# Grading Rubric — Case44 Interaction-dressed two-body diffraction under two lattice wave numbers ## 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 stated Hamiltonian, oscillator units, exchange-symmetric sector, g-specific interacting lattice-free ground states, lattice quench/time, three interactions and two lattice wave numbers. **Conditions for credit** - Instantiate the stated Hamiltonian, oscillator units, exchange-symmetric sector, g-specific interacting lattice-free ground states, lattice quench/time, three interactions and two lattice wave numbers. **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; 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Becker, Koutentakis and Schmelcher (2025), article pp. 2–4, Eqs. (1)–(7), (13)–(22). Exchange-symmetric contact-interacting initial states, CM/relative coupling, quench and exact correlated propagation. Independent-particle or mean-field replacements are not equivalent. **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 or adequately justify exact correlated two-body propagation retaining interaction-dressed states and lattice-induced centre-of-mass/relative coupling. **Conditions for credit** - Derive or adequately justify exact correlated two-body propagation retaining interaction-dressed states and lattice-induced centre-of-mass/relative coupling. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Judge this scientific obligation directly; no fixed prerequisite award. - Preserve independently valid evidence for F1, F3; 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Becker, Koutentakis and Schmelcher (2025), article pp. 2–4, Eqs. (1)–(7), (13)–(22). Exchange-symmetric contact-interacting initial states, CM/relative coupling, quench and exact correlated propagation. Independent-particle or mean-field replacements are not equivalent. **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** Define same-propagated-state reconstructible laboratory-frame normalized one-body density and momentum distribution with Fourier sign, units and common normalization. **Conditions for credit** - Define same-propagated-state reconstructible laboratory-frame normalized one-body density and momentum distribution with Fourier sign, units and common normalization. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Judge this scientific obligation directly; no fixed prerequisite award. - Preserve independently valid evidence for F1, F2; 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 4–5, Eqs. (23)–(25), (29)–(31); Appendix C. Laboratory one-body reduced density and its Fourier momentum distribution from the same state, each normalized to one particle. **Research Closure relation** Central-object role: RC0 only if the central physical identity is absent or materially invalid, independently of this atom's award. ### O1 — O — 0.5 points **Scientific requirement** Present k_lat=4/ell normalized real-space density for g=−5,0,+5, resolving attractive localization, intermediate zero interaction and repulsive broadening. **Conditions for credit** - Present k_lat=4/ell normalized real-space density for g=−5,0,+5, resolving attractive localization, intermediate zero interaction and repulsive broadening. - The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Necessary scientific premises: (F1 AND F2 AND F3). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for O2, O3, O4; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 10–11, Fig. 7(a,c), caption and interaction-state paragraphs. Smaller-wave-number real-space localization/extension across attractive, zero and repulsive interactions. **Research Closure relation** Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object. ### O2 — O — 0.5 points **Scientific requirement** Present the same k4 normalized momentum family, resolving attractive broad/flat and repulsive sharp/high-contrast structure, integer 2hbar k_lat locations and accessible weak second-order evidence. **Conditions for credit** - Present the same k4 normalized momentum family, resolving attractive broad/flat and repulsive sharp/high-contrast structure, integer 2hbar k_lat locations and accessible weak second-order evidence. - The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Necessary scientific premises: (F1 AND F2 AND F3). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for O1, O3, O4; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 11, Fig. 7(b,d), caption and momentum-sharpening paragraph. Smaller-wave-number momentum widths, contrast and integer orders; weak-order evidence is assessed only at credible resolution. **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 independently recalculated k_lat=6/ell normalized real-space density for all three interactions, resolving denser modulation and material interaction ordering. **Conditions for credit** - Present independently recalculated k_lat=6/ell normalized real-space density for all three interactions, resolving denser modulation and material interaction ordering. - The physical output object is valid and reconstructible (object_valid); the candidate presents its material structure with adequate domain, resolution and dynamic range (presentation_adequate); and the inference stays within resolved support (claim_adequate). All three are necessary; record them separately. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Necessary scientific premises: (F1 AND F2 AND F3). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for O1, O2, O4; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 11–12, Fig. 8(a,c), caption and changed-lattice-spacing paragraph. Fresh larger-wave-number real-space family and denser spatial modulation, not a relabelled smaller-wave-number dataset. **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 independently recalculated k6 normalized momentum distributions for all three interactions, with shifted first-order identity, interaction dependence and evidence-supported second-order behaviour, without a hidden predetermined suppression direction. **Conditions for credit** - The k6 Hamiltonian is freshly propagated for all three interactions and yields reconstructible normalized momentum distributions with shifted first-order identity. - Second-order statements match actual credible evidence and interaction dependence; no predetermined suppression direction or pointwise attractive-tail tolerance is imposed. - 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). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for O1, 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 11–12, Fig. 8(b,d), caption and larger-wave-number discussion. Fresh shifted first-order momentum family and actual resolved higher-order behavior. The source suppression statement is contextual; the effective task does not impose a predetermined second-order direction. **Research Closure relation** Indispensable-component role: RC1 only if the required physical object is absent or materially invalid. Presentation or claim-adequacy failure alone does not imply an absent or invalid object. ### I1 — I — 0.5 points **Scientific requirement** Identify the complementary attractive/localized/broad-momentum, intermediate and repulsive/extended/sharp-momentum ordering from the candidate's paired k4 evidence. **Conditions for credit** - Identify the complementary attractive/localized/broad-momentum, intermediate and repulsive/extended/sharp-momentum ordering from the candidate's paired k4 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: (O1 AND O2). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for I2, I3; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 11, Fig. 7(c,d), paragraphs on overlap, localization, momentum spread and phase gradients. Paired real/momentum evidence and interaction-dressed initial-state mechanism. Attractive localization/broad momentum and repulsive extension/sharp peaks are complementary, not independent arbitrary rescalings. **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 material interaction ordering in fresh paired k6 evidence and report denser modulation and momentum contrast only at the actual resolved scale. **Conditions for credit** - Identify material interaction ordering in fresh paired k6 evidence and report denser modulation and momentum contrast only at the actual resolved scale. **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 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 11–12, Fig. 8(c,d), interaction-ordering and lattice-spacing paragraphs. Interaction ordering and denser modulation in the independently recalculated larger-wave-number case, at the actual resolved scale. **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 first-order location shifts between independent k4/k6 momentum families and separate wave-number transfer from interaction ordering; identify second-order behaviour from evidence rather than a presupposed direction. **Conditions for credit** - Identify first-order location shifts between independent k4/k6 momentum families and separate wave-number transfer from interaction ordering; identify second-order behaviour from evidence rather than a presupposed direction. **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 AND O4). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for I1, I2; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 4, momentum-transfer paragraph after Eq. (22); pp. 11–12, Figs. 7(d), 8(d). Physical entailment from the standing-lattice Fourier components gives momentum transfer in integer multiples of 2 hbar k_lat; changed wave number shifts the order locations. **Research Closure relation** Supporting criterion; no independent overall RC gate. Its actual scientific facts can still be necessary to a required claim. ### M1 — M — 0.5 points **Scientific requirement** Explain the attractive branch's narrow density and broad/flat diffraction through interaction-dressed overlap/localization and larger initial momentum spread. **Conditions for credit** - Explain the attractive branch's narrow density and broad/flat diffraction through interaction-dressed overlap/localization and larger initial momentum spread. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for M2, M4, M5; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 11, Fig. 7(c,d), paragraphs on overlap, localization, momentum spread and phase gradients. Paired real/momentum evidence and interaction-dressed initial-state mechanism. Attractive localization/broad momentum and repulsive extension/sharp peaks are complementary, not independent arbitrary rescalings. **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 repulsive broad density and narrow/high-contrast momentum peaks through reduced overlap and spatial extension, placing zero interaction at the evidence-supported intermediate position. **Conditions for credit** - Explain repulsive broad density and narrow/high-contrast momentum peaks through reduced overlap and spatial extension, placing zero interaction at the evidence-supported intermediate position. **Applicability and N/A** - Applies to every assessable submission; absence or failure is scored 0, not N/A. **Scientific dependencies and independent credit** - Necessary scientific premises: I1. This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for M1, M4, M5; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 11, Fig. 7(c,d), paragraphs on overlap, localization, momentum spread and phase gradients. Paired real/momentum evidence and interaction-dressed initial-state mechanism. Attractive localization/broad momentum and repulsive extension/sharp peaks are complementary, not independent arbitrary rescalings. **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** Combine paired real/momentum signals from both wave-number families with the interaction-state account to explain how the dressed initial state reshapes short-pulse diffraction. **Conditions for credit** - Combine paired real/momentum signals from both wave-number families with the interaction-state account to explain how the dressed initial state reshapes short-pulse diffraction. **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 M1 AND M2). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for M6; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. - Judge actual cross-family synthesis here; the local denser-modulation identification in I2 is not a hard prerequisite for this mechanism. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 11–12, interaction-state and larger-wave-number mechanism paragraphs, Figs. 7(c,d), 8(c,d). Task-defined synthesis of independently established interaction-state and wave-number chains. A weak local modulation identification does not automatically erase the interaction mechanism. **Research Closure relation** Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award. ### M4 — M — 0.5 points **Scientific requirement** Connect standing-lattice momentum transfer to integer 2hbar k_lat locations and explain why changing wave number moves the first-order family rather than merely relabelling data. **Conditions for credit** - Connect standing-lattice momentum transfer to integer 2hbar k_lat locations and explain why changing wave number moves the first-order family rather than merely relabelling data. **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, M5; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 4, momentum-transfer paragraph after Eq. (22); pp. 11–12, Figs. 7(d), 8(d). Physical entailment from the standing-lattice Fourier components gives momentum transfer in integer multiples of 2 hbar k_lat; changed wave number shifts the order locations. **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 evidence-supported k6 second-order behaviour through recoil-energy spacing, pulse area and short-time accessibility; restrict claims to actual credibility of the slower-converging attractive branch without presupposing suppression. **Conditions for credit** - The candidate connects its own shifted-order/second-order evidence to recoil spacing, pulse area and short-time accessibility. - The explanation is restricted to the actual resolved credibility, especially the attractive branch, rather than assuming a hidden higher-order direction. **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, M4; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 8, Eq. (39) and recoil discussion; pp. 11–12, Figs. 7(d), 8(d); p. 5, Eq. (33). Recoil spacing, pulse area and finite-time access explain the