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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.\ntotal 32\ndrwxr-xr-x 6 simon-pierreboucher staff 192 oct. 29 23:35 .\ndrwxr-xr-x 42 simon-pierreboucher staff 1344 oct. 30 00:15 ..\n-rw-r--r-- 1 simon-pierreboucher staff 420 oct. 29 23:35 bundle.json\ndrwxr-xr-x 3 simon-pierreboucher staff 96 oct. 29 23:33 data\n-rw-r--r-- 1 simon-pierreboucher staff 1456 oct. 29 23:35 gate_spec.json\n-rw-r--r-- 1 simon-pierreboucher staff 4161 oct. 29 23:35 TASK.md\ndata:\ndesign.csv","is_error":false}]},"uuid":"f4e23d28-e260-4908-867a-bf22cb09d50e","timestamp":"2026-10-30T04:15:22.636Z","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). 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MC.\n- (c) sd_beta_exact = sqrt(sum((x_i-xbar)^2 sigma_i^2))/Sxx (analytical; exact since design fixed and errors normal).\n- (d) mean_se_conventional: MC mean of se_conv.\n- Also sd_beta_mc (ddof=1), conventional_ci_undercovers = cov_conv < 0.95 - 3*sqrt(0.95*0.05/2000).\nAssumptions: model as stated; coverage meaningful as frequentist long-run probability under repeated eps draws.\n\n**Outputs:** out/results.json (task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers, n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed); out/report.md (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json; plus protocol state files and out/exclusions.csv.\n\n**Information rules:** Only data/design.csv; do not redraw x or sigma; constants from TASK.md. Binomial MC SE uses p=0.95 (nominal) per \"binomial standard error at R=2000\"; documented choice.\n\n**Invalidity:** design file not 30 rows / missing or non-positive sigma; R != 2000; 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\":\"\"},\n\"P2\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P3\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P4\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P5\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"}}\nEOF\necho \"- 2026-10-30: P0 passed (contract written).\" > log.md","description":"Write contract, phase ledger, log"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":783,"cache_read_input_tokens":5681,"output_tokens":1125,"output_tokens_details":{"thinking_tokens":45},"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":783},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmpJtRbhJMmNEpJuS3G","requestRef":"0252b807-560a-4401-89b4-887820be415c","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"7b92a009-6302-4d6e-a28f-55dd77f6128f","timestamp":"2026-10-30T04:15:34.549Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}25{"parentUuid":"7b92a009-6302-4d6e-a28f-55dd77f6128f","isSidechain":false,"promptId":"fa071912-bf64-4a6f-928c-77961bae398c","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01MTNZzCqtT4fjFrKm6e1oBf","type":"tool_result","content":"(Bash completed with no output)","is_error":false}]},"uuid":"c743229b-e489-4419-a02c-18a093eeff0c","timestamp":"2026-10-30T04:15:34.589Z","toolUseResult":{"stdout":"","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"7b92a009-6302-4d6e-a28f-55dd77f6128f","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052/out/state","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}26{"type":"api-request-blob","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","hash":"134baed248c3e79eab1252a8c07b288cdf559c54d966b33ad998bc3e6394eafe","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01MTNZzCqtT4fjFrKm6e1oBf","name":"Bash","input":{"command":"mkdir -p out/state && cd out/state && cat > contract.md <<'EOF'\n# Contract (P0)\n**Question:** In the fixed-design model y_i = 1 + 0.5 x_i + sigma_i eps_i (n=30, x and sigma fixed from data/design.csv, eps iid N(0,1)), do nominal 95% OLS t-intervals for the slope achieve 95% coverage?\n\n**Estimands:**\n- (a) coverage_conventional: P(|beta_hat - 0.5| <= t_{.975}(28) se_conv), se_conv from s^2=RSS/(n-2). MC, R=2000.\n- (b) coverage_hc1: same with se_HC1 = sqrt(n/(n-2) * sum(w_i^2 e_i^2)), w_i=(x_i-xbar)/Sxx. MC.\n- (c) sd_beta_exact = sqrt(sum((x_i-xbar)^2 sigma_i^2))/Sxx (analytical; exact since design fixed and errors normal).\n- (d) mean_se_conventional: MC mean of se_conv.\n- Also sd_beta_mc (ddof=1), conventional_ci_undercovers = cov_conv < 0.95 - 3*sqrt(0.95*0.05/2000).\nAssumptions: model as stated; coverage meaningful as frequentist long-run probability under repeated eps draws.\n\n**Outputs:** out/results.json (task_id, coverage_conventional, coverage_hc1, sd_beta_exact, sd_beta_mc, mean_se_conventional, conventional_ci_undercovers, n_replications, n_obs_per_replication, feasibility, feasibility_reason, data_issues, n_input_rows, random_seed); out/report.md (Question; Data and exclusions; Method; Results; Uncertainty; Limitations and open issues); out/analysis.py; out/claims.json; plus protocol state files and out/exclusions.csv.\n\n**Information rules:** Only data/design.csv; do not redraw x or sigma; constants from TASK.md. Binomial MC SE uses p=0.95 (nominal) per \"binomial standard error at R=2000\"; documented choice.\n\n**Invalidity:** design file not 30 rows / missing or non-positive sigma; R != 2000; 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\":\"\"},\n\"P2\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P3\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P4\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"},\n\"P5\":{\"status\":\"pending\",\"checks\":[],\"notes\":\"\"}}\nEOF\necho \"- 2026-10-30: P0 passed (contract written).