spb/wp29_uqo

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It is the complete and only authoritative description of what is required: the scientific question, the inputs under `data/`, the required output files and their exact keys, and the information-availability rules.\n- There is no network access. Use the Python interpreter `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels, matplotlib are installed). Standard shell tools are available. Do not try to install packages.\n- Write every deliverable under `./out/`. The required deliverables are listed in `TASK.md`; at minimum they always include `out/results.json`, `out/report.md`, and a replication entry point `out/analysis.py` that regenerates `out/results.json` from `data/` when run with `~/wp29/venv/bin/python out/analysis.py` from the workspace root.\n- You are working alone and unattended; nobody will answer questions. Make reasonable, documented decisions.\n\nIntegrity rules (identical for every task):\n- Never fabricate numbers, data, results, or references. Every number in `out/results.json` and `out/report.md` must be produced by code you ran on the provided inputs.\n- Respect the information-availability rules in `TASK.md` exactly (for example, never use information that would not have been available at the stated decision time).\n- If part of the task cannot be completed as specified, say so explicitly in `out/report.md`, explain why, and still write every required file (use `null` for values you could not legitimately produce). An honest statement of a limitation is always preferred to a fabricated or guessed value.\n- Do not modify anything under `data/`.\n- When you are done, end with a brief plain-text summary of what you produced and any limitations.\n\nComplete the task described in `TASK.md` by following this research protocol. The protocol adds no scientific information to the task; it organizes how you work and what evidence you leave behind.\n\nPHASES (execute in order; a phase is eligible only when its dependencies are satisfied):\n\nP0 Scope and contract (depends on: nothing)\n- Read `TASK.md` completely. In `out/state/contract.md`, restate in your own words: the scientific question; the quantity or quantities to be estimated (the estimands) and the assumptions under which they are meaningful; every required output file and key; the information-availability rules; what would make the task impossible or the result invalid.\n- Write the initial phase ledger `out/state/phases.json`: a JSON object mapping each phase id (P0 … P5) to {\"status\": \"pending\"|\"running\"|\"passed\"|\"failed\"|\"blocked\"|\"stale\"|\"skipped_optional\", \"checks\": [], \"notes\": \"\"}. Keep it current as you work. A phase may only be set to \"passed\" when the checks listed for it in this protocol have actually been executed and their outcome recorded in \"checks\".\n\nP1 Data audit and provenance (depends on: P0)\n- Inspect every input file: compute and record its SHA-256, row and column counts, schema, units, date ranges, missing values, duplicates and anomalies in `out/state/data_audit.md`.\n- Keep an exclusion ledger in `out/exclusions.csv` (columns: step, rule, n_before, n_after, n_removed, rationale) for every filtering step you apply, so that n_raw − total removed = n_analysed.\n- Identify any input that violates the information-availability rules and decide, in writing, how you will handle it.\n\nP2 Baseline (depends on: P1)\n- Before any elaborate method, compute the simplest valid baseline that answers the question (for example a naïve or unconditional estimate), with the same evaluation rule used later. Record it in `out/results.json` under the keys the task requires, or under `baseline_*` keys when the task does not name them.\n\nP3 Method and computation (depends on: P2)\n- Implement the analysis in `out/analysis.py` so that it reads only `data/`, writes `out/results.json` deterministically (fix every random seed you use and record it), and prints nothing that contradicts the written results. Any figure or table must be generated by this script.\n\nP4 Verification (depends on: P3)\n- Re-run `out/analysis.py` from a clean state and confirm it regenerates `out/results.json` with identical values. Record the check in `out/state/phases.json` under P4.\n- Check internal consistency: identities that must hold (for example counts that must add up, standard errors that must be positive, confidence intervals that must contain their estimates, date orderings that must respect the information rules), units, and magnitudes against the baseline. Record every check and its outcome in `out/state/checks.md`. A failed check must be fixed or honestly reported, never silently ignored.\n\nP5 Reporting (depends on: P4)\n- Write `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. Every quantitative statement must cite the key of `out/results.json` it comes from.\n- Write `out/claims.json`: a list of the central claims of your report, each as {\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}. Do not state a claim in the report that is not in this ledger.\n- Set the final statuses in `out/state/phases.json`.\n\nSTOPPING RULES\n- Stop and declare the task complete only when every phase P0–P5 is \"passed\" with recorded checks. If a phase is \"blocked\" (missing information, impossible requirement, violated availability rule), stop, set the status, explain in `out/report.md` (section Limitations and open issues) and still write every required file with `null` for values you could not legitimately produce.\n- Do not spend effort on extensions that the task does not require.\n\nRECOVERY RULES\n- Append a short dated entry to `out/state/log.md` after each phase transition and after any important decision, so that work can be resumed by reading `out/state/`.\n- If you discover that an upstream input or decision changed after a downstream phase passed, mark the downstream phases \"stale\" and redo them; do not reuse stale results.