{ "version": "v1", "generated_at": "2026-06-09T21:58:12.574332Z", "notion_labels": { "fpt-pmf": { "label_en": "First-passage distribution", "label_cn": "首达时间分布" }, "survival": { "label_en": "Survival and hazard", "label_cn": "生存函数与风险率" }, "hazard": { "label_en": "Hazard interpretation", "label_cn": "风险率解释" }, "beta-scan": { "label_en": "Beta / shortcut scan", "label_cn": "beta/shortcut 扫描" }, "spectral": { "label_en": "Spectral decomposition", "label_cn": "谱分解" }, "aw-inversion": { "label_en": "AW inversion", "label_cn": "AW 反演" } }, "group_paths": [ { "group": "ring", "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" ], "step_count": 9 }, { "group": "grid2d", "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" ], "step_count": 13 }, { "group": "cross", "report_ids": [ "cross_luca_regime_map" ], "step_count": 1 }, { "group": "misc", "report_ids": [ "final_multitimescale_fpt_encounter", "exact_recursion_method_guide", "encounter_reflecting_diagonal_decomp", "encounter_reflecting_mean_validation" ], "step_count": 4 } ], "global_storyline": { "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。", "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" ] }, "reports": [ { "report_id": "ring_lazy_jump", "group": "ring", "title_en": "Lazy Ring Jump-Over Mechanism (K2 vs K4)", "title_cn": "Lazy Ring Jump-Over 机制（K2 对 K4）", "summary_en": "This report establishes the baseline jump-over mechanism for lazy rings with one directed shortcut, contrasting K=2 and K=4 under exact first-passage diagnostics. It explains why double-peak behavior is selective in shortcut strength and how pathway decomposition links peak structure to fast and delayed transport channels.", "summary_cn": "该报告建立了 lazy ring + 单向 shortcut 的 jump-over 基线机制，在精确首达诊断下比较 K=2 与 K=4。报告解释了双峰为何只在特定 shortcut 强度区间出现，并通过路径分解把峰结构与快/慢通道对应起来。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "", "next_in_group": "ring_lazy_jump_ext", "same_group_links": [ { "report_id": "ring_lazy_jump_ext", "score": 19.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 3 }, { "report_id": "ring_two_target", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_flux", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_target_double_peak", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "ring_lazy_jump_ext", "group": "ring", "title_en": "Lazy Ring Shortcut Beta Scan", "title_cn": "Lazy Ring Shortcut beta 扫描", "summary_en": "This extension quantifies how shortcut strength beta reshapes first-passage bimodality on lazy rings at fixed N=100, then checks transfer by N sweeps and Monte Carlo class decomposition. Across beta in [0,0.2], both peaks move earlier and tail decay accelerates, while K=4 preserves a wider and deeper bimodal window than K=2.", "summary_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。 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。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "ring_lazy_jump", "next_in_group": "ring_lazy_jump_ext_rev2", "same_group_links": [ { "report_id": "ring_lazy_jump", "score": 19.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 3 }, { "report_id": "ring_two_target", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_flux", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_target_double_peak", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "ring_lazy_jump_ext_rev2", "group": "ring", "title_en": "Lazy Ring Shortcut Figure-1 Revision", "title_cn": "Lazy Ring Shortcut 图1修订", "summary_en": "This revision reorganizes the lazy-shortcut extension into a publication-ready evidence flow: co-located Fig.1 overlays f(t) with window-level class bars, while threshold, window-shift/width, and MC-uncertainty analyses test robustness. The update strengthens readability and reproducibility without changing the core mechanism claims.", "summary_cn": "该修订版把 lazy-shortcut 扩展重构为出版物级证据链：图1将 f(t) 与窗口分类柱状同轴展示，并加入阈值扫描、窗口位移/宽度扰动、MC 不确定性三类敏感性分析。在不改变核心机制结论的前提下，显著提升了可读性与可复现性。", "notion_ids": [ "beta-scan" ], "previous_in_group": "ring_lazy_jump_ext", "next_in_group": "ring_lazy_flux", "same_group_links": [ { "report_id": "ring_lazy_flux", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 2 }, { "report_id": "ring_lazy_jump_ext", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 4 }, { "report_id": "ring_lazy_jump", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 3 }, { "report_id": "ring_deriv_k2", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 1 }, { "report_id": "ring_two_target", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_blackboard_bimodality", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 0 }, { "report_id": "grid2d_one_two_target_gating", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 0 }, { "report_id": "grid2d_reflecting_bimodality", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan" ], "shared_token_count": 0 } ] }, { "report_id": "ring_lazy_flux", "group": "ring", "title_en": "Lazy Ring Flux Baseline", "title_cn": "Lazy Ring Flux 基线机制", "summary_en": "This report studies first-passage distributions on a lazy ring with one directed shortcut drawn from self-loop probability. It combines Chebyshev generating-function derivation, AW/FFT inversion, and flux-recursion cross-checks to show a reproducible small-p bimodal regime, while equal4 and large shortcut strength suppress the second peak.", "summary_cn": "该报告研究 lazy ring（k=1, K=2）在单向 shortcut 且从自环分配概率时的首达分布。