\" > log.md","description":"Write contract, phase ledger, log"}}]}}27{"type":"api-request","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","id":"3c582dec-24bd-49f8-b5d2-dc149b9ef0e7","timestamp":"2026-10-30T04:15:34.592Z","version":"2.1.314","querySource":"sdk","shapeHash":"4b743b510d889c6239c0ae45b3e10d106cfc525234cc1e1d638fe3814549908d","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":"0252b807-560a-4401-89b4-887820be415c","keep":6,"tail":["134baed248c3e79eab1252a8c07b288cdf559c54d966b33ad998bc3e6394eafe","d3eb795be3af71063c120f9a4d512492749a25e67e13f51a4cae7ef6e76cfb60"],"cacheControl":[{"path":["system",0],"value":{"type":"ephemeral"}},{"path":["system",1],"value":{"type":"ephemeral"}},{"path":["messages",7,"content",0],"value":{"type":"ephemeral"}}],"requestDigest":"03dc7f83304e20ec3efcbd9394ee53fca17d8d2caa24863c93c9a301a719bf6e"}28{"parentUuid":"c743229b-e489-4419-a02c-18a093eeff0c","isSidechain":false,"message":{"model":"claude-opus-5-5","id":"msg_011CgXmq59wyQwVs6h8K8UW5","type":"message","role":"assistant","content":[{"type":"tool_use","id":"toolu_014z82goh6EMzqmgZhDUjJ82","name":"Bash","input":{"command":"~/wp29/venv/bin/python -c \"\nimport pandas as pd\nd=pd.read_csv('data/design.csv');print(d.shape,d.dtypes.to_dict());print(d.describe().T);print('na',d.isna().sum().sum(),'dupi',d.i.duplicated().sum(),'dup rows',d.duplicated(['x','sigma']).sum(),'i seq',(d.i.values==range(1,31)).all(),'sigma>0',(d.sigma>0).all(),'x sorted',d.x.is_monotonic_increasing)\"","description":"Audit design data"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1140,"cache_read_input_tokens":6464,"output_tokens":263,"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":1140},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmq4sagb5rjHYRYtR31","requestRef":"3c582dec-24bd-49f8-b5d2-dc149b9ef0e7","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"8e1592d4-a196-4428-8085-e7468bb05aac","timestamp":"2026-10-30T04:15:37.434Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052/out/state","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}29{"parentUuid":"8e1592d4-a196-4428-8085-e7468bb05aac","isSidechain":false,"promptId":"fa071912-bf64-4a6f-928c-77961bae398c","type":"user","message":{"role":"user","content":[{"type":"tool_result","content":"Exit code 1\nTraceback (most recent call last):\n File \"<string>\", line 3, in <module>\n d=pd.read_csv('data/design.csv');print(d.shape,d.dtypes.to_dict());print(d.describe().T);print('na',d.isna().sum().sum(),'dupi',d.i.duplicated().sum(),'dup rows',d.duplicated(['x','sigma']).sum(),'i seq',(d.i.values==range(1,31)).all(),'sigma>0',(d.sigma>0).all(),'x sorted',d.x.is_monotonic_increasing)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1026, in read_csv\n return _read(filepath_or_buffer, kwds)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 620, in _read\n parser = TextFileReader(filepath_or_buffer, **kwds)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1620, in __init__\n self._engine = self._make_engine(f, self.engine)\n ~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1880, in _make_engine\n self.handles = get_handle(\n ~~~~~~~~~~^\n f,\n ^^\n ...<6 lines>...\n storage_options=self.options.get(\"storage_options\", None),\n ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n )\n ^\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/common.py\", line 873, in get_handle\n handle = open(\n handle,\n ...<3 lines>...\n newline=\"\",\n )\nFileNotFoundError: [Errno 2] No such file or directory: 'data/design.csv'","is_error":true,"tool_use_id":"toolu_014z82goh6EMzqmgZhDUjJ82"}]},"uuid":"c0eb4c5b-ad68-49aa-b167-f65cbc282bd3","timestamp":"2026-10-30T04:15:37.654Z","toolUseResult":"Error: Exit code 1\nTraceback (most recent call last):\n File \"<string>\", line 3, in <module>\n d=pd.read_csv('data/design.csv');print(d.shape,d.dtypes.to_dict());print(d.describe().T);print('na',d.isna().sum().sum(),'dupi',d.i.duplicated().sum(),'dup rows',d.duplicated(['x','sigma']).sum(),'i seq',(d.i.values==range(1,31)).all(),'sigma>0',(d.sigma>0).all(),'x sorted',d.x.is_monotonic_increasing)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1026, in read_csv\n return _read(filepath_or_buffer, kwds)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 620, in _read\n parser = TextFileReader(filepath_or_buffer, **kwds)\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1620, in __init__\n self._engine = self._make_engine(f, self.engine)\n ~~~~~~~~~~~~~~~~~^^^^^^^^^^^^^^^^\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/parsers/readers.py\", line 1880, in _make_engine\n self.handles = get_handle(\n ~~~~~~~~~~^\n f,\n ^^\n ...<6 lines>...\n storage_options=self.options.get(\"storage_options\", None),\n ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n )\n ^\n File \"/Users/simon-pierreboucher/wp29/venv/lib/python3.14/site-packages/pandas/io/common.py\", line 873, in get_handle\n handle = open(\n handle,\n ...<3 lines>...\n newline=\"\",\n )\nFileNotFoundError: [Errno 2] No such file or directory: 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> out/state/data_audit.md <<'EOF'\n# Data audit (P1)\nFile: data/design.csv — SHA-256 b1a17a529973e7994bbedad5ec1bc73fe968c777301ca8745657ea300e3c74f0 (matches bundle.json).\n- 30 rows x 3 columns: i (int64, 1..30 consecutive), x (float64, dimensionless, range -1.3942..1.9919, mean 0.2823, sd 0.8212), sigma (float64, dimensionless, range 0.6034..1.7903, mean 0.8195).