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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.\ntotal 48\ndrwxr-xr-x@ 9 simon-pierreboucher staff 288 oct. 30 00:18 .\ndrwxr-xr-x 7 simon-pierreboucher staff 224 oct. 30 00:17 ..\n-rw-r--r--@ 1 simon-pierreboucher staff 3358 oct. 30 00:17 analysis.py\n-rw-r--r--@ 1 simon-pierreboucher staff 1254 oct. 30 00:18 claims.json\n-rw-r--r--@ 1 simon-pierreboucher staff 177 oct. 30 00:17 exclusions.csv\n-rw-r--r--@ 1 simon-pierreboucher staff 4516 oct. 30 00:18 report.md\n-rw-r--r--@ 1 simon-pierreboucher staff 929 oct. 30 00:17 results.json\ndrwxr-xr-x 2 simon-pierreboucher staff 64 oct. 30 00:18 review\ndrwxr-xr-x@ 8 simon-pierreboucher staff 256 oct. 30 00:18 state\n{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.26798059283028336,\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_at_nominal\": 0.004873397172404485,\n \"undercoverage_threshold\": 0.9353798084827865,\n \"mcse_coverage_conventional\": 0.00811369213120636,\n \"mcse_coverage_hc1\": 0.006648157639526908,\n \"sd_beta_homoskedastic_formula\": 0.19623471734704753,\n \"mean_se_hc1\": 0.24080977065103962,\n \"mean_beta_hat\": 0.4961965063446238,\n \"mcse_mean_beta_hat\": 0.006029498511320145,\n \"mcse_mean_se_conventional\": 0.0006978794961917418\n}[\n {\"id\":\"C1\",\"text\":\"The conventional 95% OLS interval does not attain nominal coverage: estimated coverage 0.844, below the 0.95 - 3 MCSE threshold of 0.935.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"undercoverage_threshold\",\"mcse_coverage_at_nominal\"],\"status\":\"supported\",\"caveat\":\"Single fixed design, normal errors, R=2000; MCSE ~0.008.\"},\n {\"id\":\"C2\",\"text\":\"Undercoverage arises because the mean conventional SE (0.190) understates the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mechanism attributed to sigma rising with |x - xbar| (leverage); correlation from the audit.\"},\n {\"id\":\"C3\",\"text\":\"The HC1 interval improves coverage to 0.902 but still falls short of 95% at n=30.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\",\"mean_se_hc1\"],\"status\":\"supported\",\"caveat\":\"Shortfall not formally tested with the 3-MCSE rule; descriptive.\"},\n {\"id\":\"C4\",\"text\":\"The Monte Carlo SD of the slope estimator (0.2696) agrees with the analytic exact SD (0.2680), validating the simulation.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Agreement judged against approx MC SE of a sample SD (~0.004).\"}\n]","is_error":false}]},"uuid":"74b62653-84b4-4432-b513-ee5e4525f576","timestamp":"2026-10-30T04:18:45.673Z","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). 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.\ntotal 48\ndrwxr-xr-x@ 9 simon-pierreboucher staff 288 oct. 30 00:18 .\ndrwxr-xr-x 7 simon-pierreboucher staff 224 oct. 30 00:17 ..\n-rw-r--r--@ 1 simon-pierreboucher staff 3358 oct. 30 00:17 analysis.py\n-rw-r--r--@ 1 simon-pierreboucher staff 1254 oct. 30 00:18 claims.json\n-rw-r--r--@ 1 simon-pierreboucher staff 177 oct. 30 00:17 exclusions.csv\n-rw-r--r--@ 1 simon-pierreboucher staff 4516 oct. 30 00:18 report.md\n-rw-r--r--@ 1 simon-pierreboucher staff 929 oct. 30 00:17 results.json\ndrwxr-xr-x 2 simon-pierreboucher staff 64 oct. 30 00:18 review\ndrwxr-xr-x@ 8 simon-pierreboucher staff 256 oct. 30 00:18 state\n{\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": 0.844,\n \"coverage_hc1\": 0.902,\n \"sd_beta_exact\": 0.26798059283028336,\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_at_nominal\": 0.004873397172404485,\n \"undercoverage_threshold\": 0.9353798084827865,\n \"mcse_coverage_conventional\": 0.00811369213120636,\n \"mcse_coverage_hc1\": 0.006648157639526908,\n \"sd_beta_homoskedastic_formula\": 0.19623471734704753,\n \"mean_se_hc1\": 0.24080977065103962,\n \"mean_beta_hat\": 0.4961965063446238,\n \"mcse_mean_beta_hat\": 0.006029498511320145,\n \"mcse_mean_se_conventional\": 0.0006978794961917418\n}[\n {\"id\":\"C1\",\"text\":\"The conventional 95% OLS interval does not attain nominal coverage: estimated coverage 0.844, below the 0.95 - 3 MCSE threshold of 0.935.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"undercoverage_threshold\",\"mcse_coverage_at_nominal\"],\"status\":\"supported\",\"caveat\":\"Single fixed design, normal errors, R=2000; MCSE ~0.008.