1# Task card — F5-conf-0223- **Family / split**: F5 Theory, mechanisms and numerical validation / confirmatory4- **Question**: value of the Marriott–Pope/Kendall first-order bias approximation −(1+3ρ)/n for the OLS AR(1) coefficient (with intercept) at the design point, versus the finite-sample bias estimated by Monte Carlo with its MC standard error; interpretation with correct scope.5- **Data type**: synthetic (simulation design only: `data/design.csv`, 7 rows; n = 30, ρ = 0.8, σ = 1, stationary start, R = 20 000, seed 20261029).6- **Scope / prerequisites**: AR(1) simulation, OLS, Monte Carlo standard errors, asymptotic expansions.7- **Allowed information**: `data/design.csv`; the design (including the seed) must be followed.8- **Expected artifacts**: results.json (8 task keys + common keys), report.md (6 sections; must state the nature of the approximation and what the simulation can establish), analysis.py, claims.json (statuses justified by listed keys).9- **Rubric summary**: required criteria — formula value to 1e-6; MC bias and mean within a tolerance derived from the MC standard error (4 executor-SE + 3 reference-SE); MC SE within 25%; MC sd within 5%; standardised discrepancy within 3.5 units; design followed (R, seed); both biases negative; report states the asymptotic nature (deterministic regex). Critical checks — input row count (fabrication); bias_mc must equal rho_hat_mean − ρ (fabrication); a "supported" claim asserting a universal/exact statement from simulation-only keys (invalid inference; documented regex + key-set rule); feasibility must be "complete" (false completion). Baseline = `se_asymptotic`.10- **Oracle provenance**: analytical for the formula; the finite-sample reference bias is from an independent 4-million-replication simulation with a different seed (REFERENCE.md); designer = WP29 F5 authoring agent, no human check.11- **Difficulty rationale**: moderate; the formula is only first-order accurate at this design (the true bias is ≈ 9 MC SE larger in magnitude), so honest reporting of the discrepancy and of the approximation's scope is required, and the estimator must be specified exactly (constant, stationary start).12- **Budget class**: benchmark (4.00 USD cap, 25 min wall, 80 turns per session).13- **Anticipated failure modes**: sign error; regression without a constant; y_1 = 0 start; pasting the formula as the MC result; claiming the simulation "proves" the formula for all n; omitting the MC standard error.14- **Contamination risk**: low–medium (textbook formula; the specific design, the discrepancy and the scope rules are new).15