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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\nEXECUTABLE GATES (additional, mandatory)\n- A read-only gate program is provided at `~/wp29/gates/pgar_gate.py`. Run it from the workspace root as `~/wp29/venv/bin/python ~/wp29/gates/pgar_gate.py .` It checks the output contract of `TASK.md` (required files, keys, types, finite values, report sections, the claims ledger, the phase ledger, the exclusion ledger) and reproducibility (it re-runs `out/analysis.py` in a temporary copy and compares the regenerated `out/results.json` with yours). It never checks scientific correctness and it contains no reference answers.\n- You must run the gate at the end of P4 and again at the end of P5, and record each run's receipt (the gate writes `out/state/gate_receipts/<timestamp>.json`) in `out/state/phases.json`. You may not declare the task complete while the latest gate receipt reports `\"passed\": false`. Fix the reported problems and re-run the gate.\n- The controller will run the same gate after your session ends. If it fails, you will be resumed with the gate output and asked to fix the problems; each such cycle consumes your remaining budget.\n\nSEPARATE REVIEW (additional)\n- After the gate passes, a separate reviewer session that has not seen your work in progress will examine `TASK.md` and everything under `out/` and may return a list of issues (file `out/review/review.json`). You will then be resumed once to address them: fix what is justified, explain in `out/report.md` (Limitations and open issues) what you disagree with, re-run the gate, and update `out/claims.json` and `out/state/phases.json`. The reviewer has no reference answers; its role is to challenge unsupported claims, violated information rules, unverified computations, and inconsistencies between your report and your results.\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/RC0051","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}11{"type":"tools-used","tools":["Bash"],"sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba"}12{"parentUuid":"9adb76ee-f467-407f-b842-dca052dcaaca","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmng7ZY9iFHVFFCngFk","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":4,"cache_creation_input_tokens":2811,"cache_read_input_tokens":1504,"output_tokens":142,"output_tokens_details":{"thinking_tokens":27},"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":2811},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"thinkingDurationMs":542,"thinkingDisplay":"updates","apiBlockIndex":0,"requestId":"req_011CgXmnfqC2jXhNLSUragTR","requestRef":"2d74b9a3-7a48-4320-9a92-54bc4b7f17d5","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"781b79ec-6752-445c-9d1a-5ad5126a9ab0","timestamp":"2026-10-30T04:15:03.217Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0051","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}13{"parentUuid":"781b79ec-6752-445c-9d1a-5ad5126a9ab0","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmng7ZY9iFHVFFCngFk","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_018EKXLguP9gGm5pBsnFKL3F","name":"Bash","input":{"command":"cat TASK.md; 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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.\nbundle.json\ndata\ngate_spec.json\nTASK.md\n\n./data:\ndesign.csv","is_error":true,"tool_use_id":"toolu_018EKXLguP9gGm5pBsnFKL3F"}]},"uuid":"d0793ef6-8133-4bd5-af95-dbb45a7a2c8a","timestamp":"2026-10-30T04:15:03.944Z","toolUseResult":"Error: Exit code 1\n# 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). 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 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 data/design.csv\n{\n  \"task_id\": \"F2-conf-03\",\n  \"required_files\": [\"out/results.json\", \"out/report.md\", \"out/analysis.py\", \"out/claims.json\"],\n  \"results_keys\": {\n    \"task_id\": {\"type\": \"string\", \"nullable\": false},\n    \"coverage_conventional\": {\"type\": \"number\", \"nullable\": false},\n    \"coverage_hc1\": {\"type\": \"number\", \"nullable\": false},\n    \"sd_beta_exact\": {\"type\": \"number\", \"nullable\": false},\n    \"sd_beta_mc\": {\"type\": \"number\", \"nullable\": false},\n    \"mean_se_conventional\": {\"type\": \"number\", \"nullable\": false},\n    \"conventional_ci_undercovers\": {\"type\": \"boolean\", \"nullable\": false},\n    \"n_replications\": {\"type\": \"integer\", \"nullable\": false},\n    \"n_obs_per_replication\": {\"type\": \"integer\", \"nullable\": false},\n    \"feasibility\": {\"type\": \"string\", \"nullable\": false},\n    \"feasibility_reason\": {\"type\": \"string\", \"nullable\": true},\n    \"data_issues\": 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index=False)\nfor s in excl:\n    if s[4] > 0:\n        issues.append({\"issue\": f\"{s[4]} rows removed by rule '{s[1]}'\", \"handling\": \"excluded\"})\nif len(d) != N:\n    issues.append({\"issue\": f\"design has {len(d)} usable rows, expected {N}\", \"handling\": \"reported; feasibility partial\"})\n\nd = d.sort_values(\"i\")\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float); n = len(x)\nX = np.column_stack([np.ones(n), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nH = XtX_inv @ X.T                      # 2 x n, beta_hat = H y\nw = H[1]                               # slope weights\nsd_exact = float(np.sqrt(np.sum(w**2 * sig**2)))  # exact: Var(b) = sum w_i^2 sigma_i^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 @ H.T                                           # R x 2\nresid = Y - B @ X.T\ns2 = np.sum(resid**2, axis=1) / (n - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\nhc0 = np.sum((w[None, :] ** 2) * resid**2, axis=1)    # sandwich [ (X'X)^-1 X' diag(e^2) X (X'X)^-1 ]_{11}\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\ntcrit = stats.t.ppf(0.5 + LEVEL / 2, n - 2)\nb = B[:, 1]\ncov_conv = float(np.mean(np.abs(b - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(b - BETA) <= tcrit * se_hc1))\nmcse_nominal = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n# exact expectation of conventional variance (for checking mean_se_conventional): E[s2]=tr(M Sigma)/(n-2)\nM = np.eye(n) - X @ H\nE_s2 = float(np.trace(M @ np.diag(sig**2)) / (n - 2))\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(b, ddof=1)),\n    \"mean_se_conventional\": float(np.mean(se_conv)),\n    \"conventional_ci_undercovers\": undercovers,\n    \"n_replications\": R,\n    \"n_obs_per_replication\": int(n),\n    \"feasibility\": \"complete\" if n == N else \"partial\",\n    \"feasibility_reason\": None if n == N else \"design row count differs from 30\",\n    \"data_issues\": issues,\n    \"n_input_rows\": {\"design.csv\": int(n_raw)},\n    \"random_seed\": SEED,\n    # auxiliary\n    \"t_critical_28\": float(tcrit),\n    \"mcse_coverage_conventional\": mcse_conv,\n    \"mcse_coverage_hc1\": mcse_hc1,\n    \"mcse_at_nominal_0.95\": mcse_nominal,\n    \"undercover_threshold\": float(LEVEL - 3 * mcse_nominal),\n    \"undercovers_using_estimated_p_se\": bool(cov_conv < LEVEL - 3 * mcse_conv),\n    \"mean_se_hc1\": float(np.mean(se_hc1)),\n    \"se_conv_population_sqrtEs2\": float(np.sqrt(E_s2 * XtX_inv[1, 1])),\n    \"mean_beta_mc\": float(np.mean(b)),\n    \"design_sha256\": hashlib.sha256(open(path, \"rb\").read()).hexdigest(),\n}\nwith open(\"out/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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 \"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  \"t_critical_28\": 2.048407141795244,\n  \"mcse_coverage_conventional\": 0.00811369213120636,\n  \"mcse_coverage_hc1\": 0.006648157639526908,\n  \"mcse_at_nominal_0.95\": 0.004873397172404485,\n  \"undercover_threshold\": 0.9353798084827865,\n  \"undercovers_using_estimated_p_se\": true,\n  \"mean_se_hc1\": 0.24080977065103962,\n  \"se_conv_population_sqrtEs2\": 0.19318017740129875,\n  \"mean_beta_mc\": 0.4961965063446237,\n  \"design_sha256\": 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true,\n  \"mean_se_hc1\": 0.24080977065103962,\n  \"se_conv_population_sqrtEs2\": 0.19318017740129875,\n  \"mean_beta_mc\": 0.4961965063446237,\n  \"design_sha256\": 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index=False)\nfor s in excl:\n    if s[4] > 0:\n        issues.append({\"issue\": f\"{s[4]} rows removed by rule '{s[1]}'\", \"handling\": \"excluded\"})\nif len(d) != N:\n    issues.append({\"issue\": f\"design has {len(d)} usable rows, expected {N}\", \"handling\": \"reported; feasibility partial\"})\n\nd = d.sort_values(\"i\")\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float); n = len(x)\nX = np.column_stack([np.ones(n), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nH = XtX_inv @ X.T                      # 2 x n, beta_hat = H y\nw = H[1]                               # slope weights\nsd_exact = float(np.sqrt(np.sum(w**2 * sig**2)))  # exact: Var(b) = sum w_i^2 sigma_i^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 @ H.T                                           # R x 2\nresid = Y - B @ X.T\ns2 = np.sum(resid**2, axis=1) / (n - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\nhc0 = np.sum((w[None, :] ** 2) * resid**2, axis=1)    # sandwich [ (X'X)^-1 X' diag(e^2) X (X'X)^-1 ]_{11}\nse_hc1 = np.sqrt(hc0 * n / (n - 2))\ntcrit = stats.t.ppf(0.5 + LEVEL / 2, n - 2)\nb = B[:, 1]\ncov_conv = float(np.mean(np.abs(b - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(b - BETA) <= tcrit * se_hc1))\nmcse_nominal = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n# exact expectation of conventional variance (for checking mean_se_conventional): E[s2]=tr(M Sigma)/(n-2)\nM = np.eye(n) - X @ H\nE_s2 = float(np.trace(M @ np.diag(sig**2)) / (n - 2))\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(b, ddof=1)),\n    \"mean_se_conventional\": float(np.mean(se_conv)),\n    \"conventional_ci_undercovers\": undercovers,\n    \"n_replications\": R,\n    \"n_obs_per_replication\": int(n),\n    \"feasibility\": \"complete\" if n == N else \"partial\",\n    \"feasibility_reason\": None if n == N else \"design row count differs from 30\",\n    \"data_issues\": issues,\n    \"n_input_rows\": {\"design.csv\": int(n_raw)},\n    \"random_seed\": SEED,\n    # auxiliary\n    \"t_critical_28\": float(tcrit),\n    \"mcse_coverage_conventional\": mcse_conv,\n    \"mcse_coverage_hc1\": mcse_hc1,\n    \"mcse_at_nominal_0.95\": mcse_nominal,\n    \"undercover_threshold\": float(LEVEL - 3 * mcse_nominal),\n    \"undercovers_using_estimated_p_se\": bool(cov_conv < LEVEL - 3 * mcse_conv),\n    \"mean_se_hc1\": float(np.mean(se_hc1)),\n    \"se_conv_population_sqrtEs2\": float(np.sqrt(E_s2 * XtX_inv[1, 1])),\n    \"mean_beta_mc\": float(np.mean(b)),\n    \"design_sha256\": hashlib.sha256(open(path, \"rb\").read()).hexdigest(),\n}\nwith open(\"out/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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against statsmodels, and verify reproducibility."}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":2970,"cache_read_input_tokens":7250,"output_tokens":761,"output_tokens_details":{"thinking_tokens":33},"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":2970},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmpWZeLEkNhA66vsf1c","requestRef":"4fb11cc9-1144-4252-9253-a1503ec44317","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"da1a41a6-bcc2-4794-a267-2ae2a953741f","timestamp":"2026-10-30T04:15:28.736Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0051","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}30{"parentUuid":"da1a41a6-bcc2-4794-a267-2ae2a953741f","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmpWrFmrECueW6qRXr9","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_012prVupq2TxCuryWzhAzAeu","name":"Bash","input":{"command":"cp 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x\",x.mean(),\"dup i\",d.i.duplicated().sum(),\"na\",d.isna().sum().sum())\nEOF","description":"Rerun determinism check and statsmodels 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against statsmodels, and verify reproducibility."},{"type":"tool_use","id":"toolu_012prVupq2TxCuryWzhAzAeu","name":"Bash","input":{"command":"cp out/results.json /tmp/r1.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && cmp out/results.json /tmp/r1.json && echo IDENTICAL\n~/wp29/venv/bin/python - <<'EOF'\nimport numpy as np, pandas as pd, statsmodels.api as sm\nd=pd.read_csv(\"data/design.csv\"); x=d.x.values; s=d.sigma.values\nrng=np.random.default_rng(20261030); eps=rng.standard_normal((2000,30))\nX=sm.add_constant(x)\nfor r in [0,1,1999]:\n    y=1+0.5*x+s*eps[r]\n    m=sm.OLS(y,X).fit(); h=sm.OLS(y,X).fit(cov_type=\"HC1\")\n    print(r,m.params[1],m.bse[1],h.bse[1])\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport numpy as np, pandas as pd\nd=pd.read_csv(\"data/design.csv\"); x=d.x.values; s=d.sigma.values\nrng=np.random.default_rng(20261030); eps=rng.standard_normal((2000,30))\nX=np.column_stack([np.ones(30),x]); H=np.linalg.inv(X.T@X)@X.T\nfor r in [0,1,1999]:\n    y=1+0.5*x+s*eps[r]; b=H@y; e=y-X@b\n    print(r,b[1],np.sqrt(e@e/28*np.linalg.inv(X.T@X)[1,1]), np.sqrt(np.sum(H[1]**2*e**2)*30/28))\nprint(\"mean x\",x.mean(),\"dup i\",d.i.duplicated().sum(),\"na\",d.isna().sum().sum())\nEOF","description":"Rerun determinism check and statsmodels 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statsmodels exactly. Now the state files, report, and claims."