\"},\n {\"id\":\"C2\",\"text\":\"Undercoverage arises because the mean conventional SE (0.190) understates the exact SD of the slope estimator (0.268).\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mechanism attributed to sigma rising with |x - xbar| (leverage); correlation from the audit.\"},\n {\"id\":\"C3\",\"text\":\"The HC1 interval improves coverage to 0.902 but still falls short of 95% at n=30.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\",\"mean_se_hc1\"],\"status\":\"supported\",\"caveat\":\"Shortfall not formally tested with the 3-MCSE rule; descriptive.\"},\n {\"id\":\"C4\",\"text\":\"The Monte Carlo SD of the slope estimator (0.2696) agrees with the analytic exact SD (0.2680), validating the simulation.\",\"results_keys\":[\"sd_beta_mc\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Agreement judged against approx MC SE of a sample SD (~0.004).\"}\n]","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"598e63cc-44bc-4f73-bdb1-0344331a9ddb","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}15{"type":"api-request-blob","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","hash":"bb582355bd4b6f26beb9a7bdaa84514d167e7d73235fbebc3dc71fa730d5c702","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_018wrMqwXXcgxbJdWxUU5S6f","name":"Bash","input":{"command":"cat TASK.md; 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sig = d[\"sigma\"].to_numpy(float)\n\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/exclusions.csv\", \"w\", newline=\"\") as f:\n w = csv.writer(f)\n w.writerow([\"step\", \"rule\", \"n_before\", \"n_after\", \"n_removed\", \"rationale\"])\n w.writerow([\"1\", \"drop rows with missing/invalid x or sigma\", n_raw, n_raw, 0,\n \"audit found no missing, duplicate or non-positive values; nothing removed\"])\n\nX = np.column_stack([np.ones(N), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nxc = x - x.mean(); Sxx = xc @ xc\nwts = xc / Sxx # beta_hat - beta = sum w_i sigma_i eps_i\nsd_beta_exact = float(np.sqrt(np.sum(wts**2 * sig**2)))\ntcrit = stats.t.ppf(0.5 + LEVEL / 2, N - 2)\n\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, N)) # one row per replication\nY = ALPHA + BETA * x[None, :] + sig[None, :] * eps\nA = X @ XtX_inv # (N,2): beta_hat = Y @ A\nB = Y @ A\nresid = Y - B @ X.T\ns2 = (resid**2).sum(1) / (N - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 slope variance: sum_i a_i1^2 e_i^2 (= [(X'X)^-1 X' diag(e^2) X (X'X)^-1]_{11})\nhc0 = (resid**2) @ (A[:, 1]**2)\nse_hc1 = np.sqrt(hc0 * N / (N - 2))\nbhat = B[:, 1]\ncov_conv = float(np.mean(np.abs(bhat - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bhat - BETA) <= tcrit * se_hc1))\nmcse_nominal = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_beta_exact,\n \"sd_beta_mc\": float(np.std(bhat, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercovers,\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\": int(n_raw)},\n \"random_seed\": SEED,\n # supplementary keys\n \"t_critical_28\": float(tcrit),\n \"mcse_coverage_at_nominal\": mcse_nominal,\n \"undercoverage_threshold\": float(LEVEL - 3 * mcse_nominal),\n \"mcse_coverage_conventional\": float(np.sqrt(cov_conv * (1 - cov_conv) / R)),\n \"mcse_coverage_hc1\": float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R)),\n \"sd_beta_homoskedastic_formula\": float(np.sqrt(np.mean(sig**2) / Sxx)),\n \"mean_se_hc1\": float(se_hc1.mean()),\n \"mean_beta_hat\": float(bhat.mean()),\n \"mcse_mean_beta_hat\": float(np.std(bhat, ddof=1) / np.sqrt(R)),\n \"mcse_mean_se_conventional\": float(np.std(se_conv, ddof=1) / np.sqrt(R)),\n}\nwith open(\"out/results.json\", \"w\") as f:\n json.dump(res, f, indent=2)\nprint(json.dumps(res, indent=2))\n# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic fixed design\n\n## Question\nIn y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30; x_i, σ_i fixed from `data/design.csv`; ε_i iid N(0,1)), does the nominal 95% interval β̂ ± t₀.₉₇₅(28)·se reach 95% coverage, using (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) the exact SD of β̂ and (d) E[se_conv].\n\n## Data and exclusions\nOne input, `data/design.csv` (30 rows, columns i, x, sigma; `n_input_rows`). The audit (`out/state/data_audit.md`) found no missing values, duplicates or non-positive σ; `data_issues` is empty and no rows were excluded (`out/exclusions.csv`: 30 → 30). Note on the design: σ_i increases with |x_i − x̄| (correlation 0.84, computed during the audit), so the error variance is largest at the high-leverage points.