{ "version": "v1", "generated_at": "2026-06-09T21:58:12.574332Z", "report_count": 27, "arcs": [ { "arc_id": "group-ring", "label_en": "ring progression", "label_cn": "ring 研究推进链", "summary_en": "ring track links 9 reports into one continuous argument.", "summary_cn": "ring 轨道把 9 份报告串成连续论证。", "report_ids": [ "ring_lazy_jump", "ring_lazy_jump_ext", "ring_lazy_jump_ext_rev2", "ring_lazy_flux", "ring_valley", "ring_valley_dst", "ring_deriv_k2", "ring_two_target", "ring_two_walker_encounter_shortcut" ], "claim_ids": [ "ring_lazy_jump-c1", "ring_lazy_jump-c2", "ring_lazy_jump-c3", "ring_lazy_jump-c4", "ring_lazy_jump-c5", "ring_lazy_jump_ext-c1", "ring_lazy_jump_ext-c2", "ring_lazy_jump_ext-c3", "ring_lazy_jump_ext-c4", "ring_lazy_jump_ext-c5", "ring_lazy_jump_ext_rev2-c1", "ring_lazy_jump_ext_rev2-c2", "ring_lazy_jump_ext_rev2-c3", "ring_lazy_jump_ext_rev2-c4", "ring_lazy_jump_ext_rev2-c5", "ring_lazy_flux-c1", "ring_lazy_flux-c2", "ring_lazy_flux-c3", "ring_lazy_flux-c4", "ring_lazy_flux-c5", "ring_valley-c1", "ring_valley-c2", "ring_valley-c3", "ring_valley-c4", "ring_valley-c5", "ring_valley_dst-c1", "ring_valley_dst-c2", "ring_valley_dst-c3", "ring_valley_dst-c4", "ring_valley_dst-c5", "ring_deriv_k2-c1", "ring_deriv_k2-c2", "ring_deriv_k2-c3", "ring_deriv_k2-c4", "ring_deriv_k2-c5", "ring_two_target-c1", "ring_two_target-c2", "ring_two_target-c3", "ring_two_target-c4", "ring_two_target-c5", "ring_two_walker_encounter_shortcut-c1", "ring_two_walker_encounter_shortcut-c2", "ring_two_walker_encounter_shortcut-c3", "ring_two_walker_encounter_shortcut-c4", "ring_two_walker_encounter_shortcut-c5" ], "checkpoint_count": 9, "checkpoints": [ { "report_id": "ring_lazy_jump", "title_en": "Lazy Ring Jump-Over Mechanism (K2 vs K4)", "title_cn": "Lazy Ring Jump-Over 机制（K2 对 K4）", "contribution_en": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.", "contribution_cn": "双峰只在部分 shortcut 强度区间稳定出现；在几何与停留规则对齐时，K=4 往往比 K=2 保持更强的第二峰。" }, { "report_id": "ring_lazy_jump_ext", "title_en": "Lazy Ring Shortcut Beta Scan", "title_cn": "Lazy Ring Shortcut beta 扫描", "contribution_en": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.", "contribution_cn": "随着 beta 提升，两个峰位整体提前且尾部衰减更快；在同等 beta 下，K=4 的双峰保持性优于 K=2，且精确解与 MC 在相区趋势上吻合。" }, { "report_id": "ring_lazy_jump_ext_rev2", "title_en": "Lazy Ring Shortcut Figure-1 Revision", "title_cn": "Lazy Ring Shortcut 图1修订", "contribution_en": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.", "contribution_cn": "三类敏感性分析下，机制解释在定性层面保持稳定，且不确定性区间未推翻主线中的相区判断。" }, { "report_id": "ring_lazy_flux", "title_en": "Lazy Ring Flux Baseline", "title_cn": "Lazy Ring Flux 基线机制", "contribution_en": "A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.", "contribution_cn": "自环分流且小 p 条件下可形成稳定双峰（最小 N=10 例）；equal4 或 shortcut 强度增大时，分布向单峰收敛。" }, { "report_id": "ring_valley", "title_en": "Ring Valley Regime Map", "title_cn": "Ring Valley 相区研究", "contribution_en": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.", "contribution_cn": "在扫描的偶数 N 范围内，K=2 维持单峰；K=4/6/8 则出现可复现的双峰区间，并在精确解与 MC 诊断中一致。" }, { "report_id": "ring_valley_dst", "title_en": "Destination-Scan Valley Control", "title_cn": "目的地扫描与谷值调控（N=100, K=6）", "contribution_en": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.", "contribution_cn": "dst 扫描给出了结构化的高对比区间：第二峰可被显著放大，且轨迹类别占比同步变化，确定性与 MC 结论一致。" }, { "report_id": "ring_deriv_k2", "title_en": "Ring Derivation Backbone", "title_cn": "Ring 推导主干（K=2）", "contribution_en": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.", "contribution_cn": "闭式推导明确了 shortcut 不对称性的主导项，并为后续 ring 报告提供可复用公式模块。" }, { "report_id": "ring_two_target", "title_en": "Two-Target Lazy Ring Mechanics", "title_cn": "双目标 Lazy Ring 机制", "contribution_en": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.", "contribution_cn": "无 shortcut 漂移本身即可产生稳定双峰；加入 shortcut 后路径质量重新分配，在特定几何与参数带可出现三峰行为。" }, { "report_id": "ring_two_walker_encounter_shortcut", "title_en": "1D Ring Two-Walker Encounter With Shortcut", "title_cn": "Ring 体系：ring_two_walker_encounter_shortcut 研究报告", "contribution_en": "Research report ring_two_walker_encounter_shortcut.", "contribution_cn": "Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" } ] }, { "arc_id": "group-grid2d", "label_en": "grid2d progression", "label_cn": "grid2d 研究推进链", "summary_en": "grid2d track links 13 reports into one continuous argument.", "summary_cn": "grid2d 轨道把 13 份报告串成连续论证。", "report_ids": [ "grid2d_bimodality", "grid2d_reflecting_bimodality", "grid2d_blackboard_bimodality", "grid2d_two_target_double_peak", "grid2d_two_walker_encounter_shortcut", "grid2d_rect_bimodality", "grid2d_membrane_near_target", "grid2d_one_target_base", "grid2d_one_target_exit_timing", "grid2d_one_target_valley_peak_budget", "grid2d_one_target_window_measures", "grid2d_one_two_target_gating", "grid2d_two_target_bias_radius" ], "claim_ids": [ "grid2d_bimodality-c1", "grid2d_bimodality-c2", "grid2d_bimodality-c3", "grid2d_bimodality-c4", "grid2d_bimodality-c5", "grid2d_reflecting_bimodality-c1", "grid2d_reflecting_bimodality-c2", "grid2d_reflecting_bimodality-c3", "grid2d_reflecting_bimodality-c4", "grid2d_reflecting_bimodality-c5", "grid2d_blackboard_bimodality-c1", "grid2d_blackboard_bimodality-c2", "grid2d_blackboard_bimodality-c3", "grid2d_blackboard_bimodality-c4", "grid2d_blackboard_bimodality-c5", "grid2d_two_target_double_peak-c1", "grid2d_two_target_double_peak-c2", "grid2d_two_target_double_peak-c3", "grid2d_two_target_double_peak-c4", "grid2d_two_target_double_peak-c5", "grid2d_two_walker_encounter_shortcut-c1", "grid2d_two_walker_encounter_shortcut-c2", "grid2d_two_walker_encounter_shortcut-c3", "grid2d_two_walker_encounter_shortcut-c4", "grid2d_two_walker_encounter_shortcut-c5", "grid2d_rect_bimodality-c1", "grid2d_rect_bimodality-c2", "grid2d_rect_bimodality-c3", "grid2d_rect_bimodality-c4", "grid2d_rect_bimodality-c5", "grid2d_membrane_near_target-c1", "grid2d_membrane_near_target-c2", "grid2d_membrane_near_target-c3", "grid2d_membrane_near_target-c4", "grid2d_membrane_near_target-c5", "grid2d_one_target_base-c1", "grid2d_one_target_base-c2", "grid2d_one_target_base-c3", "grid2d_one_target_base-c4", "grid2d_one_target_base-c5", "grid2d_one_target_exit_timing-c1", "grid2d_one_target_exit_timing-c2", "grid2d_one_target_exit_timing-c3", "grid2d_one_target_exit_timing-c4", "grid2d_one_target_valley_peak_budget-c1", "grid2d_one_target_valley_peak_budget-c2", "grid2d_one_target_valley_peak_budget-c3", "grid2d_one_target_valley_peak_budget-c4", "grid2d_one_target_window_measures-c1", "grid2d_one_target_window_measures-c2", "grid2d_one_target_window_measures-c3", "grid2d_one_target_window_measures-c4", "grid2d_one_target_window_measures-c5", "grid2d_one_two_target_gating-c1", "grid2d_one_two_target_gating-c2", "grid2d_one_two_target_gating-c3", "grid2d_one_two_target_gating-c4", "grid2d_one_two_target_gating-c5", "grid2d_two_target_bias_radius-c1", "grid2d_two_target_bias_radius-c2", "grid2d_two_target_bias_radius-c3", "grid2d_two_target_bias_radius-c4", "grid2d_two_target_bias_radius-c5" ], "checkpoint_count": 13, "checkpoints": [ { "report_id": "grid2d_bimodality", "title_en": "Grid2D Bimodality Baseline", "title_cn": "Grid2D 双峰基线", "contribution_en": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.", "contribution_cn": "当快通道与延迟绕行通道在同一判据下都具有可观权重时，首达分布出现稳健双峰。" }, { "report_id": "grid2d_reflecting_bimodality", "title_en": "Grid2D Reflecting-Boundary Bimodality", "title_cn": "Grid2D 全反射边界双峰", "contribution_en": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "contribution_cn": "部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。" }, { "report_id": "grid2d_blackboard_bimodality", "title_en": "Grid2D Blackboard Endpoint Case", "title_cn": "Grid2D 黑板端点构型", "contribution_en": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.", "contribution_cn": "在已扫描的端点案例中，主导行为为单峰加长尾；晚时隆起被判定为窗口边缘效应，而非稳健双峰。" }, { "report_id": "grid2d_two_target_double_peak", "title_en": "Grid2D Two-Target Double-Peak", "title_cn": "Grid2D 双目标双峰", "contribution_en": "Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.", "contribution_cn": "当近目标直达通道与远目标延迟通道都占有显著质量时，会出现稳定双峰；其相界随耦合强度呈可预测移动。" }, { "report_id": "grid2d_two_walker_encounter_shortcut", "title_en": "2D Two-Walker Encounter With Shortcut", "title_cn": "Grid2D 体系：grid2d_two_walker_encounter_shortcut 研究报告", "contribution_en": "Research report grid2d_two_walker_encounter_shortcut.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_rect_bimodality", "title_en": "Grid2D Rectangle Bimodality", "title_cn": "Grid2D 矩形域双峰", "contribution_en": "Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.", "contribution_cn": "长宽比与端点排布会改变直达/绕行通道权重平衡；稳健双峰只在特定矩形几何带中保持。" }, { "report_id": "grid2d_membrane_near_target", "title_en": "Grid2D Membrane Near Target", "title_cn": "Grid2D 体系：grid2d_membrane_near_target 研究报告", "contribution_en": "Research report grid2d_membrane_near_target.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_base", "title_en": "Grid2D One Target — Base", "title_cn": "Grid2D 体系：grid2d_one_target_base 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_base.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_exit_timing", "title_en": "Grid2D One Target — Exit Timing", "title_cn": "Grid2D 体系：grid2d_one_target_exit_timing 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_exit_timing.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_valley_peak_budget", "title_en": "Grid2D One Target — Valley/Peak Budget", "title_cn": "Grid2D 体系：grid2d_one_target_valley_peak_budget 研究报告", "contribution_en": "Research report grid2d_one_target_valley_peak_budget.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_window_measures", "title_en": "Grid2D One Target — Window Measures", "title_cn": "Grid2D 体系：grid2d_one_target_window_measures 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_window_measures.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_two_target_gating", "title_en": "Grid2D One vs Two Target — Gating", "title_cn": "Grid2D 体系：grid2d_one_two_target_gating 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_two_target_gating.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_two_target_bias_radius", "title_en": "Grid2D Two-Target Bias-Radius Scaffold", "title_cn": "Grid2D 体系：grid2d_two_target_bias_radius 研究报告", "contribution_en": "Result summary placeholder for grid2d_two_target_bias_radius.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" } ] }, { "arc_id": "group-cross", "label_en": "cross progression", "label_cn": "cross 研究推进链", "summary_en": "cross track links 1 reports into one continuous argument.", "summary_cn": "cross 轨道把 1 份报告串成连续论证。", "report_ids": [ "cross_luca_regime_map" ], "claim_ids": [ "cross_luca_regime_map-c1", "cross_luca_regime_map-c2", "cross_luca_regime_map-c3", "cross_luca_regime_map-c4", "cross_luca_regime_map-c5" ], "checkpoint_count": 1, "checkpoints": [ { "report_id": "cross_luca_regime_map", "title_en": "Cross-Model Luca Regime Map", "title_cn": "跨模型 Luca 相区图", "contribution_en": "Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric.", "contribution_cn": "在扫描工作负载上，sparse 精确解仍是 full-FPT 主基线；Luca 加速仅在部分缺陷区间显著，整体速度比接近中性。" } ] }, { "arc_id": "group-misc", "label_en": "misc progression", "label_cn": "misc 研究推进链", "summary_en": "misc track links 4 reports into one continuous argument.", "summary_cn": "misc 轨道把 4 份报告串成连续论证。", "report_ids": [ "final_multitimescale_fpt_encounter", "exact_recursion_method_guide", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "claim_ids": [ "final_multitimescale_fpt_encounter-c1", "final_multitimescale_fpt_encounter-c2", "final_multitimescale_fpt_encounter-c3", "final_multitimescale_fpt_encounter-c4", "exact_recursion_method_guide-c1", "exact_recursion_method_guide-c2", "exact_recursion_method_guide-c3", "exact_recursion_method_guide-c4", "encounter_reflecting_diagonal_decomp-c1", "encounter_reflecting_diagonal_decomp-c2", "encounter_reflecting_diagonal_decomp-c3", "encounter_reflecting_diagonal_decomp-c4", "encounter_reflecting_mean_validation-c1", "encounter_reflecting_mean_validation-c2", "encounter_reflecting_mean_validation-c3", "encounter_reflecting_mean_validation-c4", "encounter_reflecting_mean_validation-c5" ], "checkpoint_count": 4, "checkpoints": [ { "report_id": "final_multitimescale_fpt_encounter", "title_en": "Final Multitimescale FPT and Encounter Report", "title_cn": "该研究报告：final_multitimescale_fpt_encounter 研究报告", "contribution_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "exact_recursion_method_guide", "title_en": "Exact Recursion — Method Guide", "title_cn": "该研究报告：exact_recursion_method_guide 研究报告", "contribution_en": "Result summary placeholder for exact_recursion_method_guide.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "encounter_reflecting_diagonal_decomp", "title_en": "Reflecting Encounter Diagonal Decomposition", "title_cn": "该研究报告：encounter_reflecting_diagonal_decomp 研究报告", "contribution_en": "Result summary placeholder for encounter_reflecting_diagonal_decomp.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "encounter_reflecting_mean_validation", "title_en": "Reflecting Encounter Mean Validation", "title_cn": "该研究报告：encounter_reflecting_mean_validation 研究报告", "contribution_en": "Result summary placeholder for encounter_reflecting_mean_validation.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" } ] }, { "arc_id": "global-synthesis", "label_en": "Grid and ring mechanisms converge into cross-report synthesis.", "label_cn": "二维与环模型机制最终汇入跨报告综合结论。", "summary_en": "Global storyline that connects all report families from mechanism to synthesis.", "summary_cn": "连接全部报告家族的全局叙事主线，从机制到综合结论。", "report_ids": [ "ring_lazy_jump", "ring_lazy_jump_ext", "ring_lazy_jump_ext_rev2", "ring_lazy_flux", "ring_valley", "ring_valley_dst", "ring_deriv_k2", "ring_two_target", "ring_two_walker_encounter_shortcut", "grid2d_bimodality", "grid2d_reflecting_bimodality", "grid2d_blackboard_bimodality", "grid2d_two_target_double_peak", "grid2d_two_walker_encounter_shortcut", "grid2d_rect_bimodality", "grid2d_membrane_near_target", "grid2d_one_target_base", "grid2d_one_target_exit_timing", "grid2d_one_target_valley_peak_budget", "grid2d_one_target_window_measures", "grid2d_one_two_target_gating", "grid2d_two_target_bias_radius", "cross_luca_regime_map", "final_multitimescale_fpt_encounter", "exact_recursion_method_guide", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "claim_ids": [ "ring_lazy_jump-c1", "ring_lazy_jump-c2", "ring_lazy_jump-c3", "ring_lazy_jump-c4", "ring_lazy_jump-c5", "ring_lazy_jump_ext-c1", "ring_lazy_jump_ext-c2", "ring_lazy_jump_ext-c3", "ring_lazy_jump_ext-c4", "ring_lazy_jump_ext-c5", "ring_lazy_jump_ext_rev2-c1", "ring_lazy_jump_ext_rev2-c2", "ring_lazy_jump_ext_rev2-c3", "ring_lazy_jump_ext_rev2-c4", "ring_lazy_jump_ext_rev2-c5", "ring_lazy_flux-c1", "ring_lazy_flux-c2", "ring_lazy_flux-c3", "ring_lazy_flux-c4", "ring_lazy_flux-c5", "ring_valley-c1", "ring_valley-c2", "ring_valley-c3", "ring_valley-c4", "ring_valley-c5", "ring_valley_dst-c1", "ring_valley_dst-c2", "ring_valley_dst-c3", "ring_valley_dst-c4", "ring_valley_dst-c5", "ring_deriv_k2-c1", "ring_deriv_k2-c2", "ring_deriv_k2-c3", "ring_deriv_k2-c4", "ring_deriv_k2-c5", "ring_two_target-c1", "ring_two_target-c2", "ring_two_target-c3", "ring_two_target-c4", "ring_two_target-c5", "ring_two_walker_encounter_shortcut-c1", "ring_two_walker_encounter_shortcut-c2", "ring_two_walker_encounter_shortcut-c3", "ring_two_walker_encounter_shortcut-c4", "ring_two_walker_encounter_shortcut-c5", "grid2d_bimodality-c1", "grid2d_bimodality-c2", "grid2d_bimodality-c3", "grid2d_bimodality-c4", "grid2d_bimodality-c5", "grid2d_reflecting_bimodality-c1", "grid2d_reflecting_bimodality-c2", "grid2d_reflecting_bimodality-c3", "grid2d_reflecting_bimodality-c4", "grid2d_reflecting_bimodality-c5", "grid2d_blackboard_bimodality-c1", "grid2d_blackboard_bimodality-c2", "grid2d_blackboard_bimodality-c3", "grid2d_blackboard_bimodality-c4", "grid2d_blackboard_bimodality-c5", "grid2d_two_target_double_peak-c1", "grid2d_two_target_double_peak-c2", "grid2d_two_target_double_peak-c3", "grid2d_two_target_double_peak-c4", "grid2d_two_target_double_peak-c5", "grid2d_two_walker_encounter_shortcut-c1", "grid2d_two_walker_encounter_shortcut-c2", "grid2d_two_walker_encounter_shortcut-c3", "grid2d_two_walker_encounter_shortcut-c4", "grid2d_two_walker_encounter_shortcut-c5", "grid2d_rect_bimodality-c1", "grid2d_rect_bimodality-c2", "grid2d_rect_bimodality-c3", "grid2d_rect_bimodality-c4", "grid2d_rect_bimodality-c5", "grid2d_membrane_near_target-c1", "grid2d_membrane_near_target-c2", "grid2d_membrane_near_target-c3", "grid2d_membrane_near_target-c4", "grid2d_membrane_near_target-c5", "grid2d_one_target_base-c1", "grid2d_one_target_base-c2", "grid2d_one_target_base-c3", "grid2d_one_target_base-c4", "grid2d_one_target_base-c5", "grid2d_one_target_exit_timing-c1", "grid2d_one_target_exit_timing-c2", "grid2d_one_target_exit_timing-c3", "grid2d_one_target_exit_timing-c4", "grid2d_one_target_valley_peak_budget-c1", "grid2d_one_target_valley_peak_budget-c2", "grid2d_one_target_valley_peak_budget-c3", "grid2d_one_target_valley_peak_budget-c4", "grid2d_one_target_window_measures-c1", "grid2d_one_target_window_measures-c2", "grid2d_one_target_window_measures-c3", "grid2d_one_target_window_measures-c4", "grid2d_one_target_window_measures-c5", "grid2d_one_two_target_gating-c1", "grid2d_one_two_target_gating-c2", "grid2d_one_two_target_gating-c3", "grid2d_one_two_target_gating-c4", "grid2d_one_two_target_gating-c5", "grid2d_two_target_bias_radius-c1", "grid2d_two_target_bias_radius-c2", "grid2d_two_target_bias_radius-c3", "grid2d_two_target_bias_radius-c4", "grid2d_two_target_bias_radius-c5", "cross_luca_regime_map-c1", "cross_luca_regime_map-c2", "cross_luca_regime_map-c3", "cross_luca_regime_map-c4", "cross_luca_regime_map-c5", "final_multitimescale_fpt_encounter-c1", "final_multitimescale_fpt_encounter-c2", "final_multitimescale_fpt_encounter-c3", "final_multitimescale_fpt_encounter-c4", "exact_recursion_method_guide-c1", "exact_recursion_method_guide-c2", "exact_recursion_method_guide-c3", "exact_recursion_method_guide-c4", "encounter_reflecting_diagonal_decomp-c1", "encounter_reflecting_diagonal_decomp-c2", "encounter_reflecting_diagonal_decomp-c3", "encounter_reflecting_diagonal_decomp-c4", "encounter_reflecting_mean_validation-c1", "encounter_reflecting_mean_validation-c2", "encounter_reflecting_mean_validation-c3", "encounter_reflecting_mean_validation-c4", "encounter_reflecting_mean_validation-c5" ], "checkpoint_count": 27, "checkpoints": [ { "report_id": "ring_lazy_jump", "title_en": "Lazy Ring Jump-Over Mechanism (K2 vs K4)", "title_cn": "Lazy Ring Jump-Over 机制（K2 对 K4）", "contribution_en": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.", "contribution_cn": "双峰只在部分 shortcut 强度区间稳定出现；在几何与停留规则对齐时，K=4 往往比 K=2 保持更强的第二峰。" }, { "report_id": "ring_lazy_jump_ext", "title_en": "Lazy Ring Shortcut Beta Scan", "title_cn": "Lazy Ring Shortcut beta 扫描", "contribution_en": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.", "contribution_cn": "随着 beta 提升，两个峰位整体提前且尾部衰减更快；在同等 beta 下，K=4 的双峰保持性优于 K=2，且精确解与 MC 在相区趋势上吻合。" }, { "report_id": "ring_lazy_jump_ext_rev2", "title_en": "Lazy Ring Shortcut Figure-1 Revision", "title_cn": "Lazy Ring Shortcut 图1修订", "contribution_en": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.", "contribution_cn": "三类敏感性分析下，机制解释在定性层面保持稳定，且不确定性区间未推翻主线中的相区判断。" }, { "report_id": "ring_lazy_flux", "title_en": "Lazy Ring Flux Baseline", "title_cn": "Lazy Ring Flux 基线机制", "contribution_en": "A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.", "contribution_cn": "自环分流且小 p 条件下可形成稳定双峰（最小 N=10 例）；equal4 或 shortcut 强度增大时，分布向单峰收敛。" }, { "report_id": "ring_valley", "title_en": "Ring Valley Regime Map", "title_cn": "Ring Valley 相区研究", "contribution_en": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.", "contribution_cn": "在扫描的偶数 N 范围内，K=2 维持单峰；K=4/6/8 则出现可复现的双峰区间，并在精确解与 MC 诊断中一致。" }, { "report_id": "ring_valley_dst", "title_en": "Destination-Scan Valley Control", "title_cn": "目的地扫描与谷值调控（N=100, K=6）", "contribution_en": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.", "contribution_cn": "dst 扫描给出了结构化的高对比区间：第二峰可被显著放大，且轨迹类别占比同步变化，确定性与 MC 结论一致。" }, { "report_id": "ring_deriv_k2", "title_en": "Ring Derivation Backbone", "title_cn": "Ring 推导主干（K=2）", "contribution_en": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.", "contribution_cn": "闭式推导明确了 shortcut 不对称性的主导项，并为后续 ring 报告提供可复用公式模块。" }, { "report_id": "ring_two_target", "title_en": "Two-Target Lazy Ring Mechanics", "title_cn": "双目标 Lazy Ring 机制", "contribution_en": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.", "contribution_cn": "无 shortcut 漂移本身即可产生稳定双峰；加入 shortcut 后路径质量重新分配，在特定几何与参数带可出现三峰行为。" }, { "report_id": "ring_two_walker_encounter_shortcut", "title_en": "1D Ring Two-Walker Encounter With Shortcut", "title_cn": "Ring 体系：ring_two_walker_encounter_shortcut 研究报告", "contribution_en": "Research report ring_two_walker_encounter_shortcut.", "contribution_cn": "Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_bimodality", "title_en": "Grid2D Bimodality Baseline", "title_cn": "Grid2D 双峰基线", "contribution_en": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.", "contribution_cn": "当快通道与延迟绕行通道在同一判据下都具有可观权重时，首达分布出现稳健双峰。" }, { "report_id": "grid2d_reflecting_bimodality", "title_en": "Grid2D Reflecting-Boundary Bimodality", "title_cn": "Grid2D 全反射边界双峰", "contribution_en": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "contribution_cn": "部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。" }, { "report_id": "grid2d_blackboard_bimodality", "title_en": "Grid2D Blackboard Endpoint Case", "title_cn": "Grid2D 黑板端点构型", "contribution_en": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.", "contribution_cn": "在已扫描的端点案例中，主导行为为单峰加长尾；晚时隆起被判定为窗口边缘效应，而非稳健双峰。" }, { "report_id": "grid2d_two_target_double_peak", "title_en": "Grid2D Two-Target Double-Peak", "title_cn": "Grid2D 双目标双峰", "contribution_en": "Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.", "contribution_cn": "当近目标直达通道与远目标延迟通道都占有显著质量时，会出现稳定双峰；其相界随耦合强度呈可预测移动。" }, { "report_id": "grid2d_two_walker_encounter_shortcut", "title_en": "2D Two-Walker Encounter With Shortcut", "title_cn": "Grid2D 体系：grid2d_two_walker_encounter_shortcut 研究报告", "contribution_en": "Research report grid2d_two_walker_encounter_shortcut.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_rect_bimodality", "title_en": "Grid2D Rectangle Bimodality", "title_cn": "Grid2D 矩形域双峰", "contribution_en": "Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.", "contribution_cn": "长宽比与端点排布会改变直达/绕行通道权重平衡；稳健双峰只在特定矩形几何带中保持。" }, { "report_id": "grid2d_membrane_near_target", "title_en": "Grid2D Membrane Near Target", "title_cn": "Grid2D 体系：grid2d_membrane_near_target 研究报告", "contribution_en": "Research report grid2d_membrane_near_target.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_base", "title_en": "Grid2D One Target — Base", "title_cn": "Grid2D 体系：grid2d_one_target_base 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_base.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_exit_timing", "title_en": "Grid2D One Target — Exit Timing", "title_cn": "Grid2D 体系：grid2d_one_target_exit_timing 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_exit_timing.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_valley_peak_budget", "title_en": "Grid2D One Target — Valley/Peak Budget", "title_cn": "Grid2D 体系：grid2d_one_target_valley_peak_budget 研究报告", "contribution_en": "Research report grid2d_one_target_valley_peak_budget.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_target_window_measures", "title_en": "Grid2D One Target — Window Measures", "title_cn": "Grid2D 体系：grid2d_one_target_window_measures 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_target_window_measures.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_one_two_target_gating", "title_en": "Grid2D One vs Two Target — Gating", "title_cn": "Grid2D 体系：grid2d_one_two_target_gating 研究报告", "contribution_en": "Result summary placeholder for grid2d_one_two_target_gating.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "grid2d_two_target_bias_radius", "title_en": "Grid2D Two-Target Bias-Radius Scaffold", "title_cn": "Grid2D 体系：grid2d_two_target_bias_radius 研究报告", "contribution_en": "Result summary placeholder for grid2d_two_target_bias_radius.", "contribution_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "cross_luca_regime_map", "title_en": "Cross-Model Luca Regime Map", "title_cn": "跨模型 Luca 相区图", "contribution_en": "Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric.", "contribution_cn": "在扫描工作负载上，sparse 精确解仍是 full-FPT 主基线；Luca 加速仅在部分缺陷区间显著，整体速度比接近中性。" }, { "report_id": "final_multitimescale_fpt_encounter", "title_en": "Final Multitimescale FPT and Encounter Report", "title_cn": "该研究报告：final_multitimescale_fpt_encounter 研究报告", "contribution_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "exact_recursion_method_guide", "title_en": "Exact Recursion — Method Guide", "title_cn": "该研究报告：exact_recursion_method_guide 研究报告", "contribution_en": "Result summary placeholder for exact_recursion_method_guide.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "encounter_reflecting_diagonal_decomp", "title_en": "Reflecting Encounter Diagonal Decomposition", "title_cn": "该研究报告：encounter_reflecting_diagonal_decomp 研究报告", "contribution_en": "Result summary placeholder for encounter_reflecting_diagonal_decomp.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "report_id": "encounter_reflecting_mean_validation", "title_en": "Reflecting Encounter Mean Validation", "title_cn": "该研究报告：encounter_reflecting_mean_validation 研究报告", "contribution_en": "Result summary placeholder for encounter_reflecting_mean_validation.", "contribution_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" } ] } ], "claims": [ { "claim_id": "ring_lazy_jump-c1", "report_id": "ring_lazy_jump", "stage": "model", "text_en": "A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.", "text_cn": "模型采用 lazy ring 与单向 shortcut 的基线构型，在统一参数设定下比较 K=2 与 K=4 的邻接效应。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "A lazy ring with one directed shortcut is used as the baseline setting, with matched parameters across K=2 and K=4 to isolate neighborhood effects.", "snippet_cn": "模型采用 lazy ring 与单向 shortcut 的基线构型，在统一参数设定下比较 K=2 与 K=4 的邻接效应。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Model formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): A lazy ring with one directed shortcut is used as the baseline setting", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c1", "ring_lazy_flux-c1", "ring_deriv_k2-c1", "ring_lazy_jump_ext_rev2-c1", "ring_two_target-c1" ], "linked_report_ids": [ "ring_lazy_jump_ext", "ring_lazy_flux", "ring_deriv_k2", "ring_lazy_jump_ext_rev2", "ring_two_target" ] }, { "claim_id": "ring_lazy_jump-c2", "report_id": "ring_lazy_jump", "stage": "method", "text_en": "The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.", "text_cn": "方法结合 AW 精确反演与轨迹分解，将 jump-over、直达与延迟通道分离并进行量化。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "The analysis combines AW inversion for exact first-passage series with trajectory decomposition that separates jump-over, direct, and delayed path classes.", "snippet_cn": "方法结合 AW 精确反演与轨迹分解，将 jump-over、直达与延迟通道分离并进行量化。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Method formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): The analysis combines AW inversion for exact first-passage series with", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "grid2d_two_target_double_peak-c2", "ring_two_target-c2", "ring_valley-c2", "ring_valley_dst-c2", "ring_lazy_jump_ext-c2" ], "linked_report_ids": [ "grid2d_two_target_double_peak", "ring_two_target", "ring_valley", "ring_valley_dst", "ring_lazy_jump_ext" ] }, { "claim_id": "ring_lazy_jump-c3", "report_id": "ring_lazy_jump", "stage": "result", "text_en": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.", "text_cn": "双峰只在部分 shortcut 强度区间稳定出现；在几何与停留规则对齐时，K=4 往往比 K=2 保持更强的第二峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.", "snippet_cn": "双峰只在部分 shortcut 强度区间稳定出现；在几何与停留规则对齐时，K=4 往往比 K=2 保持更强的第二峰。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Result formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Bimodality appears only in selected shortcut-strength intervals", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "ring_two_target-c3", "grid2d_two_target_double_peak-c3", "ring_lazy_flux-c3", "final_multitimescale_fpt_encounter-c3", "grid2d_bimodality-c3" ], "linked_report_ids": [ "ring_two_target", "grid2d_two_target_double_peak", "ring_lazy_flux", "final_multitimescale_fpt_encounter", "grid2d_bimodality" ] }, { "claim_id": "ring_lazy_jump-c4", "report_id": "ring_lazy_jump", "stage": "finding", "text_en": "Jump-over pathways create a distinct delayed channel that is necessary for persistent second peaks.", "text_cn": "jump-over 路径会形成独立的延迟通道，是稳定第二峰的重要条件。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Jump-over pathways create a distinct delayed channel that is necessary for persistent second peaks.", "snippet_cn": "jump-over 路径会形成独立的延迟通道，是稳定第二峰的重要条件。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Finding formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Jump-over pathways create a distinct delayed channel that is", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c5", "grid2d_reflecting_bimodality-c4", "grid2d_bimodality-c2", "grid2d_blackboard_bimodality-c2", "grid2d_reflecting_bimodality-c2" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "grid2d_reflecting_bimodality", "grid2d_bimodality" ] }, { "claim_id": "ring_lazy_jump-c5", "report_id": "ring_lazy_jump", "stage": "finding", "text_en": "Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.", "text_cn": "shortcut 强度影响呈非单调：过弱或过强都会削弱稳健双峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Shortcut strength has a non-monotonic effect: too weak or too strong settings both reduce robust bimodality.", "snippet_cn": "shortcut 强度影响呈非单调：过弱或过强都会削弱稳健双峰。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Finding formula context in Lazy Ring Jump-Over Mechanism (K2 vs K4): Shortcut strength has a non-monotonic effect: too weak or too strong", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "ring_two_target-c5", "ring_lazy_flux-c4", "ring_lazy_jump_ext-c4", "ring_two_target-c3", "final_multitimescale_fpt_encounter-c2" ], "linked_report_ids": [ "ring_two_target", "ring_lazy_flux", "ring_lazy_jump_ext", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "ring_lazy_jump_ext-c1", "report_id": "ring_lazy_jump_ext", "stage": "model", "text_en": "The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.", "text_cn": "模型沿用 lazy ring 基线（ ），采用 selfloop 分流规则加入单向 shortcut，并在统一参数口径下比较 K=2 与 K=4。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "The model keeps the lazy ring baseline with one directed shortcut under the selfloop probability rule, and compares K=2 versus K=4 under matched parameter settings.", "snippet_cn": "模型沿用 lazy ring 基线（ ），采用 selfloop 分流规则加入单向 shortcut，并在统一参数口径下比较 K=2 与 K=4。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump_ext.