}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":972,"cache_read_input_tokens":10220,"output_tokens":3802,"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":972},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmq9MDMytCC753v1Yky","requestRef":"677b9b93-e3fd-4b21-b822-77c3318db780","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"80339740-acbd-478b-ac38-a08d9b1b8496","timestamp":"2026-10-30T04:15:36.948Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0051","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}35{"parentUuid":"80339740-acbd-478b-ac38-a08d9b1b8496","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmq9brK4CyotufPGjXf","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01HMj4cPdt1Jn8WTHcPrMYAx","name":"Bash","input":{"command":"cd out && cat > state/contract.md <<'EOF'\n# Contract — F2-conf-03\n**Question:** Do nominal 95% t-intervals for the OLS slope reach 95% coverage when n=30 and errors are heteroskedastic (σ_i given, fixed design)?\n**Model:** 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; only ε is redrawn.\n**Estimands:** (a) P(β ∈ β̂ ± t_.975(28) se_conv); (b) the same with se_HC1 = sqrt(HC0·n/(n−2)); (c) exact SD(β̂) = sqrt(Σ w_i² σ_i²), w = row 2 of (X'X)⁻¹X'; (d) E[se_conv]. (a),(b),(d) estimated by MC, R=2000, fixed seed; (c) analytical. These are meaningful under the stated model with the design held fixed.\n**Outputs:** out/results.json (task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers, n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed), out/report.md (6 sections incl. MC SEs), out/analysis.py, out/claims.json (incl. a nominal-coverage claim).\n**Undercoverage rule:** true iff coverage_conventional < 0.95 − 3·MCSE, where MCSE = binomial SE at R=2000. I use p=0.95 (null value) as primary: sqrt(.95·.05/2000); also report the version with estimated p.\n**Info rules:** use only design.csv; do not redraw x or σ.\n**Invalidity:** design not 30 rows / bad σ; non-reproducible simulation; R ≠ 2000.\nEOF\ncat > state/data_audit.md <<'EOF'\n# Data audit\n- data/design.csv — SHA-256 b1a17a529973e7994bbedad5ec1bc73fe968c777301ca8745657ea300e3c74f0\n- 30 rows × 3 columns: i (int 1–30), x (float, dimensionless, range −1.3942 to 1.9919, mean 0.2823), sigma (float, dimensionless, range 0.6034 to 1.7903, all > 0).\n- Missing values: 0. Duplicate i: 0. No dates (synthetic fixed design).\n- Note: σ_i increases with |x_i| (smallest near x≈0, largest at extreme x), so high-leverage points have the largest error variance. This is the structure that makes the conventional SE too small. It is a feature of the design, not a data defect.\n- Information-availability: there is only one input, it is allowed, and no violations were found.\n- Exclusions: three checks (missing, σ≤0, duplicate i) removed 0 rows; see out/exclusions.csv. n_raw=30=n_analysed.\nEOF\ncat > state/checks.md <<'EOF'\n# Verification checks (P4)\n1. Determinism: re-ran analysis.py and byte-compared results.json with the previous run: IDENTICAL. PASS\n2. Cross-check vs statsmodels OLS (nonrobust and HC1) on replications 0, 1, 1999 of the same draw: β̂, se_conv and se_HC1 agree to ~1e-15. PASS\n3. Exact vs MC SD: sd_beta_exact 0.26798 vs sd_beta_mc 0.26965. The relative difference is 0.6%, which is inside the MC sampling SD of a sample SD (≈ 1/sqrt(2·1999) ≈ 1.6%). PASS\n4. mean_se_conventional 0.19026 vs sqrt(E[s²]·[(X'X)⁻¹]₂₂) = 0.19318 (an upper bound on E[se] by Jensen). The MC mean is just below it, as expected. PASS\n5. mean_se_conventional (0.190) < sd_beta_exact (0.268): the conventional SE is biased downward, which agrees with the undercoverage. PASS\n6. Positive SEs/SDs; coverages lie in [0,1]; mean β̂ = 0.4962 ≈ 0.5 (MCSE ≈ 0.006). PASS\n7. Undercoverage: 0.844 < threshold 0.93538 (nominal-p MCSE). It also holds with the estimated-p MCSE. The conclusion is robust to that choice. PASS\n8. Exclusion ledger: 30 − 0 = 30 = n_input_rows. PASS\nEOF\ncat > report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic sample\n\n## Question\nFor the fixed design in `data/design.csv` (n = 30, y = 1 + 0.5x + σ_i ε), does the nominal 95% t-interval for the slope reach 95% coverage? We compare the conventional SE with HC1, compute the exact SD of β̂, and compute the mean conventional SE.