\n- If you are interrupted and later resumed, first read `out/state/` and continue from the last consistent state; never re-run expensive steps that already have recorded, unchanged outputs.\n"}],"tools":[{"name":"Bash","description":"execute shell commands","input_schema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"command":{"description":"The command to execute","type":"string"},"timeout":{"description":"Optional timeout in milliseconds (max 600000 for a foreground command)","type":"number"},"description":{"description":"Clear, concise description of what this command does in active voice. 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the user's own instructions about these lines, such as a CLAUDE.md or memory rule, take precedence over this reminder, but do not add attribution lines this reminder leaves out):\n- End git commit messages with:\nCo-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>\n- End pull request descriptions with:\n🤖 Generated with [Claude Code](https://claude.com/claude-code)\n</system-reminder>"}],"renderedRole":"user","renderedBesideToolResult":true,"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}11{"type":"tools-used","tools":["Bash"],"sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133"}12{"parentUuid":"11ccf594-682f-49cc-b120-268b2910a13c","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXpt1DkYUhFTR5rq8bdG","type":"message","role":"assistant","content":[{"type":"thinking","thinking":"","signature":"CAQSmgUKEAgTGAI4AUIIdGhpbmtpbmcSDGv16qfEIV1ZAN3n1BoMS1yGhf1U7waG0IfQIjBvPUHSy32kgBxKoj9R4YcJq9HOvZLVvWG2nx9Ky3iD5LOYNhnBXHEskwaq9jXolzgqtwQJIMV2AMxVAQgqXDYkAdmzz5xufDBDKMbH7AA7BqDun8vIQPr19RnjEWyyBo5wu99JM2k/4E2lmnO2hsD1QNl34A2qwFMicNW5psG3qb0p5/IV0eMh+vvP4Db/AFNIFS4UOGIii7TjVGNlEXe6gQYh4hUzBTJG5j+SioNKsgnH0pslB44oOawO8cOfAqpiFPH5uQbdyIdrkq/7D9/r4tdGOJaP28+X1PdxaWi7gYZTumBexCzE5tB18nZll4xxnVoWH3Dw3c1Ak+k2lTN6NIpOh8M7oAiAWFvxzeoALeAXO5FexhaKvvuHE2+zD26IN0CLWahVugDAF64FUfGupDBg7xb4D3IPuSjgm2S1aEfrqRH+35QRpTSl1lcurFqVH7K04Ehy8LqMZ2eAd8+fHJZ2UtqgcqROmUDRJcjq3s5S2XcNMvpVRSjwQYjuNBK/QPl/GfvKYcSCO4paLp6zTrx2r5HxnX6K2EN6jJk+uq8N5cph0FP32e/DppHZKPU+NPgAEG2W2nMv/EhfXRhhigp7+hz7VRhbDO7UFPt7I/thdgk5sZQizzPJg/idHw4Eqcu+uoYjqwGhmsuMSR/2NyzOQPMU2ZCTBHehxGXZb2h4LsdAcnfmt+48VvHXaFkvACyoJpQWXvxwJTSUTEg2RGEPZ+o9bTUS4Ya12PMLsNNU08/A9pokbcC88yjdukdtd8MD95Xl6oqRYEn3prRN87BxR/o7yPs1wGQiyKex6Nl8XZr9TZhqHagYAQ=="}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":4,"cache_creation_input_tokens":2209,"cache_read_input_tokens":1504,"output_tokens":126,"output_tokens_details":{"thinking_tokens":20},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":2209},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"thinkingDurationMs":485,"thinkingDisplay":"updates","apiBlockIndex":0,"requestId":"req_011CgXpt12qXdkQavdYKxSA9","requestRef":"c23df0d3-987f-42c0-814e-805c5ce583ee","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"41d38818-692b-4efc-ab88-cf6e802bb904","timestamp":"2026-10-30T04:55:35.990Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}13{"parentUuid":"41d38818-692b-4efc-ab88-cf6e802bb904","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXpt1DkYUhFTR5rq8bdG","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_013xrT7RK7jSyohqGzDeMT2f","name":"Bash","input":{"command":"cat TASK.md; 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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.\n\n## Inputs\n- `data/propositions.csv` — columns `id`, `statement`: the two propositions (A and B) to assess.\n- `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.\n\n## Information-availability rules\n- 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.\n- Follow the design (seed, n) for the simulated sample.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | — | \"F5-conf-03\" |\n| `statement_true` | boolean | — | verdict on proposition A |\n| `counterexample_provided` | boolean | — | true if a concrete counterexample to A is provided in `out/argument.md`, `out/analysis.py` and `out/counterexample.csv` |\n| `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. |\n| `statement_b_true` | boolean | — | verdict on proposition B (the baseline check) |\n| `feasibility` | string | — | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | — | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | — | data problems detected and how handled (empty list if none) |\n| `n_input_rows` | object | — | {\"propositions.csv\": rows read, \"design.csv\": rows read} |\n| `n_analysed` | integer or null | — | sample size of the simulated counterexample sample |\n| `random_seed` | integer or null | — | seed used |\n\n- `out/counterexample.csv` — columns `x`, `y`: the simulated sample of size n from the counterexample construction (only if a counterexample is provided).\n- `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).\n- `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.\n- `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.\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}].\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). 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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.\n\n## Inputs\n- `data/propositions.csv` — columns `id`, `statement`: the two propositions (A and B) to assess.\n- `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.\n\n## Information-availability rules\n- 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.\n- Follow the design (seed, n) for the simulated sample.