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Model formula context in Lazy Ring Shortcut Beta Scan: The model keeps the lazy ring baseline with one directed shortcut under the", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "ring_lazy_flux-c1", "ring_lazy_jump-c1", "ring_lazy_jump_ext_rev2-c1", "ring_two_target-c1", "ring_valley_dst-c1" ], "linked_report_ids": [ "ring_lazy_flux", "ring_lazy_jump", "ring_lazy_jump_ext_rev2", "ring_two_target", "ring_valley_dst" ] }, { "claim_id": "ring_lazy_jump_ext-c2", "report_id": "ring_lazy_jump_ext", "stage": "method", "text_en": "The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail diagnostics.", "text_cn": "方法链路包括精确 AW 的 beta 扫描、稳定 beta 锚点选择、N 扫描，以及基于 MC 轨迹类别与尾部诊断的交叉核验。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N sweeps, and cross-checks class composition through Monte Carlo trajectory statistics and tail", "snippet_cn": "方法链路包括精确 AW 的 beta 扫描、稳定 beta 锚点选择、N 扫描，以及基于 MC 轨迹类别与尾部诊断的交叉核验。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump_ext.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Method formula context in Lazy Ring Shortcut Beta Scan: The workflow runs exact AW beta sweeps, selects a stable beta anchor, executes N", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "ring_valley_dst-c2", "ring_two_target-c2", "ring_valley_dst-c3", "grid2d_rect_bimodality-c2", "ring_lazy_jump-c2" ], "linked_report_ids": [ "ring_valley_dst", "ring_two_target", "grid2d_rect_bimodality", "ring_lazy_jump" ] }, { "claim_id": "ring_lazy_jump_ext-c3", "report_id": "ring_lazy_jump_ext", "stage": "result", "text_en": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.", "text_cn": "随着 beta 提升，两个峰位整体提前且尾部衰减更快；在同等 beta 下，K=4 的双峰保持性优于 K=2，且精确解与 MC 在相区趋势上吻合。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.", "snippet_cn": "随着 beta 提升，两个峰位整体提前且尾部衰减更快；在同等 beta 下，K=4 的双峰保持性优于 K=2，且精确解与 MC 在相区趋势上吻合。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump_ext.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Result formula context in Lazy Ring Shortcut Beta Scan: Increasing beta advances both peaks and steepens tail decay", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "ring_valley-c3", "ring_lazy_jump_ext_rev2-c3", "cross_luca_regime_map-c3", "grid2d_blackboard_bimodality-c3", "grid2d_two_target_double_peak-c2" ], "linked_report_ids": [ "ring_valley", "ring_lazy_jump_ext_rev2", "cross_luca_regime_map", "grid2d_blackboard_bimodality", "grid2d_two_target_double_peak" ] }, { "claim_id": "ring_lazy_jump_ext-c4", "report_id": "ring_lazy_jump_ext", "stage": "finding", "text_en": "At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.", "text_cn": "在固定 N=100 时，beta 扫描显示峰位前移与尾部衰减率增大具有一致趋势。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "At fixed N=100, beta sweeps show systematic left-shifts of peak times and a larger tail-decay rate as shortcut strength increases.", "snippet_cn": "在固定 N=100 时，beta 扫描显示峰位前移与尾部衰减率增大具有一致趋势。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump_ext.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Finding formula context in Lazy Ring Shortcut Beta Scan: At fixed N=100, beta sweeps show systematic left-shifts of peak times and a", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "grid2d_one_target_base-c5", "ring_lazy_flux-c4", "ring_lazy_jump-c3", "ring_lazy_jump-c5", "ring_valley_dst-c5" ], "linked_report_ids": [ "grid2d_one_target_base", "ring_lazy_flux", "ring_lazy_jump", "ring_valley_dst" ] }, { "claim_id": "ring_lazy_jump_ext-c5", "report_id": "ring_lazy_jump_ext", "stage": "finding", "text_en": "K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.", "text_cn": "在同一 beta 计划下，K=4 相比 K=2 维持了更宽、更深的双峰区间。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched beta schedules.", "snippet_cn": "在同一 beta 计划下，K=4 相比 K=2 维持了更宽、更深的双峰区间。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Research report ring_lazy_jump_ext.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext/artifacts/tables/beta_scan_N100_K2.tex", "snippet_en": "Finding formula context in Lazy Ring Shortcut Beta Scan: K=4 keeps a broader bimodal interval and deeper valley than K=2 under matched", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext/series/scan_n_k4_beta002-probability.json", "snippet_en": "scan_N_K4_beta002 [probability]: N -> q [probability]", "snippet_cn": "scan_N_K4_beta002 [probability]: N -> q [probability]" } ], "linked_claim_ids": [ "ring_lazy_jump_ext_rev2-c1", "grid2d_blackboard_bimodality-c1", "grid2d_reflecting_bimodality-c1", "ring_lazy_jump-c1", "ring_valley-c2" ], "linked_report_ids": [ "ring_lazy_jump_ext_rev2", "grid2d_blackboard_bimodality", "grid2d_reflecting_bimodality", "ring_lazy_jump", "ring_valley" ] }, { "claim_id": "ring_lazy_jump_ext_rev2-c1", "report_id": "ring_lazy_jump_ext_rev2", "stage": "model", "text_en": "The chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.", "text_cn": "模型保持与扩展基线一致的 lazy ring + shortcut 设定，重点是在选定 beta 区间下更清晰地对齐 K=2 与 K=4 的证据表达。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "The chapter keeps the same lazy ring shortcut setup used in the extension baseline and focuses on clearer evidence alignment between K=2 and K=4 under the selected beta regime.", "snippet_cn": "模型保持与扩展基线一致的 lazy ring + shortcut 设定，重点是在选定 beta 区间下更清晰地对齐 K=2 与 K=4 的证据表达。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Research report ring_lazy_jump_ext_rev2.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Model formula context in Lazy Ring Shortcut Figure-1 Revision: The chapter keeps the same lazy ring shortcut setup used in the extension", "snippet_cn": "Overview" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c1", "ring_lazy_flux-c1", "ring_lazy_jump-c1", "ring_two_target-c1", "cross_luca_regime_map-c1" ], "linked_report_ids": [ "ring_lazy_jump_ext", "ring_lazy_flux", "ring_lazy_jump", "ring_two_target", "cross_luca_regime_map" ] }, { "claim_id": "ring_lazy_jump_ext_rev2-c2", "report_id": "ring_lazy_jump_ext_rev2", "stage": "method", "text_en": "The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation, and MC confidence intervals.", "text_cn": "流程包括标准化导出图1输入、schema 校验、同轴栈叠面板绘制，以及阈值、窗口扰动、MC 置信区间三条敏感性分析链。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "The pipeline exports standardized Fig.1 inputs, validates schema, renders stacked-bar co-located panels, and runs three sensitivity tracks: threshold sweep, window perturbation", "snippet_cn": "流程包括标准化导出图1输入、schema 校验、同轴栈叠面板绘制，以及阈值、窗口扰动、MC 置信区间三条敏感性分析链。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Research report ring_lazy_jump_ext_rev2.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Method formula context in Lazy Ring Shortcut Figure-1 Revision: The pipeline exports standardized Fig.1 inputs, validates schema, renders", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json", "snippet_en": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]", "snippet_cn": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c2", "ring_valley-c2", "cross_luca_regime_map-c2", "final_multitimescale_fpt_encounter-c2", "grid2d_bimodality-c2" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "ring_valley", "cross_luca_regime_map", "final_multitimescale_fpt_encounter", "grid2d_bimodality" ] }, { "claim_id": "ring_lazy_jump_ext_rev2-c3", "report_id": "ring_lazy_jump_ext_rev2", "stage": "result", "text_en": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.", "text_cn": "三类敏感性分析下，机制解释在定性层面保持稳定，且不确定性区间未推翻主线中的相区判断。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main", "snippet_cn": "三类敏感性分析下，机制解释在定性层面保持稳定，且不确定性区间未推翻主线中的相区判断。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Research report ring_lazy_jump_ext_rev2.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Result formula context in Lazy Ring Shortcut Figure-1 Revision: Across the three sensitivity tracks, the qualitative mechanism", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json", "snippet_en": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]", "snippet_cn": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c3", "cross_luca_regime_map-c3", "ring_valley-c3", "grid2d_two_target_double_peak-c4", "ring_two_target-c1" ], "linked_report_ids": [ "ring_lazy_jump_ext", "cross_luca_regime_map", "ring_valley", "grid2d_two_target_double_peak", "ring_two_target" ] }, { "claim_id": "ring_lazy_jump_ext_rev2-c4", "report_id": "ring_lazy_jump_ext_rev2", "stage": "finding", "text_en": "Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.", "text_cn": "同轴图1将峰谷时序与路径类别比例直接对齐，优于分面板展示。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Co-located Fig.1 directly aligns peak/valley timing with class proportions, improving interpretability over split-panel layouts.", "snippet_cn": "同轴图1将峰谷时序与路径类别比例直接对齐，优于分面板展示。" }, { "evidence_type": 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"ring_valley-c5", "ring_valley_dst-c2", "ring_valley_dst-c5" ], "linked_report_ids": [ "final_multitimescale_fpt_encounter", "ring_valley", "ring_valley_dst" ] }, { "claim_id": "ring_lazy_jump_ext_rev2-c5", "report_id": "ring_lazy_jump_ext_rev2", "stage": "finding", "text_en": "Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.", "text_cn": "经 schema 校验的 JSON/CSV 输入使图形可重建、可审计。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Schema-validated JSON/CSV inputs make figure reconstruction auditable and reproducible.", "snippet_cn": "经 schema 校验的 JSON/CSV 输入使图形可重建、可审计。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Research report ring_lazy_jump_ext_rev2.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_jump_ext_rev2/artifacts/figures/standalone/fig2_overlap_binbars_beta0.01_x1350_description_en.tex", "snippet_en": "Finding formula context in Lazy Ring Shortcut Figure-1 Revision: Schema-validated JSON/CSV inputs make figure reconstruction auditable and", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_jump_ext_rev2/series/luca_k2_fixed_shortcut_metrics-probability.json", "snippet_en": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]", "snippet_cn": "luca_k2_fixed_shortcut_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [], "linked_report_ids": [ "ring_lazy_flux", "ring_lazy_jump_ext", "ring_lazy_jump" ] }, { "claim_id": "ring_lazy_flux-c1", "report_id": "ring_lazy_flux", "stage": "model", "text_en": "The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.", "text_cn": "模型为 lazy 最近邻环并加入单向 shortcut u->v；在非 shortcut 源点处，停留/左右迁移遵循等概率基线。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "The model is a lazy nearest-neighbor ring with one directed shortcut u->v; away from the shortcut source, stay/left/right probabilities follow the equal-probability baseline.", "snippet_cn": "模型为 lazy 最近邻环并加入单向 shortcut u->v；在非 shortcut 源点处，停留/左右迁移遵循等概率基线。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Research report ring_lazy_flux.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Model formula context in Lazy Ring Flux Baseline: The model is a lazy nearest-neighbor ring with one directed shortcut u->v", "snippet_cn": "Overview" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c1", "ring_lazy_jump-c1", "ring_two_target-c1", "ring_valley_dst-c1", "final_multitimescale_fpt_encounter-c1" ], "linked_report_ids": [ "ring_lazy_jump_ext", "ring_lazy_jump", "ring_two_target", "ring_valley_dst", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "ring_lazy_flux-c2", "report_id": "ring_lazy_flux", "stage": "method", "text_en": "The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.", "text_cn": "方法链路为：解析推导生成函数，使用 AW/FFT 反演得到首达 pmf，再用 flux 递推进行独立数值核验。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "The pipeline derives the generating function analytically, inverts it via AW/FFT, and verifies the recovered pmf by independent flux recursion.", "snippet_cn": "方法链路为：解析推导生成函数，使用 AW/FFT 反演得到首达 pmf，再用 flux 递推进行独立数值核验。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Research report ring_lazy_flux.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Method formula context in Lazy Ring Flux Baseline: The pipeline derives the generating function analytically, inverts it via AW/FFT, and", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json", "snippet_en": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]", "snippet_cn": 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toward unimodality.", "snippet_cn": "自环分流且小 p 条件下可形成稳定双峰（最小 N=10 例）；equal4 或 shortcut 强度增大时，分布向单峰收敛。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Research report ring_lazy_flux.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Result formula context in Lazy Ring Flux Baseline: A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json", "snippet_en": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]", "snippet_cn": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]" } ], "linked_claim_ids": [ "grid2d_reflecting_bimodality-c3", "ring_lazy_jump-c3", "ring_valley-c5", "ring_lazy_jump_ext-c1", "ring_lazy_jump_ext-c4" ], "linked_report_ids": [ "grid2d_reflecting_bimodality", "ring_lazy_jump", "ring_valley", "ring_lazy_jump_ext" ] }, { "claim_id": "ring_lazy_flux-c4", "report_id": "ring_lazy_flux", "stage": "finding", "text_en": "A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.", "text_cn": "在 selfloop 小 p 条件下，N=10 可复现双峰首达分布。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "A minimal reproducible bimodal case appears at N=10 under small shortcut strength in the selfloop construction.", "snippet_cn": "在 selfloop 小 p 条件下，N=10 可复现双峰首达分布。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Research report ring_lazy_flux.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Finding formula context in Lazy Ring Flux Baseline: A minimal reproducible bimodal case appears at N=10 under small shortcut strength in", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json", "snippet_en": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]", "snippet_cn": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]" } ], "linked_claim_ids": [ "ring_two_target-c5", "ring_lazy_jump-c3", "ring_lazy_jump-c5", "ring_lazy_jump_ext-c4", "ring_lazy_jump_ext-c1" ], "linked_report_ids": [ "ring_two_target", "ring_lazy_jump", "ring_lazy_jump_ext" ] }, { "claim_id": "ring_lazy_flux-c5", "report_id": "ring_lazy_flux", "stage": "finding", "text_en": "AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.", "text_cn": "AW 反演与 flux 递推在数值精度内一致，验证了理论与实现的正确性。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "AW inversion and flux recursion agree to numerical precision, validating both the derivation and implementation.", "snippet_cn": "AW 反演与 flux 递推在数值精度内一致，验证了理论与实现的正确性。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Research report ring_lazy_flux.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_lazy_flux/artifacts/tables/lazy_K2_equal4_paper_geometry_summary_cn.tex", "snippet_en": "Finding formula context in Lazy Ring Flux Baseline: AW inversion and flux recursion agree to numerical precision, validating both the", "snippet_cn": "Overview" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_lazy_flux/series/lazy_k2_equal4_paper_geometry_summary_cn-binary.json", "snippet_en": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]", "snippet_cn": "lazy_K2_equal4_paper_geometry_summary_cn [tabular binary]: N -> paper [binary]" } ], "linked_claim_ids": [ "ring_valley_dst-c4", "ring_lazy_jump-c2", "ring_lazy_jump_ext-c3", "ring_two_target-c2", "ring_valley-c2" ], "linked_report_ids": [ "ring_valley_dst", "ring_lazy_jump", "ring_lazy_jump_ext", "ring_two_target", "ring_valley" ] }, { "claim_id": "ring_valley-c1", "report_id": "ring_valley", "stage": "model", "text_en": "The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.", "text_cn": "模型采用均匀 K 邻接跳转与单向 shortcut，吸收目标置于 N/2，并保持与论文图示一致的索引与几何构型。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent indexing and shortcut placement.", "snippet_cn": "模型采用均匀 K 邻接跳转与单向 shortcut，吸收目标置于 N/2，并保持与论文图示一致的索引与几何构型。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Model and graph construction (one-way shortcut): The graph is a directed-shortcut ring with uniform K-neighbor transitions and an absorbing target at N/2, using paper-consistent", "snippet_cn": "Model and graph construction (one-way shortcut): 模型采用均匀 K 邻接跳转与单向 shortcut，吸收目标置于 N/2，并保持与论文图示一致的索引与几何构型。