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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`. Below is some general advice about doing careful empirical research; it adds nothing specific to this task and you are free to ignore it.\n\nCareful empirical work usually begins by reading the task statement slowly and more than once, because the most common errors in applied research come not from difficult mathematics but from a misunderstanding of what exactly is being asked, which quantity is of interest, and which inputs are legitimately available. It is worth pausing to think about what the quantity of interest actually means, under what conditions it is well defined, and what would make an estimate of it misleading. Many published results have been overturned not because the estimator was wrong but because the sample was not what the authors believed it to be, or because information that should not have been available at the time of a decision had quietly entered the analysis.\n\nData deserve suspicion. Files assembled from several sources often carry inconsistencies: duplicated records, mixed units, shifted dates, columns whose definitions changed mid-sample, or values that were revised after the fact. Good practice is to look at the data before modelling them: distributions, ranges, missing values, identifiers, and whether counts add up. When something looks odd, it is better to document it and decide explicitly how to handle it than to let a default silently decide. The same applies to transformations: scaling, differencing, winsorising and imputing all change what is being estimated, and each deserves a sentence of justification.\n\nSimple baselines are underrated. A naïve or unconditional estimate computed with exactly the same evaluation rule as the main method tells the reader how much the method actually adds and protects against the illusion of progress created by a complicated pipeline. If the elaborate method does not beat the simple one, that is a finding, not a failure.\n\nUncertainty is part of the result. Standard errors must respect the dependence structure of the data; a confidence interval that ignores clustering, serial correlation or overlapping targets is not conservative, it is wrong. Where the sampling distribution is unknown, resampling or simulation can help, but the assumptions behind them should be stated. A result that is not statistically distinguishable from zero should be reported as such, in plain words.\n\nReproducibility is a courtesy to the reader and a safeguard for the author. Keeping the analysis in a script that reads the raw inputs and regenerates every reported number makes it possible to check the work, to find mistakes, and to rerun the analysis when the inputs change. Fixing random seeds, recording software versions and keeping intermediate files out of the final results all reduce the chance that a number in the report cannot be traced back to a computation.\n\nFinally, honesty about limitations is what distinguishes research from advocacy. Every analysis has weaknesses: assumptions that may not hold, data that may not be representative, methods that trade one bias for another. Stating them clearly, together with what would be needed to resolve them, is more valuable than a confident conclusion that the evidence does not support. 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Consider the fixed-design linear model y_i = α + β x_i + σ_i ε_i, i = 1, …, 30, with α = 1, β = 0.5, ε_i independent standard normal, and x_i and σ_i fixed at the values given in `data/design.csv` (the same 30 values in every replication; only ε is redrawn). The estimands are (a) the true coverage probability of the interval β̂ ± t_{0.975}(28) · se_conv, where se_conv is the conventional OLS standard error (residual variance with divisor n − 2); (b) the true coverage probability of the interval β̂ ± t_{0.975}(28) · se_HC1, where se_HC1 is the heteroskedasticity-robust HC1 standard error (White's HC0 sandwich multiplied by n/(n − 2)); (c) the exact finite-sample standard deviation of β̂ under this design; (d) the expected value of se_conv. Estimate (a), (b) and (d) by Monte Carlo with exactly R = 2000 replications and a fixed seed, compute (c) analytically, and conclude whether the conventional interval attains its nominal coverage.