\n\n## Method\n- Exact SD (no simulation): β̂ − β = Σ w_i σ_i ε_i with w_i = (x_i − x̄)/S_xx, so SD(β̂) = sqrt(Σ w_i² σ_i²) (`sd_beta_exact`).\n- Monte Carlo: R = 2000 replications (`n_replications`), n = 30 (`n_obs_per_replication`), numpy `default_rng` with seed 20261030 (`random_seed`). Only ε is redrawn. For each replication: OLS fit, se_conv = sqrt(s²·[(X'X)⁻¹]₂₂) with s² = RSS/(n−2), se_HC1 = sqrt(n/(n−2)·Σ a_i² e_i²) (HC0 sandwich × n/(n−2)), critical value t₀.₉₇₅(28) = 2.0484 (`t_critical_28`). Coverage = share of replications whose interval contains 0.5.\n- The baseline is the conventional interval coverage (`coverage_conventional`), which the task names as the naïve baseline.\n- Decision rule: the conventional CI undercovers iff coverage_conventional < 0.95 − 3·MCSE, with MCSE = sqrt(0.95·0.05/2000) = 0.00487 (`mcse_coverage_at_nominal`), giving a threshold of 0.9354 (`undercoverage_threshold`). I evaluated the binomial SE at the nominal 0.95 because the rule tests against that value.\n- Validation: results matched statsmodels (OLS nonrobust and HC1) for the first 5 replications to machine precision. A clean re-run reproduced the results exactly (`out/state/checks.md`).\n\n## Results\n| quantity | value | key |\n|---|---|---|\n| Coverage, conventional CI | 0.844 | `coverage_conventional` |\n| Coverage, HC1 CI | 0.902 | `coverage_hc1` |\n| Exact SD of β̂ | 0.2680 | `sd_beta_exact` |\n| MC SD of β̂ | 0.2696 | `sd_beta_mc` |\n| Mean se_conv | 0.1903 | `mean_se_conventional` |\n| Mean se_HC1 | 0.2408 | `mean_se_hc1` |\n| Mean β̂ | 0.4962 | `mean_beta_hat` |\n| Conventional CI undercovers | true | `conventional_ci_undercovers` |\n\nThe conventional interval does **not** reach its nominal 95% coverage. Its coverage of 0.844 (`coverage_conventional`) is far below the threshold of 0.9354 (`undercoverage_threshold`). The cause is that se_conv is too small on average: 0.190 (`mean_se_conventional`), compared with the true SD of 0.268 (`sd_beta_exact`), about 29% too small. The homoskedastic formula ignores that the noisiest observations are also the high-leverage ones. The HC1 interval does better at 0.902 (`coverage_hc1`) but still falls short of 95%, because its mean SE of 0.241 (`mean_se_hc1`) is still below 0.268 at n = 30. The MC SD of β̂ agrees with the exact value (0.2696 vs 0.2680), and the mean of β̂ is consistent with 0.5. This is a check that the simulation is correct.\n\n## Uncertainty\n- Monte Carlo SE of coverage_conventional, from the estimated proportion: 0.0081 (`mcse_coverage_conventional`). Approximate 95% MC interval: 0.828–0.860.\n- Monte Carlo SE of coverage_hc1: 0.0066 (`mcse_coverage_hc1`). Approximate 95% MC interval: 0.889–0.915.\n- MCSE at the nominal level used in the decision rule: 0.00487 (`mcse_coverage_at_nominal`). The decision does not change if the estimate-based MCSE is used instead: 0.95 − 3·0.0081 = 0.926 > 0.844.\n- MC SE of mean se_conv: 0.0007 (`mcse_mean_se_conventional`). MC SE of mean β̂: 0.0060 (`mcse_mean_beta_hat`).\n- `sd_beta_exact` is analytic and has no MC error.\n\n## Limitations and open issues\n- The results apply only to this one fixed design, with normal errors and R = 2000. Coverage estimates carry the MC error stated above. Other seeds would move them by about ±0.016 (2 MCSE).\n- The task does not fix how the \"binomial standard error\" in the undercoverage rule should be computed. I used the value at the nominal p = 0.95, and the conclusion is the same under the alternative.\n- The HC1 shortfall below 95% is a descriptive finding. I did not test it formally against a threshold, although 0.902 is more than 7 MCSE below 0.95.