M1 AND M3 AND M6). This identifies facts, not prerequisite half-point awards.
Preserve independently valid evidence for M2, M4, M5, M8; 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.
Correct executable branch/global-normalization identity under F2 is preserved when only the explanatory representation/global-trace mapping here is incomplete.
Accepted equivalent evidence
Accept mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 4–7, Eqs. (19), (22), (26), (30); observable definitions Eqs. (13), (15)–(16). Task-defined representation mapping to the same branch/state/order/global trace. Correct executable normalization is distinct from explaining it; no fourth route or continuum limit is inferred.
Research Closure relation
Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim.
M8 — M — 0.5 points
Scientific requirement
Identify a falsifiable method/data-emergent limitation or proportionately restrict the claim, without prescribing a unique examiner check.
Conditions for credit
- Identify a falsifiable method/data-emergent limitation or proportionately restrict the claim, without prescribing a unique examiner check.
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: M7. 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 9–11, Q denominator, narrow-frequency structure and MSA limitations; Eqs. (15)–(16), p. 3. Falsifiable candidate-arising limitations and conditional risks. Routine numerical checks are not detachable points or a prescribed method list.
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 weak-coupling finite eight-site chain, two pulses, fixed multimode object and three-route scope; do not generalize to all photonic approximations, coupling regimes or an exact continuum limit.
Conditions for credit
- State the weak-coupling finite eight-site chain, two pulses, fixed multimode object and three-route scope; do not generalize to all photonic approximations, coupling regimes or an exact continuum limit.
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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 7–8, finite Hubbard/pulse setting; p. 11, weak-coupling conclusion. Finite eight-site, weak-coupling, two-pulse three-route scope, not all approximations or an exact continuum limit.
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 Q denominators, narrow structures, branch/mode normalization, Fock tail, electronic/time propagation, representation mapping or other submitted facts threaten the hierarchy, provide discriminating evidence or restriction; otherwise record N/A.
Conditions for credit
If actual Q denominators, narrow structures, branch/mode normalization, Fock tail, electronic/time propagation, representation mapping or other submitted facts threaten the hierarchy, provide discriminating evidence or restriction; otherwise record N/A.
When triggered, award only if claim-relevant discriminating evidence resolves the threat or an evidence-based restriction makes it immaterial; merely acknowledging an unresolved material threat is not positive validation credit. With no trigger, record N/A and retain the fixed 0.5 without claiming performed validation.
Applicability and 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 mapped symmetry bases/gauges/pictures, converged propagators, Fock cutoffs, interpolation, frequency grids, numerical squeezing minimizers, lineouts, units and common normalization preserving electronic branch, physical mode set, state, approximation order and global trace.
Fixed odd physical modes define the multimode object, not all display points. Additional evaluation points are allowed and must be distinguished from physical modes. No fixed cutoff, Q mask threshold, smoothing, source peak, plot range, layout or absolute conventional spectrum scale.
Scientific source locations
- Author manuscript pp. 9–11, Q denominator, narrow-frequency structure and MSA limitations; Eqs. (15)–(16), p. 3. Falsifiable candidate-arising limitations and conditional risks. Routine numerical checks are not detachable points or a prescribed method list.
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): Instantiate the specified eight-site periodic half-filled Hubbard ring, spin-symmetric K=0 ground state, parameters and matched Nc=10/18 pulses, deriving diagonal/transition currents from common propagation for all routes. Implement the ground-transition branches, H1,H3,…,H59 physical mode product, same-branch other-mode norm factors and one global normalization, without independently normalizing modes. Derive and implement S, Q and optimized-squeezing moments from the defined local Markov-type Fock equation, preserving its eliminated-branch/state identity. Derive moments from the defined displaced-correlation MSA state, retaining W, beta, electronic expectation, higher-commutator neglect and second-order g0 identity.
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.
O3: Achieve full coverage: all six pulse×observable members are valid, reconstructible, adequately presented and mutually comparable under matched model, route, normalization, domain and 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.
M4: Synthesize observables and both pulses to give an observable-dependent duration validity boundary rather than unrelated plots.
M6: Explain how local-time elimination and higher-commutator/second-order reductions discard correlation content and produce observable-dependent agreement/failure.