candidate-established order behavior. The source direction is not a hidden mandatory prediction, and no fixed suppression factor is required. **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** Use independent k4/k6 calculations and shifted-order/recoil-accessibility identity to conclude how wave number reshapes transfer, excluding mere redraw/rescaling. **Conditions for credit** - Use independent k4/k6 calculations and shifted-order/recoil-accessibility identity to conclude how wave number reshapes transfer, excluding mere redraw/rescaling. **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 AND M4). This identifies facts, not prerequisite half-point awards. - Preserve independently valid evidence for M3; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. - M5's detailed second-order/recoil chain retains separate credit and is not a hard gate on this main wave-number-transfer conclusion. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 8, Eq. (39) and recoil discussion; pp. 11–12, Figs. 7(d), 8(d); p. 5, Eq. (33). Recoil spacing, pulse area and finite-time access explain the candidate-established order behavior. The source direction is not a hidden mandatory prediction, and no fixed suppression factor is required. **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** Integrate interaction-state and independent wave-number/recoil chains to answer how and why contact interaction and lattice wave number jointly reshape the specified two-body diffraction. **Conditions for credit** - Integrate interaction-state and independent wave-number/recoil chains to answer how and why contact interaction and lattice wave number jointly reshape the specified two-body diffraction. **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: (M3 AND M6). 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 11–12, interaction-state and larger-wave-number mechanism paragraphs, Figs. 7(c,d), 8(c,d). Task-defined synthesis of independently established interaction-state and wave-number chains. A weak local modulation identification does not automatically erase the interaction mechanism. **Research Closure relation** Principal-mechanism role: RC2 only if the specified principal physical inference is absent or incorrect, independently of this atom's award. ### M8 — M — 0.5 points **Scientific requirement** Distinguish direct six-case paired-observable evidence from interpretation conditional on the one-dimensional two-body deep-lattice short-pulse model; do not claim many-body, experimental or long-time universality. **Conditions for credit** - Distinguish direct six-case paired-observable evidence from interpretation conditional on the one-dimensional two-body deep-lattice short-pulse model; do not claim many-body, experimental or long-time universality. **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 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 2–4, model/sector/quench; pp. 12–14, sudden-comparison discussion. Six-case exact-two-body short-pulse domain and distinction between direct evidence and conditional interpretation; long-time fidelity is outside scope. **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 exchange-symmetric one-dimensional exact-two-body, deep-lattice, short-pulse, six-case paired-one-body-observable domain, distinct from many-body/absolute experimental calibration/long-time fidelity. **Conditions for credit** - State the exchange-symmetric one-dimensional exact-two-body, deep-lattice, short-pulse, six-case paired-one-body-observable domain, distinct from many-body/absolute experimental calibration/long-time 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 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 converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article pp. 2–4, model/sector/quench; pp. 12–14, sudden-comparison discussion. Six-case exact-two-body short-pulse domain and distinction between direct evidence and conditional interpretation; long-time fidelity is outside scope. **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 basis, propagation, same-state binding, normalization, unresolved orders, recycled wave-number data, anomalies or claims threaten a central conclusion, provide discriminating evidence or a proportionate restriction; otherwise record N/A. **Conditions for credit** - If actual basis, propagation, same-state binding, normalization, unresolved orders, recycled wave-number data, anomalies or claims threaten a central conclusion, provide discriminating evidence or a proportionate restriction; otherwise record N/A. - When triggered, award only if claim-relevant discriminating evidence resolves the threat or an evidence-based restriction makes it immaterial; merely acknowledging an unresolved material threat is not positive validation credit. With no trigger, record N/A