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | — | \"F5-conf-03\" |\n| `statement_true` | boolean | — | verdict on proposition A |\n| `counterexample_provided` | boolean | — | true if a concrete counterexample to A is provided in `out/argument.md`, `out/analysis.py` and `out/counterexample.csv` |\n| `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. |\n| `statement_b_true` | boolean | — | verdict on proposition B (the baseline check) |\n| `feasibility` | string | — | \"complete\", \"partial\" or \"infeasible\" |\n| `feasibility_reason` | string or null | — | explanation if not complete |\n| `data_issues` | array of {\"issue\",\"handling\"} | — | data problems detected and how handled (empty list if none) |\n| `n_input_rows` | object | — | {\"propositions.csv\": rows read, \"design.csv\": rows read} |\n| `n_analysed` | integer or null | — | sample size of the simulated counterexample sample |\n| `random_seed` | integer or null | — | seed used |\n\n- `out/counterexample.csv` — columns `x`, `y`: the simulated sample of size n from the counterexample construction (only if a counterexample is provided).\n- `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).\n- `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.\n- `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.\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}].\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). 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-p out/state && cd out/state && cat > contract.md <<'EOF'\n# Contract (P0)\n**Question.** Decide truth of two propositions on sums of normal r.v.s (data/propositions.csv):\n- A: X, Y each normal (no joint assumption) => X+Y normal (constants count as degenerate normal).\n- B: X, Y independent and each normal => X+Y normal.\n**Estimands.** Truth values of A and B; for a false proposition a concrete, verifiable counterexample (joint law of (X,Y)) plus a simulated sample (seed, n from data/design.csv) showing normal marginals but non-normal sum.\nNote: because constants count as normal, Y = -X is NOT a valid counterexample; the sum must be non-degenerate and non-normal.\n**Outputs.** out/results.json (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); out/counterexample.csv (x,y); 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Verdicts rest on mathematical proof; simulation only illustrates. Use seed and n from design.csv with numpy default_rng.\n**Invalidity.** Counterexample whose sum is constant (degenerate normal) or whose marginals are not exactly normal; using a seed/n other than the design; claiming the simulation proves anything.\nEOF\ncat > log.md <<'EOF'\n# Log\n- 2026-10-30 P0 passed: contract written.\nEOF\ncd ../..; printf 'step,rule,n_before,n_after,n_removed,rationale\\n' > out/exclusions.csv\ncat > out/state/data_audit.md <<'EOF'\n# Data audit (P1)\n| file | sha256 | rows | cols | schema |\n|---|---|---|---|---|\n| data/propositions.csv | 0e12befb7693999629fbe1a46d5243e78bd54681b05be6a8b81efe8d307db1f6 | 2 | 2 | id (str), statement (str) |\n| data/design.csv | b9c14d67b34e811cafa44a01d8d03d44f7b31a9a01c882e8bc51ac563003a9df | 2 | 3 | parameter, value, description |\n\nHashes match bundle.json. No missing values, no duplicates. Design: seed=20261029, n=100000 (dimensionless count).\nNo dates; no information-availability violations. No rows excluded (exclusions.csv has header only; n_raw = n_analysed for input files; the simulated sample has n=100000 rows with no filtering).\nEOF\necho \"- 2026-10-30 P1 passed: audit done, no issues, no exclusions.\" >> out/state/log.md","description":"Write contract, audit, log, exclusions"}}]}}27{"type":"api-request","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","id":"4941dd68-796a-48f1-9637-f6296fcf107e","timestamp":"2026-10-30T04:55:53.104Z","version":"2.1.314","querySource":"sdk","shapeHash":"c476dccff750d1df4af4bcb23bd855a178afef7ca3ecd9c0205488ebcd9a51ab","params":{"model":"claude-opus-5-5","betas":["claude-code-20250219","interleaved-thinking-2025-05-14","context-management-2025-06-27","prompt-caching-scope-2026-01-05","per-turn-control-2026-07-01","thinking-token-count-2026-05-13","mid-conversation-system-clear-at-2026-08-21","effort-2025-11-24","thinking-binding-controls-2026-08-01","thinking-display-updates-2026-08-18","cache-diagnosis-2026-04-07"],"max_tokens":128000,"thinking":{"type":"adaptive","display":"updates"},"context_management":{"edits":[{"type":"clear_thinking_20251015","keep":"all"}]},"output_config":{"effort":"medium"}},"base":"39686463-269b-40ec-b332-351446a6cc2d","keep":6,"tail":["f7363c96ed8bc71d3ac4e955cc760e6059fadb4b8fec2c372e6373588e23c249","424d98be2c02c4670a73141cf820064a8ee0711482d413844ff56bfc9f8a4da5"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",7,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"b333ee013d2eb7f6df9ac284960411f23a1edd91f3101842a24177a4219cbf40"}28{"parentUuid":"0c262558-a7db-41e7-80cf-5c1c9ad379df","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXpuMwTUUFYydz1vGrCU","type":"message","role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1512,"cache_read_input_tokens":6239,"output_tokens":1705,"output_tokens_details":{"thinking_tokens":68},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":1512},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"thinkingDurationMs":709,"thinkingDisplay":"updates","apiBlockIndex":0,"requestId":"req_011CgXpuMfLmr5L3X2oV25sn","requestRef":"4941dd68-796a-48f1-9637-f6296fcf107e","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"5b2e2ab8-7172-49b4-9b82-656da5099439","timestamp":"2026-10-30T04:55:55.324Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}29{"parentUuid":"5b2e2ab8-7172-49b4-9b82-656da5099439","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXpuMwTUUFYydz1vGrCU","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01Hg92jNX4YSyL6Hm1tZpUcd","name":"Bash","input":{"command":"cat > out/analysis.py <<'EOF'\n\"\"\"F5-conf-03: counterexample to 'sum of normals is normal' (prop. A) and check of B.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nConstruction: X ~ N(0,1), S ~ Rademacher(+-1 w.p. 1/2) independent of X, Y = S*X.\nY ~ N(0,1) (X symmetric, S independent); X+Y = (1+S)X equals 0 w.p. 1/2 and 2X otherwise\n-> has an atom at 0 yet Var = 2 > 0, so it is neither non-degenerate normal nor constant.