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality ranges detected under the Fig. 3 rule are printed to stdout.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Classification of trajectories", "snippet_cn": "Classification of trajectories" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Directed-shortcut transition kernel", "snippet_cn": "Directed-shortcut transition kernel" } ], "linked_claim_ids": [ "ring_valley_dst-c1", "ring_lazy_flux-c1", "ring_two_target-c1", "ring_deriv_k2-c1", "ring_lazy_jump-c1" ], "linked_report_ids": [ "ring_valley_dst", "ring_lazy_flux", "ring_two_target", "ring_deriv_k2", "ring_lazy_jump" ] }, { "claim_id": "ring_valley-c2", "report_id": "ring_valley", "stage": "method", "text_en": "The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.", "text_cn": "方法链路结合 AW 反演、MC 全路径模拟与 Fig.3 峰谷判据，并对 K=2 使用两步粗粒化处理奇偶微峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "The workflow combines AW inversion, MC trajectory simulation, and Fig.3 peak-valley criteria with K=2 parity coarse graining.", "snippet_cn": "方法链路结合 AW 反演、MC 全路径模拟与 Fig.3 峰谷判据，并对 K=2 使用两步粗粒化处理奇偶微峰。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality ranges detected under the Fig. 3 rule are printed to stdout.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Classification of trajectories", "snippet_cn": "Classification of trajectories" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Directed-shortcut transition kernel", "snippet_cn": "Directed-shortcut transition kernel" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley/series/bimodality_scan.json", "snippet_en": "bimodality_scan: n -> bimodal", "snippet_cn": "bimodality_scan: n -> bimodal" } ], "linked_claim_ids": [ "ring_two_target-c2", "ring_valley_dst-c2", "ring_lazy_jump-c2", "ring_lazy_jump_ext-c2", "final_multitimescale_fpt_encounter-c2" ], "linked_report_ids": [ "ring_two_target", "ring_valley_dst", "ring_lazy_jump", "ring_lazy_jump_ext", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "ring_valley-c3", "report_id": "ring_valley", "stage": "result", "text_en": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.", "text_cn": "在扫描的偶数 N 范围内，K=2 维持单峰；K=4/6/8 则出现可复现的双峰区间，并在精确解与 MC 诊断中一致。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.", "snippet_cn": "在扫描的偶数 N 范围内，K=2 维持单峰；K=4/6/8 则出现可复现的双峰区间，并在精确解与 MC 诊断中一致。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality ranges detected under the Fig. 3 rule are printed to stdout.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Classification of trajectories", "snippet_cn": "Classification of trajectories" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley/series/bimodality_scan.json", "snippet_en": "bimodality_scan: n -> bimodal", "snippet_cn": "bimodality_scan: n -> bimodal" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c3", "cross_luca_regime_map-c3", "ring_two_target-c3", "grid2d_rect_bimodality-c2", "ring_lazy_jump_ext_rev2-c3" ], "linked_report_ids": [ "ring_lazy_jump_ext", "cross_luca_regime_map", "ring_two_target", "grid2d_rect_bimodality", "ring_lazy_jump_ext_rev2" ] }, { "claim_id": "ring_valley-c4", "report_id": "ring_valley", "stage": "finding", "text_en": "Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.", "text_cn": "K=4、K=6、K=8 分别出现结构化双峰 N 区间。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.", "snippet_cn": "K=4、K=6、K=8 分别出现结构化双峰 N 区间。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality ranges detected under the Fig. 3 rule are printed to stdout.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Classification of trajectories", "snippet_cn": "Classification of trajectories" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley/series/bimodality_scan.json", "snippet_en": "bimodality_scan: n -> bimodal", "snippet_cn": "bimodality_scan: n -> bimodal" } ], "linked_claim_ids": [ "grid2d_reflecting_bimodality-c4", "ring_two_target-c5", "ring_two_target-c3", "grid2d_reflecting_bimodality-c2" ], "linked_report_ids": [ "grid2d_reflecting_bimodality", "ring_two_target" ] }, { "claim_id": "ring_valley-c5", "report_id": "ring_valley", "stage": "finding", "text_en": "Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range.", "text_cn": "在 Fig.3 判据与奇偶修正下，K=2 在扫描区间内不呈现稳定双峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range.", "snippet_cn": "在 Fig.3 判据与奇偶修正下，K=2 在扫描区间内不呈现稳定双峰。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Bimodality detected: K=2 none; Explicitly: K=2 still shows no two-peak structure under the reference figure.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley/manuscript/ring_valley_en.tex", "snippet_en": "Let the first two peaks be at t1 bimodal", "snippet_cn": "bimodality_scan: n -> bimodal" } ], "linked_claim_ids": [ "final_multitimescale_fpt_encounter-c2", "final_multitimescale_fpt_encounter-c3", "final_multitimescale_fpt_encounter-c4", "grid2d_blackboard_bimodality-c3", "grid2d_membrane_near_target-c4" ], "linked_report_ids": [ "final_multitimescale_fpt_encounter", "grid2d_blackboard_bimodality", "grid2d_membrane_near_target" ] }, { "claim_id": "ring_valley_dst-c1", "report_id": "ring_valley_dst", "stage": "model", "text_en": "The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.", "text_cn": "模型为 K=6 环 + 单向 shortcut src->dst + 吸收目标；只改变 dst，使机制差异可归因于几何落点。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "The model uses a K=6 ring with one directed shortcut src->dst and an absorbing target; only dst is varied so mechanism changes can be attributed to geometric landing location.", "snippet_cn": "模型为 K=6 环 + 单向 shortcut src->dst + 吸收目标；只改变 dst，使机制差异可归因于几何落点。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Research report ring_valley_dst.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Model formula context in Destination-Scan Valley Control: The model uses a K=6 ring with one directed shortcut src->dst and an absorbing", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "ring_two_target-c1", "ring_lazy_flux-c1", "ring_lazy_jump_ext-c1", "ring_valley-c1", "grid2d_rect_bimodality-c1" ], "linked_report_ids": [ "ring_two_target", "ring_lazy_flux", "ring_lazy_jump_ext", "ring_valley", "grid2d_rect_bimodality" ] }, { "claim_id": "ring_valley_dst-c2", "report_id": "ring_valley_dst", "stage": "method", "text_en": "The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.", "text_cn": "方法链结合确定性 flux 扫描、AW 首达诊断与按轨迹类别条件化的 Monte Carlo 统计，并采用统一峰谷判据。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "The workflow combines deterministic flux scans, AW-style first-passage diagnostics, and class-conditioned Monte Carlo paths under one peak/valley criterion.", "snippet_cn": "方法链结合确定性 flux 扫描、AW 首达诊断与按轨迹类别条件化的 Monte Carlo 统计，并采用统一峰谷判据。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Research report ring_valley_dst.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Method formula context in Destination-Scan Valley Control: The workflow combines deterministic flux scans, AW-style first-passage", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley_dst/series/scan-probability.json", "snippet_en": "scan [probability]: steps -> mass, remaining [probability]", "snippet_cn": "scan [probability]: steps -> mass, remaining [probability]" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c2", "grid2d_rect_bimodality-c2", "grid2d_two_target_double_peak-c2", "ring_two_target-c2", "ring_valley-c2" ], "linked_report_ids": [ "ring_lazy_jump_ext", "grid2d_rect_bimodality", "grid2d_two_target_double_peak", "ring_two_target", "ring_valley" ] }, { "claim_id": "ring_valley_dst-c3", "report_id": "ring_valley_dst", "stage": "result", "text_en": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.", "text_cn": "dst 扫描给出了结构化的高对比区间：第二峰可被显著放大，且轨迹类别占比同步变化，确定性与 MC 结论一致。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining", "snippet_cn": "dst 扫描给出了结构化的高对比区间：第二峰可被显著放大，且轨迹类别占比同步变化，确定性与 MC 结论一致。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Research report ring_valley_dst.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Result formula context in Destination-Scan Valley Control: Destination scanning reveals structured dst windows where the second peak is", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley_dst/series/scan-probability.json", "snippet_en": "scan [probability]: steps -> mass, remaining [probability]", "snippet_cn": "scan [probability]: steps -> mass, remaining [probability]" } ], "linked_claim_ids": [ "ring_lazy_jump_ext-c2", "ring_two_target-c2", "grid2d_blackboard_bimodality-c5", "ring_lazy_jump-c3", "ring_valley-c3" ], "linked_report_ids": [ "ring_lazy_jump_ext", "ring_two_target", "grid2d_blackboard_bimodality", "ring_lazy_jump", "ring_valley" ] }, { "claim_id": "ring_valley_dst-c4", "report_id": "ring_valley_dst", "stage": "finding", "text_en": "Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.", "text_cn": "确定性 flux/master 方程与 MC 轨迹分类在高对比 dst 区间上相互印证。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Deterministic flux/master-equation results and Monte Carlo class decomposition agree on high-contrast destination windows.", "snippet_cn": "确定性 flux/master 方程与 MC 轨迹分类在高对比 dst 区间上相互印证。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Research report ring_valley_dst.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Finding formula context in Destination-Scan Valley Control: Deterministic flux/master-equation results and Monte Carlo class decomposition", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley_dst/series/scan-probability.json", "snippet_en": "scan [probability]: steps -> mass, remaining [probability]", "snippet_cn": "scan [probability]: steps -> mass, remaining [probability]" } ], "linked_claim_ids": [ "ring_lazy_flux-c5", "ring_lazy_jump_ext-c2", "grid2d_reflecting_bimodality-c2", "grid2d_blackboard_bimodality-c2", "ring_lazy_flux-c2" ], "linked_report_ids": [ "ring_lazy_flux", "ring_lazy_jump_ext", "grid2d_reflecting_bimodality", "grid2d_blackboard_bimodality" ] }, { "claim_id": "ring_valley_dst-c5", "report_id": "ring_valley_dst", "stage": "finding", "text_en": "Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.", "text_cn": "在固定 N、K 下，仅扫描 dst 就能显著改变第二峰高度比与谷深。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Scanning dst alone can substantially change second-peak height ratio and valley depth at fixed N and K.", "snippet_cn": "在固定 N、K 下，仅扫描 dst 就能显著改变第二峰高度比与谷深。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Research report ring_valley_dst.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_valley_dst/artifacts/data/bimodality_flux_scan/N100K2_n0_1_target_50_src_1/latest/bimodal_table.tex", "snippet_en": "Finding formula context in Destination-Scan Valley Control: Scanning dst alone can substantially change second-peak height ratio and", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_valley_dst/series/scan-probability.json", "snippet_en": "scan [probability]: steps -> mass, remaining [probability]", "snippet_cn": "scan [probability]: steps -> mass, remaining [probability]" } ], "linked_claim_ids": [ "cross_luca_regime_map-c5", "grid2d_blackboard_bimodality-c5", "ring_lazy_jump_ext-c4", "ring_lazy_jump_ext_rev2-c4", "ring_lazy_jump-c3" ], "linked_report_ids": [ "cross_luca_regime_map", "grid2d_blackboard_bimodality", "ring_lazy_jump_ext", "ring_lazy_jump_ext_rev2", "ring_lazy_jump" ] }, { "claim_id": "ring_deriv_k2-c1", "report_id": "ring_deriv_k2", "stage": "model", "text_en": "The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring interpretations.", "text_cn": "模型为有限环随机游走并加入单向长程连边，采用与 lazy-reservoir 与 rewiring 两类解释兼容的统一表达。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The setting is a finite ring random walk with periodic indexing and one directed long-range connection, expressed in a form compatible with both lazy-reservoir and rewiring", "snippet_cn": "模型为有限环随机游走并加入单向长程连边，采用与 lazy-reservoir 与 rewiring 两类解释兼容的统一表达。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We consider a discrete-time random walk on a 1D ring (cycle) of size N with periodic boundary conditions (PBC). Transition matrices and convention.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Model formula context in Ring Derivation Backbone: The setting is a finite ring random walk with periodic indexing and one directed", "snippet_cn": "Notation and model" } ], "linked_claim_ids": [ "ring_lazy_jump-c1", "ring_lazy_flux-c1", "ring_lazy_jump_ext-c1", "final_multitimescale_fpt_encounter-c2", "final_multitimescale_fpt_encounter-c3" ], "linked_report_ids": [ "ring_lazy_jump", "ring_lazy_flux", "ring_lazy_jump_ext", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "ring_deriv_k2-c2", "report_id": "ring_deriv_k2", "stage": "method", "text_en": "The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.", "text_cn": "方法链从 Green 函数传播子出发，构造缺陷 resolvent 修正，再得到可数值反演的首达生成函数形式。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The report derives Green-function style propagators, constructs defect-resolvent corrections, and obtains first-passage generating forms that can be numerically inverted.", "snippet_cn": "方法链从 Green 函数传播子出发，构造缺陷 resolvent 修正，再得到可数值反演的首达生成函数形式。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Method formula context in Ring Derivation Backbone: The report derives Green-function style propagators, constructs defect-resolvent", "snippet_cn": "Notation and model" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_deriv_k2/series/note_k2-probability.json", "snippet_en": "note_k2 [tabular probability]: N -> q, p [probability]", "snippet_cn": "note_k2 [tabular probability]: N -> q, p [probability]" } ], "linked_claim_ids": [ "grid2d_bimodality-c2", "ring_lazy_flux-c2", "ring_two_target-c2", "ring_valley_dst-c2", "grid2d_rect_bimodality-c2" ], "linked_report_ids": [ "grid2d_bimodality", "ring_lazy_flux", "ring_two_target", "ring_valley_dst", "grid2d_rect_bimodality" ] }, { "claim_id": "ring_deriv_k2-c3", "report_id": "ring_deriv_k2", "stage": "result", "text_en": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.", "text_cn": "闭式推导明确了 shortcut 不对称性的主导项，并为后续 ring 报告提供可复用公式模块。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.", "snippet_cn": "闭式推导明确了 shortcut 不对称性的主导项，并为后续 ring 报告提供可复用公式模块。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Result formula context in Ring Derivation Backbone: The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and", "snippet_cn": "Notation and model" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_deriv_k2/series/note_k2-probability.json", "snippet_en": "note_k2 [tabular probability]: N -> q, p [probability]", "snippet_cn": "note_k2 [tabular probability]: N -> q, p [probability]" } ], "linked_claim_ids": [ "ring_lazy_flux-c1", "ring_lazy_jump-c1", "ring_lazy_jump_ext-c1", "ring_lazy_jump_ext_rev2-c1", "ring_two_target-c1" ], "linked_report_ids": [ "ring_lazy_flux", "ring_lazy_jump", "ring_lazy_jump_ext", "ring_lazy_jump_ext_rev2", "ring_two_target" ] }, { "claim_id": "ring_deriv_k2-c4", "report_id": "ring_deriv_k2", "stage": "finding", "text_en": "Defect-free and defect-corrected propagators can be written in a unified analytic framework.", "text_cn": "无缺陷与缺陷修正传播子可在同一解析框架下统一表示。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Defect-free and defect-corrected propagators can be written in a unified analytic framework.", "snippet_cn": "无缺陷与缺陷修正传播子可在同一解析框架下统一表示。