\n- Missing values: 0. Duplicate i: 0. Duplicate (x,sigma) rows: 0. All sigma > 0. x sorted ascending.\n- No date columns (synthetic fixed design). sigma rises with |x - ~0| (approx. quadratic pattern), i.e. largest variances at extreme x — the high-leverage points — which is the case where conventional SEs are expected to understate the variance of beta_hat.\n- Minor observation: rows 21/22 have nearly identical x (0.7123, 0.7126); not an error.\n- Information-availability: the only input is the permitted design file; no violation. No exclusions applied.\nEOF\nprintf \"step,rule,n_before,n_after,n_removed,rationale\\n0,none (all design rows retained),30,30,0,Design is clean and must not be altered\\n\" > out/exclusions.csv\ncd out/state && ~/wp29/venv/bin/python -c \"\nimport json;p=json.load(open('phases.json'));p['P1']={'status':'passed','checks':['sha256 recorded and matches bundle.json','30x3 schema verified','0 NA, 0 duplicates, sigma>0','exclusions.csv written (0 removed)'],'notes':''};json.dump(p,open('phases.json','w'),indent=1)\"\necho \"- 2026-10-30: P1 passed (data clean, no exclusions).\" >> log.md","description":"Write audit, exclusions, update ledger"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":533,"cache_read_input_tokens":8691,"output_tokens":815,"output_tokens_details":{"thinking_tokens":68},"server_tool_use":{"web_search_requests":0,"web_fetch_requests":0},"service_tier":"standard","cache_creation":{"ephemeral_1h_input_tokens":0,"ephemeral_5m_input_tokens":533},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmqXRWu2TLRTiFEuMBb","requestRef":"2df2f3e6-b84d-44ab-b114-7050ee76a08d","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"e57079d1-1eb8-4450-b654-73643075ada7","timestamp":"2026-10-30T04:15:48.933Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}39{"parentUuid":"e57079d1-1eb8-4450-b654-73643075ada7","isSidechain":false,"promptId":"fa071912-bf64-4a6f-928c-77961bae398c","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01Yanb4YvNcZS75yqF8GZxRU","type":"tool_result","content":"(Bash completed with no output)","is_error":false}]},"uuid":"e788e99a-5918-447b-8178-daa8a37b78b4","timestamp":"2026-10-30T04:15:49.007Z","toolUseResult":{"stdout":"","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"e57079d1-1eb8-4450-b654-73643075ada7","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052/out/state","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}40{"type":"api-request-blob","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","hash":"f8bf7e1ae3228633e9cfba39135c970829f4232ef18f99b7abfc430023155a1b","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01Yanb4YvNcZS75yqF8GZxRU","name":"Bash","input":{"command":"cat > out/state/data_audit.md <<'EOF'\n# Data audit (P1)\nFile: data/design.csv — SHA-256 b1a17a529973e7994bbedad5ec1bc73fe968c777301ca8745657ea300e3c74f0 (matches bundle.json).\n- 30 rows x 3 columns: i (int64, 1..30 consecutive), x (float64, dimensionless, range -1.3942..1.9919, mean 0.2823, sd 0.8212), sigma (float64, dimensionless, range 0.6034..1.7903, mean 0.8195).\n- Missing values: 0. 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No exclusions applied.\nEOF\nprintf \"step,rule,n_before,n_after,n_removed,rationale\\n0,none (all design rows retained),30,30,0,Design is clean and must not be altered\\n\" > out/exclusions.csv\ncd out/state && ~/wp29/venv/bin/python -c \"\nimport json;p=json.load(open('phases.json'));p['P1']={'status':'passed','checks':['sha256 recorded and matches bundle.json','30x3 schema verified','0 NA, 0 duplicates, sigma>0','exclusions.csv written (0 removed)'],'notes':''};json.dump(p,open('phases.json','w'),indent=1)\"\necho \"- 2026-10-30: P1 passed (data clean, no exclusions).\" >> log.md","description":"Write audit, exclusions, update 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/Users/simon-pierreboucher/wp29/runs/RC0052 && cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS slope CIs under heteroskedasticity (fixed design).\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nReads only data/design.csv; writes out/results.json deterministically.\"\"\"\nimport json\nfrom pathlib import Path\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 # fixed seed for numpy.random.default_rng (PCG64)\n\nd = pd.read_csv(\"data/design.csv\")\nn_rows = int(len(d))\nissues = []\nassert n_rows == N and d[[\"x\", \"sigma\"]].notna().all().all() and (d[\"sigma\"] > 0).all()\nx = d[\"x\"].to_numpy(float)\nsig = d[\"sigma\"].to_numpy(float)\n\nxc = x - x.mean()\nSxx = float(xc @ xc)\nw = xc / Sxx # beta_hat = beta + sum(w_i sigma_i eps_i)\nsd_beta_exact = float(np.sqrt(np.sum(w**2 * sig**2)))\nsd_beta_conv_formula = float(np.sqrt(np.mean(sig**2) / Sxx)) # naive homoskedastic SD (diagnostic only)\n\ntcrit = float(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[None, :] + sig[None, :] * eps\nX = np.column_stack([np.ones(N), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nB = Y @ X @ XtX_inv # (R,2) OLS coefficients (XtX_inv symmetric)\nbhat = B[:, 1]\nE = Y - B @ X.T # residuals\ns2 = np.sum(E**2, axis=1) / (N - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 sandwich for slope: (X'X)^-1 X' diag(e^2) X (X'X)^-1, element [1,1]; HC1 = HC0 * n/(n-2)\nA = X @ XtX_inv # row i: a_i' ; slope weight = A[:,1] (= w)\nhc0 = (E**2) @ (A[:, 1] ** 2)\nse_hc1 = np.sqrt(hc0 * N / (N - 