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 Q denominators, narrow structures, branch/mode normalization, Fock tail, electronic/time propagation, representation mapping or other submitted facts threaten the hierarchy, provide discriminating evidence or restriction; otherwise record N/A.
Before assigning RC3, establish all six elements: the concrete failure mode; its origin in the submitted method/data/conclusion; a fair recognizable scientific risk; the affected central conclusion; why the submitted evidence cannot discriminate or exclude it; and functionally equivalent remedies. Evidence-based restriction may remove unsupported breadth, but cannot repair an incorrect signal, absent component or missing mechanism.
RC4
Every indispensable and co-primary claim has all its nonempty required role slots closed; supporting claims do not independently determine overall RC. Preserve AND/OR branch logic shown above and use scientific gate outcomes, not atomic scores.
No triggered material risk remains unresolved. Untriggered risk criteria recorded N/A do not block RC4 and do not establish validation evidence.
Classify descriptively by the most severe applicable failed required role (lowest RC), without score ceilings or target distributions. A score/RC tension prompts scientific review, not forced adjustment.
Applicability and Equivalent Evidence
Use the periodic eight-site half-filled ring with 4+4 electrons, spin-symmetric K=0 ground state, a=7.5589, t0=0.0191, U=10t0, omega_L=0.005, A0=0.194, g0=4×10^−8 and Nc=10,18. H_el=−t0 sum[e^(iaAcl)c†jmu c(j+1)mu+H.c.]+U sum n_up n_down; Acl=A0 sin(omega_L t+pi/2)sin²(omega_L t/(2Nc)).
The common current is j=−ia t0 sum[e^(iaAcl)c†jmu c(j+1)mu−H.c.], and j_mn=<phi_m(t)|j|phi_n(t)> comes from common electronic propagation. A_Qlambda=(g0/sqrt(omega_lambda))[a exp(−iomega_lambda t)+a†exp(iomega_lambda t)].
Ground-transition branches obey i partial_t|chi_lambda^i>=A_Qlambda sum_m j_im|chi_lambda^m>, i partial_t|chi_lambda^m>=A_Qlambda j_mi|chi_lambda^i> for m≠i, with vacuum in i and zero other branches. Retain the diagonal i current and only initial-to-other transitions.
Use physical modes H1,H3,…,H59. For any one-mode O_lambda, numerator=sum_m <chi_lambda^m|O|chi_lambda^m> product_(kappa≠lambda)||chi_kappa^m||² and denominator=sum_m product_kappa||chi_kappa^m||². Normalize once globally. Independent per-mode normalization changes the state; F2 scores executable identity, M7 separately scores why this branch trace is required.
The local Markov-type state obeys i partial_t chi_i=A_Qlambda[j_ii−i sum_(m≠i)j_im integral_(ti)^t A_Qlambda(t')j_mi(t')dt']chi_i. Derive computable moments without replacing this branch by an unrelated reduction.
For MSA, W_lambda=integral A_Qlambda(t')[j_H(t')−j_ii(t')]dt', beta_ii=−i g0/sqrt(omega_lambda) integral j_ii(t')exp(iomega_lambda t')dt', and |chi_MS>=D(beta)exp[−<W_lambda²>_el/2]|0>. Neglect higher current-correlation commutators and retain through second order in g0; derive the required moments.
S=omega_lambda³/(g0²(2pi)²c³); Q=(<n²>−²)/−1; eta=−10log10[4 min_theta Var X(theta)], X=(a exp(−i theta)+a†exp(i theta))/2. Optimize the quadrature. Interpret Q only where photon number is resolved relative to declared uncertainty, not isolated small-denominator spikes.
The six coverage members are Nc=10/18 crossed with spectrum/Q/squeezing, each including all three routes. O1 requires at least one valid presented member; O2 requires at least three including both pulses, at least two observable types and at least one nonclassical type; O3 requires all six mutually comparable members. Coverage does not backfill named-member identifications.
I1, I2 and I3 each require both pulse members of their named observable, judged directly rather than through generic coverage awards; Q additionally respects denominator/resolution boundaries. A missing observable elsewhere does not erase independently valid evidence.
Both pulses have regular predominantly odd low-order harmonics through roughly H19 and more irregular higher-frequency structure. Spectra agree closely among routes, but do not establish emitted-state fidelity. Q can have narrow resolved differences; longer-pulse low-harmonic squeezing separates non-MSA/MSA more strongly while relevant higher-frequency structures are closer. No exact crossover or universal winner is imposed.
