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It is the complete and only authoritative description of what is required: the scientific question, the inputs under `data/`, the required output files and their exact keys, and the information-availability rules.\n- There is no network access. Use the Python interpreter `~/wp29/venv/bin/python` (numpy, pandas, scipy, statsmodels, matplotlib are installed). Standard shell tools are available. Do not try to install packages.\n- Write every deliverable under `./out/`. The required deliverables are listed in `TASK.md`; at minimum they always include `out/results.json`, `out/report.md`, and a replication entry point `out/analysis.py` that regenerates `out/results.json` from `data/` when run with `~/wp29/venv/bin/python out/analysis.py` from the workspace root.\n- You are working alone and unattended; nobody will answer questions. Make reasonable, documented decisions.\n\nIntegrity rules (identical for every task):\n- Never fabricate numbers, data, results, or references. Every number in `out/results.json` and `out/report.md` must be produced by code you ran on the provided inputs.\n- Respect the information-availability rules in `TASK.md` exactly (for example, never use information that would not have been available at the stated decision time).\n- If part of the task cannot be completed as specified, say so explicitly in `out/report.md`, explain why, and still write every required file (use `null` for values you could not legitimately produce). An honest statement of a limitation is always preferred to a fabricated or guessed value.\n- Do not modify anything under `data/`.\n- When you are done, end with a brief plain-text summary of what you produced and any limitations.\n\nComplete the task described in `TASK.md` by following this research protocol. The protocol adds no scientific information to the task; it organizes how you work and what evidence you leave behind.\n\nPHASES (execute in order; a phase is eligible only when its dependencies are satisfied):\n\nP0 Scope and contract (depends on: nothing)\n- Read `TASK.md` completely. In `out/state/contract.md`, restate in your own words: the scientific question; the quantity or quantities to be estimated (the estimands) and the assumptions under which they are meaningful; every required output file and key; the information-availability rules; what would make the task impossible or the result invalid.\n- Write the initial phase ledger `out/state/phases.json`: a JSON object mapping each phase id (P0 … P5) to {\"status\": \"pending\"|\"running\"|\"passed\"|\"failed\"|\"blocked\"|\"stale\"|\"skipped_optional\", \"checks\": [], \"notes\": \"\"}. Keep it current as you work. A phase may only be set to \"passed\" when the checks listed for it in this protocol have actually been executed and their outcome recorded in \"checks\".\n\nP1 Data audit and provenance (depends on: P0)\n- Inspect every input file: compute and record its SHA-256, row and column counts, schema, units, date ranges, missing values, duplicates and anomalies in `out/state/data_audit.md`.\n- Keep an exclusion ledger in `out/exclusions.csv` (columns: step, rule, n_before, n_after, n_removed, rationale) for every filtering step you apply, so that n_raw − total removed = n_analysed.\n- Identify any input that violates the information-availability rules and decide, in writing, how you will handle it.\n\nP2 Baseline (depends on: P1)\n- Before any elaborate method, compute the simplest valid baseline that answers the question (for example a naïve or unconditional estimate), with the same evaluation rule used later. Record it in `out/results.json` under the keys the task requires, or under `baseline_*` keys when the task does not name them.\n\nP3 Method and computation (depends on: P2)\n- Implement the analysis in `out/analysis.py` so that it reads only `data/`, writes `out/results.json` deterministically (fix every random seed you use and record it), and prints nothing that contradicts the written results. Any figure or table must be generated by this script.\n\nP4 Verification (depends on: P3)\n- Re-run `out/analysis.py` from a clean state and confirm it regenerates `out/results.json` with identical values. Record the check in `out/state/phases.json` under P4.\n- Check internal consistency: identities that must hold (for example counts that must add up, standard errors that must be positive, confidence intervals that must contain their estimates, date orderings that must respect the information rules), units, and magnitudes against the baseline. Record every check and its outcome in `out/state/checks.md`. A failed check must be fixed or honestly reported, never silently ignored.\n\nP5 Reporting (depends on: P4)\n- Write `out/report.md` with the sections: Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues. Every quantitative statement must cite the key of `out/results.json` it comes from.\n- Write `out/claims.json`: a list of the central claims of your report, each as {\"id\": \"C1\", \"text\": \"...\", \"results_keys\": [\"...\"], \"status\": \"supported\"|\"unsupported\"|\"not_evaluated\", \"caveat\": \"...\"}. Do not state a claim in the report that is not in this ledger.\n- Set the final statuses in `out/state/phases.json`.\n\nSTOPPING RULES\n- Stop and declare the task complete only when every phase P0–P5 is \"passed\" with recorded checks. If a phase is \"blocked\" (missing information, impossible requirement, violated availability rule), stop, set the status, explain in `out/report.md` (section Limitations and open issues) and still write every required file with `null` for values you could not legitimately produce.