and retain the fixed 0.5 without claiming performed validation. **Applicability and N/A** - Applies only when the submitted method, data, anomaly or conclusion creates the concrete material risk described here. If untriggered, record not_applicable and retain the fixed 0.5 contribution; this is not evidence that validation was performed, and does not block RC4. **Scientific dependencies and independent credit** - Judge this scientific obligation directly; no fixed prerequisite award. - Preserve independently valid evidence for V1; failure here does not automatically erase those criteria. - Propagate only an absent or incorrect scientific fact necessary for this obligation, not an upstream award or an unrelated display defect. Preserve independently valid work. Examiner reconstruction cannot supply candidate presentation, identification, inference or validation. **Accepted equivalent evidence** - Accept converged DVR, sparse energy-basis or algebraically equivalent exact correlated two-body routes retaining the interaction-dressed state and lattice-induced CM/relative coupling. Independent-particle or mean-field replacement is not equivalent. - Accept mapped Fourier signs, units and common normalization, widths, contrasts, order-resolved integrals, fitted scales, interval/ROI comparisons and equivalent reconstructible invariants. No fixed cutoff, projection size, RK solver, timestep, mesh, peak heights, tail values or source layout. **Scientific source locations** - Article p. 15, Appendix A, Figs. 10–11; pp. 4–5, same-state observable definitions. Bounded short-time basis/observable credibility. Conditional testing does not mandate a basis size or use unresolved long-time fidelity as gold. **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): Instantiate the stated Hamiltonian, oscillator units, exchange-symmetric sector, g-specific interacting lattice-free ground states, lattice quench/time, three interactions and two lattice wave numbers. Derive or adequately justify exact correlated two-body propagation retaining interaction-dressed states and lattice-induced centre-of-mass/relative coupling. Define same-propagated-state reconstructible laboratory-frame normalized one-body density and momentum distribution with Fourier sign, units and common normalization. ### RC1 - The central object exists, but a required indispensable component is absent or materially invalid. A failed presentation or overextended claim alone does not establish object absence or invalidity. - (O1 AND O2): Present k_lat=4/ell normalized real-space density for g=−5,0,+5, resolving attractive localization, intermediate zero interaction and repulsive broadening. Present the same k4 normalized momentum family, resolving attractive broad/flat and repulsive sharp/high-contrast structure, integer 2hbar k_lat locations and accessible weak second-order evidence. - (O3 AND O4): Present independently recalculated k_lat=6/ell normalized real-space density for all three interactions, resolving denser modulation and material interaction ordering. Present independently recalculated k6 normalized momentum distributions for all three interactions, with shifted first-order identity, interaction dependence and evidence-supported second-order behaviour, without a hidden predetermined suppression direction. ### RC2 - Required objects are scientifically valid, but a required principal mechanism or state-dependence inference is absent or incorrect. Reference evidence cannot provide omitted candidate reasoning. - M3: Combine paired real/momentum signals from both wave-number families with the interaction-state account to explain how the dressed initial state reshapes short-pulse diffraction. - M6: Use independent k4/k6 calculations and shifted-order/recoil-accessibility identity to conclude how wave number reshapes transfer, excluding mere redraw/rescaling. - M7: Integrate interaction-state and independent wave-number/recoil chains to answer how and why contact interaction and lattice wave number jointly reshape the specified two-body diffraction. ### 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 basis, propagation, same-state binding, normalization, unresolved orders, recycled wave-number data, anomalies or claims threaten a central conclusion, provide discriminating evidence or a proportionate restriction; otherwise record N/A. - Before assigning RC3, establish all six elements: the concrete failure mode; its origin in the submitted method/data/conclusion; a fair recognizable scientific risk; the affected central conclusion; why the submitted evidence cannot discriminate or exclude it; and functionally equivalent remedies. Evidence-based restriction may remove unsupported breadth, but cannot repair an incorrect signal, absent component or missing mechanism. ### RC4 - Every indispensable