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Finding formula context in Ring Derivation Backbone: Defect-free and defect-corrected propagators can be written in a unified analytic", "snippet_cn": "Notation and model" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_deriv_k2/series/note_k2-probability.json", "snippet_en": "note_k2 [tabular probability]: N -> q, p [probability]", "snippet_cn": "note_k2 [tabular probability]: N -> q, p [probability]" } ], "linked_claim_ids": [ "grid2d_bimodality-c2", "cross_luca_regime_map-c2" ], "linked_report_ids": [ "grid2d_bimodality", "cross_luca_regime_map" ] }, { "claim_id": "ring_deriv_k2-c5", "report_id": "ring_deriv_k2", "stage": "finding", "text_en": "Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.", "text_cn": "单向长程连边对首达统计的影响体现在 resolvent 级修正，而非经验拟合。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Directed long-range links alter first-passage statistics through resolvent-level corrections rather than ad-hoc fitting.", "snippet_cn": "单向长程连边对首达统计的影响体现在 resolvent 级修正，而非经验拟合。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "We validated eq:Q chebyshev, eq:S closed, and eq:F cheb closed against direct matrix computation of the resolvent.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "The FPT generating function from n0 to n is F n0-> n (z)= S n0 (n,z) S n (n,z). eq:F def Using eq:S closed both in the numerator and with n0=n in the denominator yields F n0-> n", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_deriv_k2/manuscript/extras/note_k2.tex", "snippet_en": "Finding formula context in Ring Derivation Backbone: Directed long-range links alter first-passage statistics through resolvent-level", "snippet_cn": "Notation and model" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_deriv_k2/series/note_k2-probability.json", "snippet_en": "note_k2 [tabular probability]: N -> q, p [probability]", "snippet_cn": "note_k2 [tabular probability]: N -> q, p [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c5", "grid2d_two_target_double_peak-c5", "ring_two_target-c4", "ring_two_walker_encounter_shortcut-c4", "grid2d_bimodality-c2" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "grid2d_two_target_double_peak", "ring_two_target", "ring_two_walker_encounter_shortcut", "grid2d_bimodality" ] }, { "claim_id": "ring_two_target-c1", "report_id": "ring_two_target", "stage": "model", "text_en": "The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.", "text_cn": "模型在 lazy ring 上设置两个吸收目标，并可选加入单向 shortcut，显式保留索引与几何距离定义以支持机制级比较。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "The model places two absorbing targets on a lazy ring with optional directed shortcut, keeping index conventions and distance geometry explicit for mechanism-level comparison.", "snippet_cn": "模型在 lazy ring 上设置两个吸收目标，并可选加入单向 shortcut，显式保留索引与几何距离定义以支持机制级比较。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Research report ring_two_target.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Model formula context in Two-Target Lazy Ring Mechanics: The model places two absorbing targets on a lazy ring with optional directed", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "ring_valley_dst-c1", "grid2d_bimodality-c1", "grid2d_two_target_double_peak-c1", "ring_lazy_flux-c1", "ring_lazy_jump_ext-c1" ], "linked_report_ids": [ "ring_valley_dst", "grid2d_bimodality", "grid2d_two_target_double_peak", "ring_lazy_flux", "ring_lazy_jump_ext" ] }, { "claim_id": "ring_two_target-c2", "report_id": "ring_two_target", "stage": "method", "text_en": "Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.", "text_cn": "方法结合精确生成函数/AW 反演、参数扫描与轨迹诊断，在统一判据下对峰结构进行分类。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Exact generating-function/AW inversion is combined with parameter scans and trajectory-style diagnostics to classify peak structures under consistent criteria.", "snippet_cn": "方法结合精确生成函数/AW 反演、参数扫描与轨迹诊断，在统一判据下对峰结构进行分类。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Research report ring_two_target.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Method formula context in Two-Target Lazy Ring Mechanics: Exact generating-function/AW inversion is combined with parameter scans and", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json", "snippet_en": "small_scan_metrics [probability]: beta -> q [probability]", "snippet_cn": "small_scan_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "grid2d_bimodality-c2", "ring_valley-c2", "ring_valley_dst-c2", "grid2d_blackboard_bimodality-c2", "grid2d_two_target_double_peak-c2" ], "linked_report_ids": [ "grid2d_bimodality", "ring_valley", "ring_valley_dst", "grid2d_blackboard_bimodality", "grid2d_two_target_double_peak" ] }, { "claim_id": "ring_two_target-c3", "report_id": "ring_two_target", "stage": "result", "text_en": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.", "text_cn": "无 shortcut 漂移本身即可产生稳定双峰；加入 shortcut 后路径质量重新分配，在特定几何与参数带可出现三峰行为。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and", "snippet_cn": "无 shortcut 漂移本身即可产生稳定双峰；加入 shortcut 后路径质量重新分配，在特定几何与参数带可出现三峰行为。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Research report ring_two_target.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Result formula context in Two-Target Lazy Ring Mechanics: No-shortcut drift can already produce strong bimodality, while shortcut", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json", "snippet_en": "small_scan_metrics [probability]: beta -> q [probability]", "snippet_cn": "small_scan_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "ring_lazy_jump-c3", "ring_valley-c3", "grid2d_rect_bimodality-c2", "grid2d_rect_bimodality-c3", "ring_lazy_jump-c5" ], "linked_report_ids": [ "ring_lazy_jump", "ring_valley", "grid2d_rect_bimodality" ] }, { "claim_id": "ring_two_target-c4", "report_id": "ring_two_target", "stage": "finding", "text_en": "Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.", "text_cn": "双目标几何会形成快慢通道竞争，使多峰现象具有结构机制基础。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Two-target geometry introduces competing fast and delayed channels, making multimodality a structural rather than numerical artifact.", "snippet_cn": "双目标几何会形成快慢通道竞争，使多峰现象具有结构机制基础。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Research report ring_two_target.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Finding formula context in Two-Target Lazy Ring Mechanics: Two-target geometry introduces competing fast and delayed channels, making", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json", "snippet_en": "small_scan_metrics [probability]: beta -> q [probability]", "snippet_cn": "small_scan_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c5", "grid2d_blackboard_bimodality-c5", "grid2d_bimodality-c5", "grid2d_bimodality-c4", "grid2d_membrane_near_target-c4" ], "linked_report_ids": [ "grid2d_rect_bimodality", "grid2d_blackboard_bimodality", "grid2d_bimodality", "grid2d_membrane_near_target" ] }, { "claim_id": "ring_two_target-c5", "report_id": "ring_two_target", "stage": "finding", "text_en": "Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.", "text_cn": "在无 shortcut 漂移条件下，可复现的参数窗口内可观察到稳健双峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Under no-shortcut drift, robust bimodality appears in reproducible parameter windows.", "snippet_cn": "在无 shortcut 漂移条件下，可复现的参数窗口内可观察到稳健双峰。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Research report ring_two_target.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_target/artifacts/tables/case_configs.tex", "snippet_en": "Finding formula context in Two-Target Lazy Ring Mechanics: Under no-shortcut drift, robust bimodality appears in reproducible parameter", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_target/series/small_scan_metrics-probability.json", "snippet_en": "small_scan_metrics [probability]: beta -> q [probability]", "snippet_cn": "small_scan_metrics [probability]: beta -> q [probability]" } ], "linked_claim_ids": [ "ring_lazy_flux-c4", "ring_lazy_jump-c5", "final_multitimescale_fpt_encounter-c4", "ring_lazy_jump-c3", "ring_valley-c4" ], "linked_report_ids": [ "ring_lazy_flux", "ring_lazy_jump", "final_multitimescale_fpt_encounter", "ring_valley" ] }, { "claim_id": "ring_two_walker_encounter_shortcut-c1", "report_id": "ring_two_walker_encounter_shortcut", "stage": "model", "text_en": "Research report ring_two_walker_encounter_shortcut.", "text_cn": "Ring 体系：本节说明状态空间、参数约束与边界条件设定。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节说明状态空间、参数约束与边界条件设定。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Model formula context in 1D Ring Two-Walker Encounter With Shortcut: Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "grid2d_membrane_near_target-c1", "grid2d_one_target_valley_peak_budget-c1", "grid2d_two_walker_encounter_shortcut-c1", "grid2d_membrane_near_target-c2", "grid2d_membrane_near_target-c3" ], "linked_report_ids": [ "grid2d_membrane_near_target", "grid2d_one_target_valley_peak_budget", "grid2d_two_walker_encounter_shortcut" ] }, { "claim_id": "ring_two_walker_encounter_shortcut-c2", "report_id": "ring_two_walker_encounter_shortcut", "stage": "method", "text_en": "[ring] Method chain: Research report ring_two_walker_encounter_shortcut.", "text_cn": "[ring] 方法链路：Ring 体系：本节给出解析/数值流程、反演方法与验证协议。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "[ring] Method chain: Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "[ring] 方法链路：Ring 体系：本节给出解析/数值流程、反演方法与验证协议。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Method formula context in 1D Ring Two-Walker Encounter With Shortcut: [ring] Method chain: Research report", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json", "snippet_en": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]", "snippet_cn": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]" } ], "linked_claim_ids": [ "grid2d_membrane_near_target-c2", "grid2d_one_target_valley_peak_budget-c2", "grid2d_two_walker_encounter_shortcut-c2", "encounter_reflecting_diagonal_decomp-c2", "encounter_reflecting_mean_validation-c2" ], "linked_report_ids": [ "grid2d_membrane_near_target", "grid2d_one_target_valley_peak_budget", "grid2d_two_walker_encounter_shortcut", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ] }, { "claim_id": "ring_two_walker_encounter_shortcut-c3", "report_id": "ring_two_walker_encounter_shortcut", "stage": "result", "text_en": "[ring] Result statement: Research report ring_two_walker_encounter_shortcut.", "text_cn": "[ring] 结果陈述：Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "[ring] Result statement: Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "[ring] 结果陈述：Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Result formula context in 1D Ring Two-Walker Encounter With Shortcut: [ring] Result statement: Research report", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json", "snippet_en": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]", "snippet_cn": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]" } ], "linked_claim_ids": [ "grid2d_membrane_near_target-c3", "grid2d_one_target_valley_peak_budget-c3", "grid2d_two_walker_encounter_shortcut-c3", "final_multitimescale_fpt_encounter-c3", "cross_luca_regime_map-c1" ], "linked_report_ids": [ "grid2d_membrane_near_target", "grid2d_one_target_valley_peak_budget", "grid2d_two_walker_encounter_shortcut", "final_multitimescale_fpt_encounter", "cross_luca_regime_map" ] }, { "claim_id": "ring_two_walker_encounter_shortcut-c4", "report_id": "ring_two_walker_encounter_shortcut", "stage": "finding", "text_en": "Exact first-encounter computation on 1D ring with directed shortcut.", "text_cn": "Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Exact first-encounter computation on 1D ring with directed shortcut.", "snippet_cn": "Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Finding formula context in 1D Ring Two-Walker Encounter With Shortcut: Exact first-encounter computation on 1D ring with directed shortcut.", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json", "snippet_en": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]", "snippet_cn": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]" } ], "linked_claim_ids": [ "grid2d_two_walker_encounter_shortcut-c4", "ring_deriv_k2-c5", "ring_lazy_flux-c1", "ring_lazy_jump-c1", "ring_lazy_jump_ext-c1" ], "linked_report_ids": [ "grid2d_two_walker_encounter_shortcut", "ring_deriv_k2", "ring_lazy_flux", "ring_lazy_jump", "ring_lazy_jump_ext" ] }, { "claim_id": "ring_two_walker_encounter_shortcut-c5", "report_id": "ring_two_walker_encounter_shortcut", "stage": "finding", "text_en": "Key finding for ring_two_walker_encounter_shortcut: see the equation chain, interactive panels, and source-linked assets.", "text_cn": "Ring 体系（ring_two_walker_encounter_shortcut）：关键结论见本页公式链、交互图与原始资产。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Key finding for ring_two_walker_encounter_shortcut: see the equation chain, interactive panels, and source-linked assets.", "snippet_cn": "Ring 体系（ring_two_walker_encounter_shortcut）：关键结论见本页公式链、交互图与原始资产。" }, { "evidence_type": "section_summary", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Research report ring_two_walker_encounter_shortcut.", "snippet_cn": "Ring 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/ring_two_walker_encounter_shortcut/artifacts/tables/a1a8_test_table.tex", "snippet_en": "Finding formula context in 1D Ring Two-Walker Encounter With Shortcut: Key finding for ring_two_walker_encounter_shortcut: see the", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/ring_two_walker_encounter_shortcut/series/case_summary-probability.json", "snippet_en": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]", "snippet_cn": "case_summary [probability]: detector_compare_beta -> detector_compare_peak_ratio_legacy [probability]" } ], "linked_claim_ids": [ "grid2d_two_walker_encounter_shortcut-c5", "encounter_reflecting_diagonal_decomp-c4", "exact_recursion_method_guide-c4", "grid2d_one_target_valley_peak_budget-c4", "grid2d_membrane_near_target-c2" ], "linked_report_ids": [ "grid2d_two_walker_encounter_shortcut", "encounter_reflecting_diagonal_decomp", "exact_recursion_method_guide", "grid2d_one_target_valley_peak_budget", "grid2d_membrane_near_target" ] }, { "claim_id": "grid2d_bimodality-c1", "report_id": "grid2d_bimodality", "stage": "model", "text_en": "The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.", "text_cn": "模型为二维 N×N 格点 + 吸收目标，包含各向异性漂移控制与惰性停留概率，并显式给出边界条件。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "The model is a two-dimensional N×N lattice with an absorbing target, anisotropic drift controls, and lazy waiting probability under explicit boundary assumptions.", "snippet_cn": "模型为二维 N×N 格点 + 吸收目标，包含各向异性漂移控制与惰性停留概率，并显式给出边界条件。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "\" y \" ， g y>0 。 、 （ x 、 y ）", "snippet_cn": "本文采用 \" y 轴向下为正\" 的坐标约定，因此 g y>0 表示向下偏置。边界条件支持反射、周期或混合（ x 周期、 y 反射）" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "reflecting / ： “ ”。 periodic / ， ， ， （ A）。", "snippet_cn": "reflecting 更贴近有限容器或围墙区域的扩散/搜索问题：概率守恒且不会“漏出系统”。 periodic 对应拓扑上环/环面或均匀介质的平铺近似，现实中并不常见，但可自然引入绕行慢通道，便于产生时间尺度分离（候选 A）。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Model formula context in Grid2D Bimodality Baseline: The model is a two-dimensional N×N lattice with an absorbing target, anisotropic", "snippet_cn": "结论与关键图示" } ], "linked_claim_ids": [ "grid2d_two_target_double_peak-c1", "ring_two_target-c1", "grid2d_rect_bimodality-c1", "grid2d_reflecting_bimodality-c1", "final_multitimescale_fpt_encounter-c2" ], "linked_report_ids": [ "grid2d_two_target_double_peak", "ring_two_target", "grid2d_rect_bimodality", "grid2d_reflecting_bimodality", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "grid2d_bimodality-c2", "report_id": "grid2d_bimodality", "stage": "method", "text_en": "The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.", "text_cn": "方法将无缺陷/缺陷修正传播子连接到生成函数反演，再通过参数扫描与通道诊断验证候选相区。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "The method links defect-free and defect-corrected propagators to generating-function inversion, then validates candidate regimes through parameter scans and channel diagnostics.", "snippet_cn": "方法将无缺陷/缺陷修正传播子连接到生成函数反演，再通过参数扫描与通道诊断验证候选相区。