\n- Do not spend effort on extensions that the task does not require.\n\nRECOVERY RULES\n- Append a short dated entry to `out/state/log.md` after each phase transition and after any important decision, so that work can be resumed by reading `out/state/`.\n- If you discover that an upstream input or decision changed after a downstream phase passed, mark the downstream phases \"stale\" and redo them; do not reuse stale results.\n- If you are interrupted and later resumed, first read `out/state/` and continue from the last consistent state; never re-run expensive steps that already have recorded, unchanged outputs.\n"}],"tools":[{"name":"Bash","description":"execute shell commands","input_schema":{"$schema":"https://json-schema.org/draft/2020-12/schema","type":"object","properties":{"command":{"description":"The command to execute","type":"string"},"timeout":{"description":"Optional timeout in milliseconds (max 600000 for a foreground command)","type":"number"},"description":{"description":"Clear, concise description of what this command does in active voice. 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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_01JXwq1sFHNotD7ZGF5MjTDy"}]},"uuid":"ff8811d7-adb0-442a-ab60-b9310da8e35c","timestamp":"2026-10-30T04:15:39.789Z","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). 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-p out/state && cd out/state && cat > contract.md <<'EOF'\n# Contract (P0)\n**Question.** Do nominal 95% t-intervals for the OLS slope achieve 95% coverage in a fixed-design, n=30, heteroskedastic linear model y = 1 + 0.5x + sigma_i*eps_i (eps iid N(0,1))?\n**Estimands.** (a) coverage of b̂ ± t.975(28)·se_conv; (b) coverage of b̂ ± t.975(28)·se_HC1 (HC0 × n/(n−2)); (c) exact SD of b̂ = sqrt(sum w_i^2 sigma_i^2), w_i=(x_i−x̄)/Sxx (analytic); (d) E[se_conv]. (a),(b),(d) by MC, R=2000, fixed seed. Meaningful under: x, sigma fixed (conditioning on design), normal errors, correct mean model.\n**Outputs.** out/results.json with keys 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 (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json.\n**Undercover rule.** true iff coverage_conventional < 0.95 − 3·sqrt(0.95·0.05/2000) (binomial SE evaluated at nominal p, the null hypothesis value; the SE at p̂ is also reported).\n**Information rules.** Only data/design.csv; never redraw x or sigma; only eps redrawn.\n**Invalidity.** Design file malformed/not 30 rows, non-positive sigma, constant x, wrong R/n, unseeded simulation.\nEOF\ncat > phases.json <<'EOF'\n{\"P0\":{\"status\":\"passed\",\"checks\":[\"TASK.md read fully\",\"contract.md written\"],\"notes\":\"\"},\n\"P1\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\"P2\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P3\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\"P4\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P5\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"}}\nEOF\necho \"- 2026-10-30 P0 passed: contract written.\" > log.md\ncd ../.. && ~/wp29/venv/bin/python -c \"\nimport pandas as pd;d=pd.read_csv('data/design.csv');print(d.shape,d.dtypes.to_dict());print(d.describe());print(d.isna().sum().sum(),d.duplicated().sum(),d.i.tolist()==list(range(1,31)),(d.sigma>0).all())\"","description":"Write contract, ledger, audit data"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1063,"cache_read_input_tokens":5469,"output_tokens":1196,"output_tokens_details":{"thinking_tokens":104},"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":1063},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmqaxb1wbDwAh9hzF7h","requestRef":"8a114063-5fef-4600-b989-32cb26dfbd0d","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"742d5538-16b2-4413-ace8-7d42965d43de","timestamp":"2026-10-30T04:15:52.088Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0053","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","version":"2.1.314","gitBranch":"HEAD"}25{"parentUuid":"742d5538-16b2-4413-ace8-7d42965d43de","isSidechain":false,"promptId":"a6a691d9-425a-478d-a15e-b9c617fc5580","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01XokhfT1FCUoWpMRxJn2vrC","type":"tool_result","content":"(30, 3) {'i': dtype('int64'), 'x': dtype('float64'), 'sigma': dtype('float64')}\n i x sigma\ncount 30.000000 30.000000 30.000000\nmean 15.500000 0.282297 0.819457\nstd 8.803408 0.821180 0.290442\nmin 1.000000 -1.394200 0.603400\n25% 8.250000 -0.231900 0.617025\n50% 15.500000 0.231150 0.735150\n75% 22.750000 0.765550 0.869300\nmax 30.000000 1.991900 1.790300\n0 0 True True","is_error":false}]},"uuid":"e376a0a6-9348-4c9b-bfdd-a916ee0e3b0a","timestamp":"2026-10-30T04:15:52.420Z","toolUseResult":{"stdout":"(30, 3) {'i': dtype('int64'), 'x': dtype('float64'), 'sigma': dtype('float64')}\n i x sigma\ncount 30.000000 30.000000 30.000000\nmean 15.500000 0.282297 0.819457\nstd 8.803408 0.821180 0.290442\nmin 1.000000 -1.394200 0.603400\n25% 8.250000 -0.231900 0.617025\n50% 15.500000 0.231150 0.735150\n75% 22.750000 0.765550 0.869300\nmax 30.000000 1.991900 1.790300\n0 0 True True","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"742d5538-16b2-4413-ace8-7d42965d43de","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0053","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","version":"2.1.314","gitBranch":"HEAD"}26{"type":"api-request-blob","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","hash":"b995fca29b492e2c3461861d9dc8d9a5fce54452ee6a7d6a73ca72001313c0c2","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01XokhfT1FCUoWpMRxJn2vrC","name":"Bash","input":{"command":"mkdir -p out/state && cd out/state && cat > contract.md <<'EOF'\n# Contract (P0)\n**Question.