\n\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\nD = os.path.join(ROOT, \"data\"); O = os.path.join(ROOT, \"out\")\n\nprops = pd.read_csv(os.path.join(D, \"propositions.csv\"))\ndesign = pd.read_csv(os.path.join(D, \"design.csv\"))\npar = dict(zip(design[\"parameter\"], design[\"value\"]))\nseed, n = int(par[\"seed\"]), int(par[\"n\"])\ndata_issues = []\nassert set(props[\"id\"]) == {\"A\", \"B\"}, \"unexpected proposition ids\"\n\nrng = np.random.default_rng(seed)\nx = rng.standard_normal(n)\ns = rng.choice(np.array([-1.0, 1.0]), size=n)\ny = s * x\nz = x + y\npd.DataFrame({\"x\": x, \"y\": y}).to_csv(os.path.join(O, \"counterexample.csv\"), index=False)\n\ndef ks_fit(v):\n    return float(stats.kstest(v, \"norm\", args=(v.mean(), v.std(ddof=1))).statistic)\n\ncv = {\n    \"construction\": \"X ~ N(0,1); S ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. \"\n                    \"Then Y ~ N(0,1) but X+Y = (1+S)X is 0 with prob. 1/2 and 2X otherwise (atom at 0, Var 2).\",\n    \"n\": int(n), \"seed\": seed,\n    \"mean_x\": float(x.mean()), \"sd_x\": float(x.std(ddof=1)),\n    \"mean_y\": float(y.mean()), \"sd_y\": float(y.std(ddof=1)),\n    \"mean_sum\": float(z.mean()), \"sd_sum\": float(z.std(ddof=1)),\n    \"ks_sum_vs_normal\": ks_fit(z),\n}\n# supplementary diagnostics (marginals vs fitted normal, atom share, independence-based baseline for B)\nsupp = {\n    \"ks_x_vs_normal\": ks_fit(x), \"ks_y_vs_normal\": ks_fit(y),\n    \"share_sum_exactly_zero\": float(np.mean(z == 0.0)),\n    \"ks_critical_value_5pct_approx\": float(1.358 / np.sqrt(n)),\n    \"corr_xy\": float(np.corrcoef(x, y)[0, 1]),\n}\nrng_b = np.random.default_rng(seed + 1)  # separate stream for the B illustration only\nxb, yb = rng_b.standard_normal(n), rng_b.standard_normal(n)\nbaseline = {\"baseline_b_indep_ks_sum_vs_normal\": ks_fit(xb + yb),\n            \"baseline_b_indep_sd_sum\": float((xb + yb).std(ddof=1)),\n            \"baseline_b_seed\": seed + 1}\n\nres = {\n    \"task_id\": \"F5-conf-03\",\n    \"statement_true\": False,           # by proof in out/argument.md\n    \"counterexample_provided\": True,\n    \"counterexample_values\": cv,\n    \"statement_b_true\": True,          # by proof (characteristic functions) in out/argument.md\n    \"feasibility\": \"complete\",\n    \"feasibility_reason\": None,\n    \"data_issues\": data_issues,\n    \"n_input_rows\": {\"propositions.csv\": int(len(props)), \"design.csv\": int(len(design))},\n    \"n_analysed\": int(n),\n    \"random_seed\": seed,\n    \"supplementary_diagnostics\": supp,\n    **baseline,\n}\nwith open(os.path.join(O, \"results.json\"), \"w\") as f:\n    json.dump(res, f, indent=2)\nprint(json.dumps(res, indent=2))\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis script"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1512,"cache_read_input_tokens":6239,"output_tokens":1705,"output_tokens_details":{"thinking_tokens":68},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":1512},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXpuMfLmr5L3X2oV25sn","requestRef":"4941dd68-796a-48f1-9637-f6296fcf107e","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"c785d0aa-cca0-4d58-b0bd-08cdf0cc9a59","timestamp":"2026-10-30T04:56:07.211Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}30{"parentUuid":"c785d0aa-cca0-4d58-b0bd-08cdf0cc9a59","isSidechain":false,"promptId":"a1301c1e-956d-45b7-a01c-3676354e4f64","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01Hg92jNX4YSyL6Hm1tZpUcd","type":"tool_result","content":"{\n  \"task_id\": \"F5-conf-03\",\n  \"statement_true\": false,\n  \"counterexample_provided\": true,\n  \"counterexample_values\": {\n    \"construction\": \"X ~ N(0,1); S ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. Then Y ~ N(0,1) but X+Y = (1+S)X is 0 with prob. 1/2 and 2X otherwise (atom at 0, Var 2).\",\n    \"n\": 100000,\n    \"seed\": 20261029,\n    \"mean_x\": -0.0033187423835634258,\n    \"sd_x\": 0.9982238690893596,\n    \"mean_y\": -0.00024909222432809854,\n    \"sd_y\": 0.9982293548745821,\n    \"mean_sum\": -0.0035678346078915233,\n    \"sd_sum\": 1.4058866469982523,\n    \"ks_sum_vs_normal\": 0.2508475723268392\n  },\n  \"statement_b_true\": true,\n  \"feasibility\": \"complete\",\n  \"feasibility_reason\": null,\n  \"data_issues\": [],\n  \"n_input_rows\": {\n    \"propositions.csv\": 2,\n    \"design.csv\": 2\n  },\n  \"n_analysed\": 100000,\n  \"random_seed\": 20261029,\n  \"supplementary_diagnostics\": {\n    \"ks_x_vs_normal\": 0.0020917798156042666,\n    \"ks_y_vs_normal\": 0.0019518220846299617,\n    \"share_sum_exactly_zero\": 0.50047,\n    \"ks_critical_value_5pct_approx\": 0.0042943730625086595,\n    \"corr_xy\": -0.008226889835527693\n  },\n  \"baseline_b_indep_ks_sum_vs_normal\": 0.0018749949107932373,\n  \"baseline_b_indep_sd_sum\": 1.4152986579809257,\n  \"baseline_b_seed\": 20261030\n}","is_error":false}]},"uuid":"ac57be30-0d36-41f9-a92c-7a5c3666aa86","timestamp":"2026-10-30T04:56:07.958Z","toolUseResult":{"stdout":"{\n  \"task_id\": \"F5-conf-03\",\n  \"statement_true\": false,\n  \"counterexample_provided\": true,\n  \"counterexample_values\": {\n    \"construction\": \"X ~ N(0,1); S ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. Then Y ~ N(0,1) but X+Y = (1+S)X is 0 with prob. 1/2 and 2X otherwise (atom at 0, Var 2).