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "： propagator -> propagator -> FPT -> AW 。 。 A ， A v,u = ， 1， p t+1 =A p t 。 P n0 (n,t)= e n^->p A^t e n0, P (z)= t0 A^t z^, P n0 (n,z)= e n^->p (I-zA)e-1 e n0.", "snippet_cn": "本节给出统一推导链条： 无缺陷 propagator -> 有缺陷 propagator -> FPT 生成函数 -> AW 反演。三类候选均可落在同一框架内。 矩阵形式与生成函数 设无吸收系统的转移矩阵为 A，约定 A v,u =，因此列和为 1，且 p t+1 =A p t" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "reflecting / ： “ ”。 periodic / ， ， ， （ A）。", "snippet_cn": "reflecting 更贴近有限容器或围墙区域的扩散/搜索问题：概率守恒且不会“漏出系统”。 periodic 对应拓扑上环/环面或均匀介质的平铺近似，现实中并不常见，但可自然引入绕行慢通道，便于产生时间尺度分离（候选 A）。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Method formula context in Grid2D Bimodality Baseline: The method links defect-free and defect-corrected propagators to generating-function", "snippet_cn": "结论与关键图示" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_bimodality/series/scan_candidate_b_corridor-probability.json", "snippet_en": "scan_candidate_B_corridor [probability]: l -> mass [probability]", "snippet_cn": "scan_candidate_B_corridor [probability]: l -> mass [probability]" } ], "linked_claim_ids": [ "ring_two_target-c2", "ring_deriv_k2-c2", "grid2d_blackboard_bimodality-c2", "grid2d_rect_bimodality-c2", "grid2d_two_target_double_peak-c2" ], "linked_report_ids": [ "ring_two_target", "ring_deriv_k2", "grid2d_blackboard_bimodality", "grid2d_rect_bimodality", "grid2d_two_target_double_peak" ] }, { "claim_id": "grid2d_bimodality-c3", "report_id": "grid2d_bimodality", "stage": "result", "text_en": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.", "text_cn": "当快通道与延迟绕行通道在同一判据下都具有可观权重时，首达分布出现稳健双峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.", "snippet_cn": "当快通道与延迟绕行通道在同一判据下都具有可观权重时，首达分布出现稳健双峰。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "A （ + ）：FPT ； / \" \" \" \" ；AW exact recursion ，MC （ 、 ）。 B （ + ）： (g x,g y,)=(-0.25,0.40,0.70) ， x ， fast ； （ ）。", "snippet_cn": "候选 A （周期边界 + 全局偏置）：FPT 在统一双峰判据下呈现双峰；快/慢通道由 \"逆漂移短路\" 与 \"顺漂移绕行\" 叠加产生；AW 反演与 exact recursion 在 内一致，MC 叠加吻合（图 、 ）" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "。 B （ g x,g y ）； C door + sticky， wrap-around 。", "snippet_cn": "三类候选的核心参数。候选 B 采用贴边走廊与自动调参（此表给出最终采用的 g x,g y ）；候选 C 采用 door + sticky，并保留 wrap-around 通道。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Result formula context in Grid2D Bimodality Baseline: Bimodality emerges when fast direct routes and delayed wrap-around/detour routes", "snippet_cn": "结论与关键图示" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_bimodality/series/scan_candidate_b_corridor-probability.json", "snippet_en": "scan_candidate_B_corridor [probability]: l -> mass [probability]", "snippet_cn": "scan_candidate_B_corridor [probability]: l -> mass [probability]" } ], "linked_claim_ids": [ "grid2d_two_target_double_peak-c3", "grid2d_rect_bimodality-c3", "ring_lazy_jump-c3", "grid2d_blackboard_bimodality-c5", "grid2d_rect_bimodality-c5" ], "linked_report_ids": [ "grid2d_two_target_double_peak", "grid2d_rect_bimodality", "ring_lazy_jump", "grid2d_blackboard_bimodality" ] }, { "claim_id": "grid2d_bimodality-c4", "report_id": "grid2d_bimodality", "stage": "finding", "text_en": "A unified FPT criterion distinguishes structural bimodality from numerical or visualization artifacts.", "text_cn": "统一 FPT 判据可稳定区分结构性双峰与数值/可视化伪峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "A unified FPT criterion distinguishes structural bimodality from numerical or visualization artifacts.", "snippet_cn": "统一 FPT 判据可稳定区分结构性双峰与数值/可视化伪峰。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "A （ + ）：FPT ； / \" \" \" \" ；AW exact recursion ，MC （ 、 ）。 B （ + ）： (g x,g y,)=(-0.25,0.40,0.70) ， x ， fast ； （ ）。", "snippet_cn": "候选 A （周期边界 + 全局偏置）：FPT 在统一双峰判据下呈现双峰；快/慢通道由 \"逆漂移短路\" 与 \"顺漂移绕行\" 叠加产生；AW 反演与 exact recursion 在 内一致，MC 叠加吻合（图 、 ）" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "： propagator -> propagator -> FPT -> AW 。 。 A ， A v,u = ， 1， p t+1 =A p t 。 P n0 (n,t)= e n^->p A^t e n0, P (z)= t0 A^t z^, P n0 (n,z)= e n^->p (I-zA)e-1 e n0.", "snippet_cn": "本节给出统一推导链条： 无缺陷 propagator -> 有缺陷 propagator -> FPT 生成函数 -> AW 反演。三类候选均可落在同一框架内。 矩阵形式与生成函数 设无吸收系统的转移矩阵为 A，约定 A v,u =，因此列和为 1，且 p t+1 =A p t" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Finding formula context in Grid2D Bimodality Baseline: A unified FPT criterion distinguishes structural bimodality from numerical or", "snippet_cn": "结论与关键图示" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_bimodality/series/scan_candidate_b_corridor-probability.json", "snippet_en": "scan_candidate_B_corridor [probability]: l -> mass [probability]", "snippet_cn": "scan_candidate_B_corridor [probability]: l -> mass [probability]" } ], "linked_claim_ids": [ "ring_two_target-c4", "cross_luca_regime_map-c1", "final_multitimescale_fpt_encounter-c2", "grid2d_blackboard_bimodality-c2", "ring_valley_dst-c2" ], "linked_report_ids": [ "ring_two_target", "cross_luca_regime_map", "final_multitimescale_fpt_encounter", "grid2d_blackboard_bimodality", "ring_valley_dst" ] }, { "claim_id": "grid2d_bimodality-c5", "report_id": "grid2d_bimodality", "stage": "finding", "text_en": "Candidate corridor/bias settings show that delayed channels can be amplified without changing the target definition.", "text_cn": "走廊与偏置参数可在不改目标定义的前提下放大延迟通道贡献。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Candidate corridor/bias settings show that delayed channels can be amplified without changing the target definition.", "snippet_cn": "走廊与偏置参数可在不改目标定义的前提下放大延迟通道贡献。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "A （ + ）：FPT ； / \" \" \" \" ；AW exact recursion ，MC （ 、 ）。 B （ + ）： (g x,g y,)=(-0.25,0.40,0.70) ， x ， fast ； （ ）。", "snippet_cn": "候选 A （周期边界 + 全局偏置）：FPT 在统一双峰判据下呈现双峰；快/慢通道由 \"逆漂移短路\" 与 \"顺漂移绕行\" 叠加产生；AW 反演与 exact recursion 在 内一致，MC 叠加吻合（图 、 ）" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "： 、 ； 、 （ p pass ）、 sticky ； local bias； (g x,g y) 。 ， 、 P(n,t) 、FPT ， ； B valley。", "snippet_cn": "图 给出统一符号语法：红色方块为起点、蓝色菱形为目标；粗实线为反射墙、虚线为门（标注 p pass ）、半透明阴影为 sticky 区；红色小箭头为 local bias；图外大箭头给出全局偏置方向与 (g x,g y) 数值。为便于阅读，将环境示意、 P(n,t) 切片、FPT 多尺度图与通道分解分别独立成图，避免子图拥挤" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_bimodality/manuscript/grid2d_bimodality_cn.tex", "snippet_en": "Finding formula context in Grid2D Bimodality Baseline: Candidate corridor/bias settings show that delayed channels can be amplified", "snippet_cn": "结论与关键图示" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_bimodality/series/scan_candidate_b_corridor-probability.json", "snippet_en": "scan_candidate_B_corridor [probability]: l -> mass [probability]", "snippet_cn": "scan_candidate_B_corridor [probability]: l -> mass [probability]" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c5", "ring_two_target-c4", "grid2d_membrane_near_target-c4", "grid2d_membrane_near_target-c5", "grid2d_rect_bimodality-c4" ], "linked_report_ids": [ "grid2d_rect_bimodality", "ring_two_target", "grid2d_membrane_near_target" ] }, { "claim_id": "grid2d_reflecting_bimodality-c1", "report_id": "grid2d_reflecting_bimodality", "stage": "model", "text_en": "The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).", "text_cn": "模型保持反射二维格点与吸收目标，并引入可控局部结构（绕行、孔道、轨道）。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "The model keeps a reflecting 2D lattice with absorbing target and controlled local transport structures (detours, pores, tracks).", "snippet_cn": "模型保持反射二维格点与吸收目标，并引入可控局部结构（绕行、孔道、轨道）。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "\" y \" ， g y>0 。 ： p stay 。 (1) g x>0 p left p right ， g x>0 （ g x<0 ）", "snippet_cn": "本文采用 \" y 轴向下为正\" 的坐标约定，因此 g y>0 表示向下偏置。本文仅使用 全反射边界：越界概率全部加回 p stay。 由式(1)可见 g x>0 会增大 p left 并减小 p right，因此本文约定 g x>0 表示 向左 全局偏置（ g x<0 表示向右偏置）" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "（first-passage time） PMF f(t)= ，CDF F(t)= (t) ， ， f(t) “ t ” ， f(t) 。 ， F(t) “ ”， S(t) “ ”", "snippet_cn": "定义与直观 设首达时间（first-passage time）为 其 PMF 为 f(t)=，CDF 为 F(t)= (t)，生存概率为 直观上， f(t) 描述“恰好在第 t 步首次抵达目标”的概率，因此图中的双峰就是 f(t) 在早期与晚期各有一个显著局部极大值" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Model formula context in Grid2D Reflecting-Boundary Bimodality: The model keeps a reflecting 2D lattice with absorbing target and", "snippet_cn": "模型定义与坐标约定" } ], "linked_claim_ids": [ "grid2d_two_target_double_peak-c1", "grid2d_bimodality-c1", "grid2d_blackboard_bimodality-c1", "grid2d_rect_bimodality-c1", "ring_valley_dst-c1" ], "linked_report_ids": [ "grid2d_two_target_double_peak", "grid2d_bimodality", "grid2d_blackboard_bimodality", "grid2d_rect_bimodality", "ring_valley_dst" ] }, { "claim_id": "grid2d_reflecting_bimodality-c2", "report_id": "grid2d_reflecting_bimodality", "stage": "method", "text_en": "Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.", "text_cn": "各案例在统一 PMF/hazard 诊断口径下评估，并通过通道分解与时窗比较进行机制判别。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Case families are evaluated under common PMF/hazard diagnostics and compared by channel decomposition and timing windows.", "snippet_cn": "各案例在统一 PMF/hazard 诊断口径下评估，并通过通道分解与时窗比较进行机制判别。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "（first-passage time） PMF f(t)= ，CDF F(t)= (t) ， ， f(t) “ t ” ， f(t) 。 ， F(t) “ ”， S(t) “ ”", "snippet_cn": "定义与直观 设首达时间（first-passage time）为 其 PMF 为 f(t)=，CDF 为 F(t)= (t)，生存概率为 直观上， f(t) 描述“恰好在第 t 步首次抵达目标”的概率，因此图中的双峰就是 f(t) 在早期与晚期各有一个显著局部极大值" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "： propagator -> propagator -> FPT -> AW 。 。 A ， A v,u = ， 1， p t+1 =A p t 。 P n0 (n,t)= e n^->p A^t e n0, P (z)= t0 A^t z^, P n0 (n,z)= e n^->p (I-zA)e-1 e n0.", "snippet_cn": "本节给出统一推导链条： 无缺陷 propagator -> 有缺陷 propagator -> FPT 生成函数 -> AW 反演。本文构造均可落在同一框架内。 矩阵形式与生成函数 设无吸收系统的转移矩阵为 A，约定 A v,u =，因此列和为 1，且 p t+1 =A p t" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Method formula context in Grid2D Reflecting-Boundary Bimodality: Case families are evaluated under common PMF/hazard diagnostics and", "snippet_cn": "模型定义与坐标约定" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json", "snippet_en": "cases_reflecting_summary [probability]: gx -> q [probability]", "snippet_cn": "cases_reflecting_summary [probability]: gx -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c2", "grid2d_bimodality-c2", "grid2d_rect_bimodality-c2", "grid2d_two_target_double_peak-c2", "ring_lazy_flux-c2" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "grid2d_bimodality", "grid2d_rect_bimodality", "grid2d_two_target_double_peak", "ring_lazy_flux" ] }, { "claim_id": "grid2d_reflecting_bimodality-c3", "report_id": "grid2d_reflecting_bimodality", "stage": "result", "text_en": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "text_cn": "部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "snippet_cn": "部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 \" + \" / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， \" \" fast/slow", "snippet_cn": "全反射边界下仍可稳定形成双峰：正文保留八类代表构造 R1（几何绕行）、R7（半透膜孔道 + 绕行）、C3（主动运输轨道）、R6（door 阵列）、NB4（外边界“传送带”模板）、NB5（传送带 + 全局偏置）、S1/S2（无内部 barrier/door 的软通道双轨），均在全反射外边界下产生清晰双峰" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "双峰指标汇总: Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "snippet_cn": "双峰指标汇总: 部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Result formula context in Grid2D Reflecting-Boundary Bimodality: Several reflecting cases preserve clear early/late channel separation", "snippet_cn": "模型定义与坐标约定" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json", "snippet_en": "cases_reflecting_summary [probability]: gx -> q [probability]", "snippet_cn": "cases_reflecting_summary [probability]: gx -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c3", "ring_lazy_flux-c3", "final_multitimescale_fpt_encounter-c3", "grid2d_blackboard_bimodality-c4", "final_multitimescale_fpt_encounter-c2" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "ring_lazy_flux", "final_multitimescale_fpt_encounter" ] }, { "claim_id": "grid2d_reflecting_bimodality-c4", "report_id": "grid2d_reflecting_bimodality", "stage": "finding", "text_en": "Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.", "text_cn": "在竞争路径拓扑仍可区分时，双峰可在全反射边界下持续存在。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.", "snippet_cn": "在竞争路径拓扑仍可区分时，双峰可在全反射边界下持续存在。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 \" + \" / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， \" \" fast/slow", "snippet_cn": "全反射边界下仍可稳定形成双峰：正文保留八类代表构造 R1（几何绕行）、R7（半透膜孔道 + 绕行）、C3（主动运输轨道）、R6（door 阵列）、NB4（外边界“传送带”模板）、NB5（传送带 + 全局偏置）、S1/S2（无内部 barrier/door 的软通道双轨），均在全反射外边界下产生清晰双峰" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "双峰指标汇总: Bimodality can persist under fully reflecting boundaries when competing pathways remain topologically distinct.", "snippet_cn": "双峰指标汇总: 在竞争路径拓扑仍可区分时，双峰可在全反射边界下持续存在。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Finding formula context in Grid2D Reflecting-Boundary Bimodality: Bimodality can persist under fully reflecting boundaries when competing", "snippet_cn": "模型定义与坐标约定" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json", "snippet_en": "cases_reflecting_summary [probability]: gx -> q [probability]", "snippet_cn": "cases_reflecting_summary [probability]: gx -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c4", "ring_lazy_jump-c4", "ring_valley-c4", "grid2d_bimodality-c3", "grid2d_two_target_double_peak-c3" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "ring_lazy_jump", "ring_valley", "grid2d_bimodality", "grid2d_two_target_double_peak" ] }, { "claim_id": "grid2d_reflecting_bimodality-c5", "report_id": "grid2d_reflecting_bimodality", "stage": "finding", "text_en": "Pore/track structures modify delay channels through accessibility, not only through drift magnitude.", "text_cn": "孔道与轨道结构通过可达性重塑延迟通道，而非仅靠漂移强度变化。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Pore/track structures modify delay channels through accessibility, not only through drift magnitude.", "snippet_cn": "孔道与轨道结构通过可达性重塑延迟通道，而非仅靠漂移强度变化。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "： R1（ ）、R7（ + ）、C3（ ）、R6（door ）、NB4（ “ ” ）、NB5（ + ）、S1/S2（ barrier/door ）， 。 vs ：R1/R7/C3 \" + \" / ；R6 door ；NB4/NB5 + ；S1/S2 sticky + ， \" \" fast/slow", "snippet_cn": "全反射边界下仍可稳定形成双峰：正文保留八类代表构造 R1（几何绕行）、R7（半透膜孔道 + 绕行）、C3（主动运输轨道）、R6（door 阵列）、NB4（外边界“传送带”模板）、NB5（传送带 + 全局偏置）、S1/S2（无内部 barrier/door 的软通道双轨），均在全反射外边界下产生清晰双峰" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "： propagator -> propagator -> FPT -> AW 。 。 A ， A v,u = ， 1， p t+1 =A p t 。 P n0 (n,t)= e n^->p A^t e n0, P (z)= t0 A^t z^, P n0 (n,z)= e n^->p (I-zA)e-1 e n0.", "snippet_cn": "本节给出统一推导链条： 无缺陷 propagator -> 有缺陷 propagator -> FPT 生成函数 -> AW 反演。本文构造均可落在同一框架内。 矩阵形式与生成函数 设无吸收系统的转移矩阵为 A，约定 A v,u =，因此列和为 1，且 p t+1 =A p t" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_reflecting_bimodality/manuscript/grid2d_reflecting_bimodality_cn.tex", "snippet_en": "Finding formula context in Grid2D Reflecting-Boundary Bimodality: Pore/track structures modify delay channels through accessibility, not", "snippet_cn": "模型定义与坐标约定" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_reflecting_bimodality/series/cases_reflecting_summary-probability.json", "snippet_en": "cases_reflecting_summary [probability]: gx -> q [probability]", "snippet_cn": "cases_reflecting_summary [probability]: gx -> q [probability]" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c3", "cross_luca_regime_map-c1", "final_multitimescale_fpt_encounter-c2", "grid2d_bimodality-c1", "grid2d_bimodality-c2" ], "linked_report_ids": [ "grid2d_rect_bimodality", "cross_luca_regime_map", "final_multitimescale_fpt_encounter", "grid2d_bimodality" ] }, { "claim_id": "grid2d_blackboard_bimodality-c1", "report_id": "grid2d_blackboard_bimodality", "stage": "model", "text_en": "The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.", "text_cn": "模型保持反射边界格点动力学，重点考察走廊受限明显的墙端点几何。