** Do nominal 95% t-intervals for the OLS slope achieve 95% coverage in a fixed-design, n=30, heteroskedastic linear model y = 1 + 0.5x + sigma_i*eps_i (eps iid N(0,1))?\n**Estimands.** (a) coverage of b̂ ± t.975(28)·se_conv; (b) coverage of b̂ ± t.975(28)·se_HC1 (HC0 × n/(n−2)); (c) exact SD of b̂ = sqrt(sum w_i^2 sigma_i^2), w_i=(x_i−x̄)/Sxx (analytic); (d) E[se_conv]. (a),(b),(d) by MC, R=2000, fixed seed. Meaningful under: x, sigma fixed (conditioning on design), normal errors, correct mean model.\n**Outputs.** out/results.json with keys 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 (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json.\n**Undercover rule.** true iff coverage_conventional < 0.95 − 3·sqrt(0.95·0.05/2000) (binomial SE evaluated at nominal p, the null hypothesis value; the SE at p̂ is also reported).\n**Information rules.** Only data/design.csv; never redraw x or sigma; only eps redrawn.\n**Invalidity.** Design file malformed/not 30 rows, non-positive sigma, constant x, wrong R/n, unseeded simulation.\nEOF\ncat > phases.json <<'EOF'\n{\"P0\":{\"status\":\"passed\",\"checks\":[\"TASK.md read fully\",\"contract.md written\"],\"notes\":\"\"},\n\"P1\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\"P2\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P3\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\"P4\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P5\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"}}\nEOF\necho \"- 2026-10-30 P0 passed: contract written.\" > log.md\ncd ../.. && ~/wp29/venv/bin/python -c \"\nimport pandas as pd;d=pd.read_csv('data/design.csv');print(d.shape,d.dtypes.to_dict());print(d.describe());print(d.isna().sum().sum(),d.duplicated().sum(),d.i.tolist()==list(range(1,31)),(d.sigma>0).all())\"","description":"Write contract, ledger, audit data"}}]}}27{"type":"api-request","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","id":"d4efec4f-dffd-4c1b-93e5-e88c598925c3","timestamp":"2026-10-30T04:15:52.424Z","version":"2.1.314","querySource":"sdk","shapeHash":"cbc09f252ff7e553d8e132fd82b221005b46bcaf4b710162396bab127cbc188a","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":"8a114063-5fef-4600-b989-32cb26dfbd0d","keep":6,"tail":["b995fca29b492e2c3461861d9dc8d9a5fce54452ee6a7d6a73ca72001313c0c2","9bc410f33ffee40edbf9eee45133bd96516b34a052a05fd9852fe8a016e802c2"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",7,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"23cedfe6ec87bb262c943c354dca18098ba721a79593ef2c2559c132a3f61dc8"}28{"parentUuid":"e376a0a6-9348-4c9b-bfdd-a916ee0e3b0a","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmrPP6U4CBUgSzhCk3b","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_012HN84cfpdQfsJ9pfDihkeH","name":"Bash","input":{"command":"cat > out/state/data_audit.md <<'EOF'\n# Data audit (P1)\n- File: data/design.csv, SHA-256 b1a17a529973e7994bbedad5ec1bc73fe968c777301ca8745657ea300e3c74f0 (matches bundle.json).\n- 30 rows x 3 columns: i (int, 1..30 consecutive), x (float, dimensionless, range −1.3942..1.9919, mean 0.2823), sigma (float, dimensionless, range 0.6034..1.7903, all > 0).\n- Missing values: 0. Duplicate rows: 0. No dates (fixed design, no time dimension).\n- Pattern: sigma is roughly a convex (U-shaped/quadratic-like) function of x, largest at extreme x — the classic case where conventional OLS SE understates Var(b̂) because high-leverage points have high variance.\n- No availability-rule violations: the only input is the permitted design file. No exclusions applied.\nEOF\nprintf 'step,rule,n_before,n_after,n_removed,rationale\\n0,none (all design rows used),30,30,0,design is complete and valid; task forbids altering x or sigma\\n' > out/exclusions.csv\ncat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, R, LEVEL = 1.0, 0.5, 2000, 0.95\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nif d.isna().any().any():\n issues.append({\"issue\": \"missing values\", \"handling\": \"none expected; would abort\"})\nx, sig = d[\"x\"].to_numpy(float), d[\"sigma\"].to_numpy(float)\nn = len(x)\nassert n == 30 and (sig > 0).all()\n\nX = np.column_stack([np.ones(n), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nxc = x - x.mean(); Sxx = xc @ xc\nw = xc / Sxx # beta_hat = beta + sum w_i sigma_i eps_i\nsd_exact = float(np.sqrt(np.sum(w**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))\nY = ALPHA + BETA * x + sig * eps # R x n\nB = Y @ X @ XtX_inv # (XtX)^-1 X'y for each row (XtX_inv symmetric)\nE = Y - B @ X.T # residuals\ns2 = (E**2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 sandwich slope variance: sum (row-2 of (XtX)^-1 X')_i^2 e_i^2 ; HC1 = HC0 * n/(n-2)\na = (XtX_inv @ X.T)[1] # equals w\nse_hc1 = np.sqrt((a**2 * E**2).sum(1) * n / (n - 2))\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tcrit * se_hc1))\nmcse_nom = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nom)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n \"mean_se_conventional\": float(se_conv.mean()),\n \"conventional_ci_undercovers\": 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(len(d))},\n \"random_seed\": SEED,\n # supplementary keys\n \"t_crit_28\": float(tcrit),\n \"mcse_binomial_at_nominal\": mcse_nom,\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 \"undercover_threshold\": float(LEVEL - 3 * mcse_nom),\n \"mean_se_hc1\": float(se_hc1.mean()),\n \"sd_beta_conventional_formula_homoskedastic\": float(np.sqrt(np.mean(sig**2) * XtX_inv[1, 1])),\n \"mean_beta_hat_mc\": float(bh.mean()),\n}\nos.makedirs(\"out\", exist_ok=True)\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 audit, exclusions, analysis script; 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Duplicate rows: 0. No dates (fixed design, no time dimension).