2))\n\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))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n\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(np.mean(se_conv)),\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\": n_rows},\n \"random_seed\": SEED,\n # supplementary (not required) diagnostics\n \"t_critical_28\": tcrit,\n \"mcse_binomial_nominal\": mcse_nominal,\n \"mcse_coverage_conventional\": mcse_conv,\n \"mcse_coverage_hc1\": mcse_hc1,\n \"undercoverage_threshold\": LEVEL - 3 * mcse_nominal,\n \"mean_se_hc1\": float(np.mean(se_hc1)),\n \"sd_beta_homoskedastic_formula\": sd_beta_conv_formula,\n \"mean_beta_hat_mc\": float(np.mean(bhat)),\n}\nPath(\"out\").mkdir(exist_ok=True)\nPath(\"out/results.json\").write_text(json.dumps(res, indent=2))\nfor k, v in res.items():\n print(f\"{k}: {v}\")\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis 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F2-conf-03\ncoverage_conventional: 0.844\ncoverage_hc1: 0.902\nsd_beta_exact: 0.26798059283028336\nsd_beta_mc: 0.2696473708309126\nmean_se_conventional: 0.19026231012152658\nconventional_ci_undercovers: True\nn_replications: 2000\nn_obs_per_replication: 30\nfeasibility: complete\nfeasibility_reason: None\ndata_issues: []\nn_input_rows: {'design.csv': 30}\nrandom_seed: 20261030\nt_critical_28: 2.048407141795244\nmcse_binomial_nominal: 0.004873397172404485\nmcse_coverage_conventional: 0.00811369213120636\nmcse_coverage_hc1: 0.006648157639526908\nundercoverage_threshold: 0.9353798084827865\nmean_se_hc1: 0.24080977065103962\nsd_beta_homoskedastic_formula: 0.19623471734704753\nmean_beta_hat_mc: 0.4961965063446238","is_error":false}]},"uuid":"5f34e912-c49f-4840-9e91-fdcf824b21c6","timestamp":"2026-10-30T04:16:04.270Z","toolUseResult":{"stdout":"task_id: F2-conf-03\ncoverage_conventional: 0.844\ncoverage_hc1: 0.902\nsd_beta_exact: 0.26798059283028336\nsd_beta_mc: 0.2696473708309126\nmean_se_conventional: 0.19026231012152658\nconventional_ci_undercovers: True\nn_replications: 2000\nn_obs_per_replication: 30\nfeasibility: complete\nfeasibility_reason: None\ndata_issues: []\nn_input_rows: {'design.csv': 30}\nrandom_seed: 20261030\nt_critical_28: 2.048407141795244\nmcse_binomial_nominal: 0.004873397172404485\nmcse_coverage_conventional: 0.00811369213120636\nmcse_coverage_hc1: 0.006648157639526908\nundercoverage_threshold: 0.9353798084827865\nmean_se_hc1: 0.24080977065103962\nsd_beta_homoskedastic_formula: 0.19623471734704753\nmean_beta_hat_mc: 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/Users/simon-pierreboucher/wp29/runs/RC0052 && cat > out/analysis.py <<'EOF'\n\"\"\"F2-conf-03: Monte Carlo coverage of OLS slope CIs under heteroskedasticity (fixed design).\nRun from workspace root: ~/wp29/venv/bin/python out/analysis.py\nReads only data/design.csv; writes out/results.json deterministically.\"\"\"\nimport json\nfrom pathlib import Path\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 # fixed seed for numpy.random.default_rng (PCG64)\n\nd = pd.read_csv(\"data/design.csv\")\nn_rows = int(len(d))\nissues = []\nassert n_rows == N and d[[\"x\", \"sigma\"]].notna().all().all() and (d[\"sigma\"] > 0).all()\nx = d[\"x\"].to_numpy(float)\nsig = d[\"sigma\"].to_numpy(float)\n\nxc = x - x.mean()\nSxx = float(xc @ xc)\nw = xc / Sxx # beta_hat = beta + sum(w_i sigma_i eps_i)\nsd_beta_exact = float(np.sqrt(np.sum(w**2 * sig**2)))\nsd_beta_conv_formula = float(np.sqrt(np.mean(sig**2) / Sxx)) # naive homoskedastic SD (diagnostic only)\n\ntcrit = float(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[None, :] + sig[None, :] * eps\nX = np.column_stack([np.ones(N), x])\nXtX_inv = np.linalg.inv(X.T @ X)\nB = Y @ X @ XtX_inv # (R,2) OLS coefficients (XtX_inv symmetric)\nbhat = B[:, 1]\nE = Y - B @ X.T # residuals\ns2 = np.sum(E**2, axis=1) / (N - 2)\nse_conv = np.sqrt(s2 * XtX_inv[1, 1])\n# HC0 sandwich for slope: (X'X)^-1 X' diag(e^2) X (X'X)^-1, element [1,1]; HC1 = HC0 * n/(n-2)\nA = X @ XtX_inv # row i: a_i' ; slope weight = A[:,1] (= w)\nhc0 = (E**2) @ (A[:, 1] ** 2)\nse_hc1 = np.sqrt(hc0 * N / (N - 2))\n\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))\nmcse_conv = float(np.sqrt(cov_conv * (1 - cov_conv) / R))\nmcse_hc1 = float(np.sqrt(cov_hc1 * (1 - cov_hc1) / R))\nundercovers = bool(cov_conv < LEVEL - 3 * mcse_nominal)\n\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(np.mean(se_conv)),\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\": n_rows},\n \"random_seed\": SEED,\n # supplementary (not required) diagnostics\n \"t_critical_28\": tcrit,\n \"mcse_binomial_nominal\": mcse_nominal,\n \"mcse_coverage_conventional\": mcse_conv,\n \"mcse_coverage_hc1\": mcse_hc1,\n \"undercoverage_threshold\": LEVEL - 3 * mcse_nominal,\n \"mean_se_hc1\": float(np.mean(se_hc1)),\n \"sd_beta_homoskedastic_formula\": sd_beta_conv_formula,\n \"mean_beta_hat_mc\": float(np.mean(bhat)),\n}\nPath(\"out\").mkdir(exist_ok=True)\nPath(\"out/results.json\").write_text(json.dumps(res, indent=2))\nfor k, v in res.items():\n print(f\"{k}: {v}\")\nEOF\n~/wp29/venv/bin/python out/analysis.py","description":"Write and run analysis 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-a 256 out/results.json && rm out/results.json && ~/wp29/venv/bin/python out/analysis.py >/dev/null && shasum -a 256 out/results.json && ~/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\nrng=np.random.default_rng(r['random_seed']); eps=rng.standard_normal((2000,30))\nX=sm.add_constant(x); from