\n\n## Data and exclusions\nI used one input, `design.csv`: 30 rows (`n_input_rows`), with no missing values and no duplicates. All σ are positive. No rows were excluded (`out/exclusions.csv`), and `data_issues` is empty. σ_i grows with |x_i|, so the high-leverage observations are also the noisiest.\n\n## Method\nEach replication draws ε ~ N(0, I₃₀) and computes OLS β̂, se_conv (s² with divisor 28) and se_HC1 (HC0 × 30/28). An interval counts as covering when |β̂ − 0.5| ≤ t₀.₉₇₅(28)·se, with t = 2.0484 (`t_critical_28`). There are R = 2000 replications (`n_replications`) and the numpy default_rng seed is 20261030 (`random_seed`). The exact SD is sqrt(Σ w_i² σ_i²), where w is the slope row of (X'X)⁻¹X'. The undercoverage rule is coverage < 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercover_threshold`). The binomial SE is evaluated at the nominal level.\n\n## Results\n- Conventional-interval coverage: **0.844** (`coverage_conventional`). HC1-interval coverage: **0.902** (`coverage_hc1`).\n- Exact SD(β̂) = 0.2680 (`sd_beta_exact`). The MC SD is 0.2696 (`sd_beta_mc`) and the MC mean of β̂ is 0.4962 (`mean_beta_mc`).\n- Mean se_conv = 0.1903 (`mean_se_conventional`), about 29% below the true SD. Mean se_HC1 = 0.2408 (`mean_se_hc1`), closer to the true SD but still too small.\n- **The conventional interval does not attain nominal coverage** (`conventional_ci_undercovers` = true). HC1 improves coverage but also stays below 95%. Its gap of 0.048 is about 7 MCSE (`coverage_hc1`, `mcse_coverage_hc1`); the report applies no formal test to it.\n\n## Uncertainty\nThese are binomial Monte Carlo standard errors at R = 2000:\n- Conventional: 0.0081 using the estimated p (`mcse_coverage_conventional`).\n- HC1: 0.0066 (`mcse_coverage_hc1`).\n- At p = 0.95: 0.0049 (`mcse_at_nominal_0.95`).\n\nThe conventional shortfall (0.106) is about 13 MCSE under either SE definition, so the undercoverage conclusion does not depend on which one is used (`undercovers_using_estimated_p_se` = true). The sampling uncertainty in sd_beta_mc relative to the exact value is about 1.6%; the observed difference is 0.6%.\n\n## Limitations and open issues\n- All results are conditional on this one synthetic design and normal errors. Other designs or error distributions will give different coverage.\n- The task does not say whether the binomial SE in the undercoverage rule should use p = 0.95 or the estimated p. I used p = 0.95 and checked that the alternative gives the same conclusion.\n- Monte Carlo estimates depend on the seed. Different seeds will move the coverages by roughly ±2 MCSE.\n- Small-sample corrections stronger than HC1, such as HC2/HC3 or a Bell–McCaffrey degrees-of-freedom correction, were not evaluated because the task does not require them.\nEOF\ncat > claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain nominal coverage under this design: estimated coverage is 0.844, more than 3 Monte Carlo SEs below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"mcse_at_nominal_0.95\",\"mcse_coverage_conventional\",\"undercover_threshold\"],\"status\":\"supported\",\"caveat\":\"Conditional on this fixed design, normal errors and seed; the conclusion holds with either binomial-SE convention.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95%.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"The gap is about 7 MCSE; no formal test is applied to HC1.\"},\n {\"id\":\"C3\",\"text\":\"The exact SD of the slope estimator is 0.268, and the MC SD (0.270) agrees within simulation error.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"The exact value is analytical given the design.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is about 29% below the true SD.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"The mean is a Monte Carlo estimate.\"}\n]\nEOF\ncat > state/log.md <<'EOF'\n- 2026-10-30 P0: read TASK, wrote contract. Decision: undercover rule uses binomial SE at p=0.95, with the estimated-p version also reported.\n- 2026-10-30 P1: audited design.csv; clean, 0 exclusions.\n- 2026-10-30 P2/P3: analysis.py written; the baseline (conventional coverage) and the HC1 comparison are computed in the same script. Seed 20261030.\n- 2026-10-30 P4: determinism and statsmodels cross-checks passed; running gate.