\n- Pattern: sigma is roughly a convex (U-shaped/quadratic-like) function of x, largest at extreme x — the classic case where conventional OLS SE understates Var(b̂) because high-leverage points have high variance.\n- No availability-rule violations: the only input is the permitted design file. No exclusions applied.\nEOF\nprintf 'step,rule,n_before,n_after,n_removed,rationale\\n0,none (all design rows used),30,30,0,design is complete and valid; task forbids altering x or sigma\\n' > out/exclusions.csv\ncat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS CIs under heteroskedasticity.\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\"\"\"\nimport json, os\nimport numpy as np, pandas as pd\nfrom scipy import stats\n\nSEED = 20261030\nALPHA, BETA, R, LEVEL = 1.0, 0.5, 2000, 0.95\n\nd = pd.read_csv(\"data/design.csv\")\nissues = []\nif d.isna().any().any():\n issues.append({\"issue\": \"missing values\", \"handling\": \"none expected; would abort\"})\nx, sig = d[\"x\"].to_numpy(float), d[\"sigma\"].to_numpy(float)\nn = len(x)\nassert n == 30 and (sig > 0).all()\n\nX = np.column_stack([np.ones(n), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nxc = x - x.mean(); Sxx = xc @ xc\nw = xc / Sxx # beta_hat = beta + sum w_i sigma_i eps_i\nsd_exact = float(np.sqrt(np.sum(w**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))\nY = ALPHA + BETA * x + sig * eps # R x n\nB = Y @ X @ XtX_inv # (XtX)^-1 X'y for each row (XtX_inv symmetric)\nE = Y - B @ X.T # residuals\ns2 = (E**2).sum(1) / (n - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 sandwich slope variance: sum (row-2 of (XtX)^-1 X')_i^2 e_i^2 ; HC1 = HC0 * n/(n-2)\na = (XtX_inv @ X.T)[1] # equals w\nse_hc1 = np.sqrt((a**2 * E**2).sum(1) * n / (n - 2))\nbh = B[:, 1]\ncov_conv = float(np.mean(np.abs(bh - BETA) <= tcrit * se_conv))\ncov_hc1 = float(np.mean(np.abs(bh - BETA) <= tcrit * se_hc1))\nmcse_nom = float(np.sqrt(LEVEL * (1 - LEVEL) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nom)\n\nres = {\n \"task_id\": \"F2-conf-03\",\n \"coverage_conventional\": cov_conv,\n \"coverage_hc1\": cov_hc1,\n \"sd_beta_exact\": sd_exact,\n \"sd_beta_mc\": float(np.std(bh, ddof=1)),\n 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out/results.json /tmp/r1.json && rm out/results.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && cmp /tmp/r1.json out/results.json && echo IDENTICAL\n~/wp29/venv/bin/python - <<'EOF'\nimport numpy as np,pandas as pd,statsmodels.api as sm,json\nr=json.load(open('out/results.json'))\nd=pd.read_csv('data/design.csv');x=d.x.values;s=d.sigma.values;n=30\nX=sm.add_constant(x);rng=np.random.default_rng(20261030);e=rng.standard_normal((2000,n))\ny=1+.5*x+s*e[0]\nm=sm.OLS(y,X).fit();h=sm.OLS(y,X).fit(cov_type='HC1')\nprint('sm b,se,hc1',m.params[1],m.bse[1],h.bse[1])\n# reproduce script's first rep\nXtXi=np.linalg.inv(X.T@X);b=XtXi@X.T@y;res=y-X@b\nprint('mine',b[1],np.sqrt(res@res/28*XtXi[1,1]),np.sqrt(((XtXi@X.T)[1]**2*res**2).sum()*30/28))\n# analytic E[s2]\nM=np.eye(n)-X@XtXi@X.T;Es2=np.trace(M@np.diag(s**2))/28\nprint('sqrt(E s2 * c) (upper bound for E se_conv by Jensen)',np.sqrt(Es2*XtXi[1,1]),'mc mean se',r['mean_se_conventional'])\nprint('exact sd via matrix',np.sqrt((XtXi@X.T@np.diag(s**2)@X@XtXi)[1,1]))\nprint('CI for conv cov',r['coverage_conventional']-1.96*r['mcse_coverage_conventional'],r['coverage_conventional']+1.96*r['mcse_coverage_conventional'])\nEOF","description":"Rerun for determinism; cross-check with statsmodels and 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out/results.json /tmp/r1.json && rm out/results.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && cmp /tmp/r1.json out/results.json && echo IDENTICAL\n~/wp29/venv/bin/python - <<'EOF'\nimport numpy as np,pandas as pd,statsmodels.api as sm,json\nr=json.load(open('out/results.json'))\nd=pd.read_csv('data/design.csv');x=d.x.values;s=d.sigma.values;n=30\nX=sm.add_constant(x);rng=np.random.default_rng(20261030);e=rng.standard_normal((2000,n))\ny=1+.5*x+s*e[0]\nm=sm.OLS(y,X).fit();h=sm.OLS(y,X).fit(cov_type='HC1')\nprint('sm b,se,hc1',m.params[1],m.bse[1],h.bse[1])\n# reproduce script's first rep\nXtXi=np.linalg.inv(X.T@X);b=XtXi@X.T@y;res=y-X@b\nprint('mine',b[1],np.sqrt(res@res/28*XtXi[1,1]),np.sqrt(((XtXi@X.T)[1]**2*res**2).sum()*30/28))\n# analytic E[s2]\nM=np.eye(n)-X@XtXi@X.T;Es2=np.trace(M@np.diag(s**2))/28\nprint('sqrt(E s2 * c) (upper bound for E se_conv by Jensen)',np.sqrt(Es2*XtXi[1,1]),'mc mean se',r['mean_se_conventional'])\nprint('exact sd via matrix',np.sqrt((XtXi@X.T@np.diag(s**2)@X@XtXi)[1,1]))\nprint('CI for conv cov',r['coverage_conventional']-1.96*r['mcse_coverage_conventional'],r['coverage_conventional']+1.96*r['mcse_coverage_conventional'])\nEOF","description":"Rerun for determinism; cross-check with statsmodels and 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> out/state/checks.md <<'EOF'\n# Verification checks (P4)\n1. Determinism: deleted out/results.json, re-ran analysis.py; `cmp` with previous output -> IDENTICAL. PASS\n2. Per-replication correctness: replication 1 slope, se_conv, se_HC1 match statsmodels OLS / HC1 to ~1e-15. PASS\n3. Exact SD: closed form sqrt(sum w_i^2 sigma_i^2)=0.267981 equals sandwich (X'X)^-1X'ΣX(X'X)^-1 [1,1]. PASS\n4. MC SD vs exact: 0.26965 vs 0.26798, diff 0.0017 < 1 SE of sample SD (~0.268/sqrt(2·1999)=0.0042). PASS\n5. Unbiasedness: mean b̂ 0.4962 vs 0.5, diff −0.0038 < 1 MC SE (0.27/sqrt(2000)=0.006). PASS\n6. E[se_conv]: MC mean 0.19026 ≤ Jensen upper bound sqrt(E[s²]·[(X'X)^-1]_22)=0.19318 (E[s²]=tr(MΣ)/28 analytic). Consistent. PASS\n7. Ranges: coverages in [0,1]; MC SEs >0; n_replications=2000, n_obs=30, n_input_rows=30; no exclusions (30−0=30). PASS\n8. Undercover rule: 0.844 < 0.95−3·0.004873=0.93538 -> true; also robust to using SE at p̂ (0.95−0.844=0.106 ≫ 3·0.0081). PASS\n9. Magnitude vs baseline: mean se_conv (0.190) < exact SD (0.268), ratio 0.71 — explains undercoverage; HC1 mean SE 0.241 closer but still below → HC1 also undercovers (0.902). Coherent. PASS\nEOF\ncat > out/report.md <<'EOF'\n# F2-conf-03: Monte Carlo coverage of OLS confidence intervals with heteroskedastic errors (n = 30)\n\n## Question\nThe model has a fixed design: y_i = 1 + 0.5 x_i + σ_i ε_i, with ε_i iid N(0,1), and x_i and σ_i taken from `data/design.csv` (n = 30). The question is whether the nominal 95% interval β̂ ± t_{0.975}(28)·se for the OLS slope reaches 95% coverage. Two standard errors are compared: (i) the conventional OLS standard error and (ii) the HC1 robust standard error. Two further quantities are reported: the exact SD of β̂ and E[se_conv].\n\n## Data and exclusions\nThe only input is `data/design.csv`. It has 30 rows (`n_input_rows`), SHA-256 b1a17a52…74f0, and matches the bundle manifest. It has no missing values, no duplicates and all σ_i > 0. No rows were excluded (`out/exclusions.csv`), and `data_issues` is empty. σ_i rises toward both ends of the x range, so the largest error variances sit at the high-leverage points. This is the pattern under which the conventional SE is biased downward.\n\n## Method\n- **Exact SD.** β̂ − β = Σ w_i σ_i ε_i, where w_i = (x_i − x̄)/S_xx. So SD(β̂) = sqrt(Σ w_i² σ_i²), computed analytically (`sd_beta_exact`).\n- **Monte Carlo.** R = 2000 replications (`n_replications`) with numpy `default_rng` seed 20261030 (`random_seed`). Only ε is redrawn. In each replication:\n - fit OLS;\n - compute se_conv = sqrt(s²·[(X'X)⁻¹]₂₂), with s² = RSS/28;\n - compute se_HC1 = sqrt((n/(n−2))·Σ a_i² e_i²), where a = row 2 of (X'X)⁻¹X';\n - check whether β = 0.5 lies inside β̂ ± 2.0484·se (`t_crit_28`).\n- **Undercoverage rule.** The conventional interval counts as undercovering if its coverage is below 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercover_threshold`). The binomial SE here is evaluated at the nominal p (`mcse_binomial_at_nominal`).\n- **Validation.** One replication was cross-checked against statsmodels OLS/HC1 (exact match).\n\n## Results\n- **Conventional interval coverage:** 0.844 (`coverage_conventional`). This is the naïve baseline.\n- **HC1 interval coverage:** 0.902 (`coverage_hc1`).\n- **Exact SD(β̂):** 0.2680 (`sd_beta_exact`). The Monte Carlo SD is 0.2696 (`sd_beta_mc`), which agrees.\n- **Mean se_conv:** 0.1903 (`mean_se_conventional`). That is about 71% of the true SD. The formula that assumes constant variance gives 0.1962 (`sd_beta_conventional_formula_homoskedastic`).\n- **Mean se_HC1:** 0.2408 (`mean_se_hc1`).\n- **Conclusion:** the conventional 95% interval does **not** reach its nominal coverage (`conventional_ci_undercovers` = true). It undercovers by about 10.6 percentage points. HC1 does better but still falls short of 95% at n = 30.\n\n## Uncertainty\n- **Monte Carlo SE of each coverage estimate** (binomial, evaluated at p̂):\n - conventional: 0.0081 (`mcse_coverage_conventional`), giving an approximate 95% MC interval of [0.828, 0.860];\n - HC1: 0.0066 (`mcse_coverage_hc1`), giving approximately [0.889, 0.915].\n- **At nominal p = 0.95,** the binomial SE is 0.0049 (`mcse_binomial_at_nominal`).\n- **Size of the shortfall:** both coverages lie more than 10 nominal SEs below 0.95. The conclusion does not depend on which SE is used.\n- **Other Monte Carlo checks:**\n - the MC SD of β̂ is within one SE (≈0.004) of the exact value;\n - mean β̂ = 0.4962 (`mean_beta_hat_mc`), which is within one MC SE (≈0.006) of 0.5.\n\n## Limitations and open issues\n- The results hold conditionally on this particular fixed design and on normal errors. They are not general statements about OLS.\n- With R = 2000, coverage is known only to about ±0.016 (95% MC interval). The conclusions above are well clear of that margin.\n- The task specifies the HC1 interval with t(28) critical values. Small-sample alternatives (HC2/HC3, or Satterthwaite/Bell–McCaffrey degrees of freedom) would probably come closer to 95%, but they were not evaluated because the task did not ask for them.\n- No part of the task was infeasible.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional 95% OLS interval does not attain nominal coverage: estimated coverage 0.844, more than three binomial MC standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"undercover_threshold\",\"mcse_coverage_conventional\"],\"status\":\"supported\",\"caveat\":\"Conditional on this fixed design and normal errors; MC uncertainty ±~0.016.