\",\n    \"n\": 100000,\n    \"seed\": 20261029,\n    \"mean_x\": -0.0033187423835634258,\n    \"sd_x\": 0.9982238690893596,\n    \"mean_y\": -0.00024909222432809854,\n    \"sd_y\": 0.9982293548745821,\n    \"mean_sum\": -0.0035678346078915233,\n    \"sd_sum\": 1.4058866469982523,\n    \"ks_sum_vs_normal\": 0.2508475723268392\n  },\n  \"statement_b_true\": true,\n  \"feasibility\": \"complete\",\n  \"feasibility_reason\": null,\n  \"data_issues\": [],\n  \"n_input_rows\": {\n    \"propositions.csv\": 2,\n    \"design.csv\": 2\n  },\n  \"n_analysed\": 100000,\n  \"random_seed\": 20261029,\n  \"supplementary_diagnostics\": {\n    \"ks_x_vs_normal\": 0.0020917798156042666,\n    \"ks_y_vs_normal\": 0.0019518220846299617,\n    \"share_sum_exactly_zero\": 0.50047,\n    \"ks_critical_value_5pct_approx\": 0.0042943730625086595,\n    \"corr_xy\": -0.008226889835527693\n  },\n  \"baseline_b_indep_ks_sum_vs_normal\": 0.0018749949107932373,\n  \"baseline_b_indep_sd_sum\": 1.4152986579809257,\n  \"baseline_b_seed\": 20261030\n}","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"c785d0aa-cca0-4d58-b0bd-08cdf0cc9a59","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}31{"type":"api-request-blob","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","hash":"949a625646fec6e6d20ae747bc5ab57642e674cd16b5ecbc7f7444a34f0e02f1","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01Hg92jNX4YSyL6Hm1tZpUcd","name":"Bash","input":{"command":"cat > out/analysis.py <<'EOF'\n\"\"\"F5-conf-03: counterexample to 'sum of normals is normal' (prop. A) and check of B.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nConstruction: X ~ N(0,1), S ~ Rademacher(+-1 w.p. 1/2) independent of X, Y = S*X.\nY ~ N(0,1) (X symmetric, S independent); X+Y = (1+S)X equals 0 w.p. 1/2 and 2X otherwise\n-> has an atom at 0 yet Var = 2 > 0, so it is neither non-degenerate normal nor constant.\n\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nROOT = os.path.dirname(os.path.dirname(os.path.abspath(__file__)))\nD = os.path.join(ROOT, \"data\"); O = os.path.join(ROOT, \"out\")\n\nprops = pd.read_csv(os.path.join(D, \"propositions.csv\"))\ndesign = pd.read_csv(os.path.join(D, \"design.csv\"))\npar = dict(zip(design[\"parameter\"], design[\"value\"]))\nseed, n = int(par[\"seed\"]), int(par[\"n\"])\ndata_issues = []\nassert set(props[\"id\"]) == {\"A\", \"B\"}, \"unexpected proposition ids\"\n\nrng = np.random.default_rng(seed)\nx = rng.standard_normal(n)\ns = rng.choice(np.array([-1.0, 1.0]), size=n)\ny = s * x\nz = x + y\npd.DataFrame({\"x\": x, \"y\": y}).to_csv(os.path.join(O, \"counterexample.csv\"), index=False)\n\ndef ks_fit(v):\n    return float(stats.kstest(v, \"norm\", args=(v.mean(), v.std(ddof=1))).statistic)\n\ncv = {\n    \"construction\": \"X ~ N(0,1); S ~ Rademacher (P(S=1)=P(S=-1)=1/2) independent of X; Y = S*X. \"\n                    \"Then Y ~ N(0,1) but X+Y = (1+S)X is 0 with prob. 1/2 and 2X otherwise (atom at 0, Var 2).\",\n    \"n\": int(n), \"seed\": seed,\n    \"mean_x\": float(x.mean()), \"sd_x\": float(x.std(ddof=1)),\n    \"mean_y\": float(y.mean()), \"sd_y\": float(y.std(ddof=1)),\n    \"mean_sum\": float(z.mean()), \"sd_sum\": float(z.std(ddof=1)),\n    \"ks_sum_vs_normal\": ks_fit(z),\n}\n# supplementary diagnostics (marginals vs fitted normal, atom share, independence-based baseline for B)\nsupp = {\n    \"ks_x_vs_normal\": ks_fit(x), \"ks_y_vs_normal\": ks_fit(y),\n    \"share_sum_exactly_zero\": float(np.mean(z == 0.0)),\n    \"ks_critical_value_5pct_approx\": float(1.358 / np.sqrt(n)),\n    \"corr_xy\": float(np.corrcoef(x, y)[0, 1]),\n}\nrng_b = np.random.default_rng(seed + 1)  # separate stream for the B illustration only\nxb, yb = rng_b.standard_normal(n), rng_b.standard_normal(n)\nbaseline = {\"baseline_b_indep_ks_sum_vs_normal\": ks_fit(xb + yb),\n            \"baseline_b_indep_sd_sum\": float((xb + yb).std(ddof=1)),\n            \"baseline_b_seed\": seed + 1}\n\nres = {\n    \"task_id\": \"F5-conf-03\",\n    \"statement_true\": False,           # by proof in out/argument.md\n    \"counterexample_provided\": True,\n    \"counterexample_values\": cv,\n    \"statement_b_true\": True,          # by proof (characteristic functions) in out/argument.md\n    \"feasibility\": \"complete\",\n    \"feasibility_reason\": None,\n    \"data_issues\": data_issues,\n    \"n_input_rows\": {\"propositions.csv\": int(len(props)), \"design.csv\": int(len(design))},\n    \"n_analysed\": int(n),\n    \"random_seed\": seed,\n    \"supplementary_diagnostics\": supp,\n    **baseline,\n}\nwith open(os.path.join(O, \"results.json\"), \"w\") as f:\n    json.dump(res, f, indent=2)\nprint(json.dumps(res, indent=2))\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis 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> out/state/checks.md <<'EOF'\n# Verification checks (P4)\n| # | check | outcome |\n|---|---|---|\n| 1 | Delete results.json and counterexample.csv, re-run analysis.py; results.json byte-identical (cmp), counterexample.csv same SHA-256 | PASS |\n| 2 | rows in counterexample.csv = counterexample_values.n = n_analysed = design n = 100000; columns x,y | PASS |\n| 3 | mean_x and sd_sum recomputed from CSV match results.json | PASS |\n| 4 | abs(y) == abs(x) row-wise (construction Y = S*X holds) | PASS |\n| 5 | sd_x, sd_y, sd_sum > 0 | PASS |\n| 6 | sd_sum = 1.4059 close to theoretical sqrt(2)=1.4142 (Var(X+Y)=2 since Cov(X,SX)=E[S]E[X^2]=0) | PASS |\n| 7 | Marginal KS (0.0021 x, 0.0020 y) below approx 5% critical value 0.0043; sum KS 0.251 is ~58x that value; independent-sum baseline KS 0.0019 | PASS (consistent with theory) |\n| 8 | share_sum_exactly_zero = 0.50047 vs theoretical 1/2 | PASS |\n| 9 | seed 20261029 and n from design.csv used | PASS |\n| 10 | Sum is non-degenerate (sd_sum>0), so it is not a \"degenerate normal\" exception | PASS |\nEOF\ncat > out/argument.md <<'EOF'\n# Mathematical argument\n\n## Proposition B — TRUE\n*If X and Y are independent and each normal, X+Y is normal.*\n\nProof sketch. Let X ~ N(μ₁, σ₁²), Y ~ N(μ₂, σ₂²) (σᵢ ≥ 0, σᵢ=0 meaning a constant). The characteristic function of N(μ, σ²) is φ(t) = exp(iμt − σ²t²/2). By independence, φ_{X+Y}(t) = φ_X(t)φ_Y(t) = exp(i(μ₁+μ₂)t − (σ₁²+σ₂²)t²/2), which is the characteristic function of N(μ₁+μ₂, σ₁²+σ₂²). By the uniqueness theorem for characteristic functions, X+Y ~ N(μ₁+μ₂, σ₁²+σ₂²) (degenerate if both σᵢ=0). ∎\n\n## Proposition A — FALSE\n*If X and Y are each normal, X+Y is normal (constants count as normal).*\n\nNormality of each marginal does not constrain the joint law. Note that the trivial pair Y = −X is **not** a counterexample here, since X+Y = 0 is a constant, which the proposition counts as (degenerate) normal. We need a sum that is non-constant and non-normal.