Main spectra primarily follow diagonal current; nonclassical moments probe transition-current correlations lost partly in local-time/commutator reductions. Use matched duration/region evidence to delimit observable-dependent validity, not unconnected plots. Fock tails, missing narrow structures, denominator amplification and branch normalization matter only through actually affected claims. Source Eq. (20) is not an extra duty.
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.
For the stated Hubbard ring, $H=-t_0\sum_{j,\mu}(e^{iaA(t)}c^\dagger_{j\mu}c_{j+1,\mu}+\mathrm{h.c.})+U\sum_jn_{j\uparrow}n_{j\downarrow}$ and $\hat j=-iat_0\sum_{j,\mu}(e^{iaA}c^\dagger_{j\mu}c_{j+1,\mu}-\mathrm{h.c.})$ (Eqs. (42)–(44)). $j_{mn}(t)=\langle\phi_m(t)|\hat j(t)|\phi_n(t)\rangle$ comes from the common electronic propagation, not separate fitted currents for each photonic route.
For the adopted ground-transition state $|\Psi\rangle=\sum_m|\phi_m\rangle\bigotimes_{\ell\in\mathcal K}|\chi_\ell^{(m)}\rangle$, define $N_{m\ell}=\langle\chi_\ell^{(m)}|\chi_\ell^{(m)}\rangle$ and $\mathcal Z=\sum_m\prod_{\ell\in\mathcal K}N_{m\ell}$. Then $\langle O_k\rangle=\mathcal Z^{-1}\sum_m\langle\chi_k^{(m)}|O_k|\chi_k^{(m)}\rangle\prod_{\ell\ne k}N_{m\ell}$. This is a trace derivation from the orthogonal electronic branches and product ansatz, not an independently printed source moment formula. $\mathcal K={H1,H3,\ldots,H59}$ is the fixed task physical-mode set; modewise renormalization destroys this object.
The common observables are $S(\omega_k)=\omega_k^3\sum_\sigma\langle n_{k\sigma}\rangle/[g_0^2(2\pi)^2c^3]$, $Q=\operatorname{Var}(n)/\langle n\rangle-1$, and $\eta=-10\log_{10}[4\min_\theta\operatorname{Var}(X_\theta)]$ with $X_\theta=(ae^{-i\theta}+a^\dagger e^{i\theta})/2$ (Eqs. (13), (15)–(16)). For normalized moments, $\min_\theta\operatorname{Var}(X_\theta)={1+2[\langle n\rangle-|\langle a\rangle|^2]-2|\langle a^2\rangle-\langle a\rangle^2|}/4$ is an equivalent derived minimization. Q is undefined at zero mean; no hidden photon-number cutoff is imposed.
The MSA state is $|\chi_k^{(i)}\rangle=D[\beta_{ii}^{(k)}]\exp[-\langle W_k^2\rangle_{\rm el}/2]|0\rangle$, $W_k=\int dt,A_{Q,k}(t)[j_H(t)-\langle j_H(t)\rangle]$ and $\beta_{mn}^{(k)}=-ig_0\omega_k^{-1/2}\int dt,j_{mn}(t)e^{i\omega_kt}$ for a fixed real polarization (Eqs. (28)–(31)). The approximation neglects $[\dot W,W]$ and retains second-order $g_0$ state content. The covariance is $\langle j_H(t)j_H(t\prime)\rangle-\langle j_H(t)\rangle\langle j_H(t\prime)\rangle=\sum_{m\ne i}j_{im}(t)j_{mi}(t\prime)$ (Eq. (33)); it is not reducible to the diagonal-current spectrum.
Coverage bookkeeping counts a pulse×observable member only when all three commissioned routes are valid and mutually comparable for that member. O1: at least one of six. O2: at least three including both pulses, at least two observable types and Q or optimized squeezing. O3: all six. Named-family I1/I2/I3 independently require both corresponding pulse members and candidate identification; no generic coverage milestone supplies missing spectra, Q, squeezing or reasoning.
Preserve route-local credit: a ground-transition product/normalization defect does not automatically erase correct local-Markov or MSA work. Correct executable F2 identity can remain credited when the explanation/mapping required by M7 is incomplete. Source Eq. (20) is not silently promoted to a fourth required computation.