\nstep,rule,n_before,n_after,n_removed,rationale\r\n1,drop rows with missing/invalid x or sigma,30,30,0,\"audit found no missing, duplicate or non-positive values; nothing removed\"\r\nchecks.md\ncontract.md\ndata_audit.md\ngate_receipts\nlog.md\nphases.json\ni,x,sigma\n1,-1.3942,1.1831\n2,-0.9737,0.8844\n3,-0.8373,0.8103\n4,-0.7367,0.7628\n5,-0.6908,0.7432\n6,-0.2796,0.6235\n7,-0.2431,0.6177\n8,-0.2367,0.6168\n9,-0.2175,0.6142\n10,-0.1756,0.6093\n11,-0.1682,0.6085\n12,-0.1062,0.6034\n13,0.1078,0.6035\n14,0.1216,0.6044\n15,0.2127,0.6136\n16,0.2496,0.6187\n17,0.3046,0.6278\n18,0.4577,0.6628\n19,0.5583,0.6935\n20,0.651,0.7271\n21,0.7123,0.7522\n22,0.7126,0.7523\n23,0.7832,0.784\n24,0.9147,0.851\n25,0.9582,0.8754\n26,1.2264,1.0512\n27,1.381,1.1721\n28,1.5008,1.2757\n29,1.6841,1.4509\n30,1.9919,1.7903","is_error":false}]},"uuid":"5c86dc19-f012-4922-952e-c54284fdd8db","timestamp":"2026-10-30T04:18:47.907Z","toolUseResult":{"stdout":"\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\n\"\"\"\nimport json, os, csv\nimport numpy as np\nimport pandas as pd\nfrom scipy import stats\n\nALPHA, BETA, N, R, LEVEL = 1.0, 0.5, 30, 2000, 0.95\nSEED = 20261030\n\nd = pd.read_csv(\"data/design.csv\")\nn_raw = len(d)\nissues = []\n# audit checks (no filtering needed if all pass)\nassert list(d.columns) == [\"i\", \"x\", \"sigma\"]\nassert not d.isna().any().any(), \"missing values in design\"\nassert n_raw == N, f\"expected {N} rows, got {n_raw}\"\nassert (d[\"sigma\"] > 0).all()\nassert d[\"i\"].tolist() == list(range(1, N + 1))\nassert d[\"x\"].std() > 0\nx = d[\"x\"].to_numpy(float); sig = d[\"sigma\"].to_numpy(float)\n\nos.makedirs(\"out\", exist_ok=True)\nwith open(\"out/exclusions.csv\", \"w\", newline=\"\") as f:\n w = csv.writer(f)\n w.writerow([\"step\", \"rule\", \"n_before\", \"n_after\", \"n_removed\", \"rationale\"])\n w.writerow([\"1\", \"drop rows with missing/invalid x or sigma\", n_raw, n_raw, 0,\n \"audit found no missing, duplicate or non-positive values; nothing removed\"])\n\nX = np.column_stack([np.ones(N), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nxc = x - x.mean(); Sxx = xc @ xc\nwts = xc / Sxx # beta_hat - beta = sum w_i sigma_i eps_i\nsd_beta_exact = float(np.sqrt(np.sum(wts**2 * sig**2)))\ntcrit = stats.t.ppf(0.5 + LEVEL / 2, N - 2)\n\nrng = np.random.default_rng(SEED)\neps = rng.standard_normal((R, N)) # one row per replication\nY = ALPHA + BETA * x[None, :] + sig[None, :] * eps\nA = X @ XtX_inv # (N,2): beta_hat = Y @ A\nB = Y @ A\nresid = Y - B @ X.T\ns2 = (resid**2).sum(1) / (N - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 slope variance: sum_i a_i1^2 e_i^2 (= [(X'X)^-1 X' diag(e^2) X (X'X)^-1]_{11})\nhc0 = (resid**2) @ (A[:, 1]**2)\nse_hc1 = np.sqrt(hc0 * N / (N - 2))\nbhat = B[:, 1]\ncov_conv = float(np.mean(np.abs(bhat - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bhat - BETA) <= tcrit * se_hc1))\nmcse_nominal = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_beta_exact,\n \"sd_beta_mc\": float(np.std(bhat, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": undercovers,\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\": int(n_raw)},\n \"random_seed\": SEED,\n # supplementary keys\n \"t_critical_28\": float(tcrit),\n \"mcse_coverage_at_nominal\": mcse_nominal,\n \"undercoverage_threshold\": float(LEVEL - 3 * mcse_nominal),\n \"mcse_coverage_conventional\": float(np.sqrt(cov_conv * (1 - cov_conv) / R)),\n \"mcse_coverage_hc1\": float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R)),\n \"sd_beta_homoskedastic_formula\": float(np.sqrt(np.mean(sig**2) / Sxx)),\n \"mean_se_hc1\": float(se_hc1.mean()),\n \"mean_beta_hat\": float(bhat.mean()),\n \"mcse_mean_beta_hat\": float(np.std(bhat, ddof=1) / np.sqrt(R)),\n \"mcse_mean_se_conventional\": float(np.std(se_conv, ddof=1) / np.sqrt(R)),\n}\nwith open(\"out/results.json\", \"w\") as f:\n json.dump(res, f, indent=2)\nprint(json.dumps(res, indent=2))\n# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic fixed design\n\n## Question\nIn y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30; x_i, σ_i fixed from `data/design.csv`; ε_i iid N(0,1)), does the nominal 95% interval β̂ ± t₀.₉₇₅(28)·se reach 95% coverage, using (a) the conventional OLS SE and (b) the HC1 robust SE? Also: (c) the exact SD of β̂ and (d) E[se_conv].\n\n## Data and exclusions\nOne input, `data/design.csv` (30 rows, columns i, x, sigma; `n_input_rows`). The audit (`out/state/data_audit.md`) found no missing values, duplicates or non-positive σ; `data_issues` is empty and no rows were excluded (`out/exclusions.csv`: 30 → 30). Note on the design: σ_i increases with |x_i − x̄| (correlation 0.84, computed during the audit), so the error variance is largest at the high-leverage points.\n\n## Method\n- Exact SD (no simulation): β̂ − β = Σ w_i σ_i ε_i with w_i = (x_i − x̄)/S_xx, so SD(β̂) = sqrt(Σ w_i² σ_i²) (`sd_beta_exact`).