\n\n**Construction.** Let X ~ N(0,1) and let S be independent of X with P(S=1)=P(S=−1)=1/2. Define Y = S·X.\n\n1. *Y is N(0,1).* For any Borel set B, P(Y∈B) = ½P(X∈B) + ½P(−X∈B) = P(X∈B), because −X has the same law as X (symmetry of N(0,1)) and S is independent of X.\n2. *X is N(0,1)* by definition.\n3. *X+Y is not normal.* X+Y = (1+S)X, which equals 0 when S=−1 and 2X when S=1. Hence P(X+Y=0) ≥ P(S=−1) = 1/2 (exactly 1/2, since P(X=0)=0). A non-degenerate normal distribution is absolutely continuous and gives probability 0 to every point, so X+Y is not non-degenerate normal. It is not degenerate either: P(X+Y=0)=1/2 and, given S=1, X+Y=2X is non-constant; Var(X+Y)=E[(1+S)²]E[X²]=2 > 0. Its law is the mixture ½δ₀ + ½N(0,4). ∎\n\nRemarks. X and Y are uncorrelated (Cov(X,Y)=E[S]E[X²]=0) yet dependent (|Y|=|X|), so (X,Y) is not bivariate normal; this is exactly why B's independence (or, more generally, joint normality) hypothesis is needed. The KS distance between ½δ₀+½N(0,4) and N(0,2) is large (≈0.25), so the departure is clearly visible at n = 100 000.\n\n## Numerical illustration (not a proof)\n`out/analysis.py` simulates this construction with the design seed (20261029) and n=100 000, writing `out/counterexample.csv`. See `out/results.json` → `counterexample_values` and `supplementary_diagnostics`.\nEOF\necho \"- 2026-10-30 P2 passed: baseline (independent-sum KS for B) in results.json baseline_* keys. P3 passed: analysis.py written, construction Y=S*X chosen (Y=-X rejected: degenerate sum counts as normal). 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Let X ~ N(μ₁, σ₁²), Y ~ N(μ₂, σ₂²) (σᵢ ≥ 0, σᵢ=0 meaning a constant). The characteristic function of N(μ, σ²) is φ(t) = exp(iμt − σ²t²/2). By independence, φ_{X+Y}(t) = φ_X(t)φ_Y(t) = exp(i(μ₁+μ₂)t − (σ₁²+σ₂²)t²/2), which is the characteristic function of N(μ₁+μ₂, σ₁²+σ₂²). By the uniqueness theorem for characteristic functions, X+Y ~ N(μ₁+μ₂, σ₁²+σ₂²) (degenerate if both σᵢ=0). ∎\n\n## Proposition A — FALSE\n*If X and Y are each normal, X+Y is normal (constants count as normal).*\n\nNormality of each marginal does not constrain the joint law. Note that the trivial pair Y = −X is **not** a counterexample here, since X+Y = 0 is a constant, which the proposition counts as (degenerate) normal. We need a sum that is non-constant and non-normal.\n\n**Construction.** Let X ~ N(0,1) and let S be independent of X with P(S=1)=P(S=−1)=1/2. Define Y = S·X.\n\n1. *Y is N(0,1).* For any Borel set B, P(Y∈B) = ½P(X∈B) + ½P(−X∈B) = P(X∈B), because −X has the same law as X (symmetry of N(0,1)) and S is independent of X.\n2. *X is N(0,1)* by definition.\n3. *X+Y is not normal.* X+Y = (1+S)X, which equals 0 when S=−1 and 2X when S=1. Hence P(X+Y=0) ≥ P(S=−1) = 1/2 (exactly 1/2, since P(X=0)=0). A non-degenerate normal distribution is absolutely continuous and gives probability 0 to every point, so X+Y is not non-degenerate normal. It is not degenerate either: P(X+Y=0)=1/2 and, given S=1, X+Y=2X is non-constant; Var(X+Y)=E[(1+S)²]E[X²]=2 > 0. Its law is the mixture ½δ₀ + ½N(0,4). ∎\n\nRemarks. X and Y are uncorrelated (Cov(X,Y)=E[S]E[X²]=0) yet dependent (|Y|=|X|), so (X,Y) is not bivariate normal; this is exactly why B's independence (or, more generally, joint normality) hypothesis is needed. The KS distance between ½δ₀+½N(0,4) and N(0,2) is large (≈0.25), so the departure is clearly visible at n = 100 000.\n\n## Numerical illustration (not a proof)\n`out/analysis.py` simulates this construction with the design seed (20261029) and n=100 000, writing `out/counterexample.csv`. See `out/results.json` → `counterexample_values` and `supplementary_diagnostics`.\nEOF\necho \"- 2026-10-30 P2 passed: baseline (independent-sum KS for B) in results.json baseline_* keys. P3 passed: analysis.py written, construction Y=S*X chosen (Y=-X rejected: degenerate sum counts as normal). 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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.\n\n## Data and exclusions\n`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.\n\n## Method\nVerdicts 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_*`).\n\n## Results\n- Proposition A is **false** (`statement_true` = false); a concrete counterexample is provided (`counterexample_provided` = true).\n- Proposition B is **true** (`statement_b_true` = true).\n- 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).\n- 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`).\n- 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).\n\n## Uncertainty\nThe 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.\n\n## Limitations and open issues\n- **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`.\n- Tolerance-based checks were not used for the atom: x+y = 0 holds exactly in floating point when y = −x.\n- No other open issues; feasibility is `complete`.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"Proposition A is false: with X~N(0,1), S an independent Rademacher sign, Y=S*X, both marginals are N(0,1) but X+Y has an atom of mass 1/2 at 0 and variance 2, so it is neither normal nor constant.\",\"results_keys\":[\"statement_true\",\"counterexample_provided\",\"counterexample_values.construction\"],\"status\":\"supported\",\"caveat\":\"Supported by proof in out/argument.md; the simulation only illustrates it.\"},\n {\"id\":\"C2\",\"text\":\"Proposition B is true (characteristic-function proof).\",\"results_keys\":[\"statement_b_true\"],\"status\":\"supported\",\"caveat\":\"Deductive; the independent-sum baseline KS is only an illustration.\"},\n {\"id\":\"C3\",\"text\":\"In the simulated sample (seed 20261029, n=100000) the marginals are consistent with N(0,1) (KS about 0.002) while x+y departs clearly from a fitted normal (KS 0.251, with 50.05% of rows exactly 0).