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall geometry where corridor shortcuts are strongly constrained.", "snippet_cn": "模型保持反射边界格点动力学，重点考察走廊受限明显的墙端点几何。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "\" y \" ， g y>0 ； g x>0 。 ： p stay 。", "snippet_cn": "本文采用 \" y 轴向下为正\" 的坐标约定，因此 g y>0 表示向下偏置； g x>0 表示向左偏置。外边界与内部墙均为 全反射：越界或穿墙尝试的概率质量加回 p stay。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "， transient， Q transient ， r ， p t+1 =Qp t, f(t)=r^->p p t 。 exact recursion ，AW MC 。", "snippet_cn": "首达时间为 将目标点视为吸收态，其余为 transient，令 Q 为 transient 子矩阵， r 为一步进入吸收态的流出向量，则 等价递推为 p t+1 =Qp t, f(t)=r^->p p t。本文以 exact recursion 为基准，AW 反演与 MC 作为对照与通道统计。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "Model formula context in Grid2D Blackboard Endpoint Case: The model keeps reflecting-boundary lattice dynamics and evaluates endpoint wall", "snippet_cn": "数值扫描与尝试（基于配置 Z）" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c1", "grid2d_reflecting_bimodality-c1", "grid2d_bimodality-c1", "grid2d_two_target_double_peak-c1", "ring_lazy_jump_ext-c1" ], "linked_report_ids": [ "grid2d_rect_bimodality", "grid2d_reflecting_bimodality", "grid2d_bimodality", "grid2d_two_target_double_peak", "ring_lazy_jump_ext" ] }, { "claim_id": "grid2d_blackboard_bimodality-c2", "report_id": "grid2d_blackboard_bimodality", "stage": "method", "text_en": "The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.", "text_cn": "流程包括黑板案例构建、截图式扫描与通道/路径分解诊断，并沿用统一 FPT 判据。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.", "snippet_cn": "流程包括黑板案例构建、截图式扫描与通道/路径分解诊断，并沿用统一 FPT 判据。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "数据与复现: The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.", "snippet_cn": "数据与复现: 流程包括黑板案例构建、截图式扫描与通道/路径分解诊断，并沿用统一 FPT 判据。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "参数表: The pipeline runs blackboard case builders, screenshot-style scans, and channel/path decomposition diagnostics under the same FPT criteria.", "snippet_cn": "参数表: 流程包括黑板案例构建、截图式扫描与通道/路径分解诊断，并沿用统一 FPT 判据。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "Method formula context in Grid2D Blackboard Endpoint Case: The pipeline runs blackboard case builders, screenshot-style scans, and", "snippet_cn": "数值扫描与尝试（基于配置 Z）" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json", "snippet_en": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]", "snippet_cn": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]" } ], "linked_claim_ids": [ "grid2d_reflecting_bimodality-c2", "ring_two_target-c2", "grid2d_bimodality-c2", "grid2d_two_target_double_peak-c2", "ring_valley_dst-c2" ], "linked_report_ids": [ "grid2d_reflecting_bimodality", "ring_two_target", "grid2d_bimodality", "grid2d_two_target_double_peak", "ring_valley_dst" ] }, { "claim_id": "grid2d_blackboard_bimodality-c3", "report_id": "grid2d_blackboard_bimodality", "stage": "result", "text_en": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.", "text_cn": "在已扫描的端点案例中，主导行为为单峰加长尾；晚时隆起被判定为窗口边缘效应，而非稳健双峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.", "snippet_cn": "在已扫描的端点案例中，主导行为为单峰加长尾；晚时隆起被判定为窗口边缘效应，而非稳健双峰。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "t_p2 late-window （ ）。", "snippet_cn": "t_p2 为 late-window 最大值（非真实第二峰）。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "： S=(30,45) ， T=(30,15) ， \" + \" 。 ： t_p1 102 ，valley ratio =1 ， + ；", "snippet_cn": "起止点位于内墙段上下端点： S=(30,45)， T=(30,15)，其余结构与 \"两墙 + 走廊箭头\" 基线一致。 未出现双峰： t_p1 102，valley ratio =1，仅见单峰 + 长尾；" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "Result formula context in Grid2D Blackboard Endpoint Case: For the scanned endpoint cases, dominant behavior is single-peak plus long tail", "snippet_cn": "数值扫描与尝试（基于配置 Z）" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json", "snippet_en": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]", "snippet_cn": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]" } ], "linked_claim_ids": [ "grid2d_reflecting_bimodality-c3", "final_multitimescale_fpt_encounter-c3", "cross_luca_regime_map-c3", "final_multitimescale_fpt_encounter-c2", "grid2d_rect_bimodality-c3" ], "linked_report_ids": [ "grid2d_reflecting_bimodality", "final_multitimescale_fpt_encounter", "cross_luca_regime_map", "grid2d_rect_bimodality" ] }, { "claim_id": "grid2d_blackboard_bimodality-c4", "report_id": "grid2d_blackboard_bimodality", "stage": "finding", "text_en": "Endpoint wall configurations remain primarily unimodal with a long tail under reflecting constraints.", "text_cn": "墙端点构型在反射约束下总体表现为单峰 + 长尾。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "Endpoint wall configurations remain primarily unimodal with a long tail under reflecting constraints.", "snippet_cn": "墙端点构型在反射约束下总体表现为单峰 + 长尾。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "： S=(30,45) ， T=(30,15) ， \" + \" 。 ： t_p1 102 ，valley ratio =1 ， + ；", "snippet_cn": "起止点位于内墙段上下端点： S=(30,45)， T=(30,15)，其余结构与 \"两墙 + 走廊箭头\" 基线一致。 未出现双峰： t_p1 102，valley ratio =1，仅见单峰 + 长尾；" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "： x=28 to 29 x=31 to 32 ， y=15..45 。 bias： x=29..31 ； y=15..45 core =0.95 ， y=10..14,46..50 open =0.6 。", "snippet_cn": "几何与参数 内墙： x=28 to 29 与 x=31 to 32， y=15..45。 走廊 bias： x=29..31 上向上箭头；核心段 y=15..45 取 core =0.95，开口段 y=10..14,46..50 取 open =0.6。" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json", "snippet_en": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]", "snippet_cn": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]" } ], "linked_claim_ids": [ "grid2d_reflecting_bimodality-c3", "grid2d_reflecting_bimodality-c4", "final_multitimescale_fpt_encounter-c2", "final_multitimescale_fpt_encounter-c3", "grid2d_rect_bimodality-c1" ], "linked_report_ids": [ "grid2d_reflecting_bimodality", "final_multitimescale_fpt_encounter", "grid2d_rect_bimodality" ] }, { "claim_id": "grid2d_blackboard_bimodality-c5", "report_id": "grid2d_blackboard_bimodality", "stage": "finding", "text_en": "The delayed hump in late windows is diagnostic-window edge behavior rather than a stable second structural peak.", "text_cn": "晚时窗口中的隆起主要是窗口边缘伪峰，不构成稳定第二结构峰。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "The delayed hump in late windows is diagnostic-window edge behavior rather than a stable second structural peak.", "snippet_cn": "晚时窗口中的隆起主要是窗口边缘伪峰，不构成稳定第二结构峰。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "t_p2 late-window （ ）。", "snippet_cn": "t_p2 为 late-window 最大值（非真实第二峰）。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "： S=(30,45) ， T=(30,15) ， \" + \" 。 ： t_p1 102 ，valley ratio =1 ， + ；", "snippet_cn": "起止点位于内墙段上下端点： S=(30,45)， T=(30,15)，其余结构与 \"两墙 + 走廊箭头\" 基线一致。 未出现双峰： t_p1 102，valley ratio =1，仅见单峰 + 长尾；" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_blackboard_bimodality/manuscript/grid2d_blackboard_bimodality_cn.tex", "snippet_en": "Finding formula context in Grid2D Blackboard Endpoint Case: The delayed hump in late windows is diagnostic-window edge behavior rather", "snippet_cn": "数值扫描与尝试（基于配置 Z）" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_blackboard_bimodality/series/z_scan-probability.json", "snippet_en": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]", "snippet_cn": "Z_scan [probability]: absorption -> delta_core, delta_open [probability]" } ], "linked_claim_ids": [ "ring_two_target-c4", "grid2d_two_target_double_peak-c4", "ring_deriv_k2-c5", "ring_lazy_jump-c3", "ring_lazy_jump-c4" ], "linked_report_ids": [ "ring_two_target", "grid2d_two_target_double_peak", "ring_deriv_k2", "ring_lazy_jump" ] }, { "claim_id": "grid2d_two_target_double_peak-c1", "report_id": "grid2d_two_target_double_peak", "stage": "model", "text_en": "The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.", "text_cn": "模型在反射格点中设置固定起点与两个吸收目标，并可调节目标通道耦合强度。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex", "snippet_en": "The model places two absorbing targets in a reflecting lattice with fixed start point and tunable target-channel coupling.", "snippet_cn": "模型在反射格点中设置固定起点与两个吸收目标，并可调节目标通道耦合强度。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex", "snippet_en": "Research report grid2d_two_target_double_peak.", "snippet_cn": "Grid2D 体系：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_two_target_double_peak/artifacts/tables/case_mechanism.tex", "snippet_en": "Model formula context in Grid2D Two-Target Double-Peak: The model places two absorbing targets in a reflecting lattice with fixed start", "snippet_cn": "Fallback" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c1", "grid2d_bimodality-c1", "grid2d_reflecting_bimodality-c1", "ring_two_target-c1", "grid2d_blackboard_bimodality-c1" ], "linked_report_ids": [ "grid2d_rect_bimodality", "grid2d_bimodality", "grid2d_reflecting_bimodality", 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metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.", "text_cn": "速度比 R=t_sparse/t_luca 在固定 full-FPT 公平口径下计算，MFPT 线性系统仅作对照参考。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex", "snippet_en": "The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with MFPT linear systems separated as reference only.", "snippet_cn": "速度比 R=t_sparse/t_luca 在固定 full-FPT 公平口径下计算，MFPT 线性系统仅作对照参考。" }, { "evidence_type": "section_summary", "path": "research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex", "snippet_en": "Research report cross_luca_regime_map.", "snippet_cn": "跨模型综合：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/cross_luca_regime_map/artifacts/tables/regime_anchor_baselines.tex", "snippet_en": "Finding formula context in Cross-Model Luca Regime Map: The ratio metric R=t_sparse/t_luca is computed under fixed full-FPT fairness, with", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/cross_luca_regime_map/series/runtime_raw.json", "snippet_en": "runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds", "snippet_cn": "runtime_raw: t_max -> defect_pairs, local_bias_sites, sparse_seconds" } ], "linked_claim_ids": [ "grid2d_one_two_target_gating-c4", "ring_valley_dst-c5", "grid2d_rect_bimodality-c3", "final_multitimescale_fpt_encounter-c2", "grid2d_blackboard_bimodality-c2" ], "linked_report_ids": [ "grid2d_one_two_target_gating", "ring_valley_dst", "grid2d_rect_bimodality", "final_multitimescale_fpt_encounter", "grid2d_blackboard_bimodality" ] }, { "claim_id": "final_multitimescale_fpt_encounter-c1", "report_id": "final_multitimescale_fpt_encounter", "stage": "model", "text_en": "Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).", "text_cn": "该研究报告：本节说明状态空间、参数约束与边界条件设定。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).", "snippet_cn": "该研究报告：本节说明状态空间、参数约束与边界条件设定。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "Walker (i) has mobility (q i (0,1]). At an interior site, the walker moves left/right with probabilities and stays with probability (1-q i).", "snippet_cn": "该研究报告：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "shows the actual spatial configuration for each selected case. The left panel in each row is the physical reflecting one-dimensional lattice: the two walker starts are marked", "snippet_cn": "该研究报告：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "math_block", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "Model formula context in Final Multitimescale FPT and Encounter Report: Walker (i) has mobility (q i (0,1]).", "snippet_cn": "Conclusion" } ], "linked_claim_ids": [ "ring_lazy_flux-c1", "grid2d_bimodality-c1", "ring_lazy_jump_ext-c1", "grid2d_two_target_bias_radius-c4", "ring_lazy_jump-c5" ], "linked_report_ids": [ "ring_lazy_flux", "grid2d_bimodality", "ring_lazy_jump_ext", "grid2d_two_target_bias_radius", "ring_lazy_jump" ] }, { "claim_id": "final_multitimescale_fpt_encounter-c2", "report_id": "final_multitimescale_fpt_encounter", "stage": "method", "text_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "text_cn": "该研究报告：本节给出解析/数值流程、反演方法与验证协议。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.", "snippet_cn": "该研究报告：本节给出解析/数值流程、反演方法与验证协议。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.", "snippet_cn": "该研究报告：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "Stage 2 introduced smoothing persistence, parity-pair filtering, and negative controls. 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The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "text_cn": "[misc] 结果陈述：该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "[misc] Result statement: For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was", "snippet_cn": "[misc] 结果陈述：该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found.", "snippet_cn": "该研究报告：本节整理关键证据、图表说明与复现路径。" }, { "evidence_type": "section_summary", "path": "research/reports/final_multitimescale_fpt_encounter/artifacts/encounter_search/results/final_report.tex", "snippet_en": "Stage 2 introduced smoothing persistence, parity-pair filtering, and negative controls. 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"research/reports/grid2d_two_target_bias_radius", "snippet_en": "Boundary rule: reflecting attempted-outside-stays.", "snippet_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_two_target_bias_radius", "snippet_en": "Overview: Boundary rule: reflecting attempted-outside-stays.", "snippet_cn": "Overview: Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_two_target_bias_radius", "snippet_en": "Finding formula context in Grid2D Two-Target Bias-Radius Scaffold: Boundary rule: reflecting attempted-outside-stays.", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_two_target_bias_radius/series/grid2d_bias_radius_scan_config-probability.json", "snippet_en": "grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]", "snippet_cn": "grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]" } ], "linked_claim_ids": [ "grid2d_blackboard_bimodality-c1", "final_multitimescale_fpt_encounter-c1", "final_multitimescale_fpt_encounter-c2", "grid2d_bimodality-c1", "grid2d_rect_bimodality-c1" ], "linked_report_ids": [ "grid2d_blackboard_bimodality", "final_multitimescale_fpt_encounter", "grid2d_bimodality", "grid2d_rect_bimodality" ] }, { "claim_id": "grid2d_two_target_bias_radius-c5", "report_id": "grid2d_two_target_bias_radius", "stage": "finding", "text_en": "State space: rectangular grid {0, Lx-1} x {0, Ly-1}.", "text_cn": "Grid2D 体系（grid2d_two_target_bias_radius）：关键结论见本页公式链、交互图与原始资产。", "evidence": [ { "evidence_type": "source_document", "path": "research/reports/grid2d_two_target_bias_radius", "snippet_en": "State space: rectangular grid {0, Lx-1} x {0, Ly-1}.", "snippet_cn": "Grid2D 体系（grid2d_two_target_bias_radius）：关键结论见本页公式链、交互图与原始资产。" }, { "evidence_type": "section_summary", "path": "research/reports/grid2d_two_target_bias_radius", "snippet_en": "Overview: State space: rectangular grid {0, Lx-1} x {0, Ly-1}.", "snippet_cn": "Overview: Grid2D 体系（grid2d_two_target_bias_radius）：关键结论见本页公式链、交互图与原始资产。" }, { "evidence_type": "math_block", "path": "research/reports/grid2d_two_target_bias_radius", "snippet_en": "Finding formula context in Grid2D Two-Target Bias-Radius Scaffold: State space: rectangular grid {0, Lx-1} x {0, Ly-1}.", "snippet_cn": "Fallback" }, { "evidence_type": "dataset", "path": "/data/v1/reports/grid2d_two_target_bias_radius/series/grid2d_bias_radius_scan_config-probability.json", "snippet_en": "grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]", "snippet_cn": "grid2d_bias_radius_scan_config [probability]: near_x -> q [probability]" } ], "linked_claim_ids": [ "grid2d_rect_bimodality-c1", "grid2d_rect_bimodality-c2" ], "linked_report_ids": [ "grid2d_rect_bimodality" ] } ], "report_guides": [ { "report_id": "ring_lazy_jump", "objective_en": "Bimodality appears only in selected shortcut-strength intervals; K=4 generally maintains stronger second-peak persistence than K=2 when geometry and waiting rules are aligned.", "objective_cn": "双峰只在部分 