cases/case21/sources.md

# Scientific sources - Case21

## References

I. A. Aleksandrov, M. A. Dorodnyi, and E. D. Akimkina. 2026. *Spectral fringes without subcycles in Schwinger pair production and Dirac materials*. *arXiv*, 2605.19086. [Source](https://arxiv.org/abs/2605.19086v1). Version 1.

Exact scalar-QED and gapped-Dirac spectra under smooth single-lobe fields, loss of single-contribution dominance and qualitative cross-system interference transfer. Approximate mechanism models, broader maps and far-field robustness checks have their stated conditional scope.

## Criterion-to-source crosswalk

Locations refer to the article or named preprint/manuscript page, not a figure-file index. Source examples establish scientific identity and robust structure; the task defines required reporting, case coverage and risk-triggered validation. No original source images are reproduced.

| Criteria | Source locations, scientific use and limits |
|---|---|
| F1, O1, O2, I1, M1 | main article p. 2, Fig. 1(a,b), Eqs. (1)-(2) and exact-mode paragraph; Supplemental pp. i-ii, Eqs. (S1)-(S12) Gaussian versus deformed exact scalar occupations are different production objects; display multipliers and WKB predictions do not replace unscaled exact occupations. |
| F2, O4, I4 | main article p. 3, Eqs. (6)-(7), Fig. 3(a,b); Supplemental Eqs. (S14)-(S48) Filled-valence gapped-Dirac dynamics is read as post-pulse conduction occupation, with the specified gap and durations. |
| F3, O3, I2, M2 | main article p. 2, Eqs. (2)-(5), Fig. 2 and its caption; Supplemental Eqs. (S1)-(S13) The deformed-pulse imaginary-axis turning point loses effective sole dominance near the finite critical integral; onset evidence is distinct from a complete active-weight demonstration. |
| I3 | main article p. 2, Fig. 2 caption and semiclassical-mechanism paragraphs; Supplemental Eqs. (S1)-(S13) Several physically active comparable contributions with relative actions are needed to identify interference; merely finding multiple zeros does not establish comparable weights. |
| M3 | main article p. 2, Fig. 1(b), Fig. 2 and Eqs. (3)-(5) Connect the dominance transition to central suppression, side extrema and nonadiabatic dependence in the scalar signal. |
| M4, M5, M6 | main article pp. 3-4, Fig. 3(a,b), Eqs. (6)-(7), and Fig. 4(a,b) The pulse-family/nonadiabatic crossover transfers qualitatively to Dirac carriers, not as equality of yields or critical scales. Fig. 4 is optional broad-domain context; its visibility scalar is not an exact target. |
| M7, V1 | main article p. 4, Discussion and outlook and preceding zero-area paragraph The demonstrated result is bounded to homogeneous prescribed fields and the established model/domain; source experimental suggestions are not required conclusions. |
| V2 | main article p. 2, exact-mode versus semiclassical distinction; p. 4, zero-area discussion; Supplemental p. vii, Eqs. (S55)-(S56) and Fig. S3 These supply context for actual approximation, fringe, active-weight or freely propagating-pulse threats. A compensating-tail calculation is not compulsory for an external-field-bounded claim. |

## Formula and convention notes

Source-numbered identities below are paraphrased mathematically. Explicit algebraic consequences and accepted model-conditioned representations are identified as such; they do not add a required solver, plot or validation recipe.

For scalar modes $Q_{\mathbf p}(t)=\sqrt{m^2+p_\perp^2+[p_\parallel-eA(t)]^2}$ and $\ddot u_{\mathbf p}+Q_{\mathbf p}^2u_{\mathbf p}=0$ with incoming vacuum normalization, read $n_{\mathbf p}=|\beta_{\mathbf p}|^2$ asymptotically; equivalent exact bosonic kinetic evolution is accepted. Turning points solve $Q_{\mathbf p}(t_i)=0$ and $a(t_i/\tau)=(\pm i\sqrt{m^2+p_\perp^2}-p_\parallel)\gamma/m$, with $a(z)=-\int_0^z e(z^{\prime})dz^{\prime}$ (Eqs. (2)-(3)).

For $e(z)=e^{-z^2-z^4}$, $\gamma_*=\int_0^\infty e^{z^2-z^4}dz\simeq1.381$ (Eq. (5)). This is a dominance/onset diagnostic; an interference claim additionally needs coherent relative weights/actions, not a prescribed branch count.

The Dirac Hamiltonian is $H_{\mathbf k}=v_F\boldsymbol\sigma\cdot[\hbar\mathbf k-e\mathbf A]+\Delta\sigma_z$ with energy $\varepsilon_{\mathbf k}=\sqrt{\Delta^2+v_F^2|\hbar\mathbf k-e\mathbf A|^2}$ and $\gamma_D=\Delta/(|e|v_FE_0\tau)$ (Eqs. (6)-(7)); the filled-valence initial state gives $n_{\mathbf k}=|\widetilde\beta_{\mathbf k}|^2$ after the pulse.

## Scope of the source anchors

Figure panels, displayed source coordinates, curve multipliers, numerical grids and named controls are reference examples, not compulsory submission layouts or hidden pointwise targets. Equivalent evidence must preserve the actual physical observable, case, relative scale and discriminating function. Correct data, candidate identification, causal explanation and risk resolution are separate evidential acts. Existing branch-local dependencies, applicability/N/A and descriptive Research Closure rules remain in the case rubric.