\n- Monte Carlo: R = 2000 replications (`n_replications`), n = 30 (`n_obs_per_replication`), numpy `default_rng` with seed 20261030 (`random_seed`). Only ε is redrawn. For each replication: OLS fit, se_conv = sqrt(s²·[(X'X)⁻¹]₂₂) with s² = RSS/(n−2), se_HC1 = sqrt(n/(n−2)·Σ a_i² e_i²) (HC0 sandwich × n/(n−2)), critical value t₀.₉₇₅(28) = 2.0484 (`t_critical_28`). Coverage = share of replications whose interval contains 0.5.\n- The baseline is the conventional interval coverage (`coverage_conventional`), which the task names as the naïve baseline.\n- Decision rule: the conventional CI undercovers iff coverage_conventional < 0.95 − 3·MCSE, with MCSE = sqrt(0.95·0.05/2000) = 0.00487 (`mcse_coverage_at_nominal`), giving a threshold of 0.9354 (`undercoverage_threshold`). I evaluated the binomial SE at the nominal 0.95 because the rule tests against that value.\n- Validation: results matched statsmodels (OLS nonrobust and HC1) for the first 5 replications to machine precision. A clean re-run reproduced the results exactly (`out/state/checks.md`).\n\n## Results\n| quantity | value | key |\n|---|---|---|\n| Coverage, conventional CI | 0.844 | `coverage_conventional` |\n| Coverage, HC1 CI | 0.902 | `coverage_hc1` |\n| Exact SD of β̂ | 0.2680 | `sd_beta_exact` |\n| MC SD of β̂ | 0.2696 | `sd_beta_mc` |\n| Mean se_conv | 0.1903 | `mean_se_conventional` |\n| Mean se_HC1 | 0.2408 | `mean_se_hc1` |\n| Mean β̂ | 0.4962 | `mean_beta_hat` |\n| Conventional CI undercovers | true | `conventional_ci_undercovers` |\n\nThe conventional interval does **not** reach its nominal 95% coverage. Its coverage of 0.844 (`coverage_conventional`) is far below the threshold of 0.9354 (`undercoverage_threshold`). The cause is that se_conv is too small on average: 0.190 (`mean_se_conventional`), compared with the true SD of 0.268 (`sd_beta_exact`), about 29% too small. The homoskedastic formula ignores that the noisiest observations are also the high-leverage ones. The HC1 interval does better at 0.902 (`coverage_hc1`) but still falls short of 95%, because its mean SE of 0.241 (`mean_se_hc1`) is still below 0.268 at n = 30. The MC SD of β̂ agrees with the exact value (0.2696 vs 0.2680), and the mean of β̂ is consistent with 0.5. This is a check that the simulation is correct.\n\n## Uncertainty\n- Monte Carlo SE of coverage_conventional, from the estimated proportion: 0.0081 (`mcse_coverage_conventional`). Approximate 95% MC interval: 0.828–0.860.\n- Monte Carlo SE of coverage_hc1: 0.0066 (`mcse_coverage_hc1`). Approximate 95% MC interval: 0.889–0.915.\n- MCSE at the nominal level used in the decision rule: 0.00487 (`mcse_coverage_at_nominal`). The decision does not change if the estimate-based MCSE is used instead: 0.95 − 3·0.0081 = 0.926 > 0.844.\n- MC SE of mean se_conv: 0.0007 (`mcse_mean_se_conventional`). MC SE of mean β̂: 0.0060 (`mcse_mean_beta_hat`).\n- `sd_beta_exact` is analytic and has no MC error.\n\n## Limitations and open issues\n- The results apply only to this one fixed design, with normal errors and R = 2000. Coverage estimates carry the MC error stated above. Other seeds would move them by about ±0.016 (2 MCSE).\n- The task does not fix how the \"binomial standard error\" in the undercoverage rule should be computed. I used the value at the nominal p = 0.95, and the conclusion is the same under the alternative.\n- The HC1 shortfall below 95% is a descriptive finding. I did not test it formally against a threshold, although 0.902 is more than 7 MCSE below 0.95.\nstep,rule,n_before,n_after,n_removed,rationale\r\n1,drop rows with missing/invalid x or sigma,30,30,0,\"audit found no missing, duplicate or non-positive values; nothing removed\"\r\nchecks.md\ncontract.md\ndata_audit.md\ngate_receipts\nlog.md\nphases.json\ni,x,sigma\n1,-1.3942,1.1831\n2,-0.9737,0.8844\n3,-0.8373,0.8103\n4,-0.7367,0.7628\n5,-0.6908,0.7432\n6,-0.2796,0.6235\n7,-0.2431,0.6177\n8,-0.2367,0.6168\n9,-0.2175,0.6142\n10,-0.1756,0.6093\n11,-0.1682,0.6085\n12,-0.1062,0.6034\n13,0.1078,0.6035\n14,0.1216,0.6044\n15,0.2127,0.6136\n16,0.2496,0.6187\n17,0.3046,0.6278\n18,0.4577,0.6628\n19,0.5583,0.6935\n20,0.651,0.7271\n21,0.7123,0.7522\n22,0.7126,0.7523\n23,0.7832,0.784\n24,0.9147,0.851\n25,0.9582,0.8754\n26,1.2264,1.0512\n27,1.381,1.1721\n28,1.5008,1.2757\n29,1.6841,1.4509\n30,1.9919,1.7903","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"aac829b4-ebfc-41b6-8667-77c1e421c10d","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}19{"type":"api-request-blob","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","hash":"b735aa3af5bfcced8c7e68fa719cb88ff3986f3db1228464cf28a5be6e1980c9","message":{"role":"assistant","content":[{"type":"tool_use","id":"toolu_01DRYtSKFkUX5fKf1BSNTkaw","name":"Bash","input":{"command":"cat 