scipy import stats; t=stats.t.ppf(.975,28)\ncc=ch=0; bs=[]; ses=[]\nfor k in range(2000):\n y=1+0.5*x+s*eps[k]\n f=sm.OLS(y,X).fit(); h=f.get_robustcov_results('HC1')\n cc+=abs(f.params[1]-.5)<=t*f.bse[1]; ch+=abs(h.params[1]-.5)<=t*h.bse[1]; bs.append(f.params[1]); ses.append(f.bse[1])\nprint('sm cov conv',cc/2000,'hc1',ch/2000,'sd',np.std(bs,ddof=1),'mean se',np.mean(ses))\n# exact E[s^2]\nH=X@np.linalg.inv(X.T@X)@X.T; M=np.eye(30)-H\nEs2=np.trace(M@np.diag(s**2))/28; print('sqrt(E s2 * c11)',np.sqrt(Es2*np.linalg.inv(X.T@X)[1,1]))\nprint('sd 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diff / se(sd)', (r['sd_beta_mc']-r['sd_beta_exact'])/(r['sd_beta_exact']/np.sqrt(2*1999)))\nprint('0<cov<1',0<r['coverage_conventional']<1, 'hc1>conv',r['coverage_hc1']>r['coverage_conventional'])\nEOF","description":"Re-run determinism check and statsmodels 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> out/state/checks.md <<'EOF'\n# Verification checks (P4)\n| # | check | outcome |\n|---|---|---|\n| 1 | Delete out/results.json, re-run analysis.py, compare SHA-256 | PASS — identical (8684b4a4…09f0f) both runs |\n| 2 | Independent re-implementation with statsmodels OLS + HC1 per replication, same seed/draws | PASS — coverage_conventional 0.844, coverage_hc1 0.902, sd_beta_mc 0.269647, mean_se_conventional 0.190262 all identical |\n| 3 | sd_beta_mc vs sd_beta_exact (0.26965 vs 0.26798) | PASS — difference 0.39 approx. SEs of a sample SD (sd/sqrt(2(R-1))) |\n| 4 | mean_se_conventional vs analytic sqrt(E[s^2]·c11)=0.19318 | PASS — MC 0.19026 slightly below, as expected by Jensen (E[sqrt] ≤ sqrt(E)) |\n| 5 | mean beta_hat 0.4962 vs true 0.5 | PASS — -0.63 MC SEs (0.268/sqrt(2000)=0.006), unbiased as OLS should be |\n| 6 | Coverages in (0,1); MC SEs > 0; n_replications=2000; n_obs=30; n_input_rows=30 | PASS |\n| 7 | Undercoverage flag: 0.844 < 0.95 − 3·0.004873 = 0.93538 → true | PASS — consistent with key value |\n| 8 | Magnitude vs baseline: HC1 (0.902) > conventional baseline (0.844); mean se_conv 0.190 < exact SD 0.268, explaining undercoverage | PASS — coherent |\n| 9 | Information rules: only data/design.csv read; x, sigma not redrawn | PASS (code inspection) |\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json;f='out/state/phases.json';p=json.load(open(f))\np['P2']={'status':'passed','checks':['baseline = coverage_conventional (task-named naive baseline) computed with R=2000, same evaluation rule as HC1'],'notes':'0.844'}\np['P3']={'status':'passed','checks':['out/analysis.py reads only data/, fixed seed 20261030, writes results.json'],'notes':''}\np['P4']={'status':'passed','checks':['clean re-run identical SHA-256','statsmodels cross-check identical','consistency checks 3-9 in checks.md all PASS'],'notes':''}\njson.dump(p,open(f,'w'),indent=1)\nEOF\necho \"- 2026-10-30: P2/P3 passed (seed 20261030; conv cov 0.844, HC1 0.902). P4 passed (determinism + statsmodels cross-check).\" >> out/state/log.md","description":"Record checks and update ledger"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1155,"cache_read_input_tokens":12291,"output_tokens":1200,"output_tokens_details":{"thinking_tokens":83},"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":1155},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":1,"requestId":"req_011CgXmt3GS2S228csZ6qrqY","requestRef":"111ea796-bd38-481d-8b4d-2f155f0ce59f","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"c510b3db-dcf9-45b2-90b5-bb6687465a9e","timestamp":"2026-10-30T04:16:25.472Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}56{"parentUuid":"c510b3db-dcf9-45b2-90b5-bb6687465a9e","isSidechain":false,"promptId":"fa071912-bf64-4a6f-928c-77961bae398c","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01SHEWWUhE9PdtELQLncLvcf","type":"tool_result","content":"(Bash completed with no output)","is_error":false}]},"uuid":"cbc0ec07-78f0-48d6-9adc-c38de91bf747","timestamp":"2026-10-30T04:16:25.500Z","toolUseResult":{"stdout":"","stderr":"","interrupted":false,"isImage":false,"noOutputExpected":false},"sandboxed":false,"sourceToolAssistantUUID":"c510b3db-dcf9-45b2-90b5-bb6687465a9e","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}57{"type":"api-request-blob","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","hash":"473caa4b3a37f79f3166709f1ddef09a15e9715771b60b378e1969e217afe21e","message":{"role":"assistant","content":[{"type":"thinking","thinking":"","signature":"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"},{"type":"tool_use","id":"toolu_01SHEWWUhE9PdtELQLncLvcf","name":"Bash","input":{"command":"cat > out/state/checks.md <<'EOF'\n# Verification checks (P4)\n| # | check | outcome |\n|---|---|---|\n| 1 | Delete out/results.json, re-run analysis.py, compare SHA-256 | PASS — identical (8684b4a4…09f0f) both runs |\n| 2 | Independent re-implementation with statsmodels OLS + HC1 per replication, same seed/draws | PASS — coverage_conventional 0.844, coverage_hc1 0.902, sd_beta_mc 0.269647, mean_se_conventional 0.190262 all identical |\n| 3 | sd_beta_mc vs sd_beta_exact (0.26965 vs 0.26798) | PASS — difference 0.39 approx. SEs of a sample SD (sd/sqrt(2(R-1))) |\n| 4 | mean_se_conventional vs analytic sqrt(E[s^2]·c11)=0.19318 | PASS — MC 0.19026 slightly below, as expected by Jensen (E[sqrt] ≤ sqrt(E)) |\n| 5 | mean beta_hat 0.4962 vs true 0.5 | PASS — -0.63 MC SEs (0.268/sqrt(2000)=0.006), unbiased as OLS should be |\n| 6 | Coverages in (0,1); MC SEs > 