cases/case49/sources.md

# Scientific sources - Case49

## References

Kunlong Liu, Yidian Tian, and Peng-Cheng Li. 2026. *Time-resolving the birth of photoelectrons in strong-field ionization with an isolated attosecond pulse*. *Physical Review A*, 113, 053110. [Source](https://doi.org/10.1103/x6r8-yg7s).

One hydrogen counterclockwise circular-pulse direct-SFA internal reconstruction, with three coherent spectra, independent complex comparison and physical-time/localized content. Source TDSE/Coulomb/scattering-state, angular-delay, SST and multiple-regime results are not image gold or required validation.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Liu, Tian and Li (2026), author manuscript pp. 1–3, Eqs. (1)–(6); Appendix A, p. 7, Eqs. (A2)–(A10). Pulse, coherence, common final time and continuum-normalization identities. Direct-SFA production is the visible task model; the source TDSE solver and its pictures are not direct-SFA validation. |
| F2 | Author manuscript p. 2, Eqs. (3)–(4) and Hilbert-transform discussion; p. 3, Eqs. (6)–(7); Appendix B, pp. 7–8, Eqs. (B1)–(B2) and final translation paragraph. Three-spectrum complex recovery, analytic branch, known full reference phase and physical-time mapping. Exact reference-phase restoration is a task-required refinement of the source instant-reference approximation. |
| O1, O2, I2 | Author manuscript p. 2, Eq. (2), Fig. 1(b); p. 3, Eq. (6), Fig. 2(b1–b3). Target/reference/coherent spectral identities and cross term. Two-of-three and three-of-three are task coverage milestones, not claims that source TDSE images must be reproduced. |
| O3, O4, I1 | Author manuscript p. 2, Eqs. (3)–(4), Fig. 1(c) and global-phase paragraph; p. 3, Fig. 2(c1–c3). Magnitude and energy-dependent phase comparisons motivate recovery verification. Candidate direct-SFA reference must be independently retained, not copied from the recovered array; no source overlap threshold is imposed. |
| O5, I3, M4 | Author manuscript pp. 3–4, Eq. (7), Fig. 2(d1–d3), BTD interpretation paragraphs. Birth-time transform and periodic ionization content. Source angular delays and cross-regime maxima are not direct-SFA targets; physical time may not be recentered to align peaks. |
| O6, M5 | Author manuscript pp. 4, 6–8, Fig. 3(a1–a3), Fig. 5, Appendix B Eqs. (B1)–(B2) and final time-translation paragraph. Localized energy-time support and resolution tradeoff. Source Gabor examples provide transform identity only; SST Fig. 6 is excluded and no source window width is mandatory. |
| M1, M2 | Author manuscript pp. 2–3, Eqs. (2)–(6), analytic-signal and global-phase discussion. Task-defined internal-recovery decision from candidate spectra and independent complex comparison. The cross term retains relative phase; a wrong branch or energy-dependent reference phase cannot be repaired by global alignment. |
| M3 | Author manuscript p. 3, Eqs. (5)–(7), periodic burst discussion; Fig. 2(b1–b3). Fourier entailment links ATI-scale energy spacing to cycle-scale direct emission. It does not import source Coulomb/angular-delay or multiple-regime mechanisms. |
| M6 | Author manuscript pp. 3–4, Eqs. (5)–(7); Appendix B, pp. 7–8. Task-defined synthesis of complex recovery, periodic time content and localized support. O5 independent-comparator credit is not duplicated as another hard gate on independently established time inference. |
| M7, V1 | Author manuscript Appendix A, pp. 6–7, Eqs. (A1), (A6)–(A10); Appendix B, pp. 7–8. Explicit model distinction: source uses Coulomb TDSE/scattering projection and may show SST; the commissioned result is direct-SFA internal reconstruction, not equivalence to those methods. |
| V2 | Author manuscript pp. 2–3, Eq. (3), Eq. (6) and reference-support paragraph; Appendix B, pp. 7–8, physical-time translation. Task-defined conditional risk from reference nodes/phase, carrier sampling, common-time binding or circular verification. No fixed reference mask, overlap threshold or transform window is prescribed. |

## Formula and convention notes

One admissible direct-SFA convention is $M(\mathbf p)=-i\int dt\,\mathbf d[\mathbf p+\mathbf A(t)]\cdot\mathbf E(t)e^{-iS(\mathbf p,t)}$, $S=\int_t^{t_f}\{[\mathbf p+\mathbf A(t\prime)]^2/2+I_p\}dt\prime$. This is the task-defined production model, not a formula claimed to be the source TDSE solver. The energy-normalized packet is $\psi(E,\Omega;t_f)=(2E)^{1/4}\psi(\mathbf p=\sqrt{2E},\Omega;t_f)$ (source Eq. (A8)); direction $\Omega$, normalization and common $t_f$ are preserved for both pulses.

$P_0=|\psi_0|^2$, $P_x=|\psi_x|^2$, $P=|\psi_0+\psi_x|^2$ and $W=(P-P_0-P_x)/(2\sqrt{P_x})=\operatorname{Re}[\psi_0e^{-i\arg\psi_x}]$ on the reference-supported domain (Eqs. (2)–(6), exact algebra). The correct analytic-signal branch recovers the complex carrier; multiply by the full known $e^{i\arg\psi_x(E)}$, not a constant reference phase. For the stated positive delay and source Fourier convention a positive-support analytic signal requires the conjugate branch. Only one overall constant phase is free.

$B_\Omega(\tau_0)=|(2\pi)^{-1/2}\int_0^\infty\psi(E,\Omega;t_f)e^{-i(\tau_0-t_f)E}dE|^2$ (Eq. (7), common-final-time generalization). A suitable local representation is $G(E,\tau)=|\int_0^\infty dE\prime\,\psi(E\prime;t_f)\eta(E\prime-E)e^{-i\tau(E\prime-E)}|^2$, with $\tau=\tau_0-t_f$ and declared localization window $\eta$ (Appendix B). The pulse-defined physical birth time is $\tau_0=\tau+t_f$, not a peak-recentered axis.

For cycle-separated coherent direct bursts, the energy/time Fourier relation gives ATI spacing $\Delta E\simeq\omega$ and burst spacing $T=2\pi/\omega$ in atomic units. This is a source-supported physical entailment, not an exact envelope/angle-delay prediction. Localization broadens conjugate time/energy resolution; it must be adequate for the features claimed, without a prescribed source width or SST algorithm.


## Scope of the source anchors

Source examples verify physical identity and robust structure; their numerical samples, plotting conventions and panel layout are not compulsory candidate deliverables. Reporting, reconstructibility, evidence attribution, conditional validation and Research Closure are examination-defined obligations, not additional results asserted by the article. A derivation or physical entailment is identified as such rather than presented as a directly printed source formula.