\nEOF","description":"Write state, report, claims files"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":972,"cache_read_input_tokens":10220,"output_tokens":3802,"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":972},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmq9MDMytCC753v1Yky","requestRef":"677b9b93-e3fd-4b21-b822-77c3318db780","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"9258369e-f38d-4827-a1be-25efa5be8a99","timestamp":"2026-10-30T04:16:09.329Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0051","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}36{"parentUuid":"9258369e-f38d-4827-a1be-25efa5be8a99","isSidechain":false,"promptId":"1b3001d3-05ab-4fdf-b378-6c2783290e20","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01HMj4cPdt1Jn8WTHcPrMYAx","type":"tool_result","content":"(Bash completed with no output)","is_error":false}]},"uuid":"19acf39e-b8d7-4e4b-92aa-b7c528d69ed6","timestamp":"2026-10-30T04:16:09.355Z","toolUseResult":{"stdout":"","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"9258369e-f38d-4827-a1be-25efa5be8a99","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0051/out","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","version":"2.1.314","gitBranch":"HEAD"}37{"type":"api-request-blob","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","hash":"99bbe38aed108dd9ad8ab276b31c6e35b82db4a348e18019987cce38c57bdbf5","message":{"role":"assistant","content":[{"type":"text","text":"Matches statsmodels exactly. Now the state files, report, and claims."},{"type":"tool_use","id":"toolu_01HMj4cPdt1Jn8WTHcPrMYAx","name":"Bash","input":{"command":"cd out && cat > state/contract.md <<'EOF'\n# Contract — F2-conf-03\n**Question:** Do nominal 95% t-intervals for the OLS slope reach 95% coverage when n=30 and errors are heteroskedastic (σ_i given, fixed design)?\n**Model:** 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; only ε is redrawn.\n**Estimands:** (a) P(β ∈ β̂ ± t_.975(28) se_conv); (b) the same with se_HC1 = sqrt(HC0·n/(n−2)); (c) exact SD(β̂) = sqrt(Σ w_i² σ_i²), w = row 2 of (X'X)⁻¹X'; (d) E[se_conv]. (a),(b),(d) estimated by MC, R=2000, fixed seed; (c) analytical. These are meaningful under the stated model with the design held fixed.\n**Outputs:** out/results.json (task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers, n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed), out/report.md (6 sections incl. MC SEs), out/analysis.py, out/claims.json (incl. a nominal-coverage claim).\n**Undercoverage rule:** true iff coverage_conventional < 0.95 − 3·MCSE, where MCSE = binomial SE at R=2000. I use p=0.95 (null value) as primary: sqrt(.95·.05/2000); also report the version with estimated p.\n**Info rules:** use only design.csv; do not redraw x or σ.\n**Invalidity:** design not 30 rows / bad σ; non-reproducible simulation; R ≠ 2000.\nEOF\ncat > state/data_audit.md <<'EOF'\n# Data audit\n- data/design.csv — SHA-256 b1a17a529973e7994bbedad5ec1bc73fe968c777301ca8745657ea300e3c74f0\n- 30 rows × 3 columns: i (int 1–30), x (float, dimensionless, range −1.3942 to 1.9919, mean 0.2823), sigma (float, dimensionless, range 0.6034 to 1.7903, all > 0).\n- Missing values: 0. Duplicate i: 0. No dates (synthetic fixed design).\n- Note: σ_i increases with |x_i| (smallest near x≈0, largest at extreme x), so high-leverage points have the largest error variance. This is the structure that makes the conventional SE too small. It is a feature of the design, not a data defect.\n- Information-availability: there is only one input, it is allowed, and no violations were found.\n- Exclusions: three checks (missing, σ≤0, duplicate i) removed 0 rows; see out/exclusions.csv. n_raw=30=n_analysed.\nEOF\ncat > state/checks.md <<'EOF'\n# Verification checks (P4)\n1. Determinism: re-ran analysis.py and byte-compared results.json with the previous run: IDENTICAL. PASS\n2. Cross-check vs statsmodels OLS (nonrobust and HC1) on replications 0, 1, 1999 of the same draw: β̂, se_conv and se_HC1 agree to ~1e-15. PASS\n3. Exact vs MC SD: sd_beta_exact 0.26798 vs sd_beta_mc 0.26965. The relative difference is 0.6%, which is inside the MC sampling SD of a sample SD (≈ 1/sqrt(2·1999) ≈ 1.6%). PASS\n4. mean_se_conventional 0.19026 vs sqrt(E[s²]·[(X'X)⁻¹]₂₂) = 0.19318 (an upper bound on E[se] by Jensen). The MC mean is just below it, as expected. PASS\n5. mean_se_conventional (0.190) < sd_beta_exact (0.268): the conventional SE is biased downward, which agrees with the undercoverage. PASS\n6. Positive SEs/SDs; coverages lie in [0,1]; mean β̂ = 0.4962 ≈ 0.5 (MCSE ≈ 0.006). PASS\n7. Undercoverage: 0.844 < threshold 0.93538 (nominal-p MCSE). It also holds with the estimated-p MCSE. The conclusion is robust to that choice. PASS\n8. Exclusion ledger: 30 − 0 = 30 = n_input_rows. PASS\nEOF\ncat > report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic sample\n\n## Question\nFor the fixed design in `data/design.csv` (n = 30, y = 1 + 0.5x + σ_i ε), does the nominal 95% t-interval for the slope reach 95% coverage? We compare the conventional SE with HC1, compute the exact SD of β̂, and compute the mean conventional SE.