\"},\n {\"id\":\"C2\",\"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\"],\"status\":\"supported\",\"caveat\":\"Uses t(28) critical values as specified; other small-sample corrections not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The exact finite-sample SD of the slope estimator is 0.268, confirmed by the Monte Carlo SD of 0.270.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"Analytic value is exact; MC value has sampling error ~0.004.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is about 71% of the true SD, which explains the undercoverage.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean estimated from 2000 replications.\"}\n]\nEOF\ncat > out/state/phases.json <<'EOF'\n{\"P0\":{\"status\":\"passed\",\"checks\":[\"TASK.md read fully\",\"contract.md written\"],\"notes\":\"\"},\n\"P1\":{\"status\":\"passed\",\"checks\":[\"SHA-256 matches bundle.json\",\"30x3, no NA, no duplicates, i=1..30, sigma>0\",\"exclusions.csv: 30-0=30\"],\"notes\":\"no availability violations\"},\n\"P2\":{\"status\":\"passed\",\"checks\":[\"coverage_conventional (naive baseline) computed with the same CI rule: 0.844\"],\"notes\":\"\"},\n\"P3\":{\"status\":\"passed\",\"checks\":[\"analysis.py reads only data/, fixed seed 20261030, writes results.json\"],\"notes\":\"\"},\n\"P4\":{\"status\":\"passed\",\"checks\":[\"clean rerun byte-identical\",\"statsmodels cross-check exact\",\"analytic SD matches sandwich\",\"MC SD/mean within 1 SE\",\"Jensen bound on E[se_conv] holds\",\"undercover rule verified; see checks.md\"],\"notes\":\"\"},\n\"P5\":{\"status\":\"passed\",\"checks\":[\"report.md has all six sections with key citations\",\"claims.json with 4 claims incl. nominal-coverage claim\"],\"notes\":\"\"}}\nEOF\ncat >> out/state/log.md <<'EOF'\n- 2026-10-30 P1 passed: data clean, no exclusions.\n- 2026-10-30 Decision: undercover rule uses binomial SE at nominal p=0.95 (also robust at p-hat).\n- 2026-10-30 P2/P3 passed: seed 20261030; cov_conv 0.844, cov_hc1 0.902.\n- 2026-10-30 P4 passed: deterministic rerun, cross-checks in checks.md.\n- 2026-10-30 P5 passed: report and claims written. Task complete.\nEOF\nls out out/state","description":"Write checks, report, claims, final ledger and log"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1047,"cache_read_input_tokens":10383,"output_tokens":3822,"output_tokens_details":{"thinking_tokens":129},"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":1047},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmtNZHu7KJVCA1pXxic","requestRef":"2ce1352b-118b-46b5-913a-3b7022ba3cd5","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"7646b651-9dca-40df-b4bb-ca7d56f3f3ba","timestamp":"2026-10-30T04:16:52.016Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0053","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","version":"2.1.314","gitBranch":"HEAD"}41{"parentUuid":"7646b651-9dca-40df-b4bb-ca7d56f3f3ba","isSidechain":false,"promptId":"a6a691d9-425a-478d-a15e-b9c617fc5580","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01SBDHux799ovLqWQNF93Lqo","type":"tool_result","content":"out:\nanalysis.py\nclaims.json\nexclusions.csv\nreport.md\nresults.json\nstate\n\nout/state:\nchecks.md\ncontract.md\ndata_audit.md\nlog.md\nphases.json","is_error":false}]},"uuid":"bf381b24-ac50-4a8b-a007-d9fe85e208de","timestamp":"2026-10-30T04:16:52.087Z","toolUseResult":{"stdout":"out:\nanalysis.py\nclaims.json\nexclusions.csv\nreport.md\nresults.json\nstate\n\nout/state:\nchecks.md\ncontract.md\ndata_audit.md\nlog.md\nphases.json","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"7646b651-9dca-40df-b4bb-ca7d56f3f3ba","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0053","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","version":"2.1.314","gitBranch":"HEAD"}42{"type":"api-request-blob","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","hash":"d5678bfedf8e882255763a9d6e0f05fa43c401e773657ae6ed023c107f611556","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01SBDHux799ovLqWQNF93Lqo","name":"Bash","input":{"command":"cat > out/state/checks.md <<'EOF'\n# Verification checks (P4)\n1. Determinism: deleted out/results.json, re-ran analysis.py; `cmp` with previous output -> IDENTICAL. PASS\n2. Per-replication correctness: replication 1 slope, se_conv, se_HC1 match statsmodels OLS / HC1 to ~1e-15. PASS\n3. Exact SD: closed form sqrt(sum w_i^2 sigma_i^2)=0.267981 equals sandwich (X'X)^-1X'ΣX(X'X)^-1 [1,1]. PASS\n4. MC SD vs exact: 0.26965 vs 0.26798, diff 0.0017 < 1 SE of sample SD (~0.268/sqrt(2·1999)=0.0042). PASS\n5. Unbiasedness: mean b̂ 0.4962 vs 0.5, diff −0.0038 < 1 MC SE (0.27/sqrt(2000)=0.006). PASS\n6. E[se_conv]: MC mean 0.19026 ≤ Jensen upper bound sqrt(E[s²]·[(X'X)^-1]_22)=0.19318 (E[s²]=tr(MΣ)/28 analytic). Consistent. PASS\n7. Ranges: coverages in [0,1]; MC SEs >0; n_replications=2000, n_obs=30, n_input_rows=30; no exclusions (30−0=30). PASS\n8. Undercover rule: 0.844 < 0.95−3·0.004873=0.93538 -> true; also robust to using SE at p̂ (0.95−0.844=0.106 ≫ 3·0.0081). PASS\n9. Magnitude vs baseline: mean se_conv (0.190) < exact SD (0.268), ratio 0.71 — explains undercoverage; HC1 mean SE 0.241 closer but still below → HC1 also undercovers (0.902). Coherent. PASS\nEOF\ncat > out/report.md <<'EOF'\n# F2-conf-03: Monte Carlo coverage of OLS confidence intervals with heteroskedastic errors (n = 30)\n\n## Question\nThe model has a fixed design: y_i = 1 + 0.5 x_i + σ_i ε_i, with ε_i iid N(0,1), and x_i and σ_i taken from `data/design.csv` (n = 30). The question is whether the nominal 95% interval β̂ ± t_{0.975}(28)·se for the OLS slope reaches 95% coverage. Two standard errors are compared: (i) the conventional OLS standard error and (ii) the HC1 robust standard error. Two further quantities are reported: the exact SD of β̂ and E[se_conv].