通过 Chebyshev 生成函数推导、AW/FFT 反演与 flux 递推交叉校验，报告给出可复现的小 p 双峰区间，并说明 equal4 与较大 shortcut 强度会抑制第二峰。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "previous_in_group": "ring_lazy_jump_ext_rev2", "next_in_group": "ring_valley", "same_group_links": [ { "report_id": "ring_lazy_jump", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "ring_lazy_jump_ext", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "ring_valley", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_deriv_k2", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_two_target", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "final_multitimescale_fpt_encounter", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_bimodality", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_blackboard_bimodality", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_one_two_target_gating", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 } ] }, { "report_id": "ring_valley", "group": "ring", "title_en": "Ring Valley Regime Map", "title_cn": "Ring Valley 相区研究", "summary_en": "Under Fig.3 criteria with parity-aware treatment, K=2 shows no robust two-peak regime in the scanned range. Bimodality windows emerge for K=4, K=6, and K=8 with distinct N bands.", "summary_cn": "该 valley 研究在非 lazy 的 K 邻接环上考察单向 shortcut（6->N/2+1）并采用 Fig.3 峰谷判据。通过 AW 精确反演与 MC 校验，报告给出偶数 N 与 K 的双峰区间图：K=2 在奇偶振荡修正后仍不出现稳定双峰，而 K=4、6、8 出现清晰双峰窗口。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "ring_lazy_flux", "next_in_group": "ring_valley_dst", "same_group_links": [ { "report_id": "ring_valley_dst", "score": 19.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "ring_lazy_jump", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump_ext", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_target", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_flux", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_target_double_peak", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "ring_valley_dst", "group": "ring", "title_en": "Destination-Scan Valley Control", "title_cn": "目的地扫描与谷值调控（N=100, K=6）", "summary_en": "This report fixes N=100 and K=6, then scans shortcut destination dst to control the second-peak structure of first-passage distributions. Deterministic flux/master-equation scans and Monte Carlo trajectory classes jointly identify where valley depth and second-peak prominence are maximized.", "summary_cn": "该报告固定 N=100、K=6，仅扫描 shortcut 终点 dst 来调控首达分布第二峰。通过确定性 flux/master 方程扫描与 Monte Carlo 轨迹分类联合诊断，报告识别出第二峰增强与谷值加深最显著的 dst 区间。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "ring_valley", "next_in_group": "ring_deriv_k2", "same_group_links": [ { "report_id": "ring_valley", "score": 19.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "ring_lazy_jump", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump_ext", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_target", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_deriv_k2", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_target_double_peak", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "ring_deriv_k2", "group": "ring", "title_en": "Ring Derivation Backbone", "title_cn": "Ring 推导主干（K=2）", "summary_en": "This derivation report provides the analytical backbone for ring models with one directed long-range link, including defect-free propagators, defect corrections, and first-passage generating functions. It serves as the shared mathematical base used by later shortcut and valley studies.", "summary_cn": "该推导报告给出了“环 + 单向长程连边”模型的解析主干：包括无缺陷传播子、缺陷修正与首达生成函数。它是后续 shortcut 与 valley 报告共用的数学基座。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "spectral" ], "previous_in_group": "ring_valley_dst", "next_in_group": "ring_two_target", "same_group_links": [ { "report_id": "ring_two_target", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_flux", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump_ext", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "spectral" ], "shared_token_count": 0 }, { "report_id": "grid2d_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "spectral" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "final_multitimescale_fpt_encounter", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_blackboard_bimodality", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_one_two_target_gating", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 } ] }, { "report_id": "ring_two_target", "group": "ring", "title_en": "Two-Target Lazy Ring Mechanics", "title_cn": "双目标 Lazy Ring 机制", "summary_en": "This report builds an exact two-target lazy-ring framework and compares no-shortcut versus selfloop-shortcut regimes. By scanning N, K, and beta with peak/valley diagnostics, it shows how target geometry and shortcut routing jointly control the transition among unimodal, bimodal, and trimodal first-passage behavior.", "summary_cn": "该报告建立了双目标 lazy ring 的精确分析框架，并比较无 shortcut 与 selfloop-shortcut 两类机制。通过对 N、K、beta 的系统扫描与峰谷判据，报告说明了目标几何与 shortcut 路由如何共同决定首达分布在单峰、双峰、三峰之间的转变。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "ring_deriv_k2", "next_in_group": "ring_two_walker_encounter_shortcut", "same_group_links": [ { "report_id": "ring_lazy_jump", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump_ext", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 17.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 16.