\",\"results_keys\":[\"counterexample_values.ks_sum_vs_normal\",\"supplementary_diagnostics.ks_x_vs_normal\",\"supplementary_diagnostics.ks_y_vs_normal\",\"supplementary_diagnostics.share_sum_exactly_zero\",\"random_seed\",\"n_analysed\"],\"status\":\"supported\",\"caveat\":\"Numerical illustration with Monte Carlo error around 0.003; it is not a proof.\"},\n {\"id\":\"C4\",\"text\":\"X and Y in the counterexample are uncorrelated but dependent, so (X,Y) is not bivariate normal.\",\"results_keys\":[\"supplementary_diagnostics.corr_xy\"],\"status\":\"supported\",\"caveat\":\"Zero covariance is proven analytically; the sample correlation of -0.008 is only consistent with it.\"}\n]\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json\nph={\"P0\":[\"contract.md written\"],\"P1\":[\"sha256 computed and matched bundle.json\",\"no missing/duplicates\",\"exclusions.csv written (none)\"],\n\"P2\":[\"baseline_b_indep_ks_sum_vs_normal computed with same KS rule\"],\"P3\":[\"analysis.py reads only data/, fixed seed 20261029 (+1 for baseline)\"],\n\"P4\":[\"clean rerun: results.json byte-identical, csv sha identical\",\"10 consistency checks in checks.md all PASS\"],\n\"P5\":[\"report.md with required sections citing keys\",\"claims.json written\",\"json files parse\"]}\njson.dump({k:{\"status\":\"passed\",\"checks\":v,\"notes\":\"\"} for k,v in ph.items()},open('out/state/phases.json','w'),indent=2)\njson.load(open('out/claims.json'));json.load(open('out/results.json'));print('ok')\nEOF\necho \"- 2026-10-30 P5 passed: report, claims, argument written; all phases passed.\" >> out/state/log.md; ls out out/state","description":"Write report, claims, final phase ledger"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1954,"cache_read_input_tokens":10864,"output_tokens":2492,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":1954},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXpxHqEiqL76yhAzFqoy","requestRef":"2d1f0641-4234-4e1c-b6ca-90973e40b6a3","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"f1566ea9-b826-40e7-bbdf-5a2a219ce736","timestamp":"2026-10-30T04:56:54.456Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}46{"parentUuid":"f1566ea9-b826-40e7-bbdf-5a2a219ce736","isSidechain":false,"promptId":"a1301c1e-956d-45b7-a01c-3676354e4f64","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01FtH5Bt75AoVZtn1eYdU5jZ","type":"tool_result","content":"ok\nout:\nanalysis.py\nargument.md\nclaims.json\ncounterexample.csv\nexclusions.csv\nreport.md\nresults.json\nstate\n\nout/state:\nchecks.md\ncontract.md\ndata_audit.md\nlog.md\nphases.json","is_error":false}]},"uuid":"9e74f97c-1edc-4318-94e6-f777d7ccbaa7","timestamp":"2026-10-30T04:56:54.495Z","toolUseResult":{"stdout":"ok\nout:\nanalysis.py\nargument.md\nclaims.json\ncounterexample.csv\nexclusions.csv\nreport.md\nresults.json\nstate\n\nout/state:\nchecks.md\ncontract.md\ndata_audit.md\nlog.md\nphases.json","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"f1566ea9-b826-40e7-bbdf-5a2a219ce736","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}47{"type":"api-request-blob","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","hash":"0819beab545a0ed0a8372879694fcb7fdea395428b74da301e24cd8204cd7cea","message":{"role":"assistant","content":[{"type":"tool_use","id":"toolu_01FtH5Bt75AoVZtn1eYdU5jZ","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# Report — F5-conf-03: sums of normally distributed random variables\n\n## Question\nProposition 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.\n\n## Data and exclusions\n`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.\n\n## Method\nVerdicts 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_*`).\n\n## Results\n- Proposition A is **false** (`statement_true` = false); a concrete counterexample is provided (`counterexample_provided` = true).\n- Proposition B is **true** (`statement_b_true` = true).\n- 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).\n- 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`).\n- 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).\n\n## Uncertainty\nThe 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.\n\n## Limitations and open issues\n- **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`.\n- Tolerance-based checks were not used for the atom: x+y = 0 holds exactly in floating point when y = −x.\n- No other open issues; feasibility is `complete`.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"Proposition A is false: with X~N(0,1), S an independent Rademacher sign, Y=S*X, both marginals are N(0,1) but X+Y has an atom of mass 1/2 at 0 and variance 2, so it is neither normal nor constant.\",\"results_keys\":[\"statement_true\",\"counterexample_provided\",\"counterexample_values.construction\"],\"status\":\"supported\",\"caveat\":\"Supported by proof in out/argument.md; the simulation only illustrates it.\"},\n {\"id\":\"C2\",\"text\":\"Proposition B is true (characteristic-function proof).\",\"results_keys\":[\"statement_b_true\"],\"status\":\"supported\",\"caveat\":\"Deductive; the independent-sum baseline KS is only an illustration.\"},\n {\"id\":\"C3\",\"text\":\"In the simulated sample (seed 20261029, n=100000) the marginals are consistent with N(0,1) (KS about 0.002) while x+y departs clearly from a fitted normal (KS 0.251, with 50.05% of rows exactly 0).\",\"results_keys\":[\"counterexample_values.ks_sum_vs_normal\",\"supplementary_diagnostics.ks_x_vs_normal\",\"supplementary_diagnostics.ks_y_vs_normal\",\"supplementary_diagnostics.share_sum_exactly_zero\",\"random_seed\",\"n_analysed\"],\"status\":\"supported\",\"caveat\":\"Numerical illustration with Monte Carlo error around 0.003; it is not a proof.\"},\n {\"id\":\"C4\",\"text\":\"X and Y in the counterexample are uncorrelated but dependent, so (X,Y) is not bivariate normal.\",\"results_keys\":[\"supplementary_diagnostics.corr_xy\"],\"status\":\"supported\",\"caveat\":\"Zero covariance is proven analytically; the sample correlation of -0.008 is only consistent with it.