\n\n## Inputs\n- `data/design.csv` — the fixed design (a synthetic design generated for this exercise; not observed data). Columns: `i` (observation index 1–30), `x` (regressor value, dimensionless), `sigma` (standard deviation of the error for observation i, dimensionless). 30 rows.\n\n## Information-availability rules\n- Use only `data/design.csv`; the model constants (α = 1, β = 0.5, n = 30, R = 2000, nominal level 95%, t(28) critical value) are given above. Do not redraw x or σ.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | – | \"F2-conf-03\" |\n| `coverage_conventional` | number | proportion | share of the 2000 replications in which the conventional interval contains β = 0.5 — the naïve baseline |\n| `coverage_hc1` | number | proportion | share of the 2000 replications in which the HC1 interval contains β = 0.5 |\n| `sd_beta_exact` | number | – | exact standard deviation of β̂ under the design (analytical, no simulation) |\n| `sd_beta_mc` | number | – | sample standard deviation (divisor R − 1) of the 2000 simulated β̂ |\n| `mean_se_conventional` | number | – | average of se_conv over the 2000 replications |\n| `conventional_ci_undercovers` | boolean | – | true if and only if `coverage_conventional` is below 0.95 by more than three Monte Carlo standard errors (binomial standard error at R = 2000) |\n| `n_replications` | integer | – | number of replications (2000) |\n| `n_obs_per_replication` | integer | – | sample size per replication (30) |\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 | – | {\"design.csv\": rows read} |\n| `random_seed` | integer | – | the seed used for the simulation (required; the simulation must be reproducible) |\n\n- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty (include the Monte Carlo standard error of each coverage estimate); Limitations and open issues.\n- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically (same seed, same numbers).\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}]. Include one claim about whether the conventional interval attains its nominal coverage.\n\n## Constraints\n- No network access. Python: `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels available). Do not modify `data/`.\n- This task is designed to be completed in well under 15 minutes of work; the simulation itself should take seconds.\ndesign.csv","is_error":false}]},"uuid":"a0150241-50b4-4344-9f11-068b51176e19","timestamp":"2026-10-30T05:31:10.872Z","toolUseResult":{"stdout":"# F2-conf-03 — Monte Carlo coverage of OLS confidence intervals in a small heteroskedastic sample\n\n## Scientific question\nHow well do nominal 95% confidence intervals for an OLS slope perform in a small sample with heteroskedastic errors? Consider the fixed-design linear model y_i = α + β x_i + σ_i ε_i, i = 1, …, 30, with α = 1, β = 0.5, ε_i independent standard normal, and x_i and σ_i fixed at the values given in `data/design.csv` (the same 30 values in every replication; only ε is redrawn). 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Columns: `i` (observation index 1–30), `x` (regressor value, dimensionless), `sigma` (standard deviation of the error for observation i, dimensionless). 30 rows.\n\n## Information-availability rules\n- Use only `data/design.csv`; the model constants (α = 1, β = 0.5, n = 30, R = 2000, nominal level 95%, t(28) critical value) are given above. Do not redraw x or σ.