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0.902 0.19026231012152658\n0.26798059283028336\nShell cwd was reset to /Users/simon-pierreboucher/wp29/runs/RC0058","is_error":false}]},"uuid":"125b4d48-da30-4a16-bae2-18c7a544645a","timestamp":"2026-10-30T04:18:55.040Z","toolUseResult":{"stdout":"True\n0.844 0.902 0.19026231012152658\n0.26798059283028336","stderr":"\nShell cwd was reset to 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Re-ran out/analysis.py from scratch; results.json byte-identical (cmp) -> PASS.\n2. Cross-check vs statsmodels OLS (nonrobust and cov_type=HC1) for replications 1-5 using same eps draws: beta_hat, se_conv, se_HC1 agree to ~1e-16 -> PASS.\n3. Exclusion ledger: 30 raw - 0 removed = 30 analysed = n_input_rows -> PASS.\n4. Positive SDs/SEs: sd_beta_exact=0.2680, sd_beta_mc=0.2696, mean_se_conventional=0.1903 > 0 -> PASS.\n5. sd_beta_mc vs sd_beta_exact: rel diff 0.6%; the MC SE of a sample SD at R=2000 is ~sd/sqrt(2(R-1)) ~ 0.0042, diff 0.0017 -> consistent, PASS.\n6. mean_beta_hat=0.4962, |0.4962-0.5| = 0.0038 < 2*mcse 0.012 -> unbiasedness consistent, PASS.\n7. mean_se_conventional (0.190) close to homoskedastic-formula SD sqrt(mean sigma^2/Sxx)=0.196 (below it, as expected from E[sqrt]<sqrt E and the leverage-weighting of residual variance) and well below sd_beta_exact 0.268 -> explains undercoverage, PASS.\n8. 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Also robust to using estimate-based MCSE (0.95-3*0.0081=0.926) -> same decision, PASS.\n9. coverage values in [0,1]; n_replications=2000, n_obs=30, seed recorded -> PASS.","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"f4c7bd6e-968a-4ca9-b02c-db4c71c09df4","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}31{"type":"api-request-blob","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","hash":"2bdeb9ed2c1d06a2abe52ad744b8193b7f493bbd5ae37ff548779f6015ce7172","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_017goe2drRtLxi66EnPNL1Bw","name":"Bash","input":{"command":"~/wp29/venv/bin/python -c \"\nimport pandas as pd,numpy as np;d=pd.read_csv('data/design.csv');print(np.corrcoef(d.sigma,abs(d.x-d.x.mean()))[0,1], d.duplicated().sum())\"; 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The evidence is strong ((0.95-0.902)/0.0066 ≈ 7.2 MCSE, as the report says), but the status and the caveat do not quite match.\",\n \"suggested_fix\": \"Put the 7-MCSE comparison into the claim or caveat (e.g. 0.902 < 0.95 - 3*0.00487) so that 'supported' rests on a stated test.\"\n },\n {\n \"id\": \"R2\",\n \"severity\": \"minor\",\n \"location\": \"out/report.md Data/Results; claims.json C2\",\n \"issue\": \"The mechanism in C2 depends on corr(sigma, |x - xbar|) = 0.84. I reproduced this value (0.837), but it is not stored in results.json and the code does not compute it. Also, mean se_conv < SD(beta_hat) shows that the SE is too small, but saying that leverage-weighted heteroskedasticity is the cause is an interpretation.\",\n \"suggested_fix\": \"Add the correlation to results.json as a supplementary key and word C2 as an explanation that is consistent with the design.\"\n }\n ],\n \"summary\": \"Re-running analysis.py in a scratch copy reproduced results.json exactly. An independent statsmodels loop over all 2000 replications (same seed and draws) gave coverage_conventional = 0.844, coverage_hc1 = 0.902 and mean_se_conventional = 0.19026, all matching. The analytic sd_beta_exact = 0.26798 also matches. All required keys are present with correct types. The code uses only data/design.csv, keeps x and sigma fixed, uses R = 2000 with the seed recorded, uses n-2 divisors and the t(28) critical value, and computes HC1 as HC0 times n/(n-2). The undercoverage flag follows the 3-MCSE rule, and the conclusion holds under either reading of the binomial SE. The report gives MCSEs for both coverages and states the scope limits. Two minor wording issues only.