# Contract (P0)

Question. Decide truth of two propositions on sums of normal r.v.s:

• A: X, Y each normal (no joint assumption) => X+Y normal (constants count as normal).
• B: X, Y independent and each normal => X+Y normal.

Estimands. Truth values of A and B (mathematical; the proof/counterexample is the evidence). For a false proposition: an explicit joint construction of (X,Y) with normal marginals and a non-normal sum, plus a simulated sample (seed and n from data/design.csv) illustrating it.

Outputs. out/results.json keys: task_id, statement_true, counterexample_provided, counterexample_values{construction,n,seed,mean_x,sd_x,mean_y,sd_y,mean_sum,sd_sum,ks_sum_vs_normal}, statement_b_true, feasibility, feasibility_reason, data_issues, n_input_rows, n_analysed, random_seed. Also out/counterexample.csv (x,y), out/argument.md, out/report.md (6 sections), out/analysis.py, out/claims.json, out/exclusions.csv, state files.

Information rules. Only data/; verdicts by mathematical argument, simulation is illustration only; use design seed and n with numpy default_rng.

Invalidity. Missing/inconsistent seed or n; counterexample whose sum is degenerate (e.g. Y=-X: constant sum counts as normal under A, so not a counterexample); counterexample whose marginals are not exactly normal.