\n\n## Required outputs\n- `out/results.json` with the keys:\n\n| key | type | unit | meaning |\n|---|---|---|---|\n| `task_id` | string | – | \"F2-conf-03\" |\n| `coverage_conventional` | number | proportion | share of the 2000 replications in which the conventional interval contains β = 0.5 — the naïve baseline |\n| `coverage_hc1` | number | proportion | share of the 2000 replications in which the HC1 interval contains β = 0.5 |\n| `sd_beta_exact` | number | – | exact standard deviation of β̂ under the design (analytical, no simulation) |\n| `sd_beta_mc` | number | – | sample standard deviation (divisor R − 1) of the 2000 simulated β̂ |\n| `mean_se_conventional` | number | – | average of se_conv over the 2000 replications |\n| `conventional_ci_undercovers` | boolean | – | true if and only if `coverage_conventional` is below 0.95 by more than three Monte Carlo standard errors (binomial standard error at R = 2000) |\n| `n_replications` | integer | – | number of replications (2000) |\n| `n_obs_per_replication` | integer | – | sample size per replication (30) |\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 | – | {\"design.csv\": rows read} |\n| `random_seed` | integer | – | the seed used for the simulation (required; the simulation must be reproducible) |\n\n- `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty (include the Monte Carlo standard error of each coverage estimate); Limitations and open issues.\n- `out/analysis.py` — running `~/wp29/venv/bin/python out/analysis.py` from the workspace root must regenerate `out/results.json` from `data/` only, deterministically (same seed, same numbers).\n- `out/claims.json` — a list of the central claims of the report: [{\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}]. 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data/design.csv"}}]}}19{"type":"api-request","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","id":"a6e4b67f-0989-4615-a195-c7acf7a9db31","timestamp":"2026-10-30T05:31:12.214Z","version":"2.1.314","querySource":"sdk","shapeHash":"8fc35f110db5e413807114f5bc77e0180907cfbb770dc6ee6e8b290889640e34","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":"a37fb342-eae2-4ece-b83b-21f5c9032929","keep":4,"tail":["fad834f25a5e4bc6646cf54d607d13d1ded5d2769f7b897483906fd29b56586b","f15689476ba5cbb0c6eb41e17cb8ccce0a2c83e18a3bce9467344f37f0d635d2"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",5,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"8417321ab8a2dfdd7ef2d1cb85c533bc6ce7cfa517972b88d339ec925bb4086b"}20{"parentUuid":"c62fa272-70c4-44c1-8f13-408aba5d09c0","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXsbZ7mnZ2F6TYS9CU2n","type":"message","role":"assistant","content":[{"type":"text","text":"Data look clean (30 rows, positive sigma, no dupes). Writing the script."}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":434,"cache_read_input_tokens":4832,"output_tokens":1642,"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":434},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXsbYqQCDDDbkX1sf7uw","requestRef":"a6e4b67f-0989-4615-a195-c7acf7a9db31","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"8a3cd046-48f4-4fab-9e1d-014fd3069c0d","timestamp":"2026-10-30T05:31:13.383Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RA0002","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","version":"2.1.314","gitBranch":"HEAD"}21{"parentUuid":"8a3cd046-48f4-4fab-9e1d-014fd3069c0d","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXsbZ7mnZ2F6TYS9CU2n","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01HjNPupddYPLHF2aj529B22","name":"Bash","input":{"command":"mkdir -p out; cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, R, LEVEL = 1.0, 0.5, 2000, 0.95\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nn_rows = len(d)\nassert list(d.columns) == [\"i\", \"x\", \"sigma\"]\nif d.isna().any().any(): issues.append({\"issue\": \"missing values\", \"handling\": \"none expected; would abort\"})\nif d[\"i\"].duplicated().any(): issues.append({\"issue\": \"duplicate indices\", \"handling\": \"reported\"})\nif (d[\"sigma\"] <= 0).any(): issues.append({\"issue\": \"non-positive sigma\", \"handling\": \"reported\"})\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float); n = len(x)\nassert n == 30\n\nX = np.column_stack([np.ones(n), x])\nXtXi = np.linalg.inv(X.T @ X)\nA = XtXi @ X.T # 2 x n, beta_hat = A y\n# exact variance of slope: sum a_i^2 sigma_i^2\nsd_exact = float(np.sqrt(np.sum(A[1] ** 2 * sig ** 2)))\n\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, n))\nY = ALPHA + BETA * x[None, :] + sig[None, :] * eps # R x n\nB = Y @ A.T # R x 2\nE = Y - B @ X.T # residuals\ns2 = (E ** 2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtXi[1, 1])\nhc0 = (E ** 2) @ (A[1] ** 2) # sandwich slope element\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\ntc = stats.t.ppf(0.5 + LEVEL / 2, n - 2)\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tc * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tc * se_hc1))\nmcse_nom = np.sqrt(LEVEL * (1 - LEVEL) / R)\nundercov = bool(cov_conv < LEVEL - 3 * mcse_nom)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercov,\n \"n_replications\": R,\n \"n_obs_per_replication\": n,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": None,\n \"data_issues\": issues,\n \"n_input_rows\": {\"design.csv\": n_rows},\n \"random_seed\": SEED,\n}\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/results.json\", \"w\") as f: json.dump(res, f, indent=2)\n# auxiliary diagnostics (printed, not part of results.json)\nprint(json.dumps(res, indent=2))\nprint(\"t crit\", tc, \"MCSE at 0.95\", mcse_nom,\n \"MCSE conv\", np.sqrt(cov_conv*(1-cov_conv)/R), \"MCSE hc1\", np.sqrt(cov_hc1*(1-cov_hc1)/R),\n \"mean se_hc1\", se_hc1.mean(), \"naive sd formula\", np.sqrt(np.mean(sig**2)*XtXi[1,1]),\n \"E[se_conv^2] exact sqrt\", np.sqrt(XtXi[1,1]*np.sum(sig**2*np.diag(np.eye(n)-X@XtXi@X.T))/(n-2)))\nEOF\n~/wp29/venv/bin/python out/analysis.py; md5 out/results.json; ~/wp29/venv/bin/python out/analysis.py >/dev/null; md5 out/results.json"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":434,"cache_read_input_tokens":4832,"output_tokens":1642,"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":434},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXsbYqQCDDDbkX1sf7uw","requestRef":"a6e4b67f-0989-4615-a195-c7acf7a9db31","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"883dba24-8a04-4bf5-9f4d-f2944136e988","timestamp":"2026-10-30T05:31:25.122Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RA0002","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","version":"2.1.314","gitBranch":"HEAD"}22{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"883dba24-8a04-4bf5-9f4d-f2944136e988","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f"}23{"type":"atis-latch","atis":"","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f"}24{"parentUuid":"883dba24-8a04-4bf5-9f4d-f2944136e988","isSidechain":false,"promptId":"be0f8734-95bb-4be8-9f71-514cc89e20e5","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01HjNPupddYPLHF2aj529B22","type":"tool_result","content":"{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.2679805928302833,\n \"sd_beta_mc\": 0.2696473708309126,\n \"mean_se_conventional\": 0.1902623101215266,\n \"conventional_ci_undercovers\": true,\n \"n_replications\": 2000,\n \"n_obs_per_replication\": 30,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": null,\n \"data_issues\": [],\n \"n_input_rows\": {\n \"design.csv\": 30\n },\n \"random_seed\": 20261030\n}\nt crit 2.048407141795244 MCSE at 0.95 0.004873397172404485 MCSE conv 0.00811369213120636 MCSE hc1 0.006648157639526908 mean se_hc1 0.24080977065103962 naive sd formula 0.19623471734704753 E[se_conv^2] exact sqrt 0.19318017740129875\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496","is_error":false}]},"uuid":"0f3dc83d-ad10-432c-944b-3493f9ceb900","timestamp":"2026-10-30T05:31:26.554Z","toolUseResult":{"stdout":"{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.2679805928302833,\n \"sd_beta_mc\": 0.2696473708309126,\n \"mean_se_conventional\": 0.1902623101215266,\n \"conventional_ci_undercovers\": true,\n \"n_replications\": 2000,\n \"n_obs_per_replication\": 30,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": null,\n \"data_issues\": [],\n \"n_input_rows\": {\n \"design.csv\": 30\n },\n \"random_seed\": 20261030\n}\nt crit 2.048407141795244 MCSE at 0.95 0.004873397172404485 MCSE conv 0.00811369213120636 MCSE hc1 0.006648157639526908 mean se_hc1 0.24080977065103962 naive sd formula 0.19623471734704753 E[se_conv^2] exact sqrt 0.19318017740129875\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496\nMD5 (out/results.json) = 67879644da732090cb4236ecef056496","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"883dba24-8a04-4bf5-9f4d-f2944136e988","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RA0002","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","version":"2.1.314","gitBranch":"HEAD"}25{"type":"api-request-blob","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","hash":"c23ab227663b2b98d458c65ae405ab64b5e570d4d1a166fcaf63463670c0bdb0","message":{"role":"assistant","content":[{"type":"text","text":"Data look clean (30 rows, positive sigma, no dupes). Writing the script."