# 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 (constants count as degenerate normal). Proposition B: if X and Y are independent and each normal, then X+Y is normal. Decide each; give a concrete counterexample for any false one.

## Data and exclusions
`data/propositions.csv` (2 rows) and `data/design.csv` (2 rows) were read (`n_input_rows`); SHA-256 hashes match `bundle.json` (see `out/state/data_audit.md`). No data issues (`data_issues` = []), no exclusions (`out/exclusions.csv` is header-only). Design: seed `random_seed` = 20261029, sample size `n_analysed` = 100000.

## Method
Verdicts are by mathematical argument (`out/argument.md`): B via characteristic functions; A refuted by the construction X ~ N(0,1), S Rademacher independent of X, Y = S·X (`counterexample_values.construction`). Y = −X was rejected because its sum is a constant, which the proposition counts as normal. `out/analysis.py` simulates the construction with `numpy.random.default_rng(seed)`, writes `out/counterexample.csv`, and computes sample means/SDs (ddof=1) and the KS distance of x+y to the normal with matching sample mean and SD. As a baseline, it also computes the same KS statistic for a sum of two independent N(0,1) samples (seed+1 stream; `baseline_b_*`).

## Results
- Proposition A is **false** (`statement_true` = false); a concrete counterexample is provided (`counterexample_provided` = true).
- Proposition B is **true** (`statement_b_true` = true).
- Simulated sample (`counterexample_values`): mean_x = −0.0033, sd_x = 0.9982; mean_y = −0.0002, sd_y = 0.9982; mean_sum = −0.0036, sd_sum = 1.4059 (theory: 0, 1, 0, 1, 0, √2).
- KS distance of x+y to a fitted normal: `counterexample_values.ks_sum_vs_normal` = 0.2508, compared with `supplementary_diagnostics.ks_x_vs_normal` = 0.0021, `ks_y_vs_normal` = 0.0020, and the independent-sum baseline `baseline_b_indep_ks_sum_vs_normal` = 0.0019. The approximate 5% KS critical value at this n is 0.0043 (`supplementary_diagnostics.ks_critical_value_5pct_approx`).
- Share of rows with x+y exactly 0: `supplementary_diagnostics.share_sum_exactly_zero` = 0.5005 (theory 1/2) — the atom that rules out normality. Sample correlation of x and y: `supplementary_diagnostics.corr_xy` = −0.0082 (theory 0: uncorrelated but dependent).

## Uncertainty
The verdicts are deductive and carry no statistical uncertainty. The simulation figures are subject to Monte Carlo error of order 1/√n ≈ 0.003; the sum's KS distance (0.25) exceeds this by about two orders of magnitude, while the marginal KS distances are at sampling-noise level. The KS critical value is the asymptotic one for a fully specified null; using fitted parameters (Lilliefors setting) makes it conservative, which does not affect the conclusion.

## Limitations and open issues
- **What the illustration shows and what it doesn't:** It shows numerically that, for this construction, the samples of x and y look like N(0,1) and the sample of x+y clearly doesn't look normal (half its mass sits at 0). It does not prove anything. Finite-sample checks can't establish that a distribution is exactly normal, and the verdicts rely only on the proofs in `out/argument.md`.
- Tolerance-based checks were not used for the atom: x+y = 0 holds exactly in floating point when y = −x.
- No other open issues; feasibility is `complete`.