shortcut 强度区间稳定出现；在几何与停留规则对齐时，K=4 往往比 K=2 保持更强的第二峰。", "upstream_report_ids": [], "downstream_report_ids": [ "ring_lazy_jump_ext" ], "related_report_ids": [ "ring_lazy_jump_ext", "ring_two_target", "ring_valley", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "grid2d_two_walker_encounter_shortcut" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_lazy_jump_ext", "objective_en": "Increasing beta advances both peaks and steepens tail decay; under the same beta, K=4 remains more robustly bimodal, and exact-versus-MC diagnostics agree on phase-level trends.", "objective_cn": "随着 beta 提升，两个峰位整体提前且尾部衰减更快；在同等 beta 下，K=4 的双峰保持性优于 K=2，且精确解与 MC 在相区趋势上吻合。", "upstream_report_ids": [ "ring_lazy_jump" ], "downstream_report_ids": [ "ring_lazy_jump_ext_rev2" ], "related_report_ids": [ "ring_lazy_jump", "ring_two_target", "ring_valley", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "grid2d_two_walker_encounter_shortcut" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_lazy_jump_ext_rev2", "objective_en": "Across the three sensitivity tracks, the qualitative mechanism interpretation remains stable, and uncertainty bars do not contradict the phase-level conclusions used in the main narrative.", "objective_cn": "三类敏感性分析下，机制解释在定性层面保持稳定，且不确定性区间未推翻主线中的相区判断。", "upstream_report_ids": [ "ring_lazy_jump_ext" ], "downstream_report_ids": [ "ring_lazy_flux" ], "related_report_ids": [ "ring_lazy_flux", "ring_lazy_jump_ext", "ring_lazy_jump", "grid2d_blackboard_bimodality", "grid2d_one_two_target_gating", "grid2d_reflecting_bimodality" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_lazy_flux", "objective_en": "A small-p selfloop regime yields clear two-peak structure, whereas equal4 and stronger shortcut injection collapse the distribution toward unimodality.", "objective_cn": "自环分流且小 p 条件下可形成稳定双峰（最小 N=10 例）；equal4 或 shortcut 强度增大时，分布向单峰收敛。", "upstream_report_ids": [ "ring_lazy_jump_ext_rev2" ], "downstream_report_ids": [ "ring_valley" ], "related_report_ids": [ "ring_lazy_jump", "ring_lazy_jump_ext", "ring_valley", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "final_multitimescale_fpt_encounter" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_valley", "objective_en": "Across scanned even N, K=2 remains unimodal under the study rule, while K=4/6/8 exhibit structured bimodality bands that are reproducible in both exact and MC diagnostics.", "objective_cn": "在扫描的偶数 N 范围内，K=2 维持单峰；K=4/6/8 则出现可复现的双峰区间，并在精确解与 MC 诊断中一致。", "upstream_report_ids": [ "ring_lazy_flux" ], "downstream_report_ids": [ "ring_valley_dst" ], "related_report_ids": [ "ring_valley_dst", "ring_lazy_jump", "ring_lazy_jump_ext", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "grid2d_two_walker_encounter_shortcut" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_valley_dst", "objective_en": "Destination scanning reveals structured dst windows where the second peak is amplified and trajectory-class usage shifts, with deterministic and Monte Carlo diagnostics remaining consistent.", "objective_cn": "dst 扫描给出了结构化的高对比区间：第二峰可被显著放大，且轨迹类别占比同步变化，确定性与 MC 结论一致。", "upstream_report_ids": [ "ring_valley" ], "downstream_report_ids": [ "ring_deriv_k2" ], "related_report_ids": [ "ring_valley", "ring_lazy_jump", "ring_lazy_jump_ext", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "grid2d_two_walker_encounter_shortcut" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_deriv_k2", "objective_en": "The closed-form derivation clarifies which terms govern shortcut-induced asymmetry and provides reusable formula blocks for downstream ring reports.", "objective_cn": "闭式推导明确了 shortcut 不对称性的主导项，并为后续 ring 报告提供可复用公式模块。", "upstream_report_ids": [ "ring_valley_dst" ], "downstream_report_ids": [ "ring_two_target" ], "related_report_ids": [ "ring_two_target", "ring_valley_dst", "ring_lazy_flux", "grid2d_reflecting_bimodality", "grid2d_bimodality", "cross_luca_regime_map" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_two_target", "objective_en": "No-shortcut drift can already produce strong bimodality, while shortcut activation redistributes pathway mass and can introduce trimodal behavior in selected geometry and parameter bands.", "objective_cn": "无 shortcut 漂移本身即可产生稳定双峰；加入 shortcut 后路径质量重新分配，在特定几何与参数带可出现三峰行为。", "upstream_report_ids": [ "ring_deriv_k2" ], "downstream_report_ids": [ "ring_two_walker_encounter_shortcut" ], "related_report_ids": [ "ring_lazy_jump", "ring_lazy_jump_ext", "ring_valley", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "grid2d_two_walker_encounter_shortcut" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "ring_two_walker_encounter_shortcut", "objective_en": "Research report ring_two_walker_encounter_shortcut.", "objective_cn": "Ring 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "ring_two_target" ], "downstream_report_ids": [], "related_report_ids": [ "ring_two_target", "ring_lazy_jump", "ring_lazy_jump_ext", "grid2d_two_walker_encounter_shortcut", "grid2d_reflecting_bimodality", "cross_luca_regime_map" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_bimodality", "objective_en": "Bimodality emerges when fast direct routes and delayed wrap-around/detour routes coexist at measurable weights under the same diagnostic criterion.", "objective_cn": "当快通道与延迟绕行通道在同一判据下都具有可观权重时，首达分布出现稳健双峰。", "upstream_report_ids": [], "downstream_report_ids": [ "grid2d_reflecting_bimodality" ], "related_report_ids": [ "grid2d_reflecting_bimodality", "grid2d_blackboard_bimodality", "grid2d_rect_bimodality", "ring_deriv_k2", "cross_luca_regime_map", "final_multitimescale_fpt_encounter" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_reflecting_bimodality", "objective_en": "Several reflecting cases preserve clear early/late channel separation, while others collapse toward long-tail unimodality depending on geometric bottlenecks.", "objective_cn": "部分反射案例保留了清晰的早/晚通道分离；另一些在几何瓶颈下退化为长尾单峰。", "upstream_report_ids": [ "grid2d_bimodality" ], "downstream_report_ids": [ "grid2d_blackboard_bimodality" ], "related_report_ids": [ "grid2d_two_walker_encounter_shortcut", "grid2d_bimodality", "grid2d_rect_bimodality", "ring_lazy_jump", "ring_lazy_jump_ext", "ring_two_target" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_blackboard_bimodality", "objective_en": "For the scanned endpoint cases, dominant behavior is single-peak plus long tail; the late-window hump is identified as an edge artifact instead of a robust bimodal signature.", "objective_cn": "在已扫描的端点案例中，主导行为为单峰加长尾；晚时隆起被判定为窗口边缘效应，而非稳健双峰。", "upstream_report_ids": [ "grid2d_reflecting_bimodality" ], "downstream_report_ids": [ "grid2d_two_target_double_peak" ], "related_report_ids": [ "grid2d_reflecting_bimodality", "grid2d_one_two_target_gating", "grid2d_two_walker_encounter_shortcut", "ring_deriv_k2", "ring_lazy_flux", "ring_lazy_jump" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_two_target_double_peak", "objective_en": "Double-peak regions appear when direct-to-near-target and delayed-to-far-target channels both carry substantial mass; phase boundaries shift predictably with coupling strength.", "objective_cn": "当近目标直达通道与远目标延迟通道都占有显著质量时，会出现稳定双峰；其相界随耦合强度呈可预测移动。", "upstream_report_ids": [ "grid2d_blackboard_bimodality" ], "downstream_report_ids": [ "grid2d_two_walker_encounter_shortcut" ], "related_report_ids": [ "grid2d_two_walker_encounter_shortcut", "grid2d_membrane_near_target", "grid2d_rect_bimodality", "ring_two_target", "cross_luca_regime_map", "ring_lazy_jump" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_two_walker_encounter_shortcut", "objective_en": "Research report grid2d_two_walker_encounter_shortcut.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_two_target_double_peak" ], "downstream_report_ids": [ "grid2d_rect_bimodality" ], "related_report_ids": [ "grid2d_reflecting_bimodality", "grid2d_two_target_double_peak", "grid2d_rect_bimodality", "ring_two_walker_encounter_shortcut", "ring_lazy_jump", "ring_lazy_jump_ext" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_rect_bimodality", "objective_en": "Aspect ratio and endpoint arrangement shift the balance between direct and detour channels; robust double peaks persist only in specific rectangular geometry bands.", "objective_cn": "长宽比与端点排布会改变直达/绕行通道权重平衡；稳健双峰只在特定矩形几何带中保持。", "upstream_report_ids": [ "grid2d_two_walker_encounter_shortcut" ], "downstream_report_ids": [ "grid2d_membrane_near_target" ], "related_report_ids": [ "grid2d_membrane_near_target", "grid2d_reflecting_bimodality", "grid2d_two_walker_encounter_shortcut", "cross_luca_regime_map", "ring_lazy_jump", "ring_lazy_jump_ext" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_membrane_near_target", "objective_en": "Research report grid2d_membrane_near_target.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_rect_bimodality" ], "downstream_report_ids": [ "grid2d_one_target_base" ], "related_report_ids": [ "grid2d_rect_bimodality", "grid2d_two_target_double_peak", "grid2d_reflecting_bimodality", "cross_luca_regime_map", "ring_lazy_jump", "ring_lazy_jump_ext" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "cross_luca_regime_map", "objective_en": "Across the scanned workload, sparse exact remains the dominant full-FPT baseline; Luca-mode speedups appear only in limited defect-regime subsets and are near-neutral in the aggregate ratio metric.", "objective_cn": "在扫描工作负载上，sparse 精确解仍是 full-FPT 主基线；Luca 加速仅在部分缺陷区间显著，整体速度比接近中性。", "upstream_report_ids": [], "downstream_report_ids": [], "related_report_ids": [ "grid2d_reflecting_bimodality", "ring_lazy_jump", "ring_lazy_jump_ext" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "final_multitimescale_fpt_encounter", "objective_en": "For the current finite one-dimensional reflecting lattice with synchronous lazy updates and co-location-only absorption, no robust (F 2) double peak was found. The unusual curves seen across Stage 1, Stage 2, and Stage 2b are best explained as parity artifacts, same-target long tails, or weak target-shift shoulders.", "objective_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [], "downstream_report_ids": [ "exact_recursion_method_guide" ], "related_report_ids": [ "exact_recursion_method_guide", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation", "cross_luca_regime_map", "grid2d_bimodality", "grid2d_reflecting_bimodality" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_one_target_base", "objective_en": "Result summary placeholder for grid2d_one_target_base.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_membrane_near_target" ], "downstream_report_ids": [ "grid2d_one_target_exit_timing" ], "related_report_ids": [ "grid2d_membrane_near_target", "grid2d_one_target_exit_timing", "grid2d_one_target_valley_peak_budget", "cross_luca_regime_map", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_one_target_exit_timing", "objective_en": "Result summary placeholder for grid2d_one_target_exit_timing.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_one_target_base" ], "downstream_report_ids": [ "grid2d_one_target_valley_peak_budget" ], "related_report_ids": [ "grid2d_one_target_base", "grid2d_one_target_valley_peak_budget", "grid2d_membrane_near_target", "cross_luca_regime_map", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_one_target_valley_peak_budget", "objective_en": "Research report grid2d_one_target_valley_peak_budget.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_one_target_exit_timing" ], "downstream_report_ids": [ "grid2d_one_target_window_measures" ], "related_report_ids": [ "grid2d_one_target_exit_timing", "grid2d_one_target_window_measures", "grid2d_membrane_near_target", "cross_luca_regime_map", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_one_target_window_measures", "objective_en": "Result summary placeholder for grid2d_one_target_window_measures.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_one_target_valley_peak_budget" ], "downstream_report_ids": [ "grid2d_one_two_target_gating" ], "related_report_ids": [ "grid2d_one_target_valley_peak_budget", "grid2d_one_two_target_gating", "grid2d_membrane_near_target", "cross_luca_regime_map", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_one_two_target_gating", "objective_en": "Result summary placeholder for grid2d_one_two_target_gating.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_one_target_window_measures" ], "downstream_report_ids": [ "grid2d_two_target_bias_radius" ], "related_report_ids": [ "grid2d_two_walker_encounter_shortcut", "grid2d_blackboard_bimodality", "grid2d_reflecting_bimodality", "ring_two_target", "ring_deriv_k2", "ring_lazy_flux" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "exact_recursion_method_guide", "objective_en": "Result summary placeholder for exact_recursion_method_guide.", "objective_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "final_multitimescale_fpt_encounter" ], "downstream_report_ids": [ "encounter_reflecting_diagonal_decomp" ], "related_report_ids": [ "encounter_reflecting_diagonal_decomp", "final_multitimescale_fpt_encounter", "encounter_reflecting_mean_validation", "cross_luca_regime_map", "grid2d_bimodality", "grid2d_blackboard_bimodality" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "encounter_reflecting_diagonal_decomp", "objective_en": "Result summary placeholder for encounter_reflecting_diagonal_decomp.", "objective_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "exact_recursion_method_guide" ], "downstream_report_ids": [ "encounter_reflecting_mean_validation" ], "related_report_ids": [ "encounter_reflecting_mean_validation", "exact_recursion_method_guide", "final_multitimescale_fpt_encounter", "cross_luca_regime_map", "grid2d_membrane_near_target", "grid2d_rect_bimodality" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "encounter_reflecting_mean_validation", "objective_en": "Result summary placeholder for encounter_reflecting_mean_validation.", "objective_cn": "该研究报告：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "encounter_reflecting_diagonal_decomp" ], "downstream_report_ids": [], "related_report_ids": [ "encounter_reflecting_diagonal_decomp", "exact_recursion_method_guide", "final_multitimescale_fpt_encounter", "cross_luca_regime_map", "grid2d_membrane_near_target", "grid2d_rect_bimodality" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] }, { "report_id": "grid2d_two_target_bias_radius", "objective_en": "Result summary placeholder for grid2d_two_target_bias_radius.", "objective_cn": "Grid2D 体系：本节总结峰谷结构、通道机制与跨报告对比结论。", "upstream_report_ids": [ "grid2d_one_two_target_gating" ], "downstream_report_ids": [], "related_report_ids": [ "grid2d_one_two_target_gating", "grid2d_membrane_near_target", "grid2d_one_target_base", "ring_two_target", "cross_luca_regime_map", "encounter_reflecting_diagonal_decomp" ], "verification_steps_en": [ "Read the key claims and their evidence references first.", "Verify at least one equation card and one dataset panel against source paths.", "Cross-check this report with upstream/downstream linked reports." ], "verification_steps_cn": [ "先读关键 claim 及其证据引用。", "至少核对一条公式卡与一个数据面板的来源路径。", "再与上游/下游关联报告做交叉核对。" ] } ], "consistency_checks": [ { "check": "all_reports_have_claims", "pass": true, "details": { "missing_report_ids": [], "claim_count": 130 } }, { "check": "all_claims_have_evidence", "pass": true, "details": { "claims_without_evidence": [] } }, { "check": "all_reports_have_guides", "pass": true, "details": { "missing_report_ids": [] } }, { "check": "all_reports_have_core_stage_chain", "pass": true, "details": { "missing_stage_chain": [] } }, { "check": "cross_report_claim_links", "pass": true, "details": { "linked_claim_count": 130, "claim_count": 130 } }, { "check": "duplicate_claim_signatures", "pass": true, "details": [ { "signature": "landcodefigures", "count": 2 } ] } ] }