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "ring_deriv_k2", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_target_double_peak", "score": 10.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "ring_two_walker_encounter_shortcut", "group": "ring", "title_en": "1D Ring Two-Walker Encounter With Shortcut", "title_cn": "Ring 体系：ring_two_walker_encounter_shortcut 研究报告", "summary_en": "Robust numerical verification of appendix Eq. Exact first-encounter computation on 1D ring with directed shortcut.", "summary_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。 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。", "notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "ring_two_target", "next_in_group": "", "same_group_links": [ { "report_id": "ring_two_target", "score": 16.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "ring_lazy_jump", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_lazy_jump_ext", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley_dst", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_deriv_k2", "score": 8.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 13.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 4 }, { "report_id": "grid2d_reflecting_bimodality", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_membrane_near_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_rect_bimodality", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "grid2d_two_target_double_peak", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 } ] }, { "report_id": "grid2d_bimodality", "group": "grid2d", "title_en": "Grid2D Bimodality Baseline", "title_cn": "Grid2D 双峰基线", "summary_en": "This foundational Grid2D report establishes how biased/lazy random walks generate first-passage bimodality on a square lattice. It unifies model constraints, defect-aware propagators, AW inversion, and candidate-case scans into one auditable chain that separates genuine two-channel mechanisms from plotting artifacts.", "summary_cn": "该基础 Grid2D 报告建立了偏置/惰性随机游走下首达双峰形成的统一框架。报告把模型约束、缺陷修正传播子、AW 反演与参数扫描串成可审计链路，用于区分真实双通道机制与绘图伪峰。", "notion_ids": [ "aw-inversion", "fpt-pmf", "spectral" ], "previous_in_group": "", "next_in_group": "grid2d_reflecting_bimodality", "same_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 13.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "spectral" ], "shared_token_count": 2 }, { "report_id": "grid2d_blackboard_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_rect_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_membrane_near_target", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "grid2d_one_target_base", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "grid2d_one_target_exit_timing", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "ring_deriv_k2", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "spectral" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "final_multitimescale_fpt_encounter", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_flux", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_reflecting_bimodality", "group": "grid2d", "title_en": "Grid2D Reflecting-Boundary Bimodality", "title_cn": "Grid2D 全反射边界双峰", "summary_en": "This reflecting-boundary Grid2D report tests whether bimodality survives when periodic shortcuts are removed and all walls reflect. Using representative detour, pore, and transport-track cases, it shows that multi-channel timing structure can persist under strict boundary confinement.", "summary_cn": "该反射边界 Grid2D 报告检验：在移除周期绕行、四壁全反射后，双峰是否仍能保持。通过绕行、孔道与输运轨道等代表案例，报告证明在严格边界约束下，多通道时序结构仍可出现。", "notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "spectral", "survival" ], "previous_in_group": "grid2d_bimodality", "next_in_group": "grid2d_blackboard_bimodality", "same_group_links": [ { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_bimodality", "score": 13.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "spectral" ], "shared_token_count": 2 }, { "report_id": "grid2d_rect_bimodality", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_membrane_near_target", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_target_double_peak", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_blackboard_bimodality", "score": 10.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "ring_lazy_jump", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley_dst", "score": 15.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "cross_luca_regime_map", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_blackboard_bimodality", "group": "grid2d", "title_en": "Grid2D Blackboard Endpoint Case", "title_cn": "Grid2D 黑板端点构型", "summary_en": "This blackboard-style Grid2D case studies reflecting boundaries with start and target anchored at wall endpoints. The Z/S endpoint configurations are used to test whether a visually delayed hump is a true second peak or a window-edge artifact, with diagnostics linked to channel decomposition and path geometry.", "summary_cn": "该黑板图风格 Grid2D 报告研究了反射边界下“墙端点起止”构型。通过 Z/S 端点案例，报告检验晚时隆起究竟是稳定第二峰还是窗口边缘伪峰，并将判断与通道分解和路径几何对应。", "notion_ids": [ "beta-scan", "fpt-pmf" ], "previous_in_group": "grid2d_reflecting_bimodality", "next_in_group": "grid2d_two_target_double_peak", "same_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 10.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_one_two_target_gating", "score": 8.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 8.