},{"type":"tool_use","id":"toolu_01HjNPupddYPLHF2aj529B22","name":"Bash","input":{"command":"mkdir -p out; cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, R, LEVEL = 1.0, 0.5, 2000, 0.95\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nn_rows = len(d)\nassert list(d.columns) == [\"i\", \"x\", \"sigma\"]\nif d.isna().any().any(): issues.append({\"issue\": \"missing values\", \"handling\": \"none expected; would abort\"})\nif d[\"i\"].duplicated().any(): issues.append({\"issue\": \"duplicate indices\", \"handling\": \"reported\"})\nif (d[\"sigma\"] <= 0).any(): issues.append({\"issue\": \"non-positive sigma\", \"handling\": \"reported\"})\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float); n = len(x)\nassert n == 30\n\nX = np.column_stack([np.ones(n), x])\nXtXi = np.linalg.inv(X.T @ X)\nA = XtXi @ X.T # 2 x n, beta_hat = A y\n# exact variance of slope: sum a_i^2 sigma_i^2\nsd_exact = float(np.sqrt(np.sum(A[1] ** 2 * sig ** 2)))\n\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, n))\nY = ALPHA + BETA * x[None, :] + sig[None, :] * eps # R x n\nB = Y @ A.T # R x 2\nE = Y - B @ X.T # residuals\ns2 = (E ** 2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtXi[1, 1])\nhc0 = (E ** 2) @ (A[1] ** 2) # sandwich slope element\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\ntc = stats.t.ppf(0.5 + LEVEL / 2, n - 2)\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tc * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tc * se_hc1))\nmcse_nom = np.sqrt(LEVEL * (1 - LEVEL) / R)\nundercov = bool(cov_conv < LEVEL - 3 * mcse_nom)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercov,\n \"n_replications\": R,\n \"n_obs_per_replication\": n,\n \"feasibility\": \"complete\",\n \"feasibility_reason\": None,\n \"data_issues\": issues,\n \"n_input_rows\": {\"design.csv\": n_rows},\n \"random_seed\": SEED,\n}\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/results.json\", \"w\") as f: json.dump(res, f, indent=2)\n# auxiliary diagnostics (printed, not part of results.json)\nprint(json.dumps(res, indent=2))\nprint(\"t crit\", tc, \"MCSE at 0.95\", mcse_nom,\n \"MCSE conv\", np.sqrt(cov_conv*(1-cov_conv)/R), \"MCSE hc1\", np.sqrt(cov_hc1*(1-cov_hc1)/R),\n \"mean se_hc1\", se_hc1.mean(), \"naive sd formula\", np.sqrt(np.mean(sig**2)*XtXi[1,1]),\n \"E[se_conv^2] exact sqrt\", np.sqrt(XtXi[1,1]*np.sum(sig**2*np.diag(np.eye(n)-X@XtXi@X.T))/(n-2)))\nEOF\n~/wp29/venv/bin/python out/analysis.py; md5 out/results.json; ~/wp29/venv/bin/python out/analysis.py >/dev/null; md5 out/results.json"}}]}}26{"type":"api-request","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","id":"b4dc0834-e3d1-40fc-b398-bed1ca6a12c6","timestamp":"2026-10-30T05:31:26.559Z","version":"2.1.314","querySource":"sdk","shapeHash":"8fc35f110db5e413807114f5bc77e0180907cfbb770dc6ee6e8b290889640e34","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":"a6e4b67f-0989-4615-a195-c7acf7a9db31","keep":6,"tail":["c23ab227663b2b98d458c65ae405ab64b5e570d4d1a166fcaf63463670c0bdb0","f952e577761501625b7bb421fb7f17055f7c6e8c5c2e03ad6193d653d1e15e0d"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",7,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"aa8b679422e96fd7a9fce5177887c96e9fbedb1f34b60151f84b4b0c683244e6"}27{"parentUuid":"0f3dc83d-ad10-432c-944b-3493f9ceb900","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXsccRyizdfwRyD8PVRR","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":2060,"cache_read_input_tokens":5266,"output_tokens":2345,"output_tokens_details":{"thinking_tokens":61},"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":2060},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"thinkingDurationMs":430,"thinkingDisplay":"updates","apiBlockIndex":0,"requestId":"req_011CgXsccA6ZGwYszTgpkyV3","requestRef":"b4dc0834-e3d1-40fc-b398-bed1ca6a12c6","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"fa372a22-3167-41ff-9419-3d6997b15369","timestamp":"2026-10-30T05:31:28.128Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RA0002","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","version":"2.1.314","gitBranch":"HEAD"}28{"parentUuid":"fa372a22-3167-41ff-9419-3d6997b15369","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXsccRyizdfwRyD8PVRR","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_016iQEKEBayHCVt1QNfdzwzQ","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS slope CIs, n = 30, heteroskedastic fixed design\n\n## Question\nIn y_i = 1 + 0.5 x_i + σ_i ε_i (ε_i iid N(0,1), x_i and σ_i fixed from `data/design.csv`, n = 30), what is the true coverage of the nominal 95% intervals β̂ ± t_{0.975}(28)·se for (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) exact SD of β̂, (d) E[se_conv]. Does the conventional interval attain 95%?\n\n## Data and exclusions\n`data/design.csv`: 30 rows read, columns i, x, sigma. No missing values, no duplicate indices, all σ_i > 0; no exclusions, no data issues. The design is strongly heteroskedastic: σ_i rises with |x_i| (from ≈0.60 near x = 0 to 1.79 at x = 1.99), i.e. the error variance is largest exactly at the high-leverage points.\n\n## Method\n- (c) Analytically: β̂ = a'y with a the slope row of (X'X)⁻¹X', so Var(β̂) = Σ a_i² σ_i²; sd_beta_exact = √ of that.\n- Monte Carlo: R = 2000 replications, numpy `default_rng(20261030)`, only ε redrawn. In each replication: OLS fit; se_conv = √(s²·[(X'X)⁻¹]₂₂) with s² = RSS/(n−2); se_HC1 = √(n/(n−2) · Σ a_i² e_i²) (HC0 sandwich × n/(n−2)). Critical value t_{0.975}(28) = 2.0484. Coverage = share of replications with |β̂ − 0.5| ≤ t·se.\n- Undercoverage decision: coverage_conventional < 0.95 − 3·√(0.95·0.05/2000) = 0.95 − 3·0.00487 = 0.9354.\n- `out/analysis.py` regenerates `out/results.json` deterministically (verified by identical checksums over two runs).\n\n## Results\n| quantity | value |\n|---|---|\n| Coverage, conventional interval | 0.844 |\n| Coverage, HC1 interval | 0.902 |\n| Exact SD of β̂ (analytical) | 0.2680 |\n| MC SD of β̂ (ddof = 1) | 0.2696 |\n| Mean se_conv | 0.1903 |\n| (diagnostic, not in results.json) mean se_HC1 | 0.2408 |\n\nThe conventional SE is on average about 29% too small relative to the true SD of β̂ (0.190 vs 0.268), because it assumes a constant variance while the largest variances sit at the high-leverage x values. Consequently the conventional interval covers β only 84.4% of the time: **it does not attain its nominal 95% coverage** (`conventional_ci_undercovers` = true). HC1 helps substantially (90.2%) but is still clearly below 95% in this small sample, since HC1 is itself downward biased under high leverage (mean se_HC1 0.241 < 0.268). The MC SD of β̂ agrees with the analytical value within MC error, a check on the simulation.\n\n## Uncertainty\nBinomial Monte Carlo standard errors (R = 2000):\n- coverage_conventional: √(0.844·0.156/2000) = 0.0081 (≈ 95% MC interval 0.828–0.860).\n- coverage_hc1: √(0.902·0.098/2000) = 0.0066 (≈ 95% MC interval 0.889–0.915).\n- At the nominal 0.95, MCSE = 0.0049; the conventional estimate is ≈ 22 such SEs below 0.95, and the HC1 estimate ≈ 10 below, so both shortfalls are far beyond MC noise.\n- The MC SD of β̂ has relative MC error ≈ 1/√(2(R−1)) ≈ 1.6%; observed deviation from exact is 0.6%.\n\n## Limitations and open issues\n- Results are specific to this fixed design (heteroskedasticity aligned with leverage) and to Gaussian errors; with other designs the conventional interval may be fine.\n- Coverage is estimated with a single seed and R = 2000 as prescribed; MC error (~0.7–0.8 pp) is reported above.\n- Better small-sample alternatives (HC2/HC3, wild bootstrap, Bell–McCaffrey df corrections) were not evaluated because the task did not ask for them; HC1 with t(28) still undercovers here.\n- E[se_conv] is estimated by MC only, as requested (an exact closed form for E[√s²] is not simple; √E[se_conv²] = 0.1932 analytically, consistent with the MC mean 0.1903 by Jensen's inequality).