# F5-conf-03 — Prove or refute: sums of normally distributed random variables

# Scientific question

Two propositions about sums of normal random variables are given in data/propositions.csv. For each proposition, decide whether it is true (give a proof sketch) or false (give an explicit counterexample). For a false proposition, the counterexample must be made concrete: specify the joint construction of (X, Y), and produce a simulated sample from it that illustrates numerically that the marginals are normal while the sum is not (choose a construction whose departure from normality is clearly visible at the design sample size). The estimands are the truth values of propositions A and B and, for a false proposition, a verifiable counterexample.

# Inputs

data/propositions.csv — columns id, statement: the two propositions (A and B) to assess.
data/design.csv — columns parameter, value, description: the random seed to use (numpy.random.default_rng) and the sample size n for the numerical illustration.

# Information-availability rules

Use only the files under data/. The verdicts must rest on mathematical argument; the simulation is an illustration and a numerical check, not the proof.
Follow the design (seed, n) for the simulated sample.

# Required outputs

out/results.json with the keys:

key	type	unit	meaning
`task_id`	string	—	"F5-conf-03"
`statement_true`	boolean	—	verdict on proposition A
`counterexample_provided`	boolean	—	true if a concrete counterexample to A is provided in `out/argument.md`, `out/analysis.py` and `out/counterexample.csv`
`counterexample_values`	object	—	summary of the simulated counterexample sample: `construction` (string describing the joint law of (X, Y)), `n` (rows in `out/counterexample.csv`), `seed`, `mean_x`, `sd_x`, `mean_y`, `sd_y`, `mean_sum`, `sd_sum` (sample means and sample standard deviations, ddof = 1, of x, y and x + y), and `ks_sum_vs_normal` (Kolmogorov–Smirnov distance between the empirical distribution of x + y and the normal distribution with the sample mean and sample sd of x + y). If no counterexample is provided, an empty object.
`statement_b_true`	boolean	—	verdict on proposition B (the baseline check)
`feasibility`	string	—	"complete", "partial" or "infeasible"
`feasibility_reason`	string or null	—	explanation if not complete
`data_issues`	array of {"issue","handling"}	—	data problems detected and how handled (empty list if none)
`n_input_rows`	object	—	{"propositions.csv": rows read, "design.csv": rows read}
`n_analysed`	integer or null	—	sample size of the simulated counterexample sample
`random_seed`	integer or null	—	seed used

out/counterexample.csv — columns x, y: the simulated sample of size n from the counterexample construction (only if a counterexample is provided).
out/argument.md — the mathematical argument: the proof sketch for the true proposition(s) and the explicit counterexample (with the reasoning that its marginals are normal and its sum is not) for the false proposition(s).
out/report.md with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. State explicitly what the numerical illustration establishes and what it does not.
out/analysis.py — running ~/wp29/venv/bin/python out/analysis.py from the workspace root must regenerate out/results.json and out/counterexample.csv from data/ only, deterministically.
out/claims.json — a list of the central claims of the report: [{"id": "C1", "text": "...", "results_keys": ["..."], "status": "supported"|"unsupported"|"not_evaluated", "caveat": "..."}].

# Constraints

No network access. Python: ~/wp29/venv/bin/python (numpy, pandas, scipy, statsmodels available). Do not modify data/.
This task is designed to be completed in well under 20 minutes of work.

spb/wp29_uqo