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "grid2d_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_rect_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_two_target_double_peak", "score": 6.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "ring_deriv_k2", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_flux", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_two_target_double_peak", "group": "grid2d", "title_en": "Grid2D Two-Target Double-Peak", "title_cn": "Grid2D 双目标双峰", "summary_en": "This report studies two absorbing targets in 2D and maps when the total first-passage distribution develops visible double peaks. Under reflecting boundaries and corridor-style bias design, it provides phase maps over target-coupling parameters and isolates how competing destinations create multi-timescale structure.", "summary_cn": "该报告研究二维双吸收目标情形下总首达分布何时出现可见双峰。在反射边界与走廊偏置设计下，报告给出目标耦合参数的相图，并解析“近目标直达”与“远目标延迟”竞争如何形成多时间尺度结构。", "notion_ids": [ "fpt-pmf", "hazard", "survival" ], "previous_in_group": "grid2d_blackboard_bimodality", "next_in_group": "grid2d_two_walker_encounter_shortcut", "same_group_links": [ { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 13.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_membrane_near_target", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_rect_bimodality", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_reflecting_bimodality", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_blackboard_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "grid2d_one_target_base", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "ring_two_target", "score": 10.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "cross_luca_regime_map", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_two_walker_encounter_shortcut", "group": "grid2d", "title_en": "2D Two-Walker Encounter With Shortcut", "title_cn": "Grid2D 体系：grid2d_two_walker_encounter_shortcut 研究报告", "summary_en": "A robust numerical routine verifying equivalence between appendix Eq. Exact time-domain first-encounter computation for two independent walkers.", "summary_cn": "Grid2D 体系（grid2d_two_walker_encounter_shortcut）：关键结论见本页公式链、交互图与原始资产。 Grid2D 体系（grid2d_two_walker_encounter_shortcut）：关键结论见本页公式链、交互图与原始资产。", "notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "grid2d_two_target_double_peak", "next_in_group": "grid2d_rect_bimodality", "same_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 14.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_target_double_peak", "score": 13.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_rect_bimodality", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_membrane_near_target", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_one_two_target_gating", "score": 9.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_blackboard_bimodality", "score": 8.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "ring_two_walker_encounter_shortcut", "score": 13.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 4 }, { "report_id": "ring_lazy_jump", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_valley", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley_dst", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "beta-scan", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_rect_bimodality", "group": "grid2d", "title_en": "Grid2D Rectangle Bimodality", "title_cn": "Grid2D 矩形域双峰", "summary_en": "This report extends Grid2D bimodality from square to rectangular domains, testing how aspect ratio, reflecting walls, and endpoint geometry reshape first-passage channels. It emphasizes reproducible two-target constructions and identifies when double peaks remain structural under anisotropic geometry.", "summary_cn": "该报告把 Grid2D 双峰分析从方形域扩展到矩形域，系统考察长宽比、反射边界与端点几何如何重塑首达通道。报告强调可复现的双目标构型，并识别在各向异性几何下双峰何时仍保持结构稳定。", "notion_ids": [ "fpt-pmf", "hazard", "survival" ], "previous_in_group": "grid2d_two_walker_encounter_shortcut", "next_in_group": "grid2d_membrane_near_target", "same_group_links": [ { "report_id": "grid2d_membrane_near_target", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_reflecting_bimodality", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_target_double_peak", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_blackboard_bimodality", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "cross_luca_regime_map", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_membrane_near_target", "group": "grid2d", "title_en": "Grid2D Membrane Near Target", "title_cn": "Grid2D 体系：grid2d_membrane_near_target 研究报告", "summary_en": "one-target corridor with symmetric/asymmetric semi-permeable membranes; no-corridor two-target setting with one target near the start.", "summary_cn": "Grid2D 体系（grid2d_membrane_near_target）：关键结论见本页公式链、交互图与原始资产。 Grid2D 体系（grid2d_membrane_near_target）：关键结论见本页公式链、交互图与原始资产。", "notion_ids": [ "fpt-pmf", "hazard", "survival" ], "previous_in_group": "grid2d_rect_bimodality", "next_in_group": "grid2d_one_target_base", "same_group_links": [ { "report_id": "grid2d_rect_bimodality", "score": 12.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_target_double_peak", "score": 12.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 2 }, { "report_id": "grid2d_reflecting_bimodality", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_two_walker_encounter_shortcut", "score": 11.