0; n_replications=2000; n_obs=30; n_input_rows=30 | PASS |\n| 7 | Undercoverage flag: 0.844 < 0.95 − 3·0.004873 = 0.93538 → true | PASS — consistent with key value |\n| 8 | Magnitude vs baseline: HC1 (0.902) > conventional baseline (0.844); mean se_conv 0.190 < exact SD 0.268, explaining undercoverage | PASS — coherent |\n| 9 | Information rules: only data/design.csv read; x, sigma not redrawn | PASS (code inspection) |\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json;f='out/state/phases.json';p=json.load(open(f))\np['P2']={'status':'passed','checks':['baseline = coverage_conventional (task-named naive baseline) computed with R=2000, same evaluation rule as HC1'],'notes':'0.844'}\np['P3']={'status':'passed','checks':['out/analysis.py reads only data/, fixed seed 20261030, writes results.json'],'notes':''}\np['P4']={'status':'passed','checks':['clean re-run identical SHA-256','statsmodels cross-check identical','consistency checks 3-9 in checks.md all PASS'],'notes':''}\njson.dump(p,open(f,'w'),indent=1)\nEOF\necho \"- 2026-10-30: P2/P3 passed (seed 20261030; conv cov 0.844, HC1 0.902). 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Two more targets: (c) the exact SD of β̂ and (d) E[se_conv].\n\n## Data and exclusions\nI read one file, `data/design.csv`: 30 rows, columns i, x, sigma (`n_input_rows`). It has no missing values or duplicates, and every σ is positive. Its SHA-256 matches the bundle manifest (see `out/state/data_audit.md`). I excluded nothing (`out/exclusions.csv`: 30 → 30) and found no data issues (`data_issues` = []). σ_i is largest at the extreme values of x, i.e. at the high-leverage points.\n\n## Method\n- **Exact SD of β̂:** β̂ − β = Σ w_i σ_i ε_i with w_i = (x_i − x̄)/Sxx. So SD(β̂) = sqrt(Σ w_i² σ_i²). This is exact because the design is fixed.\n- **Simulation:** R = 2000 replications (`n_replications`) of n = 30 (`n_obs_per_replication`). Only ε is redrawn, using numpy `default_rng` (PCG64) with seed 20261030 (`random_seed`).\n- **SEs and intervals:** In each replication I fit OLS and compute two SEs. se_conv = sqrt(s²·[(X'X)⁻¹]₂₂) with s² = RSS/28. se_HC1 = sqrt(n/(n−2) · Σ w_i² e_i²), i.e. the HC0 sandwich slope element times 30/28. The critical value is t_{0.975}(28) = 2.0484 (`t_critical_28`).\n- **Coverage:** the share of replications with |β̂ − 0.5| ≤ t·se.\n- **Undercoverage rule:** the conventional interval is flagged as undercovering if its coverage is below 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercoverage_threshold`). The binomial SE in this rule is evaluated at the nominal p = 0.95; that choice is documented here.\n- **Code:** `out/analysis.py` regenerates every number.\n\n## Results\n| quantity | value | key |\n|---|---|---|\n| Coverage, conventional interval | 0.844 | `coverage_conventional` |\n| Coverage, HC1 interval | 0.902 | `coverage_hc1` |\n| Exact SD of β̂ | 0.2680 | `sd_beta_exact` |\n| MC SD of β̂ (ddof 1) | 0.2696 | `sd_beta_mc` |\n| Mean se_conv | 0.1903 | `mean_se_conventional` |\n| Mean se_HC1 (supplementary) | 0.2408 | `mean_se_hc1` |\n| Conventional interval undercovers? | true | `conventional_ci_undercovers` |\n\nThe conventional interval does **not** reach its nominal 95% coverage. Its estimated coverage of 0.844 is far below the 0.9354 threshold (`coverage_conventional`, `undercoverage_threshold`). The reason is that se_conv averages 0.190 (`mean_se_conventional`), while the true SD of β̂ is 0.268 (`sd_beta_exact`). The conventional SE assumes equal error variances, so it understates the variance here, where the large-variance errors fall at high-leverage x values. The HC1 interval does better at 0.902 (`coverage_hc1`) but still falls short of 0.95. This is consistent with the known downward bias and extra variability of HC0/HC1 standard errors when n is small and leverage is high. The MC SD agrees with the exact SD (difference ≈ 0.4 sampling SEs; `out/state/checks.md`).\n\n## Uncertainty\n- **Conventional coverage:** binomial MC SE sqrt(p̂(1−p̂)/R) = 0.0081 (`mcse_coverage_conventional`). Approximate 95% MC interval 0.828–0.860.\n- **HC1 coverage:** MC SE 0.0066 (`mcse_coverage_hc1`). Approximate 95% MC interval 0.889–0.915.\n- **Binomial SE at the nominal level:** 0.0049 (`mcse_binomial_nominal`).\n- **How decisive the results are:** the conventional estimate is about 21 nominal SEs below 0.95. The HC1 estimate is also more than 9 SEs below 0.95, so neither shortfall is plausibly Monte Carlo noise.\n- **Exact quantities:** `sd_beta_exact` carries no simulation uncertainty.\n\n## Limitations and open issues\n- The results hold only for this synthetic fixed design, with normal errors and n = 30. Coverage depends on the specific pattern of σ against leverage.\n- The MC estimates depend on the seed. Another seed would change them within roughly the MC SEs reported above.\n- I did not evaluate HC2/HC3 or wild-bootstrap intervals; the task does not require them. So I make no claim about which robust method would reach 95%.\n- The binomial SE in the undercoverage rule uses p = 0.95. Using p̂ instead (0.0081) gives the same conclusion.\n- No data issues or blocked steps; the task is complete (`feasibility` = \"complete\").\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\",\"undercoverage_threshold\",\"mcse_binomial_nominal\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed design, normal errors, and seed 20261030; MC SE about 0.008.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95%.