# Ancillary files: what the paper's statements point to

This folder accompanies "Elliptic master integrals and the one-loop wavefunction of the universe" (A. McCune, M. D. Schwartz), version with the box section (Section 6, Appendix B). It maps each statement in the text that refers to ancillary material to the file that carries it.

| statement in the text | file(s) |
|---|---|
| Sec. 3-5: the closed form of the triangle coefficient, its dilogarithm representation and evaluators (general triangles; the off-slice line; the compact form) | `eval_closed_form_general.py`, `eval_closed_form_offslice.py`, `eval_closed_form_offslice_compact.py`, `eval_closed_form_V_all.py`; `li2_representation.json`, `compact_representation.json`, `vertex_blocks.json`; `closed_form_V_offslice.tex`, `closed_form_V_offslice_compact.tex`, `vertex_functions_blocks.tex` |
| Sec. 6 and App. B: the three j-invariants of the box curves | `box_j_invariants.tex` |
| App. A and App. B: the numerical checks behind the stated accuracies; statements about the box that support Sec. 6 and App. B | `numerical_checks.md`, `box_details.md` |
| Sec. 6.6, App. B.6-B.8 (equilateral line): the exact differential systems for C4 and psi4, the boundary values of u and v, the large-X series through X^-44, the singular points, and a stand-alone evaluator with a self-test | `box_line/` (see `box_line/README.md`): `eq_box_evaluate_standalone.py` (+ `_ratparse.py`), `system_C4_28.json`, `system_psi4_34.json`, `model_C4.json` (u(3/4), v(3/4), the Omega values, alpha_n and beta_n for n = 4..44), `model_psi4.json`, `singular_points.json`, `selftest_reference.json` |
| Sec. 6.2, App. B.3: an evaluator for the threshold discontinuity D_0 of eq. (D0closed) | `box_D0/d0_eval.py` (the evaluator) with `box_D0/d0_closed.py` (its library); formulas in `box_D0/D0_closed_form.tex`. Run `python d0_eval.py` for the built-in comparison with quadrature |
| Sec. 2: the numerator of the integrand | `numerator_A.txt` |
| Sec. 4: the operator L_0 and the coefficients c_i of degree 132 to 152 | `operator_L0/operator_L0.json`; `operator_L0/check_operator_L0.py` tests it on the closed form |

`box_line/` runs with python >= 3.8 and mpmath only: `python3 box_line/eq_box_evaluate_standalone.py --selftest-quick` (seconds) or `--selftest` (adds two transported points, minutes); exit code 0 means every check passed. `box_D0/` and `operator_L0/` need the same (python >= 3.8 and mpmath): `cd box_D0 && python3 d0_eval.py` prints the relative difference between the closed form (D0closed) and a quadrature of eq. (D0) for four choices of the pivot root (about 1e-38 at 40 digits, at the shape of figure 7 with P_2 = 3/2); `cd operator_L0 && python3 check_operator_L0.py [lambda]` prints the relative residual of L_0 on the closed form (about 1e-29) and a planted 1e-6 control (about 1e-12); it imports `eval_closed_form_general.py` from this folder, so keep the layout.