# Report — F5-conf-03: sums of normally distributed random variables

## Question
Proposition A: if X and Y are each normal, then X+Y is normal (a constant counts as degenerate normal). Proposition B: if X and Y are independent and each normal, then X+Y is normal. Decide each; give a proof sketch or a concrete counterexample.

## Data and exclusions
Inputs: `data/propositions.csv` (2 rows) and `data/design.csv` (2 rows: seed 20261029, n 100000) — `n_input_rows`, `random_seed`. No data issues (`data_issues` is empty). No exclusions: all simulated rows are analysed (`n_analysed` = 100000; `out/exclusions.csv`). Hashes in `out/state/data_audit.md`.

## Method
Verdicts are by mathematical argument (`out/argument.md`). B: characteristic-function product + uniqueness. A: explicit counterexample X ~ N(0,1), S independent Rademacher, Y = S·X (`counterexample_values.construction`). The trivial Y = −X is rejected because its constant sum counts as normal under A's convention. The illustration simulates n draws with `numpy.random.default_rng(seed)` (`out/analysis.py`, `out/counterexample.csv`), and reports sample means/SDs (ddof=1) and the KS distance between the empirical law of x+y and the normal with its sample mean and SD. As a baseline, the same KS statistic is computed for the sum of two independent N(0,1) samples (`baseline_indep_ks_sum_vs_normal`).

## Results
- Proposition A is **false** (`statement_true` = false); counterexample provided (`counterexample_provided` = true).
- Proposition B is **true** (`statement_b_true` = true).
- Simulated marginals: mean_x ≈ −0.003, sd_x ≈ 0.998, mean_y ≈ −0.0002, sd_y ≈ 0.998 (`counterexample_values.mean_x`, `.sd_x`, `.mean_y`, `.sd_y`); KS distances to fitted normals ≈ 0.0021 and 0.0020 (`diagnostics.ks_x_vs_normal`, `diagnostics.ks_y_vs_normal`), below the asymptotic 5% critical value ≈ 0.0043 (`diagnostics.ks_critical_5pct`).
- Sum: mean ≈ −0.004, sd ≈ 1.406 (`counterexample_values.mean_sum`, `.sd_sum`; theory 0 and √2); KS distance to fitted normal ≈ 0.251 (`counterexample_values.ks_sum_vs_normal`), driven by the atom: 50.05% of sums are exactly 0 (`diagnostics.frac_sum_exactly_zero`; theory 1/2).
- Baseline: independent normal sum has KS ≈ 0.0020 (`baseline_indep_ks_sum_vs_normal`), sd ≈ 1.412 (`baseline_indep_sd_sum`).
- Sample correlation of x and y ≈ −0.008 (`diagnostics.corr_xy`): uncorrelated yet dependent, so uncorrelated normal marginals are not enough.

## Uncertainty
The verdicts are deterministic mathematical results and carry no statistical uncertainty. Simulated statistics carry Monte Carlo error of order 1/√n ≈ 0.003; the sum's KS distance (≈ 0.25) is about 60 times the 5% critical value, and theory gives a lower bound of 1/4 for any fitted normal because of the jump of size 1/2 at 0. (KS p-values with estimated parameters would be anti-conservative—Lilliefors issue—so they are not used as evidence.)

## Limitations and open issues
- What the illustration establishes: on this seeded sample, x and y look N(0,1) while x+y has a 50% atom at 0 and is far from any normal—consistent with the proof.
- What it does not establish: a finite sample cannot prove that the marginals are exactly normal or that the sum is non-normal; those facts rest on the argument in `out/argument.md`. The B baseline simulation likewise does not prove B.
- The KS-based marginal check is a single test; it would not detect tiny departures, which is irrelevant here since marginal normality of Y is proved by symmetry.