\"\n}\nEOF\n~/wp29/venv/bin/python -m json.tool out/review/review.json >/dev/null && echo ok","description":"Write review JSON and validate"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1182,"cache_read_input_tokens":10405,"output_tokens":1007,"output_tokens_details":{"thinking_tokens":103},"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":1182},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXn6FK4UnnYddAm4sATN","requestRef":"37e697b3-5ebe-49d2-92b4-2ea10abd0b0f","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"64066bdc-2716-4b62-9fda-b3a27c3a1075","timestamp":"2026-10-30T04:19:09.905Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}35{"parentUuid":"64066bdc-2716-4b62-9fda-b3a27c3a1075","isSidechain":false,"promptId":"a33ac33d-7d89-4a21-b056-cb176d98b183","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01MfarUuAugA7gw7syhuKQ2o","type":"tool_result","content":"ok","is_error":false}]},"uuid":"cd7cc04a-6726-483e-8022-e8b981aea5a0","timestamp":"2026-10-30T04:19:09.973Z","toolUseResult":{"stdout":"ok","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"64066bdc-2716-4b62-9fda-b3a27c3a1075","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}36{"type":"api-request-blob","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","hash":"0ae5ebcaf93fed99e5f7bf19c28720d4f1ef32ebfa0e3ae25250a649b647eca0","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01MfarUuAugA7gw7syhuKQ2o","name":"Bash","input":{"command":"cat > out/review/review.json <<'EOF'\n{\n \"verdict\": \"accept\",\n \"issues\": [\n {\n \"id\": \"R1\",\n \"severity\": \"minor\",\n \"location\": \"out/claims.json C3\",\n \"issue\": \"C3 ('HC1 still falls short of 95%') is marked 'supported', but its own caveat says the shortfall was not formally tested. The evidence is strong ((0.95-0.902)/0.0066 ≈ 7.2 MCSE, as the report says), but the status and the caveat do not quite match.\",\n \"suggested_fix\": \"Put the 7-MCSE comparison into the claim or caveat (e.g. 0.902 < 0.95 - 3*0.00487) so that 'supported' rests on a stated test.\"\n },\n {\n \"id\": \"R2\",\n \"severity\": \"minor\",\n \"location\": \"out/report.md Data/Results; claims.json C2\",\n \"issue\": \"The mechanism in C2 depends on corr(sigma, |x - xbar|) = 0.84. I reproduced this value (0.837), but it is not stored in results.json and the code does not compute it. Also, mean se_conv < SD(beta_hat) shows that the SE is too small, but saying that leverage-weighted heteroskedasticity is the cause is an interpretation.\",\n \"suggested_fix\": \"Add the correlation to results.json as a supplementary key and word C2 as an explanation that is consistent with the design.\"\n }\n ],\n \"summary\": \"Re-running analysis.py in a scratch copy reproduced results.json exactly. An independent statsmodels loop over all 2000 replications (same seed and draws) gave coverage_conventional = 0.844, coverage_hc1 = 0.902 and mean_se_conventional = 0.19026, all matching. The analytic sd_beta_exact = 0.26798 also matches. All required keys are present with correct types. The code uses only data/design.csv, keeps x and sigma fixed, uses R = 2000 with the seed recorded, uses n-2 divisors and the t(28) critical value, and computes HC1 as HC0 times n/(n-2). The undercoverage flag follows the 3-MCSE rule, and the conclusion holds under either reading of the binomial SE. The report gives MCSEs for both coverages and states the scope limits. Two minor wording issues only.\"\n}\nEOF\n~/wp29/venv/bin/python -m json.tool out/review/review.json >/dev/null && echo ok","description":"Write review JSON and validate"}}]}}37{"type":"api-request","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","id":"729c2252-0c86-4ffc-a746-c9bd3a36ea37","timestamp":"2026-10-30T04:19:09.977Z","version":"2.1.314","querySource":"sdk","shapeHash":"3ffa9d59477758b20a8c06d5f90f2753b98ca588cce1c36614044842832b711b","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":"37e697b3-5ebe-49d2-92b4-2ea10abd0b0f","keep":10,"tail":["0ae5ebcaf93fed99e5f7bf19c28720d4f1ef32ebfa0e3ae25250a649b647eca0","cd785aa966b106a446f7b029957dde1745f96eb81e33ffb583e06e229b3fa2cd"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",11,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"17e631b83aa537587f155b9b3d1010df64a70eb3122897f292b2ab421846b397"}38{"parentUuid":"cd7cc04a-6726-483e-8022-e8b981aea5a0","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXn6wosW7H3mSpvRzQEF","type":"message","role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