\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain its nominal coverage: estimated coverage is 0.844, more than three Monte Carlo standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed heteroskedastic design with Gaussian errors; MC SE ≈ 0.008.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95% in this n = 30 design.\",\"results_keys\":[\"coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"MC SE ≈ 0.0066; HC2/HC3 or bootstrap alternatives not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is well below the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean se_conv is a Monte Carlo estimate.\"},\n {\"id\":\"C4\",\"text\":\"The simulated SD of the slope estimator (0.2696) matches the analytical exact SD (0.2680) within Monte Carlo error.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Relative MC error of the SD is about 1.6%.\"}\n]\nEOF\n~/wp29/venv/bin/python -c \"import json;json.load(open('out/claims.json'))\"; 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Also: (c) exact SD of β̂, (d) E[se_conv]. Does the conventional interval attain 95%?\n\n## Data and exclusions\n`data/design.csv`: 30 rows read, columns i, x, sigma. No missing values, no duplicate indices, all σ_i > 0; no exclusions, no data issues. The design is strongly heteroskedastic: σ_i rises with |x_i| (from ≈0.60 near x = 0 to 1.79 at x = 1.99), i.e. the error variance is largest exactly at the high-leverage points.\n\n## Method\n- (c) Analytically: β̂ = a'y with a the slope row of (X'X)⁻¹X', so Var(β̂) = Σ a_i² σ_i²; sd_beta_exact = √ of that.\n- Monte Carlo: R = 2000 replications, numpy `default_rng(20261030)`, only ε redrawn. In each replication: OLS fit; se_conv = √(s²·[(X'X)⁻¹]₂₂) with s² = RSS/(n−2); se_HC1 = √(n/(n−2) · Σ a_i² e_i²) (HC0 sandwich × n/(n−2)). Critical value t_{0.975}(28) = 2.0484. Coverage = share of replications with |β̂ − 0.5| ≤ t·se.\n- Undercoverage decision: coverage_conventional < 0.95 − 3·√(0.95·0.05/2000) = 0.95 − 3·0.00487 = 0.9354.\n- `out/analysis.py` regenerates `out/results.json` deterministically (verified by identical checksums over two runs).\n\n## Results\n| quantity | value |\n|---|---|\n| Coverage, conventional interval | 0.844 |\n| Coverage, HC1 interval | 0.902 |\n| Exact SD of β̂ (analytical) | 0.2680 |\n| MC SD of β̂ (ddof = 1) | 0.2696 |\n| Mean se_conv | 0.1903 |\n| (diagnostic, not in results.json) mean se_HC1 | 0.2408 |\n\nThe conventional SE is on average about 29% too small relative to the true SD of β̂ (0.190 vs 0.268), because it assumes a constant variance while the largest variances sit at the high-leverage x values. Consequently the conventional interval covers β only 84.4% of the time: **it does not attain its nominal 95% coverage** (`conventional_ci_undercovers` = true). HC1 helps substantially (90.2%) but is still clearly below 95% in this small sample, since HC1 is itself downward biased under high leverage (mean se_HC1 0.241 < 0.268). The MC SD of β̂ agrees with the analytical value within MC error, a check on the simulation.\n\n## Uncertainty\nBinomial Monte Carlo standard errors (R = 2000):\n- coverage_conventional: √(0.844·0.156/2000) = 0.0081 (≈ 95% MC interval 0.828–0.860).\n- coverage_hc1: √(0.902·0.098/2000) = 0.0066 (≈ 95% MC interval 0.889–0.915).\n- At the nominal 0.95, MCSE = 0.0049; the conventional estimate is ≈ 22 such SEs below 0.95, and the HC1 estimate ≈ 10 below, so both shortfalls are far beyond MC noise.\n- The MC SD of β̂ has relative MC error ≈ 1/√(2(R−1)) ≈ 1.6%; observed deviation from exact is 0.6%.\n\n## Limitations and open issues\n- Results are specific to this fixed design (heteroskedasticity aligned with leverage) and to Gaussian errors; with other designs the conventional interval may be fine.\n- Coverage is estimated with a single seed and R = 2000 as prescribed; MC error (~0.7–0.8 pp) is reported above.\n- Better small-sample alternatives (HC2/HC3, wild bootstrap, Bell–McCaffrey df corrections) were not evaluated because the task did not ask for them; HC1 with t(28) still undercovers here.\n- E[se_conv] is estimated by MC only, as requested (an exact closed form for E[√s²] is not simple; √E[se_conv²] = 0.1932 analytically, consistent with the MC mean 0.1903 by Jensen's inequality).