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "grid2d_one_target_base", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_one_target_exit_timing", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "cross_luca_regime_map", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 1 }, { "report_id": "ring_two_walker_encounter_shortcut", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley", "score": 9.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "cross_luca_regime_map", "group": "cross", "title_en": "Cross-Model Luca Regime Map", "title_cn": "跨模型 Luca 相区图", "summary_en": "This cross-report study benchmarks full-FPT solvers under fixed-horizon fairness, comparing sparse exact recursion against Luca defect-reduced routes while keeping linear-system MFPT only as reference. The core output is a reproducible speed-ratio map R=t_sparse/t_luca and a regime classification that distinguishes where acceleration is real versus negligible.", "summary_cn": "该跨报告研究在固定时域公平口径下比较 full-FPT 求解器：以 sparse 精确递推为基线，对照 Luca 缺陷约化路径，并把线性系统 MFPT 仅作为参考。核心产出是可复现的速度比 R=t_sparse/t_luca 及其相区分类。", "notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "previous_in_group": "", "next_in_group": "", "same_group_links": [], "cross_group_links": [ { "report_id": "grid2d_reflecting_bimodality", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump_ext", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_two_target", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 }, { "report_id": "ring_valley_dst", "score": 12.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf", "hazard", "survival" ], "shared_token_count": 0 } ] }, { "report_id": "final_multitimescale_fpt_encounter", "group": "misc", "title_en": "Final Multitimescale FPT and Encounter Report", "title_cn": "该研究报告：final_multitimescale_fpt_encounter 研究报告", "summary_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, the reflecting walls are drawn at sites (1) and (N), and the dominant early/late encounter sites are shown directly on the lattice.", "summary_cn": "该研究报告（final_multitimescale_fpt_encounter）：关键结论见本页公式链、交互图与原始资产。", "notion_ids": [ "aw-inversion", "fpt-pmf" ], "previous_in_group": "", "next_in_group": "exact_recursion_method_guide", "same_group_links": [ { "report_id": "exact_recursion_method_guide", "score": 6.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "encounter_reflecting_diagonal_decomp", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 }, { "report_id": "encounter_reflecting_mean_validation", "score": 5.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 1 } ], "cross_group_links": [ { "report_id": "cross_luca_regime_map", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_bimodality", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "grid2d_reflecting_bimodality", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_deriv_k2", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_flux", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "ring_lazy_jump", "score": 6.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "aw-inversion", "fpt-pmf" ], "shared_token_count": 0 } ] }, { "report_id": "grid2d_one_target_base", "group": "grid2d", "title_en": "Grid2D One Target — Base", "title_cn": "Grid2D 体系：grid2d_one_target_base 研究报告", "summary_en": "**Status: WIP scaffold (not yet started).** Tracking: [#4] This is the **mother report** for the `grid2d_one_target_*` sub-series. It defines the shared problem setup, notation, and conventions used by the four sibling reports listed below.", "summary_cn": "Grid2D 体系（grid2d_one_target_base）：关键结论见本页公式链、交互图与原始资产。 Grid2D 体系（grid2d_one_target_base）：关键结论见本页公式链、交互图与原始资产。", "notion_ids": [ "fpt-pmf" ], "previous_in_group": "grid2d_membrane_near_target", "next_in_group": "grid2d_one_target_exit_timing", "same_group_links": [ { "report_id": "grid2d_membrane_near_target", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_one_target_exit_timing", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_one_target_valley_peak_budget", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_one_target_window_measures", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_one_two_target_gating", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_two_target_bias_radius", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 } ], "cross_group_links": [ { "report_id": "cross_luca_regime_map", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "encounter_reflecting_diagonal_decomp", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 0 }, { "report_id": "encounter_reflecting_mean_validation", "score": 3.0, "same_group": false, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 0 }, { 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"next_in_group": "grid2d_one_target_valley_peak_budget", "same_group_links": [ { "report_id": "grid2d_one_target_base", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_one_target_valley_peak_budget", "score": 7.0, "same_group": true, "adjacent_in_track": true, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_membrane_near_target", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 2 }, { "report_id": "grid2d_one_target_window_measures", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_one_two_target_gating", "score": 6.0, "same_group": true, "adjacent_in_track": false, "shared_notion_ids": [ "fpt-pmf" ], "shared_token_count": 3 }, { "report_id": "grid2d_two_target_bias_radius", "score": 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