and co-primary claim has all its nonempty required role slots closed; supporting claims do not independently determine overall RC. Preserve AND/OR branch logic shown above and use scientific gate outcomes, not atomic scores. - No triggered material risk remains unresolved. Untriggered risk criteria recorded N/A do not block RC4 and do not establish validation evidence. - Classify descriptively by the most severe applicable failed required role (lowest RC), without score ceilings or target distributions. A score/RC tension prompts scientific review, not forced adjustment. ## Applicability and Equivalent Evidence - Use r=(x1−x2)/sqrt(2), R=(x1+x2)/sqrt(2), H_CM=−hbar² partial_R²/(2m)+m omega²R²/2 and H_rel=−hbar² partial_r²/(2m)+m omega²r²/2+g delta(r). The lattice is U0[cos(sqrt(2)k_lat r)cos(sqrt(2)k_lat R)+1], which couples relative and CM motion. - An equivalent even-sector interacting basis may use Gamma(1/2−epsilon_n)/Gamma(−epsilon_n)=−g/2 and E_(2n)=hbar omega(2epsilon_n+1/2) in oscillator units. Initialize each g in its own lattice-free interacting ground state, then apply the stated quench and propagate to t0. - The one-body reduced density matrix of the laboratory-frame two-body state defines n(x,t); its Fourier representation defines n(k,t). For these symmetric cases each one-particle distribution is even and normalized to unity, or an explicitly equivalent mapped convention. Both observables must derive from the same propagated state. - For k_lat=4/ell and g=−5,0,+5, attraction localizes density and broadens/flattens momentum features; zero interaction is intermediate; repulsion broadens density and sharpens momentum contrast. Diffraction transfers occur at integer 2hbar k_lat. - The k_lat=6/ell family must be a fresh changed-Hamiltonian calculation, with denser spatial modulation and shifted first-order family; not rescaled/relabelled k4 data. Judge second-order behaviour from the actual resolved evidence and recoil/pulse/time-domain account; do not impose a hidden suppression direction as a duty. - The attractive higher-wave-number momentum branch can converge more slowly; accept structural/interval evidence proportionate to its actual reliability rather than strict pointwise tolerance. M3's joint state mechanism does not hard-gate on I2's local denser-modulation identification; M6 retains independent main transfer credit when M5's second-order chain is incomplete. - Separate interaction-dressed initial overlap/width/momentum spread from changed recoil spacing and order accessibility. Scope is the six-case one-dimensional exchange-symmetric deep-lattice short-pulse two-body benchmark. The source long-time sudden/exact fidelity map is not a closed gold object or hidden requirement. - 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. - The laboratory Hamiltonian is $H=\sum_{i=1}^2[p_i^2/(2m)+m\omega^2x_i^2/2+U_0\cos^2(k_{\rm lat}x_i)]+\sqrt2g\delta(x_1-x_2)$ (Eq. (1)). With $R=(x_1+x_2)/\sqrt2$, $r=(x_1-x_2)/\sqrt2$ and $\ell=\sqrt{\hbar/(m\omega)}$, $H_{\rm rel}=p_r^2/(2m)+m\omega^2r^2/2+g\delta(r)$ and $H_{\rm lat}=U_0[1+\cos(\sqrt2k_{\rm lat}r)\cos(\sqrt2k_{\rm lat}R)]$. The latter couples CM and relative motion; post-quench factorization is not generally valid. - The initial state for each $g$ is $|\Psi_g(0)\rangle=|0_{\rm CM}\rangle|0_{{\rm rel},g}\rangle$, and $|\Psi_g(t_0)\rangle=e^{-iH(g,k_{\rm lat})t_0/\hbar}|\Psi_g(0)\rangle$. An equivalent Busch or numerical exact correlated representation is accepted in the exchange-even sector; it must not reuse the same interacting ground state for all three couplings. - For a unit-normalized two-body wavefunction, $\rho^{(1)}(x,x')=\int dx_2\,\Psi(x,x_2)\Psi^*(x',x_2)$, $n(x)=\rho^{(1)}(x,x)$ and $n(k)=(2\pi)^{-1}\int dx\,dx'\,e^{-ik(x-x')}\rho^{(1)}(x,x')$ (Eqs. (24)–(25), (29)). Thus $\int n(x)dx=\int n(k)dk=1$; physical momentum is $p=\hbar k$. A total-density normalization of two is equivalent only with an explicit common conversion; the unrelated source two-body-density prefactor is not imported. - The lattice contains Fourier transfers $\Delta p=\pm2\hbar k_{\rm lat}$, with order locations $p_n=2n\hbar k_{\rm lat}$ and free recoil scale $E_n=p_n^2/(2m)=4n^2E_R$, $E_R=\hbar^2k_{\rm lat}^2/(2m)$ (physical entailment, recoil discussion and Eq. (39)). Pulse area $K=\hbar^{-1}\int U_0(t)dt$ is Eq. (33). Larger recoil spacing alone does not license imposing an unreported numerical second-order suppression direction; judge the actual short-time calculation and supported scope. - O4/I3/M5 require the actual credible second-order behavior and a physically consistent explanation, not guaranteed k6 suppression or any predetermined hidden direction. A slower-converging attractive tail is not automatically wrong; preserve independently valid density, first-order and other-interaction evidence.