\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain its nominal coverage: estimated coverage is 0.844, more than three Monte Carlo standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed heteroskedastic design with Gaussian errors; MC SE ≈ 0.008.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95% in this n = 30 design.\",\"results_keys\":[\"coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"MC SE ≈ 0.0066; HC2/HC3 or bootstrap alternatives not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is well below the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean se_conv is a Monte Carlo estimate.\"},\n {\"id\":\"C4\",\"text\":\"The simulated SD of the slope estimator (0.2696) matches the analytical exact SD (0.2680) within Monte Carlo error.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Relative MC error of the SD is about 1.6%.\"}\n]\nEOF\n~/wp29/venv/bin/python -c \"import json;json.load(open('out/claims.json'))\"; 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The conventional 95% interval for the slope does **not** reach its nominal coverage: it contains the true β = 0.5 only 84.4% of the time. All required files are in `out/`, and running `out/analysis.py` twice gave byte-identical results.\n\n| Quantity | Value |\n|---|---|\n| (a) Coverage, conventional interval | 0.844 (Monte Carlo standard error 0.008) |\n| (b) Coverage, HC1 robust interval | 0.902 (Monte Carlo standard error 0.007) |\n| (c) Exact SD of β̂ (worked out analytically) | 0.2680 |\n| Simulated SD of β̂ | 0.2696 |\n| (d) Average conventional standard error | 0.1903 |\n\n- **Conventional interval:** it undercovers by far more than the three-standard-error threshold the task sets (0.9354), so `conventional_ci_undercovers` = true. Its standard error averages 0.190 against a true SD of 0.268, about 29% too small. The cause is that the error variance is largest at the extreme x values, which have the most influence on the slope.\n- **HC1 robust interval:** it improves coverage but still falls well short of 95%, because the robust standard error is also too small in a sample of 30 (it averages 0.241).\n- **Check on the simulation:** the simulated SD of β̂ is within 0.6% of the exact value, inside the expected Monte Carlo error of about 1.6%.\n\nThe design file was clean, so nothing was excluded and no data issues were recorded.\n\n**Limitations:**\n- The results hold only for this particular design and normally distributed errors.\n- Other small-sample fixes (HC2/HC3 standard errors, a wild bootstrap) were not tested because the task didn't ask for them.\n- The average conventional standard error comes from the simulation only, as the task specified.\n\nFiles are in `out/`:\n- results.json\n- report.md\n- analysis.py\n- claims.json"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":2368,"cache_read_input_tokens":7326,"output_tokens":747,"output_tokens_details":{"thinking_tokens":77},"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":2368},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXse85TnHQv55DgKvX2f","requestRef":"27d4c1da-eaab-4032-a4cd-780d4044d841","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"f4dd488b-d843-42f2-8f6b-7c847f17205a","timestamp":"2026-10-30T05:31:54.404Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RA0002","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","version":"2.1.314","gitBranch":"HEAD"}34{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"f4dd488b-d843-42f2-8f6b-7c847f17205a","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f"}35{"type":"cost-state","sessionId":"4277bebd-6798-4c72-bce4-f2898534282f","totalCostUSD":0.1423808,"totalAPIDuration":43887,"totalAPIDurationWithoutRetries":43881,"totalToolDuration":1567,"totalLinesAdded":0,"totalLinesRemoved":0,"totalDuration":48646,"startTime":1793338265766,"modelUsage":{"claude-opus-5-5":{"inputTokens":12,"outputTokens":4848,"thinkingTokens":138,"cacheReadInputTokens":22114,"cacheCreationInputTokens":8190,"cacheCreationEphemeral5mInputTokens":8190,"cacheCreationEphemeral1hInputTokens":0,"webSearchRequests":0,"costUSD":0.1423808}},"hasUnknownModelCost":false}36