\n\n## Data and exclusions\nI used one input, `design.csv`: 30 rows (`n_input_rows`), with no missing values and no duplicates. All σ are positive. No rows were excluded (`out/exclusions.csv`), and `data_issues` is empty. σ_i grows with |x_i|, so the high-leverage observations are also the noisiest.\n\n## Method\nEach replication draws ε ~ N(0, I₃₀) and computes OLS β̂, se_conv (s² with divisor 28) and se_HC1 (HC0 × 30/28). An interval counts as covering when |β̂ − 0.5| ≤ t₀.₉₇₅(28)·se, with t = 2.0484 (`t_critical_28`). There are R = 2000 replications (`n_replications`) and the numpy default_rng seed is 20261030 (`random_seed`). The exact SD is sqrt(Σ w_i² σ_i²), where w is the slope row of (X'X)⁻¹X'. The undercoverage rule is coverage < 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercover_threshold`). The binomial SE is evaluated at the nominal level.\n\n## Results\n- Conventional-interval coverage: **0.844** (`coverage_conventional`). HC1-interval coverage: **0.902** (`coverage_hc1`).\n- Exact SD(β̂) = 0.2680 (`sd_beta_exact`). The MC SD is 0.2696 (`sd_beta_mc`) and the MC mean of β̂ is 0.4962 (`mean_beta_mc`).\n- Mean se_conv = 0.1903 (`mean_se_conventional`), about 29% below the true SD. Mean se_HC1 = 0.2408 (`mean_se_hc1`), closer to the true SD but still too small.\n- **The conventional interval does not attain nominal coverage** (`conventional_ci_undercovers` = true). HC1 improves coverage but also stays below 95%. Its gap of 0.048 is about 7 MCSE (`coverage_hc1`, `mcse_coverage_hc1`); the report applies no formal test to it.\n\n## Uncertainty\nThese are binomial Monte Carlo standard errors at R = 2000:\n- Conventional: 0.0081 using the estimated p (`mcse_coverage_conventional`).\n- HC1: 0.0066 (`mcse_coverage_hc1`).\n- At p = 0.95: 0.0049 (`mcse_at_nominal_0.95`).\n\nThe conventional shortfall (0.106) is about 13 MCSE under either SE definition, so the undercoverage conclusion does not depend on which one is used (`undercovers_using_estimated_p_se` = true). The sampling uncertainty in sd_beta_mc relative to the exact value is about 1.6%; the observed difference is 0.6%.\n\n## Limitations and open issues\n- All results are conditional on this one synthetic design and normal errors. Other designs or error distributions will give different coverage.\n- The task does not say whether the binomial SE in the undercoverage rule should use p = 0.95 or the estimated p. I used p = 0.95 and checked that the alternative gives the same conclusion.\n- Monte Carlo estimates depend on the seed. Different seeds will move the coverages by roughly ±2 MCSE.\n- Small-sample corrections stronger than HC1, such as HC2/HC3 or a Bell–McCaffrey degrees-of-freedom correction, were not evaluated because the task does not require them.\nEOF\ncat > claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional OLS 95% interval does not attain nominal coverage under this design: estimated coverage is 0.844, more than 3 Monte Carlo SEs below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"mcse_at_nominal_0.95\",\"mcse_coverage_conventional\",\"undercover_threshold\"],\"status\":\"supported\",\"caveat\":\"Conditional on this fixed design, normal errors and seed; the conclusion holds with either binomial-SE convention.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95%.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"The gap is about 7 MCSE; no formal test is applied to HC1.\"},\n {\"id\":\"C3\",\"text\":\"The exact SD of the slope estimator is 0.268, and the MC SD (0.270) agrees within simulation error.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"The exact value is analytical given the design.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is about 29% below the true SD.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"The mean is a Monte Carlo estimate.\"}\n]\nEOF\ncat > state/log.md <<'EOF'\n- 2026-10-30 P0: read TASK, wrote contract. Decision: undercover rule uses binomial SE at p=0.95, with the estimated-p version also reported.