\n\n## Data and exclusions\nThe only input is `data/design.csv`. It has 30 rows (`n_input_rows`), SHA-256 b1a17a52…74f0, and matches the bundle manifest. It has no missing values, no duplicates and all σ_i > 0. No rows were excluded (`out/exclusions.csv`), and `data_issues` is empty. σ_i rises toward both ends of the x range, so the largest error variances sit at the high-leverage points. This is the pattern under which the conventional SE is biased downward.\n\n## Method\n- **Exact SD.** β̂ − β = Σ w_i σ_i ε_i, where w_i = (x_i − x̄)/S_xx. So SD(β̂) = sqrt(Σ w_i² σ_i²), computed analytically (`sd_beta_exact`).\n- **Monte Carlo.** R = 2000 replications (`n_replications`) with numpy `default_rng` seed 20261030 (`random_seed`). Only ε is redrawn. In each replication:\n - fit OLS;\n - compute se_conv = sqrt(s²·[(X'X)⁻¹]₂₂), with s² = RSS/28;\n - compute se_HC1 = sqrt((n/(n−2))·Σ a_i² e_i²), where a = row 2 of (X'X)⁻¹X';\n - check whether β = 0.5 lies inside β̂ ± 2.0484·se (`t_crit_28`).\n- **Undercoverage rule.** The conventional interval counts as undercovering if its coverage is below 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercover_threshold`). The binomial SE here is evaluated at the nominal p (`mcse_binomial_at_nominal`).\n- **Validation.** One replication was cross-checked against statsmodels OLS/HC1 (exact match).\n\n## Results\n- **Conventional interval coverage:** 0.844 (`coverage_conventional`). This is the naïve baseline.\n- **HC1 interval coverage:** 0.902 (`coverage_hc1`).\n- **Exact SD(β̂):** 0.2680 (`sd_beta_exact`). The Monte Carlo SD is 0.2696 (`sd_beta_mc`), which agrees.\n- **Mean se_conv:** 0.1903 (`mean_se_conventional`). That is about 71% of the true SD. The formula that assumes constant variance gives 0.1962 (`sd_beta_conventional_formula_homoskedastic`).\n- **Mean se_HC1:** 0.2408 (`mean_se_hc1`).\n- **Conclusion:** the conventional 95% interval does **not** reach its nominal coverage (`conventional_ci_undercovers` = true). It undercovers by about 10.6 percentage points. HC1 does better but still falls short of 95% at n = 30.\n\n## Uncertainty\n- **Monte Carlo SE of each coverage estimate** (binomial, evaluated at p̂):\n - conventional: 0.0081 (`mcse_coverage_conventional`), giving an approximate 95% MC interval of [0.828, 0.860];\n - HC1: 0.0066 (`mcse_coverage_hc1`), giving approximately [0.889, 0.915].\n- **At nominal p = 0.95,** the binomial SE is 0.0049 (`mcse_binomial_at_nominal`).\n- **Size of the shortfall:** both coverages lie more than 10 nominal SEs below 0.95. The conclusion does not depend on which SE is used.\n- **Other Monte Carlo checks:**\n - the MC SD of β̂ is within one SE (≈0.004) of the exact value;\n - mean β̂ = 0.4962 (`mean_beta_hat_mc`), which is within one MC SE (≈0.006) of 0.5.\n\n## Limitations and open issues\n- The results hold conditionally on this particular fixed design and on normal errors. They are not general statements about OLS.\n- With R = 2000, coverage is known only to about ±0.016 (95% MC interval). The conclusions above are well clear of that margin.\n- The task specifies the HC1 interval with t(28) critical values. Small-sample alternatives (HC2/HC3, or Satterthwaite/Bell–McCaffrey degrees of freedom) would probably come closer to 95%, but they were not evaluated because the task did not ask for them.\n- No part of the task was infeasible.\nEOF\ncat > out/claims.json <<'EOF'\n[\n {\"id\":\"C1\",\"text\":\"The conventional 95% OLS interval does not attain nominal coverage: estimated coverage 0.844, more than three binomial MC standard errors below 0.95.\",\"results_keys\":[\"coverage_conventional\",\"conventional_ci_undercovers\",\"undercover_threshold\",\"mcse_coverage_conventional\"],\"status\":\"supported\",\"caveat\":\"Conditional on this fixed design and normal errors; MC uncertainty ±~0.016.\"},\n {\"id\":\"C2\",\"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\"],\"status\":\"supported\",\"caveat\":\"Uses t(28) critical values as specified; other small-sample corrections not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The exact finite-sample SD of the slope estimator is 0.268, confirmed by the Monte Carlo SD of 0.270.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"Analytic value is exact; MC value has sampling error ~0.004.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward: its mean (0.190) is about 71% of the true SD, which explains the undercoverage.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean estimated from 2000 replications.\"}\n]\nEOF\ncat > out/state/phases.json <<'EOF'\n{\"P0\":{\"status\":\"passed\",\"checks\":[\"TASK.md read fully\",\"contract.md written\"],\"notes\":\"\"},\n\"P1\":{\"status\":\"passed\",\"checks\":[\"SHA-256 matches bundle.json\",\"30x3, no NA, no duplicates, i=1..30, sigma>0\",\"exclusions.csv: 30-0=30\"],\"notes\":\"no availability violations\"},\n\"P2\":{\"status\":\"passed\",\"checks\":[\"coverage_conventional (naive baseline) computed with the same CI rule: 0.844\"],\"notes\":\"\"},\n\"P3\":{\"status\":\"passed\",\"checks\":[\"analysis.py reads only data/, fixed seed 20261030, writes results.json\"],\"notes\":\"\"},\n\"P4\":{\"status\":\"passed\",\"checks\":[\"clean rerun byte-identical\",\"statsmodels cross-check exact\",\"analytic SD matches sandwich\",\"MC SD/mean within 1 SE\",\"Jensen bound on E[se_conv] holds\",\"undercover rule verified; see checks.md\"],\"notes\":\"\"},\n\"P5\":{\"status\":\"passed\",\"checks\":[\"report.md has all six sections with key citations\",\"claims.json with 4 claims incl. nominal-coverage claim\"],\"notes\":\"\"}}\nEOF\ncat >> out/state/log.md <<'EOF'\n- 2026-10-30 P1 passed: data clean, no exclusions.