1014,"cache_read_input_tokens":11587,"output_tokens":751,"output_tokens_details":{"thinking_tokens":5},"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":1014},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"thinkingDurationMs":6133,"thinkingDisplay":"updates","apiBlockIndex":0,"requestId":"req_011CgXn6wcDbacSMPzmuqfeu","requestRef":"729c2252-0c86-4ffc-a746-c9bd3a36ea37","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"f5215d8e-92fa-4d7e-84c4-7d2468fbef52","timestamp":"2026-10-30T04:19:17.639Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}39{"parentUuid":"f5215d8e-92fa-4d7e-84c4-7d2468fbef52","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXn6wosW7H3mSpvRzQEF","type":"message","role":"assistant","content":[{"type":"text","text":"My verdict is **accept**: I found nothing critical or major, only two minor wording issues. The review is in `out/review/review.json`.\n\n**Numbers reproduce.** I re-ran `out/analysis.py` in a scratch copy and got the same `results.json` exactly. I also checked it independently with statsmodels, running all 2000 replications with the same seed. That gave the same results:\n- Coverage of the conventional interval: 0.844\n- Coverage of the robust (HC1) interval: 0.902\n- Average conventional standard error: 0.19026\n- Exact standard deviation of the slope estimate: 0.26798\n\n**Rules respected.** The code reads only `data/design.csv` and never redraws x or σ. It uses exactly 2000 replications with a recorded seed, the n − 2 divisor and the t(28) critical value. HC1 is computed correctly as the HC0 sandwich times n/(n − 2).\n\n**Claims hold up.** The conventional interval is flagged as undercovering using the required three-standard-error rule: 0.844 is well below the 0.935 threshold. The conclusion doesn't change if the standard error is taken at the estimated coverage instead of at 0.95. The report gives the Monte Carlo standard error for both coverages and is honest about its scope. Other checks also pass:\n- The simulated spread of the slope (0.2696) agrees with the exact value (0.2680).\n- The average slope estimate (0.4962) is consistent with 0.5.\n- The 0.84 correlation the report cites (between σ and distance of x from its mean) recomputes to 0.837.\n\n**Minor issues:**\n1. **Claim C3:** the claim that HC1 still falls short of 95% is marked \"supported\", but its own caveat says the shortfall wasn't formally tested. The evidence is strong (about 7 standard errors below 0.95), so the fix is to state that comparison in the claim.\n2. **Claim C2:** the explanation for the undercoverage rests on the 0.84 correlation, which the code doesn't compute and `results.json` doesn't store. Also, saying the noisy high-leverage points *cause* the undercoverage is an interpretation, not something the numbers show directly. It would be better worded as an explanation consistent with the design."}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1014,"cache_read_input_tokens":11587,"output_tokens":751,"output_tokens_details":{"thinking_tokens":5},"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":1014},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXn6wcDbacSMPzmuqfeu","requestRef":"729c2252-0c86-4ffc-a746-c9bd3a36ea37","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"3edd07bc-44ae-495e-9a22-f31ea9e65506","timestamp":"2026-10-30T04:19:17.857Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0058","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","version":"2.1.314","gitBranch":"HEAD"}40{"type":"last-prompt","lastPrompt":"Review","leafUuid":"3edd07bc-44ae-495e-9a22-f31ea9e65506","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb"}41{"type":"atis-latch","atis":"","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb"}42{"type":"cost-state","sessionId":"e9514cb5-35d0-4e09-8cfe-bd45500d4cdb","totalCostUSD":0.1249874,"totalAPIDuration":32011,"totalAPIDurationWithoutRetries":32006,"totalToolDuration":2609,"totalLinesAdded":0,"totalLinesRemoved":0,"totalDuration":34795,"startTime":1793333923069,"modelUsage":{"claude-opus-5-5":{"inputTokens":14,"outputTokens":3064,"thinkingTokens":452,"cacheReadInputTokens":40832,"cacheCreationInputTokens":11097,"cacheCreationEphemeral5mInputTokens":11097,"cacheCreationEphemeral1hInputTokens":0,"webSearchRequests":0,"costUSD":0.1249874}},"hasUnknownModelCost":false}43