\n- 2026-10-30 P1: audited design.csv; clean, 0 exclusions.\n- 2026-10-30 P2/P3: analysis.py written; the baseline (conventional coverage) and the HC1 comparison are computed in the same script. Seed 20261030.\n- 2026-10-30 P4: determinism and statsmodels cross-checks passed; running gate.\nEOF","description":"Write state, report, claims files"}}]}}38{"type":"api-request","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba","id":"d6ffdf99-9827-4763-a605-e1a81ef6f6e1","timestamp":"2026-10-30T04:16:09.359Z","version":"2.1.314","querySource":"sdk","shapeHash":"41c654b2f75aa9005673aa85ed544686c25f8acc758b9c3765bbe482075ab38a","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":"677b9b93-e3fd-4b21-b822-77c3318db780","keep":10,"tail":["99bbe38aed108dd9ad8ab276b31c6e35b82db4a348e18019987cce38c57bdbf5","e928e300117ba5ba2daa49870905014d42dd23b1787b7ba121ecab722a7e3674"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",11,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"84da7296d3d2fd5a7242456d1a985bd0a2795d0120f78322bf7100323cc76eeb"}39{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"19acf39e-b8d7-4e4b-92aa-b7c528d69ed6","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba"}40{"type":"atis-latch","atis":"","sessionId":"b8471e5f-bce8-49fe-ba5c-4c2f6d65cfba"}41{"parentUuid":"19acf39e-b8d7-4e4b-92aa-b7c528d69ed6","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmsf6Bwibm9naXcoh8N","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_01ARTqbMvhyYsSwERY3CCCta","name":"Bash","input":{"command":"cat > /tmp/ph.py <<'EOF'\nimport json,sys,glob,os\nrec=sorted(glob.glob(\"out/state/gate_receipts/*.json\"))\nlast=rec[-1] if rec else None\nP={\n\"P0\":[\"contract.md written covering estimands, outputs, rules\"],\n\"P1\":[\"sha256 recorded\",\"schema/missing/duplicates checked\",\"exclusions.csv sums 30-0=30\"],\n\"P2\":[\"baseline coverage_conventional computed with same coverage rule\"],\n\"P3\":[\"analysis.py reads only data/, fixed seed 20261030, writes results.json\"],\n\"P4\":[\"rerun byte-identical\",\"statsmodels cross-check\",\"consistency checks in checks.md\"]+([f\"gate receipt {last}\"] if last else []),\n\"P5\":[\"report.md 6 sections with key citations\",\"claims.json written\"]+([f\"gate receipt {last}\"] if last and sys.argv[1]==\"final\" else []),\n}\nst={k:{\"status\":\"passed\",\"checks\":v,\"notes\":\"\"} for k,v in P.items()}\nif sys.argv[1]!=\"final\": st[\"P5\"][\"status\"]=\"running\"\njson.dump(st,open(\"out/state/phases.json\",\"w\"),indent=1)\nEOF\n~/wp29/venv/bin/python /tmp/ph.py p4; 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With n = 30 and this heteroskedastic design, the conventional 95% interval for the slope does not reach its nominal coverage: it covers β in 84.4% of the 2000 replications. The final gate run passed all 32 checks, including the reproducibility re-run.\n\n| Quantity | Value |\n|---|---|\n| Coverage, conventional interval (`coverage_conventional`) | 0.844 (Monte Carlo SE 0.008) |\n| Coverage, HC1 interval (`coverage_hc1`) | 0.902 (Monte Carlo SE 0.007) |\n| Exact SD of β̂ (`sd_beta_exact`) | 0.2680 |\n| Simulated SD of β̂ (`sd_beta_mc`) | 0.2696 |\n| Mean conventional SE (`mean_se_conventional`) | 0.1903, about 29% below the true SD |\n| Undercoverage flag (`conventional_ci_undercovers`) | true |\n\n- **Why it undercovers:** the error SD is largest at the extreme x values, which are also the points that pull hardest on the slope. The conventional SE ignores this, so it is too small.\n- **HC1 helps but isn't enough:** it raises coverage to 0.902, but that is still below 95% by about 7 Monte Carlo SEs. I didn't run a formal test on it.\n- **Undercoverage rule:** the task doesn't say whether the Monte Carlo SE should use p = 0.95 or the estimated coverage. I used p = 0.95, giving a cut-off of 0.9354. The estimated coverage gives the same answer, since the shortfall is about 13 SEs either way.\n- **Checks:** for three replications, β̂ and both SEs match statsmodels to about 1e-15. Re-running the script gives byte-identical results with seed 20261030. The design file is clean and no rows were excluded.\n\n**Limitations:** the results hold only for this one synthetic design with normal errors. A different seed would move the coverage figures by roughly ±2 SEs. 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