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"Small-sample HC1 bias with high-leverage heteroskedastic points; other HC variants not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The exact SD of the OLS slope is 0.268, and the Monte Carlo SD (0.270) agrees with it.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"Exact SD is analytic under the stated model.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward on average (mean 0.190 vs true SD 0.268), which explains the undercoverage.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean of se_conv is an MC estimate.\"}\n]\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json;json.load(open('out/claims.json'))\nf='out/state/phases.json';p=json.load(open(f))\np['P5']={'status':'passed','checks':['report.md has all 6 required sections with results keys cited','claims.json valid JSON with 4 claims incl. nominal-coverage claim C1'],'notes':''}\njson.dump(p,open(f,'w'),indent=1)\nEOF\necho \"- 2026-10-30: P5 passed; task complete.\" >> out/state/log.md; ls -R out","description":"Write report, claims, finalize ledger"},"caller":{"type":"direct"}}],"container":null,"stop_reason":"tool_use","stop_sequence":null,"stop_details":null,"usage":{"input_tokens":2,"cache_creation_input_tokens":1186,"cache_read_input_tokens":13446,"output_tokens":2776,"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":1186},"inference_geo":"global","iterations":[],"speed":"standard","fallback_credit":null},"input_transformations":[],"diagnostics":null,"context_management":null},"apiBlockIndex":0,"requestId":"req_011CgXmtpV1zTvAHH52JvWCX","requestRef":"ab2fc803-c717-41f2-a495-69e15c38f910","requestedModel":"claude-opus-5-5","type":"assistant","uuid":"d961d151-0553-41d9-a695-80cef6aedd21","timestamp":"2026-10-30T04:16:49.040Z","effort":"medium","perTurnEffort":"medium","userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}62{"parentUuid":"d961d151-0553-41d9-a695-80cef6aedd21","isSidechain":false,"promptId":"fa071912-bf64-4a6f-928c-77961bae398c","type":"user","message":{"role":"user","content":[{"tool_use_id":"toolu_01FBTsjhiYs1A8LuwfHCB8A9","type":"tool_result","content":"analysis.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":"6a38c570-acf2-4887-85e5-e3f52ed5e32f","timestamp":"2026-10-30T04:16:49.123Z","toolUseResult":{"stdout":"analysis.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":"d961d151-0553-41d9-a695-80cef6aedd21","permissionDecision":{"decision":"accept","source":"config","reasonType":"rule","ruleSource":"cliArg"},"userType":"external","entrypoint":"sdk-cli","cwd":"/Users/simon-pierreboucher/wp29/runs/RC0052","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","version":"2.1.314","gitBranch":"HEAD"}63{"type":"api-request-blob","sessionId":"7276070a-810c-496e-a82a-c4c9d33f9910","hash":"cfb73e3ef23a4a5af52f2ba479413bab28a6bf0dc8fa4d1c79c967a19c12d103","message":{"role":"assistant","content":[{"type":"tool_use","id":"toolu_01FBTsjhiYs1A8LuwfHCB8A9","name":"Bash","input":{"command":"cat > out/report.md <<'EOF'\n# F2-conf-03 — Coverage of OLS slope CIs in a small heteroskedastic fixed design\n\n## Question\nIn the fixed design y_i = 1 + 0.5 x_i + σ_i ε_i (n = 30, x_i and σ_i from `data/design.csv`, ε_i iid N(0,1)), do nominal 95% intervals β̂ ± t_{0.975}(28)·se reach 95% coverage when se is (a) the conventional OLS SE or (b) the HC1 robust SE? Two more targets: (c) the exact SD of β̂ and (d) E[se_conv].\n\n## Data and exclusions\nI read one file, `data/design.csv`: 30 rows, columns i, x, sigma (`n_input_rows`). It has no missing values or duplicates, and every σ is positive. Its SHA-256 matches the bundle manifest (see `out/state/data_audit.md`). I excluded nothing (`out/exclusions.csv`: 30 → 30) and found no data issues (`data_issues` = []). σ_i is largest at the extreme values of x, i.e. at the high-leverage points.\n\n## Method\n- **Exact SD of β̂:** β̂ − β = Σ w_i σ_i ε_i with w_i = (x_i − x̄)/Sxx. So SD(β̂) = sqrt(Σ w_i² σ_i²). This is exact because the design is fixed.\n- **Simulation:** R = 2000 replications (`n_replications`) of n = 30 (`n_obs_per_replication`). Only ε is redrawn, using numpy `default_rng` (PCG64) with seed 20261030 (`random_seed`).\n- **SEs and intervals:** In each replication I fit OLS and compute two SEs. se_conv = sqrt(s²·[(X'X)⁻¹]₂₂) with s² = RSS/28. se_HC1 = sqrt(n/(n−2) · Σ w_i² e_i²), i.e. the HC0 sandwich slope element times 30/28. The critical value is t_{0.975}(28) = 2.0484 (`t_critical_28`).\n- **Coverage:** the share of replications with |β̂ − 0.5| ≤ t·se.\n- **Undercoverage rule:** the conventional interval is flagged as undercovering if its coverage is below 0.95 − 3·sqrt(0.95·0.05/2000) = 0.9354 (`undercoverage_threshold`). The binomial SE in this rule is evaluated at the nominal p = 0.95; that choice is documented here.\n- **Code:** `out/analysis.py` regenerates every number.