\"}\n]\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json\nph={\"P0\":[\"contract.md written\"],\"P1\":[\"sha256 computed and matched bundle.json\",\"no missing/duplicates\",\"exclusions.csv written (none)\"],\n\"P2\":[\"baseline_b_indep_ks_sum_vs_normal computed with same KS rule\"],\"P3\":[\"analysis.py reads only data/, fixed seed 20261029 (+1 for baseline)\"],\n\"P4\":[\"clean rerun: results.json byte-identical, csv sha identical\",\"10 consistency checks in checks.md all PASS\"],\n\"P5\":[\"report.md with required sections citing keys\",\"claims.json written\",\"json files parse\"]}\njson.dump({k:{\"status\":\"passed\",\"checks\":v,\"notes\":\"\"} for k,v in ph.items()},open('out/state/phases.json','w'),indent=2)\njson.load(open('out/claims.json'));json.load(open('out/results.json'));print('ok')\nEOF\necho \"- 2026-10-30 P5 passed: report, claims, argument written; all phases passed.\" >> out/state/log.md; ls out out/state","description":"Write report, claims, final phase ledger"}}]}}48{"type":"api-request","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","id":"ae6d63cb-c932-4af1-b95d-469a40af281b","timestamp":"2026-10-30T04:56:54.499Z","version":"2.1.314","querySource":"sdk","shapeHash":"c476dccff750d1df4af4bcb23bd855a178afef7ca3ecd9c0205488ebcd9a51ab","params":{"model":"claude-opus-5-5","betas":["claude-code-20250219","interleaved-thinking-2025-05-14","context-management-2025-06-27","prompt-caching-scope-2026-01-05","per-turn-control-2026-07-01","thinking-token-count-2026-05-13","mid-conversation-system-clear-at-2026-08-21","effort-2025-11-24","thinking-binding-controls-2026-08-01","thinking-display-updates-2026-08-18","cache-diagnosis-2026-04-07"],"max_tokens":128000,"thinking":{"type":"adaptive","display":"updates"},"context_management":{"edits":[{"type":"clear_thinking_20251015","keep":"all"}]},"output_config":{"effort":"medium"}},"base":"2d1f0641-4234-4e1c-b6ca-90973e40b6a3","keep":14,"tail":["0819beab545a0ed0a8372879694fcb7fdea395428b74da301e24cd8204cd7cea","c7d9437c3e369d17528904944d3490efad9f3c2a80735af74b11439bb5b35ea6"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",15,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"b52f2af0e9a9e029a4c38d56719ddb89af136f8f125e1fd7611ee3ba2c989c26"}49{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"9e74f97c-1edc-4318-94e6-f777d7ccbaa7","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133"}50{"type":"atis-latch","atis":"","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133"}51{"parentUuid":"9e74f97c-1edc-4318-94e6-f777d7ccbaa7","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXpytfWsads7Xk9epV8f","type":"message","role":"assistant","content":[{"type":"text","text":"I've proved that Proposition B is true and that Proposition A is false, with a concrete counterexample. All required files are in `out/`, and re-running `out/analysis.py` from scratch reproduces the results and sample file exactly.\n\n- **Proposition B (X and Y independent and each normal, then X+Y is normal): true.** Multiplying the two normal characteristic functions gives the characteristic function of N(μ₁+μ₂, σ₁²+σ₂²), and a distribution is determined by its characteristic function.\n- **Proposition A (X and Y each normal, then X+Y is normal): false.**\n  - **Construction:** take X ~ N(0,1) and an independent random sign S (+1 or −1, each with probability ½), and set Y = S·X.\n  - **Why it works:** Y is exactly N(0,1) because the normal distribution is symmetric. But X+Y is 0 half the time and 2X otherwise. A non-constant normal variable never puts positive probability on a single point, so X+Y is not normal.\n  - **Why not the obvious example:** I didn't use Y = −X, because its sum is the constant 0, which the proposition counts as normal. This X+Y is not constant (its variance is 2).\n  - **Side point:** X and Y are uncorrelated but not independent, which is why B needs its independence assumption.\n- **Simulation (seed 20261029, n = 100,000, as the design specifies):**\n  - x and y each look like N(0,1): KS distances to a fitted normal are about 0.002, below the 5% critical value of about 0.0043.\n  - x+y does not: its KS distance is 0.251, and 50.05% of rows are exactly 0.\n  - For comparison, the sum of two independent normal samples has a KS distance of 0.0019.\n\nThe simulation only illustrates the counterexample; the verdicts rest on the proofs in `out/argument.md`. There were no data issues and no exclusions, and nothing is left incomplete.\n\nFiles are in `out/`:\n- results.json\n- counterexample.csv\n- argument.md\n- report.md\n- analysis.py\n- claims.json\n- exclusions.csv\n- state/ (working notes and checks)"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":2575,"cache_read_input_tokens":12818,"output_tokens":720,"output_tokens_details":{"thinking_tokens":0},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":2575},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXpytBk3xSU1nnXPt7Dq","requestRef":"ae6d63cb-c932-4af1-b95d-469a40af281b","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"d356fa90-378d-4bc0-949f-0b90bb30e690","timestamp":"2026-10-30T04:57:02.170Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0191","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","version":"2.1.314","gitBranch":"HEAD"}52{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"d356fa90-378d-4bc0-949f-0b90bb30e690","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133"}53{"type":"cost-state","sessionId":"46b00530-fc25-44c8-a952-fa4a0b4f9133","totalCostUSD":0.2658714,"totalAPIDuration":86261,"totalAPIDurationWithoutRetries":86255,"totalToolDuration":1924,"totalLinesAdded":0,"totalLinesRemoved":0,"totalDuration":88344,"startTime":1793336133832,"modelUsage":{"claude-opus-5-5":{"inputTokens":18,"outputTokens":9232,"thinkingTokens":690,"cacheReadInputTokens":58572,"cacheCreationInputTokens":13889,"cacheCreationEphemeral5mInputTokens":13889,"cacheCreationEphemeral1hInputTokens":0,"webSearchRequests":0,"costUSD":0.2658714}},"hasUnknownModelCost":false}54