artifacts/evidence_bundle.jsonl

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{"bytes": 6392, "path": "artifacts/five_claim_audit.json", "payload": {"anchored_claims": [{"claim_number": 1, "text": "Theorem 6 proves a constructive Omega(1/sqrt(epsilon)) lower bound on the number of iterations for Frank-Wolfe with short steps or exact line search when both the objective and the constraint set are smooth and strongly convex (Theorem 6)."}, {"claim_number": 2, "text": "The lower-bound construction uses the specific instance of minimizing f(x) = ||x-p||_2^2 over the Euclidean unit ball B_1(0) with the unconstrained minimizer p on the boundary (||p||=1), showing this hard instance matches the known O(1/sqrt(epsilon)) upper bound of Garber et al. (2015) (Theorem 6)."}, {"claim_number": 3, "text": "The lower bound is established via a 'backward reconstruction' technique that builds worst-case trajectories by working backward from terminal iterates, producing precise sequences over horizons of 10,000+ iterations (Section on Backward Reconstruction Strategy)."}, {"claim_number": 4, "text": "The analysis reduces the dynamics to a two-dimensional invariant subspace parametrized by residual magnitude r and contraction factor s, revealing stable phases of monotonically increasing contraction followed by intermittent 'jumps' that explain the convergence stalling behavior (Section on Dynamical Analysis)."}, {"claim_number": 5, "text": "The result shows strong convexity of the constraint set alone cannot yield convergence rates uniformly faster than O(1/sqrt(epsilon)), and this conclusion holds dimension-independently and extends to ellipsoidal constraints via affine invariance (Theorem 6, Implications)."}], "automated_test_suites_run": false, "claim_evidence": [{"claim_number": 1, "evidence": ["long_horizon", "direct_vector_replay"], "scope": "Finite-precision execution corroborates the constructive witness; the theorem remains an analytical result.", "status": "numerically corroborated"}, {"claim_number": 2, "evidence": ["long_horizon", "direct_vector_replay"], "scope": "The same hard instance is bracketed by matching residual envelopes and has a terminal-gap slope near -2.", "status": "numerically corroborated"}, {"claim_number": 3, "evidence": ["long_horizon"], "scope": "The largest generated trajectory exceeds 10,000 transitions and every adjacent pair is forward/backward checked.", "status": "reproduced"}, {"claim_number": 4, "evidence": ["rs_dynamics"], "scope": "The lower-bound witness is jump-free; intermittent jumps are demonstrated on a separate feasible trajectory and are not conflated with the witness.", "status": "reproduced with qualification"}, {"claim_number": 5, "evidence": ["dimension_and_affine"], "scope": "Numerical embeddings and three ellipsoid maps corroborate the theorem's dimension and affine consequences.", "status": "numerically corroborated"}], "evidence": {"dimension_and_affine": {"dimensions": [2, 3, 8, 32], "ellipsoid_maps": ["diag(0.25,4)", "diag(3,0.125)", "rot30_diag(0.5,2)"], "max_affine_positive_control_error": "7.18396235681e-265", "max_dimension_embedding_error": "0.0", "negative_control_naive_objective_deviation": "0.310523195816", "steps_per_positive_configuration": 128, "table": "affine_dimension_summary.csv"}, "direct_vector_replay": {"checkpoint_table": "direct_vector_replay_checkpoints.csv", "max_feasibility_violation": "0.0", "max_recurrence_vs_cartesian_residual_error": "2.70638628502e-152", "steps": 512}, "long_horizon": {"checkpoint_table": "trajectory_16384_checkpoints.csv", "global_c_range": ["1.72401368791111733", "2.0"], "horizon_count": 9, "largest_actual_forward_steps": 16384, "largest_trajectory_monotonicity_violations": 0, "max_forward_inverse_error": "5.01703856035e-297", "max_upper_violation": "0.0", "min_lower_margin": "0.0", "precision_decimal_digits": 300, "states_audited": 32713, "summary_table": "long_horizon_summary.csv", "terminal_gap_log_log_slope_last_five": -1.9952621757403202}, "negative_controls": {"low_precision_direct_replay": {"failure": null, "max_direct_replay_error_before_failure": "0.0114825303087", "precision_decimal_digits": 16, "rejected": true, "steps_requested": 1000}, "naive_affine_objective": "0.310523195816"}, "rs_dynamics": {"contraction_decreases": 10, "honesty_note": "The backward lower-bound witness is jump-free. 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