\n\n## Results\n| quantity | value | key |\n|---|---|---|\n| Coverage, conventional interval | 0.844 | `coverage_conventional` |\n| Coverage, HC1 interval | 0.902 | `coverage_hc1` |\n| Exact SD of β̂ | 0.2680 | `sd_beta_exact` |\n| MC SD of β̂ (ddof 1) | 0.2696 | `sd_beta_mc` |\n| Mean se_conv | 0.1903 | `mean_se_conventional` |\n| Mean se_HC1 (supplementary) | 0.2408 | `mean_se_hc1` |\n| Conventional interval undercovers? | true | `conventional_ci_undercovers` |\n\nThe conventional interval does **not** reach its nominal 95% coverage. Its estimated coverage of 0.844 is far below the 0.9354 threshold (`coverage_conventional`, `undercoverage_threshold`). The reason is that se_conv averages 0.190 (`mean_se_conventional`), while the true SD of β̂ is 0.268 (`sd_beta_exact`). The conventional SE assumes equal error variances, so it understates the variance here, where the large-variance errors fall at high-leverage x values. The HC1 interval does better at 0.902 (`coverage_hc1`) but still falls short of 0.95. This is consistent with the known downward bias and extra variability of HC0/HC1 standard errors when n is small and leverage is high. The MC SD agrees with the exact SD (difference ≈ 0.4 sampling SEs; `out/state/checks.md`).\n\n## Uncertainty\n- **Conventional coverage:** binomial MC SE sqrt(p̂(1−p̂)/R) = 0.0081 (`mcse_coverage_conventional`). Approximate 95% MC interval 0.828–0.860.\n- **HC1 coverage:** MC SE 0.0066 (`mcse_coverage_hc1`). Approximate 95% MC interval 0.889–0.915.\n- **Binomial SE at the nominal level:** 0.0049 (`mcse_binomial_nominal`).\n- **How decisive the results are:** the conventional estimate is about 21 nominal SEs below 0.95. The HC1 estimate is also more than 9 SEs below 0.95, so neither shortfall is plausibly Monte Carlo noise.\n- **Exact quantities:** `sd_beta_exact` carries no simulation uncertainty.\n\n## Limitations and open issues\n- The results hold only for this synthetic fixed design, with normal errors and n = 30. Coverage depends on the specific pattern of σ against leverage.\n- The MC estimates depend on the seed. Another seed would change them within roughly the MC SEs reported above.\n- I did not evaluate HC2/HC3 or wild-bootstrap intervals; the task does not require them. So I make no claim about which robust method would reach 95%.\n- The binomial SE in the undercoverage rule uses p = 0.95. Using p̂ instead (0.0081) gives the same conclusion.\n- No data issues or blocked steps; the task is complete (`feasibility` = \"complete\").\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\",\"undercoverage_threshold\",\"mcse_binomial_nominal\"],\"status\":\"supported\",\"caveat\":\"Specific to this fixed design, normal errors, and seed 20261030; MC SE about 0.008.\"},\n {\"id\":\"C2\",\"text\":\"The HC1 interval improves coverage to 0.902 but still undercovers relative to 95%.\",\"results_keys\":[\"coverage_hc1\",\"mcse_coverage_hc1\"],\"status\":\"supported\",\"caveat\":\"Small-sample HC1 bias with high-leverage heteroskedastic points; other HC variants not evaluated.\"},\n {\"id\":\"C3\",\"text\":\"The exact SD of the OLS slope is 0.268, and the Monte Carlo SD (0.270) agrees with it.\",\"results_keys\":[\"sd_beta_exact\",\"sd_beta_mc\"],\"status\":\"supported\",\"caveat\":\"Exact SD is analytic under the stated model.\"},\n {\"id\":\"C4\",\"text\":\"The conventional SE is biased downward on average (mean 0.190 vs true SD 0.268), which explains the undercoverage.\",\"results_keys\":[\"mean_se_conventional\",\"sd_beta_exact\"],\"status\":\"supported\",\"caveat\":\"Mean of se_conv is an MC estimate.\"}\n]\nEOF\n~/wp29/venv/bin/python - <<'EOF'\nimport json;json.load(open('out/claims.json'))\nf='out/state/phases.json';p=json.load(open(f))\np['P5']={'status':'passed','checks':['report.md has all 6 required sections with results keys cited','claims.json valid JSON with 4 claims incl. nominal-coverage claim C1'],'notes':''}\njson.dump(p,open(f,'w'),indent=1)\nEOF\necho \"- 2026-10-30: P5 passed; 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Its Monte Carlo standard error is 0.008, so the shortfall is far too large to be simulation noise.\n\n| Quantity | Value |\n|---|---|\n| Coverage, conventional interval | 0.844 (MC standard error 0.008) |\n| Coverage, HC1 robust interval | 0.902 (MC standard error 0.007) |\n| Exact standard deviation of β̂ (analytical) | 0.2680 |\n| Simulated standard deviation of β̂ | 0.2696 |\n| Average conventional standard error | 0.190 |\n| Conventional interval undercovers? | yes |\n\n- **Why it undercovers:** the conventional standard error averages 0.190, well below the true 0.268. The error variances are largest at the extreme x values, where those points have the most pull on the slope, and the conventional formula assumes equal variances, so it understates the uncertainty.\n- **The robust interval:** HC1 does better, but at 90.2% it still falls short of 95%. This is consistent with its known downward bias when the sample is small and a few points dominate the slope.\n- **Test for \"undercovers\":** the task's rule flags undercoverage if coverage is more than three binomial standard errors below 0.95, which puts the threshold at 0.9354. I computed that standard error at the nominal 0.95. Using the estimated coverage instead gives the same answer.\n\nI ran the script twice from a clean state and got an identical `out/results.json`. A separate per-replication fit using statsmodels, on the same random draws, reproduced every number exactly. The design file was clean and nothing was excluded.\n\n**Limitations:** the results apply only to this synthetic design, with normal errors and n = 30. A different seed would shift the coverage figures by roughly their standard errors. I did not test other robust methods (HC2, HC3 or bootstrap intervals), so the report makes no claim about which method would reach 95%.\n\nEverything is in `out/`. Open `report.md` first; `analysis.py` regenerates the numbers using seed 20261030. 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