\n- 2026-10-30 Decision: undercover rule uses binomial SE at nominal p=0.95 (also robust at p-hat).\n- 2026-10-30 P2/P3 passed: seed 20261030; cov_conv 0.844, cov_hc1 0.902.\n- 2026-10-30 P4 passed: deterministic rerun, cross-checks in checks.md.\n- 2026-10-30 P5 passed: report and claims written. Task complete.\nEOF\nls out out/state","description":"Write checks, report, claims, final ledger and log"}}]}}43{"type":"api-request","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","id":"fd073806-9974-431b-b8a2-334a3bc2ecfa","timestamp":"2026-10-30T04:16:52.090Z","version":"2.1.314","querySource":"sdk","shapeHash":"cbc09f252ff7e553d8e132fd82b221005b46bcaf4b710162396bab127cbc188a","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":"2ce1352b-118b-46b5-913a-3b7022ba3cd5","keep":12,"tail":["d5678bfedf8e882255763a9d6e0f05fa43c401e773657ae6ed023c107f611556","b62afe0a5881d7c04f502dd4e26136ccb3ccd8aef1b402b4e52fc607d2908b7b"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",13,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"4ec9edba7f4e7bf352df6ffb07c14440e428d3fa13af6352ac657402a5d95ab7"}44{"parentUuid":"bf381b24-ac50-4a8b-a007-d9fe85e208de","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmvnPZk8gxiAXHKpM5g","type":"message","role":"assistant","content":[{"type":"text","text":"The conventional 95% interval for the OLS slope does not reach its nominal coverage in this design: it covered the true β = 0.5 in only 84.4% of the 2,000 replications. Re-running the script from a clean state reproduces `out/results.json` byte for byte, and all four required files are written.\n\n| Quantity | Result |\n|---|---|\n| Coverage, conventional interval | 0.844 (Monte Carlo standard error 0.008) |\n| Coverage, HC1 robust interval | 0.902 (Monte Carlo standard error 0.007) |\n| Exact standard deviation of β̂ (analytical) | 0.2680 |\n| Standard deviation of β̂ across simulations | 0.2696 |\n| Average conventional standard error | 0.190 |\n\n- **Conventional interval falls short:** 0.844 is far below the task's cutoff of 0.9354 (95% minus three binomial standard errors at 2,000 replications), so `conventional_ci_undercovers` is true. For that cutoff I took the binomial standard error at the nominal 95% rather than at the estimated coverage. The conclusion is the same either way.\n- **Why:** the conventional standard error averages about 71% of the true spread of β̂. In the design file, the error variance is largest at the extreme x values, which have the most influence on the slope, so the usual formula understates the uncertainty.\n- **Robust interval:** HC1 does better but still covers only 90.2%, short of 95% at n = 30.\n\n**Checks:**\n- One replication matches statsmodels exactly.\n- The simulated spread of β̂ is within one Monte Carlo standard error of the exact value, and the average estimate (0.4962) is within one of 0.5.\n- The design file matches its recorded checksum and needed no exclusions.\n\n**Limitations:**\n- These results hold for this fixed design with normally distributed errors, not for OLS in general.\n- With 2,000 replications, each coverage figure is uncertain by roughly ±0.016.\n- I did not test other small-sample corrections (HC2/HC3, or adjusted degrees of freedom) because the task didn't ask for them. They would likely come closer to 95%.\n\nThe seed is 20261030. To regenerate, run `~/wp29/venv/bin/python out/analysis.py` from the workspace root.\n\nFiles are in `out/`:\n- results.json\n- report.md\n- analysis.py\n- claims.json\n- exclusions.csv\n- state/ (contract, data audit, checks, phase ledger, log)"}],"container":null,"stop_reason":"end_turn","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":3889,"cache_read_input_tokens":11430,"output_tokens":831,"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":3889},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmvn7h5vyiDHnRw3zg2","requestRef":"fd073806-9974-431b-b8a2-334a3bc2ecfa","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"20dc9519-7420-4901-a1df-78e2bbc3cb42","timestamp":"2026-10-30T04:17:00.073Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0053","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","version":"2.1.314","gitBranch":"HEAD"}45{"type":"last-prompt","lastPrompt":"Begin","leafUuid":"20dc9519-7420-4901-a1df-78e2bbc3cb42","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50"}46{"type":"cost-state","sessionId":"3f36d2b8-8ec4-45bd-bb82-22fcb6f0ce50","totalCostUSD":0.25839219999999996,"totalAPIDuration":80280,"totalAPIDurationWithoutRetries":80277,"totalToolDuration":2811,"totalLinesAdded":0,"totalLinesRemoved":0,"totalDuration":83258,"startTime":1793333736822,"modelUsage":{"claude-opus-5-5":{"inputTokens":16,"outputTokens":8993,"thinkingTokens":340,"cacheReadInputTokens":46966,"cacheCreationInputTokens":13815,"cacheCreationEphemeral5mInputTokens":13815,"cacheCreationEphemeral1hInputTokens":0,"webSearchRequests":0,"costUSD":0.25839219999999996}},"hasUnknownModelCost":false}47