# Figure caption

**Figure. Angle-energy dependence and geometric energy delivery in a fixed absorbing trench.** Original synthetic finite-table counterexample; no measured data. (a) The prescribed angle and energy marginals each allocate probabilities 0.25, 0.50 and 0.25 to their three categories. Angles are 0, arctan(1/20) and arctan(3/40), displayed as 0.00°, 2.86° and 4.29°; the energies are 20, 60 and 100 eV. The trench depth-to-width ratio is 10, entry position is independent of the angle-energy pair, and geometric direct-bottom acceptance is 1, 0.5 and 0.25 for the respective angle categories. Consequently, the direct-bottom fraction is fixed at 56.25% and mean incident energy at 60 eV. (b,c) Joint probability tables attain the minimum (26.25 eV) and maximum (41.25 eV) bottom-delivered projectile energy per incident population member while preserving both marginals. Cell labels denote probability mass; both panels use the same 0-50% colour scale. (d) Adding the hypothetical joint constraint that the narrowest-angle, highest-energy cell contains 12.5% of the population narrows the admissible energy range to 32.5-38.75 eV per incident member. These are admissible finite-table bounds, not confidence intervals. The independent-product assumption gives 33.75 eV in the marginals-only case (point), but its underlying joint table violates the added constraint; it is therefore omitted from the constrained row. Six existing LP extrema were checked against exact rational vertices and primal/dual certificates in the source study. This figure reuses those stored results without new optimization, simulation trajectories or measurements. The observable is projectile kinetic energy carried to the bottom per incident population member, not deposited heat, etch rate, flux, material yield or physical model validation. Static first-absorbing-contact geometry excludes reflection, charging, chemistry and moving surfaces.

## 한국어 설명

**같은 입사각 분포와 에너지 분포라도 두 변수의 결합 관계에 따라 트렌치 바닥에 도달하는 에너지는 달라집니다.** (a) 입력 분포를 고정한 상태에서 (b,c) 가능한 결합 분포의 양끝을 비교하고, (d) 결합 정보 한 개를 추가하면 허용 범위가 좁아지는 모습을 보여 줍니다. 원래 연구에서 검증한 합성 수치 예시를 재구성했으며 실측 결과가 아닙니다. 수치는 입사 입자군의 구성원 한 개당 바닥에 전달되는 운동에너지이고, 식각률이나 열 침적량을 뜻하지 않습니다.

## Website copy

Title: **When marginals hide the mechanism**
Subtitle: Four-panel research figure built from verified original numerical results.
Description: Fixed input distributions, contrasting joint tables, and exact admissible energy bounds show why angle-energy dependence matters in a static trench model.
Evidence label: **Original synthetic study · Numerically verified · No measured data**
Alt text: Four-panel figure comparing equal angle and energy marginals, minimum and maximum joint-probability tables, and the narrowed bottom-energy range after one hypothetical joint constraint.

## Source and reproducibility

Source packet: `07_assets/process_lab/experiments/SA-PROC-ETCH-HAR-011/research/angle-energy-joint-identification-20260927/`.
Numerical output: `evidence/results.json`. Frozen inputs: `contract.json`.
Verification: `verification/normal.json`, `verification/optimized.json`; interpretation: `DOSSIER.md`, `RESULTS.md`.
Primary solver documentation linked by the original packet: https://docs.scipy.org/doc/scipy-1.15.3/reference/optimize.linprog-highs.html and https://ergo-code.github.io/HiGHS/dev/options/definitions/ . These support the solver interface, not material calibration.

Portable redraw: run `python -B reproplot.py --data-dir . --out new-figure` with an existing NumPy/Matplotlib environment from the folder containing the three CSVs. This mode needs only those CSVs and this script; it does not access the canonical research folder. Internal provenance audit is separately available via `--source-root <project>` instead of `--data-dir`; that mode verifies the original sealed inputs/results before plotting. Neither mode runs the original LP solver. Display-only CSVs accompany the figure and preserve every plotted numerical value. The SVG keeps editable text and vector cells; the PDF embeds fonts and uses vector graphics. The main PNG is 360 dpi at 180 × 144 mm, and the web preview is 1200 pixels wide.