# Scientific sources - Case43

## References

Xiaodan Mao, Feng He, and Pei-Lun He. 2026. *Anomalous radiation pressure in strong-field ionization driven by quantum light*. *arXiv*, 2608.15859. [Source](https://arxiv.org/abs/2608.15859v1). Version 1.

Photon-traced H/Na coherent/BSV scaling plus one reduced low-field Na neighboring conditioned pair. Full channel scans, high-field reversal and exact source line shapes are excluded; common-object cross-term alternatives remain accepted.

## Criterion-to-source crosswalk

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1 | Mao, He and He (2026), main pp. 1–3, Eqs. (1)–(7). Photon-traced phase-space probability, direct light-front amplitude, common intensity convention and target-dependent offset. |
| F2 | Main pp. 3–4, Eqs. (8)–(10); SI pp. 9–11, Eqs. (S59)–(S82). Fixed-number amplitude projection and opposite same-radius field paths; finite-pulse two-point production is the visible bounded task approximation, not the full phase-space integral. |
| O1, I1, M1, M2 | Main pp. 2–3, Fig. 1(a,b), Eqs. (5)–(7); SI pp. 7–8, Eqs. (S46)–(S58). Coherent linear versus BSV two-thirds field law, nonlinear amplitude selection, target prefactor and offset. Leading-order curves are not pointwise source-data gold. |
| O2, I2 | Main p. 4, Fig. 3(b), low-effective-intensity paragraph. Neighboring conditioned-channel relative-yield ordering at low field, not the excluded full-channel/high-field reversal. |
| O3, I3 | Main p. 4, Fig. 3(a), low-effective-intensity paragraph. Positive mean-longitudinal-momentum ordering for the same channel pair; adequate tails and common measure remain necessary. |
| O4, I4 | Main pp. 4–5, Fig. 4(b1,b2), Eqs. (10)–(12); SI pp. 11–14, Eqs. (S82)–(S101). Low-field complementary spectral redistribution and sign-sensitive cross term. Fig. 4(a1,a2) is the unrequired high-field contrast; exact peak widths/heights are not targets. |
| M3, M6 | Main p. 3, Eq. (8), completeness paragraph; p. 4, Eq. (10); SI p. 11, Eqs. (S82)–(S85). Independent conceptual distinction between photon tracing and fixed-number phase retention. It does not backfill correct numerical action, phase or candidate-identified redistribution. |
| M4, M5 | Main pp. 4–5, Eq. (10), Fig. 4(b1,b2) and low-energy-weight interpretation paragraph. Connect the same conditioned redistribution to both integrated yield and mean momentum; theoretical parity alone is not candidate-computed quantitative evidence. |
| M7 | Main pp. 3–5, Eqs. (8)–(12), Figs. 3–4 and conclusion. Task-defined bounded relation between regimes; the one low-field pair cannot establish a complete channel or high-field survey. |
| V1 | Main p. 4, Eq. (10), Fig. 4(b1,b2); SI pp. 11–14, Eqs. (S82)–(S101). Matched incoherent deletion is one cross-term discriminator, not an exclusive control. A phase intervention or explicit term reconstruction must preserve field points, overlaps, radius, parameters, domain, normalization and observable. |
| V2 | Main pp. 2–5, Eqs. (3)–(4), (7)–(12); SI pp. 9–14, conditioned construction. Task-defined conditional threats from wrong phase/action, mean tails, offset fits, channel/radius or normalization, without a prescribed validation recipe. |
| V3 | Main pp. 2–5, leading-order direct-ionization qualification, Fig. 3 and Fig. 4 captions. Direct-model low-field neighboring-pair scope; full scan, reversal, width study and continuum/over-barrier extensions are not commissioned calculations. |

## Formula and convention notes

Photon tracing uses $P(\mathbf p)=\int dA\,e^{-A^2/(2A_{\rm rms}^2)}|M_A(\mathbf p)|^2/(\sqrt{2\pi}A_{\rm rms})$, $I=2\omega^2A_{\rm rms}^2$ (Eqs. (1)–(2)). The leading adiabatic circular saddle is $A_s\simeq\pm\sqrt{2I_p}[4A_{\rm rms}^2/(3\omega)]^{1/3}$, so $A_s^2\propto I^{2/3}$, not a linear law.

The leading mean shifts are $\langle p_z\rangle_{\rm coh}\simeq I/(4\omega^2c)+I_p/(3c)$ and $\langle p_z\rangle_{\rm BSV}\simeq\frac{I_p}{\omega^2c}\left(\frac{2}{3}\right)^{2/3}I^{2/3}+I_p/(3c)$ (Eq. (7)). $I_p$ is target binding energy; $\omega$ carrier frequency, $c$ light speed and $I$ the declared atomic-unit mean intensity. Exponents concern the field-dependent part after offset handling; these are leading-order adiabatic relations, not exact pointwise spectra.

For fixed final photon number $n$, $P_n(\mathbf p)=|\int d^2\alpha\,\langle n|\alpha\rangle\langle\alpha|\xi\rangle M_\alpha(\mathbf p)/\pi|^2$. In the bounded two-point task approximation, write $a_\pm=C_\pm M_\pm$, $P_n=|a_++a_-|^2$, $P_n^{\rm inc}=|a_+|^2+|a_-|^2$ and $P_n-P_n^{\rm inc}=2\operatorname{Re}(a_+a_-^*)$. Here $C_\pm$ contain the exact Fock/squeezed-vacuum overlaps at the common-radius opposite points. Computing the cross term requires the full finite-pulse complex phases, not probability addition.

With $\widetilde p_z=p_z-(p^2/2+I_p)/c$, $\Lambda=1-\widetilde p_z/c$, $\kappa=\sqrt{2I_p}$, the source leading branch phase is $\Delta\phi_n=\pi(\widetilde p^2+\kappa^2+A_n^2)/(2\omega\Lambda)$, $A_n=2\epsilon_V\sqrt n$ (Eqs. (9)–(12)). The term $(-1)^n\cos\Delta\phi_n$ distinguishes neighboring parity channels; this analytic phase is mechanism support, not a substitute for the task full-pulse amplitudes. $Y_n=\int P_n d^3p$ and $\langle p_z\rangle_n=\int p_zP_n d^3p/Y_n$ must come from the same arrays and measure.


## 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.

The accompanying Supplemental Material to Mao, He and He (2026) supplies the explicitly cited S-equations; [article and supplementary-file entry](https://arxiv.org/abs/2608.15859v1). SI page numbers are distinguished from main-article pages.